TOPICS IN THE THEORY OF BOSE-EINSTEIN CONDENSATES WITH A SPINDEGREE OF FREEDOM
BY
SAHEL SHAFIQ ASHHAB
B.S., University of Jordan, 1996M.S., University of Illinois at Urbana-Champaign, 1998
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2002
Urbana, Illinois
To the memory of my father and to my mother
ii
Acknowledgments
I would like to thank my thesis advisor, Anthony J. Leggett, for his guidance and encourage-
ment during my Ph.D. work. I am deeply indebted to him for all his insightful suggestions,
from which I have learned how to think about physics research. I would also like to thank
Gordon Baym for his constant support and for many useful discussions. I would like to thank
Munir Nayfeh for giving me the opportunity to spend one summer at his research lab and
learn about experimental work. I would like to thank all those named above, as well as Paul
G. Kwiat, for their patience in numerous discussions where I learned different subjects in
physics. I would like to thank my fellow students Vivek Aji, Osman Akcakir, Boris Fine, Ab-
delhamid Galal, Richard Kassman, Simon Kos, Minqiang Li, Carlos Lobo, Vladimir Lukic,
Erich Mueller, Sorin Paraoanu, Michael Turlakov and Geoffrey Warner for several discus-
sions. During my Ph.D. work I spent one semester at NTT Basic Research Laboratories
(Atsugi-Japan), where I benefitted greatly from working under the supervision of Noriyuki
Hatakenaka in the research group of Hideaki Takayanagi. I would like to thank both of them
and their research group for their hospitality during my stay in Atsugi. I also had several
fruitful discussions with Yvan Castin, Nir Gov, Tin-Lun Ho, Markus Holzmann, Kei Iida,
Sigmund Kohler, Susumu Kurihara, Anna Minguzzi, Revaz Ramazashvili, Alice Sinatra, Fer-
nando Sols, Ivan Vartaniants, Sahng-Kyoon Yoo and Ivar Zapata. I acknowledge financial
support from the National Science Foundation under grant number NSF-DMR-99-86199, the
University of Illinois at Urbana-Champaign and NTT corporation.
iii
Contents
Chapter
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Bose-Einstein Condensates of Spin 1 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Ground State I: Mean-Field Solution (Coherent States) . . . . . . . . . . . . 102.4 Ground State II: Law-Pu-Bigelow Solution (Singlet State) . . . . . . . . . . 142.5 Mean-Field vs Law-Pu-Bigelow . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Conservation of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Ground State with Fixed-Spin Constraint . . . . . . . . . . . . . . . . . . . 212.8 Excitation Spectrum of Quasiparticles . . . . . . . . . . . . . . . . . . . . . 272.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Measurement Theory of Spinor Bose-Einstein Condensates . . . . . . . . . . . . . 343.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Single-Particle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Two-Particle Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Discussion of the Results and Possible Experimental Realizations . . . . . . 463.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Interference Between Spinor Bose-Einstein Condensates . . . . . . . . . . . . . . . . 514.1 Interference Between Two Scalar Condensates with Equal Numbers of Atoms 524.2 Interference Between Two Scalar Condensates with Different Numbers of Atoms 564.3 Interference Between Two Spinor Condensates I: The Antiferromagnetic Case 594.4 Interference Between Two Spinor Condensates II: The Ferromagnetic Case . 624.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Bose Condensation of Spin 1/2 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Spin 1/2 Bosons? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 A Misleading Argument (Spontaneous Symmetry Breaking) . . . . . . . . . 68
5.2.1 Paradox 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2.2 Paradox 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 The Role of Interactions in Forming the Fragmented State . . . . . . . . . . 805.4 Extracting Entangled Atom Pairs . . . . . . . . . . . . . . . . . . . . . . . . 81
iv
5.5 Possible Spin-Relaxation Mechanisms . . . . . . . . . . . . . . . . . . . . . . 835.6 The Gross-Pitaevskii Equations . . . . . . . . . . . . . . . . . . . . . . . . . 905.7 Detection of the Fragmented State . . . . . . . . . . . . . . . . . . . . . . . 945.8 Condensation in a Toroidal Geometry . . . . . . . . . . . . . . . . . . . . . . 1015.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 External Josephson Effect in Bose-Einstein Condensates with a Spin De-gree of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.1 The Spinless Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Spin 1/2 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.1 Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2.2 Anisotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2.3 Discussion of the Equilibrium Points of the Motion . . . . . . . . . . 1156.2.4 Experimental Considerations . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Spin 1 Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendix
A Quasiorthogonality and Overcompleteness of Coherent States . . . . . . . . . . . 126A.1 Phase States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126A.2 Spinor Condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B Calculation of the Equilibrium Points and Oscillation Frequencies for theTwo-Component Josephson Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.1 Isotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134B.2 Anisotropic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
v
Chapter 1
Introduction
Since Bose-Einstein condensation (BEC) was first proposed by ? until its realization in the
alkali atomic gases by ?, it has been surrounded by controversy. Since the first treatment
considered a perfectly noninteracting gas, the idea was dismissed as an artifact of the model.
The idea of BEC was revived by ?, who conjectured that the superfluid phase of 4He is a
Bose-Einstein condensate. However, liquid 4He is far from being a noninteracting gas, and
to date there has not been a conclusive microscopic theory of BEC in 4He. Another example
of a BEC that has been known for a long time is the laser, which is a condensate of photons,
all in the same momentum state. Photons are almost perfectly noninteracting. However,
they can be created and annihilated at no cost (i.e. their chemical potential is equal to zero).
Other examples of systems involving BEC can be found in the book by ?. However, BEC
was not realized in a system with close resemblance to Einstein’s model until it was achieved
in atomic gases of 87Rb by ? and of 23Na by ?.
Since 1995 there has been a lot of interest in understanding BEC in the alkali gases
both theoretically and experimentally. Due to the diluteness of these gases, the microscopic
theory predicts the transition temperature and the properties of condensates remarkably ac-
curately [see e.g. ?]. Bogoliubov quasiparticles, first proposed by ?, have also been observed
directly for the first time by ?. The coherent nature of condensates was demonstrated by
? through the appearance of interference fringes between two independently-prepared con-
1
densates. The ability to tune the effective interaction parameter using Feshbach resonances
has also attracted a lot of attention. It is now possible to explore the weakly and strongly
interacting limits in a single system, even in a single experiment. It is also possible to switch
from repulsive to attractive interactions and watch the collapse of the condensate. There are
now several comprehensive reviews of the theory [???] and experiment [?] of BEC in the
alkali gases. We refer the reader to those for a detailed discussion of the theory of BEC.
In the early experiments on BEC in the alkali gases, magnetic traps were used. Since
the trapping depended on the interaction of the atomic magnetic moment with the external
field, atoms in only one internal state could be trapped. For example, out of the three states
in the 23Na atomic ground state triplet (F = 1), ? were able to trap only the Fz = −1 state.
The spin degrees of freedom are then frozen, and the atoms can be treated as possessing no
internal degrees of freedom. ? were able to trap two different hyperfine states in a magnetic
trap. However, since in the absence of external laser fields the number of atoms in each
component was conserved, the spin degree of freedom was not fully exploited. With the
advent of optical traps, it became possible to trap all the different hyperfine states of an
atom in the same trap [see ?].
Theoretical studies on spinor Bose-Einstein condensates1 of the alkali atomic gases began
with the pioneering work of ? and ?, who mainly studied the ground state and elementary
excitations of this kind of condensates. Law, Pu and Bigelow suggested a different ground
state that sparked several discussions of the rotational symmetry of spinor condensates and
its possible breaking [see e.g. ???]. Later studies included, among other topics, spin mixing
dynamics [?], measurement theory [??], topological excitations [???] and the possible use of
spinor condensates as a source of entangled atoms [?].2
In this Thesis we study phenomena in Bose-Einstein condensates that are associated with
1The term spinor is used inaccurately in this context, since states of integer spin do not transform as
spinors. However, after being used by several authors, it has become acceptable to describe condensates of
atoms with a spin degree of freedom as “spinor condensates”.2Recently ? have produced macroscopic pair-correlated atomic beams with spinless condensates.
2
internal spin degrees of freedom. In Chapter 2 we consider the ground state and elementary
excitations of a condensate of spin 1 atoms. We review the basic results regarding the ground
state of spinor condensates. In the case of antiferromagnetic interatomic interactions two
particular states have been suggested as the ground state of the condensate, the so-called
coherent and singlet states. We study the energetic competition between them. We then
argue that the problem must be treated with the additional constraint of a fixed total spin.
We carry out that calculation, and we find that the spin density is distributed in a nontrivial
manner in the condensate. We also find that in some cases there will be phase separation
between the different hyperfine states. In Chapter 3 we investigate the possibility of dis-
tinguishing between the coherent and singlet states introduced in Chapter 2. We find that
extracting a small part of the condensate and meauring its spin structure cannot be used to
distinguish between the two states in question, and that this type of measurement projects
the condensate closer and closer to a coherent state. Our treatment lays down certain guid-
ing principles in the study of measurement theory of spinor condensates. Our results suggest
that in order to distinguish between the two states using spin structure measurements, all the
atoms in the condensate must be probed. In Chapter 4 we analyze several different spatial
interference problems in BEC with and without spin. First we demonstrate analytically the
appearance of interference fringes between two overlapping spinless condensates. We then
consider the case when the two condensates are in the antiferromagnetic ground state. We
find that the coherent and singlet states are indistinguishable in interference experiments.
In both cases we find that both the offset and the visibility of the interference fringes vary
randomly from shot to shot. We estimate the effect of magnetic field inhomogeneities on
the visibility in the interference experiment by ?, and we find that the reduction in visibility
is negligible. The results of Chapters 3 and 4 have been published in Ref. [?]. In Chapter
5 we study the problem of condensing a Bose gas of effectively spin 1/2 atoms. Under the
assumption of small spin relaxation, where the Hamiltonian is SU(2) symmetric, we find an
interesting fragmented state where two different orbital wave functions are macroscopically
3
occupied. The formation of this fragmented state is not caused by any interatomic inter-
actions, but rather by the symmetry of the initial many-body wave function. We derive
a set of Gross-Pitaevskii equations describing the two occupied wave functions and discuss
methods to detect the formation of the fragmented state. We then consider the consequences
of forming the fragmented state in a toroidal trap, and we find the interesting result that it
is possible to create a circulating current in a trap without rotating the trap. In Chapter 6
we study certain aspects of the Josephson effect in spinless condensates, and investigate the
new phenomena that a spin degree of freedom adds to the problem. The results of Chapter
6 have been published in Ref. [?].
Using notation consistently and unambiguously is almost an impossible task in current
physics writing. In order to make this Thesis as homogeneous as possible, we shall be
using certain non-standard terminology and notation. For example, any internal degree of
freedom will be referred to as spin and denoted by the letter S (unless the meaning is clear
from the context). That will include the total atomic angular momentum, which is normally
denoted by F (Chapters 2,3,4 and 6), nuclear spin and pseudospin (Chapters 5 and 6). The
z component of the spin will be denoted by Sz, and the letter m will be limited, as much as
possible, to denote the mass (In Chapters 3 and 4, where it does not cause any confusion,
the letter m will be used to mean Sz). Relative phases will be denoted by the letter χ.
The letter φ will be used sometimes to denote the wave function and sometimes to denote
the polar angle in spherical polar coordinates. However, the meaning will be clear from the
context. The term coherent state will be used differently from the quantum optics definition
(The reason for our use of that term will become clear in Chapter 3). We shall always use
units where = 1 in spin space, and in Chapter 6 we use units where = 1 in both spin
and configuration space.
4
Chapter 2
Bose-Einstein Condensates of Spin 1
Atoms
In this Chapter we give a review of the literature on the ground state of a Bose-Einstein
condensate (BEC) of spin 1 atoms. We start by describing the physical system in Section
2.1. We derive the Hamiltonian in Section 2.2. In Section 2.3 we find the mean-field (MF)
ground state, mainly following the work of ?. In Section 2.4 we give the Law-Pu-Bigelow
(LPB) solution, which under certain conditions gives lower energy than the MF solution [?].
We study the energetic competition between the two states in Section 2.5 [?]. In Section
2.6 we argue that the problem must be treated with an additional constraint, namely the
conservation of the total spin of the condensate. We carry out that calculation in Section
2.7 [?]. Finally, in Section 2.8 we derive the quasiparticle spectrum of such a condensate
[?]. Although the analysis in Sections 2.7 and 2.8 was carried out independently by the
author, the larger part of this Chapter is a review of other authors’ work. However, since
the different ground states suggested in the literature are the starting point for Chapters 3
and 4, we discuss them in some detail here to have them at our disposal later on.
5
2.1 The System
The system that we consider in this Chapter, as well as Chapters 3 and 4 and Section 6.4, is a
Bose-Einstein condensate of an alkali atomic gas, with atomic spin S = 1 (We shall use units
where = 1 in spin space).1 The two most commonly used elements in experiment, 23Na
and 87Rb, have hyperfine states with S = 1. These states are formed by adding the nuclear
spin, which has the value 3/2, with the electronic spin, which has the value 1/2. In the
ground state, the valence electron has no orbital angular momentum (L = 0). The hyperfine
interaction between nuclear and electronic spins favours forming the smaller of the possible
values of the total spin, which in this case gives S = 3/2 − 1/2 = 1. Current experiments
typically have N ∼ 104 − 107 atoms with central densities ρ ∼ 1014cm−3. The interaction
is described by two s-wave scattering lengths, a0 and a2, corresponding to the scattering
channels Stotal = 0 and Stotal = 2, as will be explained in Section 2.2 below. Typical values
are a0 ≈ a2 ≈ 26A and a2−a0 ≈ 2A for 23Na, and a0 ≈ a2 ≈ 55A and a0−a2 ≈ 2A for 87Rb.
In order to avoid discussing condensates with attractive interactions, which is a research area
in its own right, we shall always take both a0 and a2 to be positive. The critical temperature
Tc is of the order of 200 to 600nK. The minimum temperature reached in experiment can be
as low as ∼ 5nK( Tc). Therefore, we shall make the approximation that T = 0.
An alkali atomic gas with a free spin degree of freedom cannot be realized using purely
magnetic traps. That is because it is not possible to have all three hyperfine states (Sz =
1, 0,−1) being low-field seeking, and therefore at least one of the three states will be expelled
from the center of a magnetic trap. That difficulty is avoided by using an optical trap. In
optical traps the three hyperfine states feel the same trapping potential, up to corrections of
the order of the ratio of the hyperfine splitting to the detuning of the laser beam from the
1As mentioned in Chapter 1, we shall use the letter S to denote the total atomic angular momentum, and
we shall refer to all internal degrees of freedom as spin throughout this Thesis. Note that this is different
from the usual atomic physics notation, where S usually denotes the intrinsic electronic spin, and F is used
to denote the total atomic angular momentum.
6
atomic transition [see ?]. This ratio can be made as small as ∼ 10−3, and the optical trap
can therefore be treated as spin independent.
2.2 The Hamiltonian
The Hamiltonian describing the system of Section 2.1 will include kinetic, potential and
interaction energy terms:
H = HKE + HPE + Hint. (2.1)
In second-quantized language, the kinetic energy term takes the form:
HKE =∑
α
∫drψ†
α(r)
(−
2
2m∇2
)ψα(r), (2.2)
where the index α refers to the different Sz states (1, 0 and −1). The most general form of
the potential energy term is:
HPE =∑α,α′
∫drψ†
α(r)Vαα′(r)ψα′(r). (2.3)
We shall assume that the potential energy is due to a combination of external magnetic and
optical fields. Furthermore, we shall assume that all magnetic fields point in the z direction.
The potential Vαα′(r) then becomes diagonal and can be decomposed into two parts, namely
the trapping potential, which is equal to zero at the center of the trap for all three hyperfine
states, and the relative chemical potential (usually produced by a uniform external magnetic
field). If the trapping potentials are different for the different hyperfine states, we obtain
partial or total phase separation, depending on the interaction parameters, as demonstrated
theoretically by ? and experimentally by ?. However, since we are mostly interested in
phenomena that are related to the interatomic interaction properties, we shall assume that
the trapping potential is the same for all hyperfine states. In general, we shall assume that
a uniform external magnetic field is applied to the gas. The potential energy term can then
be expressed as:
7
HPE =∑
α
∫dr ψ†
α(r)
(U(r) − gµµBBα
)ψα(r), (2.4)
where U(r) is the spin-independent trapping potential, gµ is the atomic g-factor, µB is the
Bohr magneton and B is the magnetic field strength, taken to be along the z axis.
Now we turn to the interatomic interaction term. If we consider a collision between two
spin 1 atoms, the total spin of the atom pair will be 0, 1 or 2. From rotational symmetry
we know that the interatomic interaction potential between two atoms can only depend on
their total spin S, but not on the z component of the total spin Sz. The most general form
of the interatomic interaction Hamiltonian then takes the form:
Hint =∑
α,α′,β,β′
∫dr1dr2ψ
†α(r1)ψ
†β(r2)hint(r1 − r2)ψβ′(r2)ψα′(r1), (2.5)
where
hint(r1 − r2) =∑S,Sz
PS,SzvS(r1 − r2), (2.6)
PS,Sz is the projection operator that projects out the part of the two-atom wave function
with total spin S and z component Sz, and vS(r1 − r2) is the interaction potential between
two atoms at r1 and r2 (with total spin S).2 At low temperatures, and therefore low energies,
the interaction potential can usually be replaced by the pseudopotential [see ??]:
vS(r1 − r2) ≈ gSδ(r1 − r2), (2.7)
where gS = 4π2aS/m, aS is the s-wave scattering length in the S channel, m is the atomic
mass and δ(r1−r2) is the Dirac δ-function. The above approximation works only to first order
in the energy, and one must neglect the suppression of the wave function at small interatomic
distances. It can most intuitively be understood in the spirit of the Born approximation from
scattering theory, but the criteria for its applicability are less stringent than those of the
2Note that in second-quantized language it is necessary to apply the projection operator to both left
and right sides. This is in contrast to first-quantized language, where it is sufficient to apply the projection
operator to one side only.
8
Born approximation. Since the two-body wave function must be symmetric for bosons, the
orbital (i.e. spatial) and spin states must be either both symmetric or both antisymmetric.
Therefore, if the atom pair have total spin Stotal = 1 (antisymmetric), their relative orbital
wave function must also be antisymmetric, and therefore the latter must vanish when the
atoms overlap. For this reason we can neglect the interaction term corresponding to Stotal =
1. Another justification for neglecting that term is that at low temperatures, most of the
atoms will occupy the same orbital wave function, namely the one with the lowest energy.
Therefore, the relative orbital wave function of any two atoms will be symmetric, and that
cannot correspond to Stotal = 1. The interaction Hamiltonian then reduces to:
hint(r1 − r2) =
(P0,0g0 +
∑Sz
P2,Szg2
)δ(r1 − r2). (2.8)
The projection operators can alternatively be expressed in terms of the total spin operators
as follows:
P0,0 = 1 − S2total
6
=1 − S1 · S2
3, (2.9)
∑Sz
P2,Sz =S2
total
6
=2 + S1 · S2
3. (2.10)
The interaction Hamiltonian can then be expressed as:
Hint =1
2
∫dr1dr2
∑α,α′,β,β′
∑S,Sz
ψ†α(r1)ψ
†β(r2)〈α, β|S, Sz〉
(1 − S1 · S2
3g0 +
2 + S1 · S2
3g2
)
δ(r1 − r2)〈S, Sz|α′, β ′〉ψβ′(r2)ψα′(r1)
=
∫dr
∑α,α′,β,β′
ψ†α(r)ψ†
β(r)(c0
2δα,α′δβ,β′ +
c22
Sαα′ · Sββ′)ψβ′(r)ψα′(r), (2.11)
where c0 = (2g2 + g0)/3, c2 = (g2 − g0)/3 and
S =1√2
0 1 0
1 0 1
0 1 0
x +
1√2
0 −i 0
i 0 −i0 i 0
y +
1 0 0
0 0 0
0 0 −1
z. (2.12)
9
In conclusion, the Hamiltonian (containing kinetic, trapping-potential, first-order Zeeman
and interaction terms) can be written as:
H =
∫dr∑
α
ψ†α
(−
2
2m∇2 + U(r) − gµµBBα
)ψα(r) (2.13)
+c02
∑α,β
ψ†α(r)ψ†
β(r)ψβ(r)ψα(r) +c22
∑α,α′,β,β′
ψ†α(r)ψ†
β(r)Sαα′ · Sββ′ψβ′(r)ψα′(r).
This is the Hamiltonian we shall be using in most of the remainder of this Chapter. In Section
2.7 we shall add one more term to the Hamiltonian to study the effect of the second-order
Zeeman effect.
2.3 Ground State I: Mean-Field Solution (Coherent
States)
The mean-field (MF) assumption in the context of a BEC ground state is that all the atoms
occupy the same single-particle state.3 The many-body quantum state of the condensate
can then be expressed as:
|Ψ〉 =
(a†)N
√N !
|0〉, (2.14)
where
a† =∑
α
∫drφα(r)ψ†
α(r), (2.15)
and the normalization is taken such that∑
α
∫drφ∗
α(r)φα(r) = 1. In first-quantized (i.e.
wave-function) language the MF assumption is expressed as:
Ψ(r1;α1, ..., rN ;αN) =
N∏j=1
φαj(rj). (2.16)
3In the following Chapters we shall also describe MF states as coherent states. Note that this definition
of coherent states is different from the one normally used in quantum optics.
10
If we calculate the expectation value of the Hamiltonian (Eq. 2.13) with the above ansatz
for the ground state, we find the free energy:
K ≡ 〈H − µN〉
=
∫drN
∑α
φ∗α(r)
(−
2
2m∇2 + U(r) − gµµBBα− µ
)φα(r)
+N(N − 1)∑
α,α′,β,β′φ∗
α(r)φ∗β(r)
(c02δαα′δββ′ +
c22
Sαα′ · Sββ′)φβ′(r)φα′(r). (2.17)
We now try to find the set of functions φα(r) that minimizes the free energy by differentiating
Eq. (2.17) with respect to φ∗α(r) and equating the derivative to zero, which gives the following
set of Gross-Pitaevskii (GP) equations describing the ground state:
(−
2
2m∇2 + U(r) − gµµBBα
)φα(r) + c0(N − 1)
∑β
|φβ(r)|2φα(r)
+c2(N − 1)∑
α′,β,β′
(φ∗
β(r)Sββ′φβ′(r)) · Sαα′φα′(r) = µφα(r). (2.18)
One can try to struggle his way through the set of Eqs. (2.18). However, it is easier to start
the search for the ground state by decomposing the wave function φα(r) into a density part
and a spin part. The advantage of this approach is that it will allow us to consider these
two parts separately. We go back to Eq. (2.14) and re-express a† as:
a† =∑
α
∫dr
√ρ(r)
Nζα(r)eiθ(r)ψ†
α(r), (2.19)
where the normalization is taken such that∫drρ(r) = N and ζ†(r)ζ(r) = 1. ρ(r) is the total
density of the condensate at point r, and ζ(r) is the vector describing the spin state of the
condensate at point r. The phase factor eiθ(r) will turn out to be constant in the ground
state and can therefore be neglected whenever we use the above notation. If it varied in
space, it would contribute to the kinetic energy. The free energy is now expressed as:
K =
∫dr
2
2m(∇√
ρ)2 +
2
2mρ(∇ζ)2 +
2
2mρ(∇θ)2 + U(r)ρ
−µρ− gµµBBρ〈Sz〉 +ρ2
2
(c0 + c2〈S〉2
), (2.20)
11
where the dependence on r has been left implicit and
〈S〉 = ζ†Sζ. (2.21)
In the free energy (Eq. 2.20) we have not included the cross derivative term of the form
∇ζ∇θ, since that term can always be eliminated by an appropriate redefinition of ζ and θ.
We start by treating the case of zero external magnetic field, i.e. B = 0. There is only one
term in the free energy (Eq. 2.20) containing θ, and it is minimized by making θ constant in
space, i.e. independent of r, as we have already anticipated. There are two terms containing
the vector ζ(r). The term containing |∇ζ(r)|2 is minimized by making ζ(r) independent of
r. The other term in Eq. (2.20) containing ζ(r) is the last one on the right-hand side, since
〈S(r)〉 = ζ†(r)Sζ(r). (2.22)
Depending on the sign of c2 there are two different cases:
1) The ferromagnetic case: When c2 < 0, the free energy is minimized by maximizing
|〈S〉|. The maximum value that |〈S〉| can have is 1, when ζ(r) corresponds to S ·d = 1 along
some direction d:
ζ(r) = e−iS·Ω
1
0
0
, (2.23)
where Ω is some vector.
2) The antiferromagnetic case: When c2 > 0, the free energy is minimized by minimizing
|〈S〉|. The minimum value that |〈S〉| can have is 0, when ζ(r) corresponds to S ·d = 0 along
some direction d:
ζ(r) = e−i(θ+S·Ω)
0
1
0
, (2.24)
where θ is an overall phase factor and Ω is some vector.
12
One can understand the above results in the following way: When c2 < 0, an atom pair
with Stotal = 2 has lower energy than one with Stotal = 0. Therefore, atom pairs try to be
in an Stotal = 2 state as much as they can. In fact, if all the atoms have S · d = 1 (with
the same d for all the atoms), then any atom pair is in an Stotal = 2 state, and the lowest
possible energy is achieved. The antiferromagnetic case is somewhat trickier. The atom
pairs now try to be in an Stotal = 0 state as much as they can. However, it is not possible
to make all combinations of atom pairs pair up in Stotal = 0 states. In particular, a product
of two unentangled states, which is the case in the MF solution, cannot with certainty be
in an Stotal = 0 state. If we take a general vector (ζ1, ζ0, ζ−1), we can always find a system
of coordinates where it has the form (ζ ′1, ζ′0, 0). The probability for two such (equal) vectors
to be in an Stotal = 0 state is |ζ ′0|2/3, which is maximized by making |ζ ′0| = 1. Even with
that choice of ζ , the probability of being in the desired Stotal = 0 state is only 1/3. One
might expect that the LPB state, which we shall introduce in Section 2.3, will increase the
Stotal = 0 pair probability, since it gives a lower energy than the MF state. However, as
we shall see in Section 2.5 and in Chapter 3, this probability is still equal to 1/3 in the
thermodynamic limit.
Once the value of |〈S〉| is determined (either 0 or 1), we can turn to evaluating the density
ρ(r). Minimizing the free energy with respect to ρ(r) we find the GP equation:
(−
2
2m∇2 + U(r) + cρ(r)
)√ρ(r) = µ
√ρ(r), (2.25)
where c = c0 + c2 = g2 in the ferromagnetic case, and c = c0 in the antiferromagnetic case.
In the Thomas-Fermi (TF) approximation, when the interaction energy is much larger than
the kinetic energy so that the latter can be neglected, we find that
ρ(r) =µ− U(r)
c,where c =
c0 + c2 , c2 < 0,
c0 , c2 > 0.(2.26)
When c2 c0, the density distribution of the condensate is approximately independent of
c2, including its sign. This result will be used in justifying the procedure in Section 2.4.
13
Now we turn to the case of nonzero magnetic field. In the ferromagnetic case, the ground
state is that with 〈S〉 parallel to B, which we are assuming to be in the z direction. That
case is a simple example of symmetry breaking, where one solution is picked out of a set of
degenerate solutions due to the application of an external field. In the absence of external
fields, we have the well-known phenomenon of spontaneous symmetry breaking, where a
direction is picked purely randomly. Alternatively, one can say that the small external fields
that we have neglected (and sometimes cannot measure) determine the chosen direction of
the magnetization. The antiferromagnetic case is more interesting and will be treated in
Section 2.7 for reasons that will be explained in that Section.
2.4 Ground State II: Law-Pu-Bigelow Solution (Sin-
glet State)
The mean-field assumption, which we used in Section 2.3, was inspired by the success of
that assumption in the spinless case. A few months after the original papers by ? and ?,
? (LPB) noted that the MF state does not obey the SO(3) symmetry of the Hamiltonian.
Therefore, they reconsidered the same problem using a different approach. They used the
same assumption regarding the spatial part of the wave function, namely that all the atoms
occupy the same orbital wave function. However, they did not impose any constraint on
the spin part of the many-body wave function. The many-body quantum state can then be
expressed as:
|Ψ〉 =∑
N1,N0,N−1
cN1,N0,N−1
(a†1)N1
(a†0)N0
(a†−1
)N−1
√N1!N0!N−1!
|0〉, (2.27)
where the constraint N1 +N0 +N−1 = N is implicitly understood, and
a†α =
∫drφ(r)ψ†
α(r). (2.28)
The orbital wave function φ(r) does not depend on the hyperfine index α and is normalized
14
by the condition∫drφ∗(r)φ(r) = 1. If the interaction coefficients in the Hamiltonian satisfy
the condition c2 c0, we find the following approximate GP equation for φ(r):
(−
2
2m∇2 + U(r) − gµµBB〈Sz〉 + c0N |φ(r)|2
)φ(r) = µφ(r). (2.29)
The term gµµBB〈Sz〉 depends on the spin part of the many-body wave function, which we
have not determined yet. We shall not worry about that point in this Section, since 〈Sz〉and φ(r) can be determined self-consistently. We shall see in Section 2.7, however, that this
self-consistent approach is in fact incorrect. If we solve Eq. (2.29), which is the usual spinless
GP equation, and substitute the solution in the Hamiltonian (Eq. 2.13), we find the effective
Hamiltonian:
H = H0 + gµµBBStotal,z +c′22
: S2total :
= H0 + gµµBBStotal,z +c′22S2
total − c′2N, (2.30)
where H0 is a constant given by:
H0 =
∫drNφ∗(r)
(−
2
2m∇2 + U(r)
)φ(r)
+c′02N(N − 1), (2.31)
and
c′i = ci
∫dr|φ(r)|4 ,where i = 0, 2. (2.32)
The interaction term in the effective Hamiltonian can be derived in a more intuitive
way. First we consider only two atoms in the trap and assume that both of them occupy
the orbital wave function φ(r) and have a definite value of the total spin (0 or 2). Their
interaction energy is then given by:
Eint = (c0 + c2S1 · S2)
∫dr|φ(r)|4
= c′0 + c′2S1 · S2. (2.33)
From Eq. (2.33) we infer the effective two-atom interaction Hamiltonian:
15
H2−atom int = c′0 + c′2S1 · S2. (2.34)
Now we can find the total interaction Hamiltonian of the N -atom system:
Hint =1
2
∑i,j(i=j)
(c′0 + c′2Si · Sj
)
=c′02N(N − 1) +
c′22
(∑i,j
Si · Sj −∑
i
S2i
)
=c′02N(N − 1) +
c′22
(∑
i
Si
)2
−∑
i
S2i
=c′02N(N − 1) +
c′22
(S2
total − 2N), (2.35)
since S2i = 2 for spin 1 atoms. This derivation shows that once we assume that all the atoms
occupy the same orbital wave function, the problem is equivalent to that of a magnetic
material with infinite-range interactions, since all pairs of atoms interact with the same
strength.
The Hamiltonian in Eq. (2.30) is now used as the starting point for the LPB treatment.
It is a function of the total spin and the z component of the total spin of the system. The
eigenstates of the Hamiltonian will then be eigenstates of S2total and Stotal,z. Therefore, we
now look for the possible eigenvalues of these operators. We assume that N is even.4 Any
state of this system can be produced by the following formula [?]:
|N, S, Sz〉 = (S−)S−Sz(a†1)S(a†0a
†0 − 2a†1a
†−1)
N/2−S |0〉, (2.36)
where S− is the spin-lowering operator. Spin operators are given by:
Si =∑α,β
a†αSi,αβaβ , (2.37)
where Si,αβ is the αβ matrix element of the Si matrix (i = x, y, z,+,−). For example,
4The behaviour of the system cannot depend on whether the number of atoms is odd or even, if that
number is ∼ 106. However, assuming an even number slightly simplifies the appearance of some of the
following expressions. The assumption of even N will be used throughout this Thesis.
16
Sx =1√2
(a†1a0 + a†0a1 + a†0a−1 + a†−1a0
),
S− =√
2(a†0a1 + a†−1a0
), (2.38)
and so on. The operator (a†0a†0 − 2a†1a
†−1) creates a pair of atoms in a spin singlet state, i.e.
a state with total spin equal to zero.5 Therefore, these pairs do not contribute to the total
spin of the condensate. With these elements we can now understand the structure of any
state |N, S, Sz〉. The operator (a†1)S creates S atoms with total spin S and z component
Sz = S. Then the operator (S−)S−Sz lowers the z component to the appropriate value Sz,
leaving S unchanged. The operator (a†0a†0 − 2a†1a
†−1)
N/2−S then creates extra spinless pairs
to match the required number of atoms in the condensate N . The fact that our reasoning is
ordered in a different way from Eq. (2.36) should not worry us, since the operators whose
order we have reversed do in fact commute. It is clear that one can create a condensate with
S = N . It is not possible, however, to create a condensate with S = N − 1. The reason is
that if S < N , we have to create at least one spinless pair, which would make S ≤ N − 2.
Therefore, the possible S values are N,N − 2, N − 4, ..., 0. That gives a total of
(2N + 1) + (2(N − 2) + 1) + ... + 1 =(N + 1)(N + 2)
2
different possible values of (S, Sz) for a given N . The number of different states that can
be formed by distributing N bosons among three single-particle states is (N + 1)(N + 2)/2.
The two numbers match exactly. That means that the states defined by N , S and Sz are
unique, and no extra quantum numbers are needed.
We can now turn to determining the state that minimizes the energy. In the ferromagnetic
case (ε2 < 0), the ground state is the one that maximizes S. That state corresponds to S = N
and is (2N + 1)-fold degenerate. These states have the same energy as the MF solution. In
fact, the LPB states can be written as superpositions of the MF states. Therefore, there is
nothing new in the LPB approach in the ferromagnetic case. In the antiferromagnetic case
5The minus sign in the operator (a†0a
†0 − 2a†
1a†−1) is in accordance with the phase convention of ?, which
we shall follow in this Thesis.
17
(ε2 > 0), the LPB ground state is that with S = 0, i.e. the spin singlet state. It is constructed
by creating N/2 spinless pairs of atoms. We can find that state in the |N1, N0, N−1〉 basis
by using the fact that it is destroyed by the spin-lowering operator:
S−|N, 0, 0〉 = (a†0a1 + a†−1a0)∑
N1,N0,N−1
cN1,N0,N−1 |N1, N0, N−1〉 = 0. (2.39)
However, since that calculation is rather complicated and uninspiring, we shall not carry it
out here [see ? for solution]. Instead, we now express the singlet state in a form that will
prove useful in Chapters 3 and 4. In the context of interference between two condensates,
it is well known that a product of Fock states, where there is a definite number of atoms
in each of the two condensates, can be expressed as a superposition of phase states, where
there is a definite relative phase between the two condensates (see Appendix A). By analogy,
we can expect that the LPB states can be expressed as superpositions of MF states. In fact,
that turns out to be true, and the states are given by (see Appendix A for proof):
|N, S, Sz〉 = const.×∫dΩYS,Sz(θ, φ)|N, θ, φ〉, (2.40)
where YS,Sz(θ, φ) are the spherical harmonics and |N, θ, φ〉 is the MF state with N atoms
and S · d = 0 with d being the direction defined by the angles θ and φ in spherical polar
coordinates. In particular Y0,0(θ, φ) = const. and we find that:
|N, 0, 0〉 = const.×∫dΩ|N, θ, φ〉. (2.41)
Eq. (2.41) will be very useful in the treatment in Chapters 3 and 4.
2.5 Mean-Field vs Law-Pu-Bigelow
It is clear from Section 2.4 that in the antiferromagnetic case, the LPB state has a lower
energy than the MF state of Section 2.3. However, this is not the end of the story. The
energy difference between the two states is given only by the S2total term in the Hamiltonian:
EMF −ELPB =c′22〈S2
total〉MF ∼ c2ρ. (2.42)
18
This energy difference does not increase with N , assuming the density ρ is held fixed. The
fact that the energy difference does not increase with N should alert us that we have to
make sure that we did not neglect any terms in the Hamiltonian that are proportional to
any positive power of N , no matter how small their coefficient. The reason for that is that
in the thermodynamic limit (i.e. when N → ∞), the terms with the highest power of N
determine which state lowers the energy. In fact, if we include a magnetic field gradient
term, we find that such a term favours the MF state [see ?]. The Hamiltonian describing
the field gradient can be expressed as:
HB−grad. =∑i,j
cija†iaj. (2.43)
If we take the difference in expectation value of this new term in the MF and LPB states we
find that:
〈HB−grad.〉LPB − 〈HB−grad.〉MF = N
c11 + c00 + c−1−1
3−
∑i,j(i=j)
cijζ†i ζj
. (2.44)
And there will always be some ζ that makes that difference positive, and since it scales as N ,
it will always win over the interaction energy difference. As a result the true ground state
in the presence of a magnetic field gradient will in fact be the MF state. For example, if we
take the Hamiltonian in Eq. (2.43) to contain only one term of the form ca†1a0+h.c., we find
that
〈HB−grad.〉LPB = 0,
〈HB−grad.〉MF = − c√2N, (2.45)
for ζ = (−12, 1√
2, 1
2), which corresonds to the MF state with Sz = 0 along the direction
with spherical polar coordinates θ = π/4, φ = 0. We can make a rough estimate of the
value of the magnetic field gradient required to break the symmetry by using the relations
〈HB−grad.〉 ∼ B′RgµµBN and 〈Hint〉 ∼ c2N/R3, where B′ is the value of the magnetic
field gradient, and R is a typical length scale of the condensate. From the typical values
19
R ∼ 10−5m, µB ∼ 10−23J/T and c2 ∼ 10−52Jm3, we find that B′ ∼ 10−9T/m. We suspect
that the above calculation underestimates B′, since the length scale that we should have used
in 〈HB−grad.〉 is the one describing the asymmetry in the shape of the condensate, whereas
we have used the entire size of the condensate R.
Another point that has not been studied yet is the effect of depletion on the LPB state.
It is generally believed that the energy gained by going from the MF ground state to the
Bogoliubov ground state scales as N .6 Therefore, it sounds reasonable that even in the
absence of any external fields, one has to treat depletion effects before trying to make the
ground state rotationally invariant. We suspect that if this approach is to be used (i.e. MF,
Bogoiubov and then LPB), the ground state will turn out to be:
|Ψ〉 =
∫dΩ
2π|Bog. GS(Ω)〉. (2.46)
The states inside the integral are the Bogoliubov ground states starting from the MF ground
state with Sz = 0 along the direction Ω and including depletion [?]. Taking the integral over
all angles ensures that the state is rotationally invariant.
2.6 Conservation of Spin
We begin this Section by an experimental argument from the results of ?. We ask the
question: What is the magnetic field strength that is needed to win over the antiferromagnetic
interactions and completely polarize the condensate? The spin-dependent potential felt by
one atom due to the interaction with the other atoms is given by:
Esp.−dep. int. = c2ρ〈Sz〉2, (2.47)
6Although this statement sounds almost obvious, it cannot be made rigorous, since the MF ground state
is not defined properly for small interatomic distances. In fact, in the commonly used expressions for the
energy, it appears as if the Bogoliubov ground state has a higher energy than the MF ground state. This
paradox was pointed out recently by ?.
20
where 〈Sz〉 is the polarization per atom (taken to be in the z direction). The first-order
Zeeman energy is given by:
EZ = gµµBB〈Sz〉. (2.48)
And by a simple calculation one can find that for typical experimental parameters (ρ ∼1014cm−3, a2 − a0 ∼ 1A, µB ∼ 10−23J/T), the magnetic field strength needed to polarize a
condensate with antiferromagnetic interactions is of the order ∼ 0.1mG. This might not look
too bad, since under good magnetic isolation conditions, one can reach such low magnetic
fields. The only problem is that in current experiments the magnetic field strength is usually
higher than 20mG [?], which should be enough to polarize the condensate. However, this does
not happen in the experiment. The reason behind that phenomenon is that the condensate is
almost perfectly isolated from the surrounding environment. This poses several constraints
on the system, one of which is the conservation of total spin. Dipole-dipole interactions can
change the spins of two colliding atoms. However, this is not a point to worry about, since this
process happens slowly enough that it can be neglected. That can be seen by realizing the fact
that dipole-dipole interactions cause loss of the colliding atoms. Therefore, as long as one can
talk about a fixed number of atoms in the system, one should be able to talk about fixed total
spin. Interactions can change the Sz values of the individual atoms, keeping the total spin
unchanged. We can also see that the total spin is conserved by going back to the Hamiltonian
(Eq. 2.13). That Hamiltonian commutes with both S2total ≡:
∑∫ψ†
α(r)Sαβψβ(r)2 : and
Sz ≡ ∑∫αψ†
α(r)ψα(r), which means that if the evolution of the condensate is governed by
that Hamiltonian alone, the total spin of the condensate is conserved.
2.7 Ground State with Fixed-Spin Constraint
We have shown in Section 2.6 that the problem of the ground state of a spinor BEC must be
treated with the constraint of conserving the total spin of the condensate. ? have addressed
the same problem numerically. Here we give an analytical treatment of that problem. Our
21
treatment and results are similar to those obtained by ?.
The constraint of fixed total spin can be added to the problem by introducing a term in
the free energy which is equal to the total spin of the system times a Lagrange multiplier:
K = 〈H − µN − λ · Stotal〉. (2.49)
The new term has the same form as a first-order Zeeman term, and since the Lagrange
multiplier λ is only determined from the value of the constrained total spin, the first-order
Zeeman term can be absorbed completely in this new term. We assume that the finite spin
is in the z direction (and therefore λx = λy = 0). Now we look for a mean-field solution
of the form φ(r) =√ρ(r)/Nζ(r). Since the MF solution does not have a definite value of
Sz, the true solution will, in general, be a superposition of such states, in the same way
that the LPB states are superpositions of MF states. First we consider the relation between
the different energy scales in the problem. We always assume that the spin-independent
interaction energy scale (c0ρ) is larger than the spin-dependent one (c2ρ). If the kinetic
energy scale (2/mL2) is much larger than both interaction energy scales, the ground state
is given by a uniform polarization (〈Sz〉 = Stotal/N) and a density given by the square of
the single-particle ground-state wave function. If the kinetic energy scale is much smaller
than the spin-independent interaction energy scale and much larger than the spin-dependent
interaction energy, we can use the Thomas-Fermi approximation for the density and the
weakly interacting limit for the spin part of the wave function. The ground state then has a
uniform polarization per particle (〈Sz〉 = Stotal/N), and a density approximately given by:
ρ(r) =µ− U(r)
c0. (2.50)
If the kinetic energy scale is much smaller than both interaction energy scales, we can use
the full Thomas-Fermi limit, which requires more analysis than the other two cases. As
before, we have two cases depending on the sign of c2. In the ferromagnetic case (c2 < 0),
we find that in order to minimize the interaction term, the local spin of the condensate has
to be polarized along some direction, i.e. for every point in space there is a d such that
22
〈S · d〉 = 1. This does not violate the constraint of fixed total spin, since the polarization
direction can bend in space to satisfy that constraint. The amount of bending is determined
by the kinetic energy term. Obviously it costs less energy to bend the wave function at points
where the density is small. However, regions with small density give small contributions to
the total spin. Therefore, the GP equation has to be solved in more detail to find the exact
shape of the wave function. One point to note here is that the state described above is
not time-independent. In that state there are points where the polarization direction is not
parallel to the magnetic field (assuming there is one). At those points the spins will generally
precess around the magnetic field. This means that this state cannot be the ground state,
which has to be an eigenstate of the Hamiltonian and, therefore, stationary. This in turn
means that the true ground state of this system cannot be expressed simply as a product
state. However, by using a superposition of product states, but with a different direction of
bending in each state, one can construct a stationary state. This might be the true ground
state of the system. This point needs to be looked at more closely for a more rigorous proof.
Now we turn to the antiferromagnetic case. First we make the following observation:
The miscibility or immiscibility of two condensates is determined by the condition:
a12 <√a11a22 or a12 >
√a11a22, (2.51)
respectively. If we calculate the scattering lengths between the different Sz states, we find
that the state Sz = 0 does not mix with either of the two other states Sz = ±1. The states
Sz = ±1, however, are miscible and tend to mix unless there is no contact between them
(e.g. because of an Sz = 0 region between them). The immiscibility of the Sz = 0 state
with the other two Sz states simplifies the problem, since there can be at most two different
components of the condensate at one point in space.7 It follows that at any point with
finite spin density, there cannot be any Sz = 0 component. In fact, it turns out that in the
absence of external magnetic fields, the energy is minimized by distributing the atomic spins
7That also automatically gives 〈Sx〉 = 〈Sy〉 = 0, in agreement with our assumption that the total spin is
in the z direction.
23
between the states Sz = ±1, without populating the Sz = 0 state at all. In order to give a
better quantitative description of the spatial structure of the ground state, we take another
look at the expression for the free energy (Eq. 2.20, plus the total-spin constraint term). In
the Thomas-Fermi approximation (i.e. neglecting kinetic energy) minimizing the free energy
with respect to ρ and 〈Sz〉 gives the following set of equations for the ground state:
c2ρ2〈Sz〉 − λρ = 0, (2.52)
− (µ− U(r)) + ρ(c0 + c2〈Sz〉2) − λ〈Sz〉 = 0. (2.53)
The first of these equations has the solution:
〈Sz〉 =λ
c2ρ. (2.54)
Keeping in mind that 〈Sz〉 cannot exceed the value 1, one can see that this equation cannot
be satisfied everywhere. In fact, it will be satisfied as long as it gives a value for 〈Sz〉 that
is smaller than 1 (at the center of the trap). There is an equal density surface inside the
condensate where the equation gives 〈Sz〉 = 1. Outside that region only the state Sz = +1
is present (assuming that the total spin is in the positive z direction). The polarization per
atom is given by:
〈Sz〉 =
ρc
ρ, ρ > ρc,
1 , ρ < ρc,
(2.55)
and the density is given by:
ρ =
µ− U(r)
c0, ρ > ρc,
µ− U(r) + c2ρc
c0 + c2, ρ < ρc,
(2.56)
where the boundary density ρc is determined by the value of the total spin and the other
parameters in the problem. Since the spin density is given by the product ρ〈Sz〉, we find
that the spin density is constant in the inner region of the condensate. The above results
are shown in Fig. 2.1.
24
Now we turn to the case of an applied uniform external magnetic field. The term de-
scribing the first-order Zeeman effect can be absorbed in the term constraining the total
spin to its fixed value, since both terms contain the same dependence on density and spin,
i.e.∫drρ〈Sz〉. And since the total spin is constrained to a fixed value, the first-order Zee-
man effect does not have any effect on the problem (neglecting spin relaxation processes,
i.e. taking T1, T2 → ∞). The second-order Zeeman effect, however, introduces new physical
phenomena. The second-order Zeeman Hamiltonian is given by:
〈HZ2〉 =
∫drqB2ρ(r)〈S2
z 〉(r). (2.57)
The coefficient q is positive [?], and therefore, this new term in the Hamiltonian raises the
energy of the hyperfine states Sz = ±1 relative to the Sz = 0 state. Therefore, it favours
the conversion of Sz = ±1 pairs into Sz = 0 pairs. In the strong field limit, we would expect
that as many atoms as possible will occupy the Sz = 0 state, that number being limited by
the fixed-spin constraint. We now make the argument more quantitative.
One normally expects to see a small change as soon as the magnetic field is turned on.
However, it turns out that the condensate is not affected at all until a certain critical value
is reached. This value is determined by the energetic competition between the interaction
energy, which does not favour the formation of any Sz = 0 component, and the second-order
Zeeman energy, which does. In a region with given values of the density ρ and total spin
N〈Sz〉, the energy change of converting fN atoms from the Sz = ±1 states into the Sz = 0
state is:
∆E =
∫Sz=0
drc02ρ2
0 +
∫Sz=±1
drc0 + c2〈Sz〉2/(1 − f)2
2ρ2±1 + qB2ρ±1
−∫
all
drc0 + c2〈Sz〉2
2ρ2 + qB2ρ
= fN
c0ρ
2
(√1 +
c2c0〈Sz〉2 − 1
)− qB2
+O(f 2)
≈ fNc2ρ
4〈Sz〉2 − qB2
. (2.58)
In the above expression we have assumed that the new phase separates from the original
25
one and that the system is maintained at constant pressure. We have also used the TF
approximation, where kinetic energy is neglected. Finally, in the last step we have assumed
that c2 c0, an assumption that we do not have to make. The value of the magnetic field
at which it becomes favourable to create an Sz = 0 component is given by:
B1 =
√c2ρ0
4q〈Sz〉0, (2.59)
where ρ0 and 〈Sz〉0 are the density and polarization (per atom) at the center of the conden-
sate, and q is the coefficient of the second-order Zeeman energy term. The reason why the
nucleation of the new component starts at the center is that√ρ〈Sz〉 =
√ρc〈Sz〉 and is there-
fore smallest at the center [see Fig. 2.1(a)]. After that point a core of atoms with Sz = 0 is
formed. There the second-order Zeeman energy wins over the interaction energy and causes
the formation of the Sz = 0 component. The core is surrounded by a region consisting of
a superposition of the states Sz = ±1 and, finally, a shell of atoms with Sz = +1 at the
outside of the condensate. This new core region keeps on growing with increasing magnetic
field until another critical value of the field is reached:
B2 =
√c2ρext
4q, (2.60)
where ρext is the density of the region just outside the Sz = 0 core. At that point there are
only two regions in the condensate: the inner region with Sz = 0 and the outer region with
Sz = +1. The condensate is unaffected by increasing the magnetic field beyond that point.
The magnetic field dependence is shown in Fig. 2.2.
For typical experimental parameters c0 ∼ 10−50Jm3, c2 ∼ 10−52Jm3, ρ ∼ 1020m−3,
L ∼ 10−5m and q ∼ 10−24JT−2, we find the following relation between the different energy
scales:
Ekinetic ∼ 0.1 − 1Esp.−dep. int ∼ 10−3 − 10−2Esp.−indep. int, (2.61)
which suggests that the full Thomas-Fermi approximation can be used to describe the system.
However, the closeness between the kinetic and the spin-dependent interaction energy scales
26
causes the boundaries between the different domains in the experiment to have a finite width
[see ?]. The magnetic fields B1 and B2, between which the Sz = 0 component forms in the
condensate, are ∼ 10-100mG.
2.8 Excitation Spectrum of Quasiparticles
In scalar BEC, there are two types of elementary excitations. There are collective excita-
tions, where the characteristic length scale associated with the excitation is of the order of
the size of the condensate, and there are single-particle excitations associated with a shorter
length scale. The latter type is also called quasiparticles and can be treated locally as plane
waves propagating in a uniform condensate, a problem which was treated by ?. Finding
the spectrum of quasiparticles can be done systematically and requires less guessing abilities
than finding the frequencies of collective excitations. In the short-wavelength limit, where
the wavelength of the quasiparticles is much shorter than the length scale of the conden-
sate, we can use the local-density approximation. In that approximation each point in the
condensate is treated as a uniform condensate, and the local spectrum of quasiparticles is de-
termined. We shall use the quasiparticle equations-of-motion method to find the spectrum.
The reasoning behind that method goes as follows: Let us assume that the operator b†k,α
creates a quasiparticle with momentum k, and if necessary, α represents all internal degrees
of freedom of the quasiparticle. When the operator b†k,α is applied to the ground state of the
condensate |GS〉 it can increase or decrease the energy by the chemical potential µ (if the
creation of the quasiparticle involves an increase or decrease in the total number of atoms),
and in addition it increases the energy by the excitation energy of the quasiparticle Ek,α:(H −
∑β
µβNβ
)b†k,α − b†k,α
(H −
∑β
µβNβ
)|GS〉 = (K0 + Ek,α) b†k,α|GS〉 −K0b
†k,α|GS〉
= Ek,αb†k,α|GS〉, (2.62)
from which we infer that:
27
[H −
∑β
µβNβ, b†k,α
]= Ek,αb
†k,α. (2.63)
In order to prevent the formulae from becoming incredibly large, we now make the Bogoliubov
approximation. Without any loss of generality, we assume that only the Sz = ±1 states are
macroscopically occupied.8 Therefore, we take the general case where a0,1 ≈ a†0,1 ≈ √N1,
a0,−1 ≈ a†0,−1 ≈√N−1. The energy of the condensate is given by:
E0 =c0 + c2
2V
(N2
1 +N2−1
)+c0 − c2V
N1N−1. (2.64)
The chemical potentials of the ±1 can now be determined as:
µ±1 =∂E0
∂N±1= (c0 + c2)ρ±1 + (c0 − c2)ρ∓1 = c0ρ± c2ρ〈Sz〉. (2.65)
And to create two Sz = 0 atoms, one has to destroy an atom in the Sz = 1 state and one in
the Sz = −1 state, which gives µ0 = (µ1 + µ−1)/2 = c0ρ. We now find the commutator of
the Hamiltonian with each of the particle creation and annihilation operators, keeping only
terms that contain at least two condensate operators. For example:
[H, a†k,1
]≈ εka
†k,1 +c0
(2ρ1a
†k,1 + ρ1a−k,1 + ρ−1a
†k,1 +
√ρ1ρ−1a
†k,−1 +
√ρ1ρ−1a−k,−1
)+c2
(2ρ1a
†k,1 + ρ1a−k,1 − ρ−1a
†k,1 −
√ρ1ρ−1a
†k,−1 −
√ρ1ρ−1a−k,−1
), (2.66)
where εk is the kinetic energy of a free particle with momentum k and is given by 2k2/(2m).
In the above expression, we have assumed that there are no external magnetic fields. If that
were not the case, the magnetic field terms can be included straightforwardly. Similarly
we can find the commutators of the Hamiltonian with the other operators. Interestingly,
we find that the commutator of the Hamiltonian with any one of the operators a†k,1, a−k,1,
a†k,−1 and a−k,−1 is a linear superposition of only those operators. Therefore, the operation
of commuting the Hamiltonian with the particle creation and annihilation operators can be
8The case where the condensate contains only an Sz = 0 component can be obtained by a π/2 rotation
of the case where N1 = N−1 = N/2.
28
expressed in the language of linear algebra. The basis vectors are the operators mentioned
above. The commutation operation is then expressed as:
[H −
∑β
µβNβ , b†k,±1
]=
εk + A1 −A1 B −BA1 − (εk + A1) B −BB −B εk + A−1 −A−1
B −B A−1 − (εk + A−1)
b†k,±1, (2.67)
where A±1 = (c0 + c2)ρ±1, B = (c0− c2)√ρ1ρ−1, and on the right-hand side b†k,±1 is a column
matrix describing a four dimensional vector in the basis (a†k,1, a−k,1, a†k,−1, a−k,−1). This prob-
lem is identical to that of finding the quasiparticle spectrum of two overlapping condensates,
which has already been treated by several authors, e.g. ? and ?. The quasiparticle spectrum
can be found by solving the eigenvalue problem in Eq. (2.63), which is done by diagonalizing
the matrix in Eq. (2.67). The solution is:
E2k,±1 = ε2k + εk(c0 + c2)ρ± εk
√(c0 + c2)2(ρ1 − ρ−1)2 + 4(c0 − c2)2ρ1ρ−1
= ε2k + εkρ((c0 + c2) ±
√(c0 − c2)2 + 4c0c2〈Sz〉2
), (2.68)
where ρ = ρ1 + ρ−1, and in the last step we have used the fact that in a uniform condensate
ρ1 − ρ−1 = ρ〈Sz〉. For spin 0 quasiparticles, we find that in the basis (a†k,0, a−k,0):
[H − µ0N0, b
†k,0
]=
εk + c2ρ −2c2
√ρ1ρ−1
2c2√ρ1ρ−1 −(εk + c2ρ)
b†k,0, (2.69)
which gives the energy:
E2k,0 = ε2k + 2εkc2ρ+ c22ρ
2〈Sz〉2. (2.70)
Our results reduce to those obtained by ? in the limits of the ferromagnetic (〈Sz〉 = 1) and
antiferromagnetic (〈Sz〉 = 0) ground states. The apparent discrepancies in energy gaps in
the ferromagnetic case are due to our different accounting for the energy required to remove
a particle from the condensate. The energy spectrum of quasiparticles is shown in Fig. 2.3.
29
2.9 Conclusions
In this Chapter we have presented two different treatments to find the ground state of a Bose-
Einstein condensate of spin 1 atoms. We have studied the energetic competition between
the mean-field state and the Law-Pu-Bigelow state, which are the candidates for being the
ground state in the case of antiferromagnetic interactions. Then we argued that the problem
must be treated with the additional constraint of fixed total spin of the condensate. We
have carried out that calculation and found that, in general, the condensate forms domains
with different polarization. Finally, we have calculated the excitation energy spectrum of
quasiparticles for a uniform condensate with any value of total spin.
30
a)
r
Sz
b)
r
ρ Sz
c)
r
ρ
Figure 2.1: Atomic polarization 〈Sz〉 (a), spin density ρ〈Sz〉 (b) and number density ρ (c)as functions of distance from the center of the condensate. In Fig. (c) the solid, dashed anddotted lines correspond to the total density, density of the Sz = 1 component and density ofthe Sz = −1 component, respectively.
31
a)B < B1
r
ρ
b)B1 < B < B2
r
ρ
c)B > B2
r
ρ
Figure 2.2: Density distribution of the three components Sz = 1, 0,−1 as functions ofthe distance from the center of the condensate with external magnetic field B < B1 (a),B1 < B < B2 (b) and B > B2 (c). The solid, dashed and dotted lines correspond to thedensitites of the Sz = 0, Sz = 1 and Sz = −1 components, respectively.
32
a)Sz = 0
k
E
b)Sz = 0.5
k
E
c)Sz = 1
k
E
Figure 2.3: Quasiparticle spectrum in a uniform condensate with 〈Sz〉 = 0 (a), 〈Sz〉 = 0.5(b) and 〈Sz〉 = 1 (c).
33
Chapter 3
Measurement Theory of Spinor
Bose-Einstein Condensates
In this Chapter we study the problem of devising a method to distinguish between the
coherent (i.e. mean-field) and singlet states introduced in Chapter 2. We use a model where
atoms are allowed to escape from the condensate one by one, and their spins are measured.
We find that such a measurement cannot distinguish between the two states in question,
unless all the atoms are detected. We also find that any initial state (from a certain class of
states) will evolve into a coherent state as a result of the measurement. We then consider the
possiblity of allowing pairs of atoms to escape from the condensate instead of single atoms.
In spite of the deceptive appearance of the singlet state (which is constructed by creating
entangled pairs), it turns out that two-particle measurements have the same qualitative
effect as single-particle measurements. We then conclude by discussing some theoretical
implications of our results. We compare our model with optical imaging techniques, and we
consider the effect of atom loss on the quantum state of the condensate.
34
3.1 Introduction
In Chapter 2 we discussed the problem of the ground state of a Bose-Einstein condensate
of spin 1 atoms. In particular, in the case of antiferromagnetic interatomic interactions, we
found two different candidates for the ground state. In the mean-field approximation we
found that the ground state is given by [??]:
|C〉 =
(a†0)N
√N !
|0〉, (3.1)
where a†0 creates a particle in the lowest orbital wave function (which can be calculated from
the Gross-Pitaevskii equation) with Sz = 0 along some arbitrary direction. |C〉 stands for
(antiferromagnetic) “coherent state” and |0〉 is the true vacuum state containing no particles.
The state |C〉 is a singly-condensed state, i.e. all the particles occupy the same single-particle
state. The other candidate for the ground state was the Law-Pu-Bigelow state. With the
assumption that the orbital degree of freedom is the same for all the atoms, but allowing the
spin state to take its most general form, we found that in the absence of external magnetic
fields, the ground state of the system is not a singly-condensed state, but rather a spin-singlet
state, i.e. a state with total spin of the condensate Stotal = 0 [?]:
|S〉 = const.×(a†0a
†0 − 2a†1a
†−1
)N/2
|0〉, (3.2)
where, as in Chapter 2, N is assumed to be even for simplicity.
We also discussed the energy difference between the states (3.1) and (3.2) and showed
that in the absence of spin-dependent external fields the energies per particle of the two
states are very close, with a difference of ∼ N−1. In this Chapter we address the question
of how to devise an experiment that will be able to distinguish between the two states
[Partial treatments of this question have been carried out by ???]. In other words, we
consider an experiment where an ensemble of such condensates is produced. We know that
this ensemble is comprised either of a uniform distribution of coherent states pointing in
all possible directions, or of a collection of condensates all of which are in the singlet state.
35
An important point to note here is that we assume that there is no preferred direction for
the coherent state, and all directions appear with equal probability. Otherwise, the problem
would be rather trivial and the distinguishability between the two states is obvious [see ??].
From examining the two expressions (3.1) and (3.2), it is obvious that there are physical
differences between them. For example, the singlet state has 〈S2total〉 = 0, whereas 〈S2
total〉 =
2N for the coherent state ( = 1). Also, if one could measure the occupation numbers of the
three hyperfine states (n1, n0 and n−1), a clear difference would be that in the singlet state
one always gets n1 = n−1, even though that value may vary from shot to shot. However,
in the above examples the entire condensate has to be probed, and measurements of atom
numbers have to be accurate at least to relative order N−1/2 in order to distinguish between
the two states. As an illustration we plot the probability distribution of the different values
of n0 for the two states in Fig. 3.1. To avoid the difficulty of having to probe the entire
condensate, we shall examine measurements in which only a small number of atoms is probed.
a)N=10
n0
P
b)N=1000
n0
P
Figure 3.1: Probability of having a certain value of n0 in the coherent and singlet statesfor N = 10 (a) and N = 1000 (b).
36
One important aspect of measurement theory is the evolution of the quantum state as a
result of the measurement. An example which is related to the present problem is that of
measuring the relative phase between two condensates [For that problem see ???, see also
Chapter 4]. ? demonstrated how an experiment measuring the relative phase between two
condensates “builds up” the relative phase between them. A detection measurement that
does not give information about which condensate the atom came from creates an uncertainty
in the relative number of atoms between the two condensates. Using appropriate definitions,
it can be shown that the relative phase quantum operator and the relative number quantum
operator are, to a good approximation, canonically conjugate operators [see ??]. Therefore,
an increase in the uncertainty in relative number allows for a reduction in the uncertainty
in relative phase, which is the case in the Castin-Dalibard scheme. In the context of spinor
BEC, the coherent states are analogous to the phase states, and the singlet state is analogous
to the unbroken-symmetry number state. Therefore, we shall examine whether a similar
phenomenon exists in the measurement of a spinor BEC.
We shall show that a measurement based on the detection of a small number of atoms
from the condensate and measuring their spin structure cannot distinguish between a coher-
ent state and a singlet state. We also show that a spinor BEC in a singlet state evolves into
a coherent state as a result of the measurement. These results will be obtained using the
same method that was used by ?. In Section 3.2 we study the outcome probabilities and
the evolution of the state of the condensate following a sequence of single-particle detection
measurements. In Section 3.3 we study two-particle detection measurements and show that
the effect of the measurement process is qualitatively similar to that of a single-particle mea-
surement. In Section 3.4 we discuss some properties and implications of particle-detection
measurements. We also discuss the quantum mechanical description of absorption and phase-
contrast imaging.
37
3.2 Single-Particle Measurements
In this Section we study the outcome of a sequence of single-particle measurements. We
use a model, also used by ?, where in each step of the measurement one atom leaves the
trap and the z component of the spin of that atom is measured. In order to neglect the
spin dynamics of the condensate during the measurement, we shall assume that the time
over which the measurement process is completed is much smaller than the inverse of the
energy difference between the two initial states in question [see ?]. We also assume that the
spin-spin interaction energy is small enough that the escape rate of atoms is independent of
their spin state. The Hamiltonian describing the escape of atoms from the condensate can
be expressed as:
H =∑f,m
εfb†f,mam (3.3)
where the index f refers to the unbound states that the atom can end up in, m refers to the
z component of the atomic spin, εf is some matrix element (independent of m), b†f,m is the
creation operator that corresponds to the different (empty) unbound states, and am is the
annihilation operator that removes an atom (with Sz = m) from the condensate. We shall
not go into the details of the evolution of the many-body quantum state here. However, we
note that the condensate operator in the above Hamiltonian is am, with a coefficient that is
independent of m, so that the escape rate is spin independent. We denote the initial state
of the system by |Ψ〉0, and the state of the system after n measurements by |Ψ〉n. Each
measurement gives Sz=+1,0 or −1. According to Fermi’s golden rule, the escape rate of
Sz = m atoms is proportional to∑
f〈Ψi|H|Ψf〉〈Ψf |H|Ψi〉, where i and f are the initial and
possible final states, respectively. The condensate in the final state must have one fewer atom
with Sz = m than the initial state. By separating the condensate and escaping-atom parts
of the states |Ψi〉 and |Ψf〉, we find that the escape rate of Sz = m atoms is proportional to
〈Ψ′i|a†m|Ψ′
f〉〈Ψ′f |am|Ψ′
i〉 (which is proportional to 〈Ψi|a†mam|Ψi〉), where in |Ψ′i,f〉 we only keep
the condensate part of the quantum state. It is then plausible to expect that the probability
38
of obtaining the value Sz = m in the n-th measurement is given by:
Pn(m) =n−1〈Ψ|a†mam|Ψ〉n−1∑l
n−1〈Ψ|a†lal|Ψ〉n−1
, (3.4)
It is also plausible that if the outcome of the n-th measurement is Sz = m, the state of the
system is projected as follows:
|Ψ〉n =am|Ψ〉n−1√
n−1〈Ψ|a†mam|Ψ〉n−1
. (3.5)
Using Eqs. (3.4) and (3.5) we now analyze the measurement process for the coherent and
singlet states. Our treatment can be straightforwardly generalized to the more general class
of states with Stotal N . Note that in contrast to the work of ? and ?, we do not assume
any a priori knowledge about the direction of the broken-symmetry state (Eq. 3.1).
(1) Coherent States:
Let us assume that the initial state of the system is given by Eq. (3.1). We rewrite it in
a form that contains the preferred direction, defined by the angles θ and φ in spherical polar
coordinates, explicitly:
|Ψ〉0 =1√N !
(d1
1,0(θ)e−iφa†1 + d1
0,0(θ)a†0 + d1
−1,0(θ)eiφa†−1
)N
|0〉 ≡ |N, θ, φ〉, (3.6)
where d1m,0(θ) are the single-particle rotation matrix elements:
d1m,0(θ) = 〈m|e−iθSy |0〉 =
− 1√2sin θ , if m = 1,
cos θ , if m = 0,
1√2sin θ , if m = −1.
(3.7)
Eq. (3.6) is the state that contains N atoms, all of which occupy the same single-particle
state with Sz = 0 along the direction (θ, φ). Substituting Eq. (3.6) into Eqs. (3.4) and (3.5),
one easily finds that
Pn(m) = (d1m,0)
2(θ), (3.8)
39
and
|Ψ〉n = |N − n, θ, φ〉, (3.9)
up to an overall phase factor. This means that the probabilities in the n-th measurement
are independent of the previous measurements, and that after each measurement only the
number of particles in the condensate changes, whereas θ and φ are not affected. In fact, it
is because θ and φ are not affected at all by the application of the annihilation operator that
we refer to MF states as coherent states. This definition is different from the quantum optics
definition, where coherent states are eigenstates of the annihilation operator. However, MF
states are the closest we can get to a coherent states while constraining the system to have
a well-defined number of particles. As one would expect from rotational symmetry about
the z axis, the measurement is insensitive to the angle φ. The probability P (m1, ..., mn|θ)of finding a certain sequence of values (m1, ..., mn) for a given value of θ is just the product
of the single-measurement probabilities:
P (m1, ..., mn|θ) =∏n
Pn(m)
=(sin θ)2(k1+k−1)
2k1+k−1(cos θ)2k0 (3.10)
≈n1 en sin2 θ0 ln(
sin2 θ02
)+n cos2 θ0 ln(cos2 θ0)−2n(θ−θ0)2 , (3.11)
where
tan2 θ0 =k1 + k−1
k0, (3.12)
and km is the number of times the value Sz = m appears in the sequence (m1, ..., mn).
Since the probability in Eq. (3.10) is peaked at the value θ0 = θ, the experimental
procedure described above can be considered a measurement of θ. However, it does not give
any information about the polar angle φ. A straightforward method to measure both θ and φ
is to make measurements along three (or more) different axes, not necessarily perpendicular
to one another. Let us assume that one makes n1( 1) measurements along z1. This will
40
determine that the direction (θ, φ) lies in a ring defined by a certain value of θ relative to
z1. Making n2( 1) measurements along z2 will determine that (θ, φ) lies in a ring defined
by a certain value of θ relative to z2. Combining the two measurements, the direction (θ, φ)
will be specified by the two points where the two rings intersect.1 If (θ, φ) is determined
to be one of two directions, one can then make n3( 1) measurements along z3, and the
outcome distribution will pick one of the two directions. This phenomenon is reminiscent of
the uncertainty between χ and −χ in measuring the relative phase between two condensates
[see ?]. That uncertainty can be removed by making additional measurements with a phase
shift γ given to the atoms leaving one of the two condensates. Even if γ is different for each
escaping atom, the system is driven closer and closer to a coherent (i.e. phase) state.
Since we are assuming that we do not have any a priori knowledge about the direction
(θ, φ) of the coherent state, the probability of a certain outcome is given by the average of
Eq. (3.10) over all possible directions:
P (m1, ..., mn) =
∫dΩ
2π
(sin θ)2(k1+k−1)
2k1+k−1(cos θ)2k0 (3.13)
=Γ(k1 + k−1 + 1)Γ(k0 + 1/2)
2k1+k−1+1Γ(n+ 3/2), (3.14)
where dΩ = sin θdθdφ, and the integral covers the upper hemisphere (θ ≤ π/2), as we explain
in Appendix A.
(2) Singlet State:
Now we analyze the same measurement process explained above starting with the initial
state |Stotal = 0〉 (Eq. 3.2). The singlet state can be rewritten as (see Appendix A for
derivation):
|S〉 =
√N
2π
∫dΩ |N, θ, φ〉. (3.15)
This form allows us to straightforwardly evaluate the probabilities and projected state due
to the measurement. If we define cn(θ, φ) by:1It is highly unlikely that the rings will not intersect, and therefore we do not examine that possibility.
41
|Ψ〉n ≡∫dΩ cn(θ, φ)|N − n, θ, φ〉, (3.16)
we can make use of the quasiorthogonality of coherent states (see Appendix A) and find that
Pn(m) ≈∫dΩ(d1
m,0)2(θ)|cn−1(θ, φ)|2∫
dΩ|cn−1(θ, φ)|2 , (3.17)
and
|Ψ〉n ≈∫dΩd1
m,0(θ)e−imφcn−1(θ, φ)|N − n, θ, φ〉√
2πN
∫dΩ(d1
m,0)2(θ)|cn−1(θ, φ)|2
. (3.18)
This finally leads to:
P (m1, ..., mn) =n−1〈Ψ|a†mn
amn |Ψ〉n−1∑ln
n−1〈Ψ|a†lnaln |Ψ〉n−1
n−2〈Ψ|a†mn−1amn−1 |Ψ〉n−2∑
ln−1
n−2〈Ψ|a†ln−1aln−1 |Ψ〉n−2
· · ·
· · · 0〈Ψ|a†m1am1 |Ψ〉0∑
l1
0〈Ψ|a†l1al1 |Ψ〉0
=(N − n)!
N !
0〈Ψ|a†m1· · ·a†mn
amn · · ·am1 |Ψ〉00〈Ψ|Ψ〉0
≈∫dΩ (sin θ)2(k1+k−1)
2k1+k−1(cos θ)2k0 |c0(θ, φ)|2∫
dΩ|c0(θ, φ)|2 , (3.19)
and
cn(θ, φ) ∝ (sin θ)k1+k−1(cos θ)k0ei(k−1−k1)φc0(θ, φ). (3.20)
Note that Eqs. (3.17-3.20) are valid for a general state, as long as Stotal N . For the singlet
state c0(θ, φ) =√N/2π, which gives:
P (m1, ..., mn) ≈∫dΩ
2π
(sin θ)2(k1+k−1)
2k1+k−1(cos θ)2k0 , (3.21)
cn(θ, φ) ≈(
N√n
23/2π5/2 sin θ0
)1/2
e−n(θ−θ0)2ei(k−1−k1)φ, (3.22)
where θ0 is given by Eq. (3.12). Corrections to Eq. (3.21) are of the order O(n/N)+O((k1−k−1)
2/N) relative to the value of P . The expression for the probability is (almost) exactly
the same as that for a uniform statistical distribution of coherent states (Eq. 3.13).
42
We note that although the coefficient cn(θ, φ) becomes peaked around a certain value
of θ with increasing n, the φ dependence is only affected through a phase factor. As was
discussed in the case of coherent states, making measurements along three different axes
determines both θ and φ, and therefore in each single run it projects the singlet state into a
coherent state along a well-defined direction. It is tempting to think that changing the axis
of measurement for each atom would have the effect of washing out the localization of cn in
the coordinates θ and φ. By multiplying two factors, each of which has the form (3.22) in
a different system of coordinates, one can see that this washing-out effect does not occur.
Moreover, changing the axis of measurement rids us of having to worry about conservation of
Sz of the whole condensate. In other words, if all the measurements are made along the same
axis, the total Sz of the escaping atoms and the remaining condensate is equal to zero for
the singlet state. This constraint is lifted if we use several different axes for different atoms.
Then, the condensate is projected closer to a coherent state in both θ and φ directions.
In conclusion, a measurement performed on a few single atoms leaving the condensate
cannot be used to determine whether the condensate was in a coherent or singlet state before
the measurement. In each single run, the condensate behaves like a coherent-state conden-
sate, even if it were in a singlet state before the measurement. This phenomenon follows
immediately from realizing that the model used in this Section describes a measurement of
the direction of a coherent state. A collection of such measurements gives the probabilities
for the system to be in each state of the basis (of coherent states). In a single realization of
the meaurement the system behaves as if it were in one of the basis states, regardless of its
initial state.
3.3 Two-Particle Measurements
Let us take another look at the expression for the singlet state (Eq. 3.2). This state is
constructed by creating N/2 pairs of atoms, each in a total spin singlet state (S = 0).
43
A naive argument might say that if one takes a pair of atoms out of the condensate and
measures their total spin, one would always get the value S = 0. However, the pairs of atoms
are not bound molecules, and the two atoms that were taken out of the condensate could
have come from two different pairs. Thus both values (S = 0 and S = 2) are possible.
We shall now carry out a calculation to show the above result in more detail. The
annihilation operators of a pair of atoms are given by:
A0,0 =1√6
(a0a0 − 2a1a−1) ,
A2,0 =1√3
(a0a0 + a1a−1) ,
A2,±1 = a0a±1, (3.23)
A2,±2 =1√2a±2a±2,
where the indices refer to the total spin and the z component of the total spin of the pair.
As in Section 3.2, we can now calculate the probabilities and projected states:
(1) Coherent states:
The probability of finding a total spin S and z component m for the n-th pair is given
by:
Pn(S,m) =n−1〈Ψ|A†
S,mAS,m|Ψ〉n−1∑L,l
n−1〈Ψ|A†L,lAL,l|Ψ〉n−1
(3.24)
= r2n(θ),
where
r(θ) = 〈S,m|e−iθSy |m1 = 0, m2 = 0〉 =
1√3
, if S = m = 0,√23
(32cos2 θ − 1
2
), if S = 2, m = 0,
12sin 2θ , if S = 2, m = ±1,
12sin2 θ , if S = 2, m = ±2.
(3.25)
It follows that the probability of finding a certain sequence (S1, m1; ...;Sn, mn) is simply:
P (S1, m1; ...;Sn, mn|θ) =∏
l
r2l (θ). (3.26)
44
As with single-particle measurements, the condensate remains in a coherent state regardless
of the outcome of the measurement:
|Ψ〉n = |N − 2n, θ, φ〉, (3.27)
up to an overall phase factor. Since we are assuming a uniform distribution of coherent
states, the probability of finding a certain sequence is given by:
P (S1, m1; ...;Sn, mn) =
∫dΩ
2π
∏l
r2l (θ). (3.28)
(2) Singlet state:
Applying Eq. (3.24) to the singlet state gives:
Pn(S,m) ≈∫dΩr2
n(θ)|cn−1(θ, φ)|2∫dΩ|cn−1(θ, φ)|2 ,
which in turn gives
P (S1, m1; ...;Sn, mn) ≈∫dΩ
∏l r
2l (θ)|c0(θ, φ)|2∫
dΩ|c0(θ, φ)|2
=
∫dΩ
2π
∏l
r2l (θ). (3.29)
And using the notation |Ψ〉n ≡ ∫dΩcn(θ, φ)|N − 2n, θ, φ〉,
cn(θ, φ) ∝(
3
2cos2 θ − 1
2
)k2,0
(sin 2θ)k2,1+k2,−1(sin θ)2(k2,2+k2,−2)ei(2k2,−2+k2,−1−k2,1−2k2,2)φ,
(3.30)
where kS,m is the number of times the value (S,m) appears in the measurement sequence.
Obviously, cn(θ, φ) = c0(θ, φ) unless all measurements give the value S = 0, which has a
vanishingly small probability for n 1. In general, the function cn(θ, φ) is peaked around
two values of θ, one of them between 0 and cos−1(1/√
3) ≈ 0.3π and the other between
cos−1(1/√
3) and π/2.2 However, apart from a small region in the parameter space kS,m
that corresponds to a small probability, one of these maxima will be much greater than the
other, and we find that, as in Section 3.2, the singlet state evolves into a coherent state as
the measurement proceeds.2The physical significance of the angle cos−1(1/
√3) is that a pair of atoms, each with spin projection 0
along a direction making this angle with the z axis, cannot be in a S = 2, Sz = 0 state.
45
3.4 Discussion of the Results and Possible Experimen-
tal Realizations
We have shown in Section 3.3 that measuring S and Sz of pairs of atoms gives the same
results for both a uniform distribution of coherent states and the singlet state. Starting from
the singlet state, we found that the system will evolve, because of the measurement, into a
coherent state. We found that if in a single measurement the value S = 0 is obtained, the
spin state of the system is not affected, apart from taking two atoms out of the condensate.
It was the measurement of S = 2, m = 0,±1,±2 that caused the evolution from a singlet to
a coherent state. One may then ask the question: Is it possible to measure only S (without
measuring m) and leave the singlet state unchanged? This would be achieved if one could
perform a measurement that is described by a rotationally invariant projection operator.
Since a rotationally invariant operator cannot change c(θ, φ), it would not change a singlet
state into a coherent state. However, it can be easily shown that such an operator does
not exist for S = 0. Any operator that removes two atoms from the condensate can be
written as a linear superposition of the annihilation operators AS,m defined in Eq. (3.23).
The operators AS,m transform under the same rotation group as the spherical harmonics
Y0,0, Y2,m. The operator A0,0 possesses the desired property of transforming into itself under
any rotation and, therefore, does not affect c(θ, φ). Among the other five operators there is
none that transforms into itself under an arbitrary rotation. Therefore, when a pair with
S = 2 is detected, the angular dependence of the state of the condensate will, in general,
change.
The impossibility of preserving the singlet state can be understood from the nature of
the measurement process. In a measurement based on the detection of a particle (or group
of particles), one assumes that at the moment that the particle is detected, it ceases to be
part of the system. Before the measurement, the spin states of the escaping atom and the
rest of the condensate are entangled, i.e. neither of them is determined independently of the
46
other. For example, when the first atom leaves from a singlet state, the composite system is
described by the Stotal = 0 state
|Ψ〉 =1√6
(| + 1〉a|1,−1〉C + | − 1〉a|1,+1〉C − 2|0〉a|1, 0〉C
), (3.31)
where the subscripts a and C refer to the atom and the condensate, respectively. If the atom
hits a detector, the entanglement has to remain between the atom and the condensate, or
it is transmitted to other degrees of freedom in the environment such that the total spin
is conserved. This would result in an entangled state of the condensate and the degrees
of freedom of the envirenment, provided the environment started in a state of well-defined
total spin. As long as we do not allow for such entanglement, it is necessary to specify the
exact state of the atom at detection, i.e. to project out the appropriate component of Eq.
(3.31). As a result, the condensate is left with S = 1 and a definite m value (along some
direction). In the case of a pair of atoms leaving the condensate simultaneously, arguments
similar to those given in Section 3.3 lead to the same conclusion. If one does not “read
out” all the information obtained by the measurement, the condensate will be described by
a mixed state, which simply reflects the ignorance of the experimenter of the well-defined
outcome of the measurement. We stress, though, that one cannot speak of the difference
between a mixed and pure state unless that difference can be measured experimentally.
We note that since the environment acts as a measurement device, our arguments apply
to the interaction between the condensate and its environment, even without any human
intervention. An escaping atom will interact with the environment, and its spin component
along some axis will be measured. The important point here is that regardless of the direction
of the measurement axis, the state of the condensate will be projected closer to a coherent
state, even if a different direction of this axis is chosen for each escaping atom.3 Another
point worth mentioning is that even if the atoms remain entangled with the condensate until
a measurement is performed on the condensate, tracing over the states of the escaping atoms
3Although we have not demonstrated this result explicitly, it is overwhelmingly plausible from the analogy
with phase measurement [see ?].
47
gives the same results as above. Therefore, just by losing atoms from the condensate, a singlet
state will evolve into a coherent state. This looks like spontaneous symmetry breaking, except
that it is not caused by any residual external fields, but rather by continuous monitoring of
the system. The chosen direction is determined entirely by chance, just like the outcome of
any measurement in quantum mechanics. In the absence of any symmetry-breaking external
fields, the competition between this measurement mechanism and stochastization due to
interatomic interactions, which was discussed by ?, will determine the state of the system.
Typically, the energy difference between the coherent and singlet state is of the order of
100s−1, whereas atom loss rates are of the order of 104s−1. This suggests a highly coherent
state. ? found a similar result for a harmonic oscillator in contact with its environment.
One should keep in mind that although the above argument is conceptually convenient, an
ensemble describing a uniform distribution of coherent states is equivalent to an ensemble of
definite Stotal states with the appropriate weights.
Now we turn to the question of how realistic is our model for describing the measurement
process. We have assumed that the condensate is not tightly bound inside the trap so that
occasionally an atom will escape and will be detected. In practice, however, condensates
are probed using optical imaging techniques. In order to be able to give the correct de-
sciption of the measurement, one has to understand the quantum mechanical description of
the interaction between the condensate and the imaging laser beam [???]. Near-resonance
absorption imaging is closest to our model in the sense that an atom is removed from the
condensate upon detection. In other words, the condensate operator in that kind of mea-
surement is the annihilation operator am. When imaging a scalar (i.e. spinless) condensate
a scattered photon projects the state of the condensate |Ψ〉 into a|Ψ〉, and a nonscattered
photon projects it into
√1 − γNeiδ(N)|Ψ〉, where γ is the absorption probability per photon
per atom, and normalization constants are implicitly understood. This expression can be
obtained by considering the probability for a photon not to be scattered by a condensate
with N atoms (Probability= 1 − γN in the dilute limit) and the phase shift δ(N) given to
48
such a photon, which is proportional to N in the dilute limit. Here we are assuming that
an excited atom leaves the trap before the next photon arrives at the condensate. When
imaging a spinor condensate, using appropriate probe beam frequency and polarization one
could choose to image each of the three hyperfine states separately. The initial absorption
rate is proportional to the number of atoms in the imaged hyperfine state and cannot be
used to distinguish between a coherent and singlet state, in agreement with our result. One
has to measure the population of the +1 and −1 states to relative accuracy better than
N−1/2 in order to distinguish between the coherent and singlet states, since in the singlet
state n1 = n−1. However, that measurement requires counting essentially all the atoms in
the different hyperfine states, which is exactly the difficulty we are trying to avoid.
Phase-contrast imaging differs from our model in a more fundamental way: the projection
operator associated with this type of measurement, which does not remove atoms from the
condensate, is a†mam rather than am. Another less fundamental difference occurs if the
linewidths of the atomic states are comparable to or greater than the hyperfine splitting
energy scale. In that case it is not possible to image each one of the three states separately.
Assuming the linewidths are small enough, we find that, as in absorption imaging, in order
to distinguish between the coherent and singlet states, one has to measure the different n’s
to relative accuracy better than N−1/2.4 The difference between the two methods is that
in absorption imaging the condensate is destroyed after the measurement due to heating,
whereas in phase-contrast imaging the condensate ends up in a state of definite n1, n0 and
n−1 without losing atoms from the condensate (neglecting heating effects).
4An amusing idea is that if one could find a laser beam that is phase-shifted by δ per Sz = 1 atom,
by −δ per Sz = −1 atom and is not affected by Sz = 0 atoms, one would be able to measure n1 − n−1,
without measuring n1 and n−1 separately. In that case the difficulty of relative accuracy of ∼ N−1/2 would
be avoided. Unfortunately, we do not have a realistic experimental suggestion that would realize the above
description.
49
3.5 Conclusions
We have shown that a measurement based on the detection of a small number of atoms leaving
a Bose-Einstein condensate cannot be used to distinguish between an antiferromagnetic
coherent state and a spin singlet state. A singlet state will be projected, because of the
measurement, closer and closer to a coherent state. Atom loss from the condensate has
the same effect as a detection measurement, and therefore it provides a mechanism for
spontaneous symmetry breaking. We have neglected the effect of spin-dependent interatomic
interactions on the dynamics of the condensate. Those effects can be significant if the relevant
energies are large compared to the inverse of the measurement time. However, they do not
affect our result that one needs to measure occupation numbers to relative accuracy better
than N−1/2 in order to distinguish between a coherent and a singlet state.
50
Chapter 4
Interference Between Spinor
Bose-Einstein Condensates
In this Chapter we discuss the appearance of interference fringes between two overlapping
Bose-Einstein condensates. In Section 4.1 we rederive the well-known result that if two
scalar (i.e. spinless) condensates of equal size are made to overlap, interference fringes will
be observed in the overlap region. The method that we use here is similar to that used in
Chapter 3. In Section 4.2 we discuss the same problem for two condensates with different
numbers of atoms. The result agrees with the intuitive prediction that interference fringes
will be abserved. However, since the algebra involved in deriving that result is somewhat
more complicated than the previous case, we treat it in some detail. In Section 4.3 we study
the interference patterns arising from the overlap of two condensates each of which is in
the antiferromagnetic ground state. We analyze two cases, namely when the overlapping
condensates are in coherent or singlet states. The purpose of considering these two cases is
to study the possibility of using interference experiments to distinguish between them. It will
turn out that the two different states are indistinguishable in such an experiment. In Section
4.4 we study interference between two condensates each of which is in the ferromagnetic
ground state.
51
4.1 Interference Between Two Scalar Condensates with
Equal Numbers of Atoms
We begin this Chapter by outlining an analytical method to predict the emergence of an
interference pattern when two scalar condensates are made to overlap. The results of this
section are well-known from analytical [????], experimental [?] and numerical [?] studies. It
mainly serves as an introduction to the following Sections, where we generalize the calculation
to more complicated cases.
In the original MIT interference experiment by ?, two Bose-Einstein condensates were
prepared independently on the two sides of a double-well potential. The potential was formed
by a combination of a magnetic trap with a blue-detuned laser sheet dividing the magnetic
trap into two regions with negligible tunneling between them. After the two condensates
were formed, the trap was turned off, and the two condensates expanded. After waiting
for period of time to allow the two condensates two overlap, the condensates were imaged
and interference fringes were observed. In every run of the experiment interference fringes
with an estimated visibility of 100% were observed, but the position of the maxima varied
from shot to shot (with the distance between fringes being constant). Interference fringes
are expected to be observed between two condensates if the relative phase between them is
well-defined. In the experiment mentioned above, however, the two condensates were created
independently, and they did not initially have a well-defined relative phase. In this Section
we explain how overlapping two independently prepared condensates produces interference
fringes.
For simplicity we shall treat the problem in one dimension. We shall neglect interatomic
interactions.1 We shall also assume that the two condensates have equal numbers of atoms
and are pushed towards each other such that at the time of imaging they occupy the single-
1Interatomic interactions add quantitative complications to the problem. However, they do not affect the
appearance or absence of interference fringes. For example, see ?.
52
particle wave functions θ(1/2 + x)θ(1/2 − x)e±ipx respectively, where θ(x) is the Heaviside
step function and p is a small multiple of π, so that at most a few interference fringes are
formed. The creation operators for the two condensates are a†(±p) =∫ 1/2
−1/2dxe±ipxψ†(x).2
The state of the system can be expressed in the form [?]:
|Ψ〉0 =
(a†(p)
)N/2 (a†(−p))N/2
(N/2)!|0〉
=
(πN
2
)1/4 ∫ π
−π
dχ
2π√N !
(eiχ/2
√2a†(p) +
e−iχ/2
√2a†(−p)
)N
|0〉
=
(πN
2
)1/4 ∫ π
−π
dχ
2π√N !
(√2
∫ 1/2
−1/2
dx cos(px+ χ/2)ψ†(x)
)N
|0〉
=
(πN
2
)1/4 ∫ π
−π
dχ
2π|χ〉, (4.1)
where the phase states |χ〉 are defined by the last step of the above equation. Eq. (4.1)
says that a product of two Fock states, which has a well-defined number of atoms in each
condensate, is a linear superposition of all possible phase states between −π and π, each of
which would produce interference fringes. The imaging device is modelled by a large number
of adjacent atom detectors covering the entire spatial extent of the condensates. In each
detection event, an atom is removed from the condensate at the position of the respective
detector. The interference pattern is then obtained by plotting a histogram of the number
of detected atoms as a function of position. In a real experiment it is the time of shining the
imaging laser beam that is controlled and not the number of detected atoms. However, our
method demonstrates the emergence of the interference pattern without going through the
details of how the image is produced.
The probability density of finding the n-th detected atom at position x is given by:
2We restrict the position x to values between -1/2 and 1/2, so that the momentum states are defined
with periodic boundary conditions between x = −1/2 and x = 1/2.
53
ρn(x) =n−1〈Ψ|ψ†(x)ψ(x)|Ψ〉n−1
n−1〈Ψ|N − n+ 1|Ψ〉n−1
=2∫dχ1dχ2 cos(px+ χ1/2) cos(px+ χ2/2)c∗(χ1)c(χ2)〈χ1|χ2〉∫
dχ1dχ2c∗(χ1)c(χ2)〈χ1|χ2〉≈ 2
∫dχ cos2(px+ χ/2)|cn−1(χ)|2∫
dχ|cn−1(χ)|2 , (4.2)
where cn(χ) is defined implicitly by |Ψ〉n ≡ ∫dχ cn(χ)
(∫dx
√2 cos(px+ χ/2)ψ†(x)
)N−n |0〉,and in the last step we used the quasiorthogonality of phase states [?, see also Appendix A]:
〈χ1|χ2〉 =1
N !〈0|
(eiχ1/2
√2a(p) +
e−iχ1/2
√2
a(−p))N (
eiχ2/2
√2a†(p) +
e−iχ2/2
√2
a†(−p))N
|0〉
= cosN
(χ1 − χ2
2
)
≈√
8π
Nδ(χ1 − χ2). (4.3)
The state of the system after n measurements is:
|Ψ〉n =ψ(xn)|Ψ〉n−1√
n−1〈Ψ|ψ†(xn)ψ(xn)|Ψ〉n−1
∝∫dχ cos(pxn + χ/2)cn−1(χ)
(∫dx
√2 cos(px+ χ/2)ψ†(x)
)N−n
|0〉. (4.4)
Thus the probability density of finding the first n detected atoms at x1, ..., xn is:
ρ(x1, ..., xn) =0〈Ψ|ψ†(x1) · · ·ψ†(xn)ψ†(xn) · · ·ψ(x1)|Ψ〉0
0〈Ψ|N(N − 1) · · · (N − n+ 1)|Ψ〉0≈∫dχ
2π2n∏
l
cos2(pxl + χ/2). (4.5)
This expression is what one would expect to find for a uniform distribution of initial states
with well-defined relative phases. In fact, Eq. (4.5) is sufficient to show that the interference
pattern that arises from the spatial overlap of two condensates is the same whether the
initial state is a product of two independent Fock states or a uniform statistical distribution
of phase states. However, we shall go on and find explicit expressions for the probability of
finding a certain density distribution.
A given density distribution is defined by the occupation of K cells of width 1/K and
centred at Xl = l/K (corresponding to the detectors). Some manipulation of Eq. (4.5) gives
the probability of finding the distribution kl of atoms in the cells labelled by l:
54
P (k−K/2, ..., kK/2) ≈∫dχ
2πF (k−K/2, ..., kK/2|χ), (4.6)
where
F (k−K/2, ..., kK/2|χ) =n!
Kn
∏l
R(pXl + χ/2)kl
kl!, (4.7)
and
R(pXl + χ/2) = 2 cos2(pXl + χ/2) +O( pK
)2
. (4.8)
The function R(pXl + χ/2) describes the interference pattern one finds for a well-defined
value of the relative phase χ. For large K the second term in Eq. (4.8) can be neglected
for most values of Xl, but we keep it in mind to avoid divergences that would arise in our
approximate expressions if R(pXl + χ/2) = 0. A straightforward variational calculation
shows that for a given value of χ, the function F takes its maximum value when
kl =n
KR(pXl + χ/2). (4.9)
We also find that at the point of maximum value
F ≈√
n
(2π)K−1∏
l kl, (4.10)
and
d2 lnF
dkldkm≈ − K
nR(pXl + χ/2)δl,m. (4.11)
It should be noted that these expressions are good approximations only when all the kl’s are
large. We shall make that assumption, since it does not affect the essential physics of the
interference process. F (k−K/2, ..., kK/2|χ) can now be approximated by:
F (k−K/2, ..., kK/2|χ) ≈√
n
(2π)K−1∏
l kl
e−nG, (4.12)
where
55
G =K
n2
∑l
1
R(pXl + χ/2)
(kl − n
KR(pXl + χ/2)
)2
≈∫dx
(f(x) −R(px+ χ/2))2
R(px+ χ/2), (4.13)
and f(x) is a smooth function defined between x = −1/2 and x = 1/2 such that f(Xl) =
Kkl/n.
The probability of finding a certain (normalized) density distribution f(x) takes the
maximum value if f(x) = R(px + χ/2) for some value of χ, and drops to negligibly small
values when G K/n for all values of χ. Thus, in any single interference experiment one
expects to find:
f(x) = R(px+ χ/2) +
√K
nε(x), (4.14)
where ε(x) is a random function with |ε(x)| 1. The function R(px+χ/2) describes a smooth
sinusoidal density distribution, whereas ε(x) describes shot-to-shot fluctuations around that
distribution (Note that these fluctuations are not a result of the fact that the initial state
is a product of Fock states, and they exist even if the initial state is a phase state). The
visibility of the function R(px+ χ/2) is:
V ≡ Rmax − Rmin
Rmax +Rmin= 1 −O(
p
K)2. (4.15)
In any single run, an interference pattern with almost 100% visibility is observed (correspond-
ing to a randomly chosen value of the relative phase). We now generalize the treatment given
in this Section to more complicated cases.
4.2 Interference Between Two Scalar Condensates with
Different Numbers of Atoms
From the arguments in Section 4.1, one would intuitively expect that if two condensates
with unequal numbers of atoms are made to overlap, they should still produce interference
56
fringes, except that the visibility is reduced. In fact, in the MIT interference experiment by
?, the numbers are expected to differ by about 10% due to experimental fluctuations. The
calculation to demonstrate the interefernce effect, however, is not as simple as one might
intuitively expect. Therefore we carry it out in detail here.
First we try to use the same phase-state basis as in Section 4.1 to treat this problem. In
principle, that approach should give the correct results, since that basis is overcomplete and
can therefore be used to describe any state of the condensate where only two single-particle
states are occupied. The initial state is then given by (see Appendix A for derivation):
|Ψ〉0 =
(a†(p)
)Np(a†(−p))N−p√
Np!(N −Np)!|0〉
= const.×∫ π
−π
dχ
2π√N !ei(N−p−Np)χ/2
(eiχ/2
√2a†(p) +
e−iχ/2
√2a†(−p)
)N
|0〉
= const.×∫ π
−π
dχ
2π√N !eiφ(N−p−Np)χ/2
(√2
∫ 1/2
−1/2
dx cos(px+ χ/2)ψ†(x)
)N
|0〉. (4.16)
Although the above rewriting of the initial state is exact and it appears as if the calculation
will go through smoothly as in Section 4.2, the phase factor ei(N−p−Np)χ/2 will turn out to
complicate it. When we used the quasiorthogonality property of phase states (Eq. 4.3) in
deriving Eq. (4.5), we assumed that the prefactor of the phase states inside the integral
varies slowly as a function of χ, which is clearly not the case with this new phase factor. The
unsuitability of that basis to describe the general problem can also be seen from the fact
that the basis with equal prefactors of a†(p) and a†(−p) intuitively suggests that one would
always see an interference pattern with visibility 100%. If we overlap one big condensate
with a small one, however, the resulting interference pattern cannot possibly have 100%
visibility. That leads us to consider the following basis of phase states:
|χ〉 ≡ 1√N !
(√Np
Neiχ/2a†(p) +
√N−p
Ne−iχ/2a†(−p)
)N
|0〉. (4.17)
When the basis states in Eq. (4.17) are expressed in the number-state basis, one can see
that they are peaked at the desired numbers Np and N−p. That is an initial indication that
57
they might be better suited to treat the problem at hand. The initial state |Np, N−p〉 can
now be expressed as:
|Ψ〉0 =
(a†p)Np
(a†−p
)N−p
√Np!N−p!
|0〉
=
(2πNpN−p
N
)1/4∫ π
−π
dχ
2π√N !e−i(Np−N−p)χ/2
(√γeiχ/2a†p +
√1 − γe−iχ/2a†−p
)N
|0〉,(4.18)
where γ = Np/N . The rapidly varying phase factor has not disappeared with the new
definition of phase states. At first sight this point seems somewhat discouraging. However,
let us look at the inner product of two phase states in the new basis (see Appendix A for
the details):
〈χ1|χ2〉 ≈ e−N(γ−γ2)(χ2−χ1)2/2eiN(2γ−1)(χ2−χ1)/2. (4.19)
The phase factor in the above equation will turn out to cancel the varying phase factor in
the initial state, since N(2γ − 1) = Np −N−p, and that will make this basis useful. Now we
proceed as in in Section 4.1 and find that for the initial state in Eq. (4.18)
ρ(x1, ..., xn) =0〈Ψ|ψ†(x1) · · ·ψ†(xn)ψ†(xn) · · ·ψ(x1)|Ψ〉0
0〈Ψ|N(N − 1) · · · (N − n+ 1)|Ψ〉0≈√N(γ − γ2)
8π3
∫dχ1dχ2e
−N(γ−γ2)(χ2−χ1)2/2ein(2γ−1)(χ2−χ1)/2
∏l
(√γe−i(pxl+χ1/2) +
√1 − γei(pxl+χ1/2)
)(√
γei(pxl+χ2/2) +√
1 − γe−i(pxl+χ2/2))
≈∫dχ
2π
∏l
∣∣∣√γei(pxl+χ/2) +√
1 − γe−i(pxl+χ/2)∣∣∣2 . (4.20)
Following similar analysis to that in Section 4.1 we conclude that the function describing the
interference pattern is given by:
R(x) =∣∣∣√γei(px+χ/2) +
√1 − γe−i(px+χ/2)
∣∣∣2= 1 + 2
√γ(1 − γ) cos(2px+ χ). (4.21)
This function agrees with our intuitive guess that interference fringes will be observed, even
if the two condensates have different numbers of atoms. The visibility, which is given by
58
V = 2√γ(1 − γ), (4.22)
is equal to 100% when γ = 1/2, i.e. when the two condensates have equal numbers of atoms,
and gradually drops to reach the value 0 when γ = 0 or γ = 1, as one would intuitively
expect.
4.3 Interference Between Two Spinor Condensates I:
The Antiferromagnetic Case
We now generalize the method of Section 4.1 to the case of spinor condensates. In spinor
condensates, atoms belonging to different hyperfine states do not interfere. Therefore, the
appearance or absence of interference fringes depends on the spin structure of the overlapping
condensates. In this Section we shall study the interference patterns that would arise from
the spatial overlap of two spinor condensates each of which is in the antiferromagnetic ground
state, and we shall show that such an experiment gives the same results whether we start
with coherent or singlet states.
At the time of imaging we assume that the state can be approximated by one of the
following two possibilities:
(1) Coherent states:
|Ψ〉0 =1
(N/2)!
(a†(p, θ1, φ1)
)N/2 (a†(−p, θ2, φ2)
)N/2 |0〉
=
(πN
2
)1/4 ∫dχ
2π√N !
(eiχ/2
√2a†(p, θ1, φ1) +
e−iχ/2
√2a†(−p, θ2, φ2)
)N
|0〉. (4.23)
(2) Singlet states: From Eq. (2.41) we can see that:
|Ψ〉0 ∝∫dΩ1dΩ2
(a†(p, θ1, φ1)
)N/2 (a†(−p, θ2, φ2)
)N/2 |0〉
∝∫dΩ1dΩ2dχ
(eiχ/2
√2a†(p, θ1, φ1) +
e−iχ/2
√2a†(−p, θ2, φ2)
)N
|0〉, (4.24)
where
59
a†(±p, θ, φ) = −sin θe−iφ
√2
a†1(±p) + cos θa†0(±p) +sin θeiφ
√2
a†−1(±p), (4.25)
and the creation operators a†m(±p) are the (obvious) generaliztion of the a†(±p)’s used in
Section 4.1.3
Assuming that the detectors measure the spin of the atom as well as its position, we find
that the probability density for finding n atoms at x1, ..., xn with Sz values m1, ..., mn is:
(1) Coherent states:
ρ(x1, m1; ...; xn, mn) ≈∫
dχ
2π
∏l
Rml(pxl + χ/2, θ1, θ2, φ1 − φ2). (4.26)
(2) Singlet states:
ρ(x1, m1; ...; xn, mn) ≈∫dχdΩ1dΩ2
(2π)3
∏l
Rml(pxl + χ/2, θ1, θ2, φ1 − φ2), (4.27)
where, to zeroth order in p/K,
Rm(pxl + χ/2, θ1, θ2, φ1 − φ2) =(d1
m,0)2(θ1) + (d1
m,0)2(θ2)
2(4.28)
+d1m,0(θ1)d
1m,0(θ2) cos(2pxl + χ+m(φ2 − φ1)).
By averaging Eq. (4.26) over all directions for both condensates and comparing that expres-
sion with Eq. (4.27), the indistinguishability between the coherent and the singlet states
becomes obvious. What that means is that in a single run of the experiment, the interfering
condensates behave as if they were in coherent states, regardless of their initial state. If the
initial state is a product of two singlet states, a coherent-state direction is chosen randomly
for each condensate, and from the arguments of Section 4.1 a relative phase between the
condensates is also chosen randomly.
Now we calculate the probability of finding a certain density distribution (which is now
a three-component quantity). The results apply for both a uniform distribution of coherent3In the interference experiment by ?, the condensates were not pushed towards each other, but rather let
to expand until they overlapped. A theoretical description of such an experiment requires an understanding
of the hydrodynamics of expansion, which to our knowledge has not been carried out explicitly for spinor
condensates. A related problem was treated by ?.
60
states and a product of two singlet states. The calculation parallels that of Section 4.1.
Therefore, we shall skip some of the intermediate steps. The probability of finding the
density distribution kl,m (where l and m are orbital and spin indices, respectively) is given
by:
P (kl,m) =
∫dχdΩ1dΩ2
(2π)3F (kl,m|χ, θ1, θ2, φ1, φ2), (4.29)
where
F (kl,m|χ, θ1, θ2, φ1, φ2) ≈√
n
(2π)3K−1∏
l,m kl,me−n
∑m Gm , (4.30)
and
Gm =K
n2
∑l
(kl − n
KRm(pXl + χ/2, θ1, θ2, φ1 − φ2)
)2
Rm(pXl + χ/2, θ1, θ2, φ1 − φ2)
≈∫dx
(fm(x) − Rm(px+ χ/2, θ1, θ2, φ1 − φ2))2
Rm(px+ χ/2, θ1, θ2, φ1 − φ2), (4.31)
and fm(x) is a smooth function defined between x = −1/2 and x = 1/2 such that fm(Xl) =
Kkl,m/n. The probability is maximized when G1 = G0 = G−1 = 0, i.e. when fm(x) =
Rm(px+ χ/2, θ1, θ2, φ1 − φ2) for some χ, θ1, θ2, φ1 and φ2.4
Unlike scalar condensates where interference fringes are obtained in every run of the
experiment, the appearance or absence of interference fringes has a probabilistic nature in
the case of spinor condensates. From Eq. (4.28) we can see that unless θ1 = θ2, the visibility
of the interference fringes will be less than 100%. The visibility of the m component of the
interference pattern is given by:
Vm ≡ Rmaxm − Rmin
m
Rmaxm +Rmin
m
=2d1
m,0(θ1)d1m,0(θ2)
(d1m,0)
2(θ1) + (d1m,0)
2(θ2). (4.32)
4The prefactor in Eq. (4.30) is largest for those interference patterns having Rminm
∼= 0. However, this
does not mean that these patterns have higher probability than others, because that factor is cancelled out
by another factor corresponding to the spread in kl,m around its optimum value.
61
In particular, if θ1 = 0, θ2 = π/2 (or vice versa), the visibility is 0 and there are no inter-
ference fringes. Note, however, that if the z axis of the detectors is rotated to a direction
perpendicular to both (θ1, φ1) and (θ2, φ2), one sees interference fringes with 100% visibility.
Thus, the interference pattern depends not only on the relative angle between (θ1, φ1) and
(θ2, φ2), but also on the quantization axis of the detectors. This is a well-known phenomenon
in neutron interference experiments [see e.g. ?].
The total density ρ(x) = (n/K)∑
mRm(px + χ/2, θ1, θ2, φ1 − φ2), however, must be
rotationally invariant (under rotations in spin space), and that gives the total visibility:
Vρ ≡ ρmax − ρmin
ρmax + ρmin= | cos θ|, (4.33)
where θ is the angle between (θ1, φ1) and (θ2, φ2). If θ = 0, the total density shows interfer-
ence fringes with vanishing minima (100% visibility), whereas if θ = π/2, the total density
is constant in space (0% visibility). In an ensemble of measurements, one finds a uniform
distribution of all values between 0 and 100%. There is no other system, to our best knowl-
edge, where in repeated experiments interference fringes are observed with varying visibility,
provided the setup (including external fields) are not changed. In single particle interference
experiments, for example, both the offset and the visibility are determined by the experimen-
tal setup. In the spinless-condensate interference experiment the offset of the fringes varies
from shot-to-shot, but the visibility is always 100%. We shall see the same phenomenon of
randomly varying visibility in a similar problem with spin 1/2 atoms in Chapter 5.
4.4 Interference Between Two Spinor Condensates II:
The Ferromagnetic Case
Our motivation to study interference between spinor condensates was to investigate the
possibility of using that kind of experiment to distinguish between the coherent and the
singlet states of the antiferromagnetic case [see ?]. However, for completeness we analyze
62
the ferromagnetic case as well. There is no analog of the singlet state in this case. Therefore,
our analysis will be restricted to the interference between two condensates, each of which is
in the ferromagnetic ground state along some direction. At the end of the calculation, we
shall treat two cases. First we take each of the two directions to be determined randomly,
so that they are completely uncorrelated. The second application of this analysis is to the
interference experiment by ?, where we study the effect of inhomogeneities of the trapping
magnetic field on the visibility of the observed interference fringes. Since in that experiment
a magnetic trap was used, the condensate had the form of one with ferromagnetic interactions
with the total spin aligned along the external magnetic field. If the direction of the field were
different in the two traps, the visibility would be reduced, as we shall show in this Section.
In the real experiment, however, the visibility was only about 40% due to other experimental
difficulties with the imaging device. Therefore, the effect that we shall calculate here would
not be observable in their setup.
As before, we assume that the two occupied orbital wave functions are θ(x+1/2)θ(1/2−x)e±ipx, where θ(x) is the Heaviside step function. We also assume that each one of the two
wave functions contains N/2 atoms. The initial state of the system is then given by:
|Ψ〉0 =
(a†(p, θ1, φ1)
)N/2 (a†(−p, θ2, φ2)
)N/2
(N/2)!|0〉
=
(πN
2
)1/4 ∫dχ
2π√N !
(eiχ/2
√2a†(p, θ1, φ1) +
e−iχ/2
√2a†(−p, θ2, φ2)
)N
|0〉, (4.34)
where in the ferromagnetic case
a†(p, θ, φ) = cos2(θ/2)e−iφa†1(p) +1√2
sin θa†0(p) + sin2(θ/2)eiφa†−1(p). (4.35)
In the ferromagnetic case, the state |N, θ, φ〉 (defined as 1√N !
(a†(p, θ, φ)
)N |0〉) is not related
to the state pointing in the opposite direction, i.e. |N, π − θ, φ± π〉. In fact, the two states
are orthogonal to one another. Therefore, unlike the antiferromagnetic case, where the angle
θ was restricted to values between 0 and π/2, in this case θ ranges from 0 to π. One can
go through similar steps as Sections 1 and 3 to find that in any run of the experiment, the
observed interference pattern is described by the set of functions:
63
R1(x) =∣∣cos2(θ1/2)ei(−φ1+px+χ/2) + cos2(θ2/2)ei(−φ2−px−χ/2)
∣∣2= cos4(θ1/2) + cos4(θ2/2) + 2 cos2(θ1/2) cos2(θ1/2) cos(2px+ χ + φ2 − φ1) (4.36)
R0(x) =1
2
(sin2 θ1 + sin2 θ2 + 2 sin θ1 sin θ2 cos(2px+ χ)
)(4.37)
R−1(x) = sin4(θ1/2) + sin4(θ2/2) + 2 sin2(θ1/2) sin2(θ2/2) cos(2px+ χ+ φ1 − φ2), (4.38)
where the relative phase χ is determined randomly during the experiment.
The total visibility, i.e. the visibility of the total density, is given by:
Vρ ≡ ρmax − ρmin
ρmax + ρmin= cos2(θ/2), (4.39)
where θ is the angle between the polarization vectors of the of the two condensates. As in
the antiferromagnetic case, the total visibility does not depend on the polarization axis of
the detectors, whereas the visibility of each component does. If we take the polarizations of
the two condensates to be pointing opposite to each other (θ = π), the total visibility is zero.
Depending on the polarization axis, however, the visibilities of the individual components
can take any value from 0 to 100%.
We now consider two applications of the above calculation to interference experiments.
First, if we take two independently prepared condensates and assume a uniform distribution
of states where each condensate is polarized along a random direction, we find a uniform
distribution in the visibility, i.e. in an ensemble of experiments all the different values of
V occur with equal probability. The second application is estimating the reduction in the
visibility in the MIT interference experiment due to inhomogeneities in the magnetic field.
In order to reduce the visibility of the interference fringes to 95%, the angle θ must be ≈ 26.
Therefore, this effect should not be observable at all in the real experiment.
4.5 Conclusions
We have analyzed the interference patterns arising from the overlapping of two conden-
sates. We have derived the well-known result that when two spinless condensates overlap,
64
interference fringes are observed, even if in the initial state the condensates did not have
a well-defined relative phase. We have generalized the result to the case when the two
condensates have different numbers of atoms. We have investigated the possibility of us-
ing interference experiments to distinguish between antiferromagnetic coherent and singlet
states. We found that the two states in question are indistinguishable in interference exper-
iments. In both cases we found that the appearance or absence of interference fringes has
a probabilistic nature, a phenomenon that is not observed in any other system known to
us. Another amusing result is that the interference pattern depends not only on the relative
orientation of the spin states of the overlapping condenstes, but also on the quantization
axis of the atom detectors. Finally, we analyzed the interference between two condensates
each of which is in the ferromagnetic ground state, and we showed that the reduction in the
visibility due to the possible misalignment of the magnetic field in the two condensates is
too small to be observed in the MIT interference experiment.
65
Chapter 5
Bose Condensation of Spin 1/2 Atoms
In this Chapter we study the structure of a condensate of spin 1/2 atoms. In Section 5.1
we describe briefly the physical system. In Section 5.2 we present an argument based on the
concept of spontaneous symmetry breaking of the U(1) gauge symmetry in Bose-Einstein
condensates. We discuss two paradoxes related to the predictions of that argument, and
we derive the main result of this Chapter, namely the formation of a fragmented-state
condensate. In order to study a clean problem where the new suggested phenomenon is
stressed without worrying about the other physics involved in real experiments, we assume
that the Hamiltonian is SU(2) symmetric. Recently, the fragmented state of Section 5.2
was also obtained by ?. In Sections 5.3 and 5.4 we analyze some aspects about the new
fragmented state. In Section 5.5 we discuss the validity of the approximation made in
Section 5.2, and we study the effects of some spin-relaxation mechanisms that occur in the
real system. In Section 5.6 we derive and analyze a set of Gross-Pitaevskii equations that
describe the two populated wave functions in the fragmented state. In Section 5.7 we discuss
the possible detection of the fragmented state and the evolution of the quantum state as a
result of imaging the condensate. In Section 5.8 we analyze the possible realization of the
experiment in a toroidal trap and the interesting consequences of using that geometry.
66
5.1 Spin 1/2 Bosons?
At first sight it looks as if the problem treated in this Chapter is a mathematical toy and
physically irrelevant, since spin 1/2 particles are fermions. Therefore, they will obey Fermi-
Dirac statistics and cannot form a Bose-Einstein condensate. However, the important point
to realize here is that the spin of the particles enters a physical problem in two places, namely
the statistics and the Hamiltonian (including both kinematics and dynamics). In condensed
matter systems, one usually deals with effective Hamiltonians rather than true Hamiltonians
as a result of the separation of the different energy scales. The statistics and the effective
Hamiltonian are then not necessarily related any more. We are interested in systems where
the atomic spin is integer so that the atoms obey Bose-Einstein statistics, but the effective
Hamiltonian describes atoms with half-integer spin (in particular, spin 1/2). We give two
examples where these conditions are met:
1- Spin-polarized hydrogen: Atomic hydrogen has electronic spin 1/2 and nuclear spin
1/2, adding up to total spin 0 in the ground state in the absence of external fields. However,
if a gas of atomic hydrogen is placed in a strong magnetic field (typically 1-10 Tesla), the
electronic spin is effectively frozen, since only those atoms with electronic spin parallel to the
field are trapped. On the other hand, both nuclear-spin states can be trapped, and therefore
nuclear spin can be treated as free.1 If we constrain all the electronic spins to point parallel
to the external field, we obtain an effective Hamiltonian where the only degrees of freedom
left in the problem are the orbital motion of the atoms and nuclear spin. Thus we effectively
have spin 1/2 bosons [see ?].
2- 87Rb binary mixtures: In experiment several hyperfine states of 87Rb can be trapped
simultaneously, usually the |F = 2, mF = 1〉 and |F = 1, mF = −1〉 states [?]. The
Hamiltonian describing this system contains terms describing two kinematically independent
fields (i.e. two different hyperfine species), possibly interacting though. If we treat one field
as the +1/2 component and the other field as the −1/2 component of a spin 1/2 spinor field,
1The energy difference between the two nuclear-spin states can be eliminated by going to a rotating frame.
67
the Hamiltonian looks like that describing a system of spin 1/2 particles. In this case we talk
about pseudospin 1/2 rather than real spin 1/2. Formally, however, there is no difference
between the two. One complication that arises when using this kind of mapping is that, in
general, the trapping potentials and the interaction strengths between different combinations
of the two species are substantially different, and the desirable SU(2) symmetry is not present
in the Hamiltonian.
5.2 A Misleading Argument (Spontaneous Symmetry
Breaking)
We give here the “historical” path that we took in treating the problem at hand. We start
by introducing a result that attracted our attention from a paper by ?. We present two
paradoxes related to their result that pave the way to the new phenomenon that is the
main subject of this Chapter. We shall follow the general theoretical approach of Siggia
and Ruckenstein. However, we shall purposely be using a misleading argument in order to
stress the paradoxes. The argument goes somewhat as follows: Let us take a gas of N spin-
polarized hydrogen atoms above the Bose-condensation temperature Tc. The electronic spin
degrees of freedom are essentially frozen once we are close to Tc. Furthermore let us assume
that the coupling between the nuclear spin (S=1/2 per atom) and any external fields and the
spin-dependent interactions are negligible.2 In other words, the nuclear spin does not appear
anywhere in the Hamiltonian, except as a summation index. Therefore, above Tc the spin
state of the system will be that of N randomly oriented spin 1/2’s. The z component of each
nuclear spin takes one of the two values ±1/2. By a simple counting calculation, it can be
shown that in this system 〈S2z 〉1/2 ∼ N1/2. Now let us cool down the gas past the transition
temperature Tc and down to T = 0, where all the atoms will occupy the lowest orbital wave
2To simplify the problem and clarify the flow of the following arguments, one can even neglect interactions
altogether.
68
function (assuming that the orbital ground state is nondegenerate). Using spontaneous
symmetry breaking arguments, we can say that when the transition temperature is crossed,
the U(1) symmetries of the Hamiltonian will be broken. The quantities 〈ψ↑,↓〉 will then take
finite values, which comprise the order parameter of the condensate [see ?]. At T = 0 the
system is described by a spinor order parameter, which is the natural consequence of the
spinor nature of a spin 1/2 wave function:
Ψ =
ψ↑
ψ↓
, (5.1)
where ψ↑ and ψ↓ are the Sz = ±1/2 components of the order parameter. The number of
atoms in the Sz = +1/2 is equal to |ψ↑|2, and the number of atoms in the Sz = −1/2 is
equal to |ψ↓|2. Let us assume that the interaction between the system and the heat bath
is spin independent, so that while an atom experiences a slowing interaction, its spin does
not change. In that case, since initially at high temperatures the net magnetization of the
system is ∼ √N , we would expect that |ψ↑|2 ≈ |ψ↓|2 to relative order N−1/2. To a very good
approximation, the order parameter can then be expressed as:
Ψ = eiΦ√N/2
eiχ/2
e−iχ/2
, (5.2)
where Φ is an irrelevant overall phase factor, and χ is the relative phase between the Sz =
±1/2 components. Since there is no term in the Hamiltonian to determine the value of χ,
its value will be determined by the usual spontaneous symmetry breaking. For a given value
of χ, we find that the total magnetization of the condensate is given by:
〈Sx〉 =N
2cosχ,
〈Sy〉 =N
2sinχ, (5.3)
〈Sz〉 = 0.
We can therefore conclude that in the experiment suggested above, a macroscopic nuclear
magnetization will appear in the x-y plane below Tc (with magnitude equal to N/2 at T =
69
0). This magnetization should be easily detectable in experiment due to its macroscopic
magnitude. Below (in Paradox 2) we shall show that the above argument is flawed.
5.2.1 Paradox 1
In this paragraph we follow an argument given by ?. If we look at the initial state of the
system above Tc, we find that the net magnetization is ∼ √N . This state can be described by
a thermal mixture of states, each of which has a definite number of atoms in the Sz = ±1/2
states, strongly peaked around the state with N↑ = N↓ = N/2. When we cool down to
T = 0, all the atoms will condense into the same orbital wave function, but the number of
atoms in the Sz = ±1/2 will remain unchanged. The state of the system will then be a
mixture of states of the form:
|Ψ〉 =(a†↑)
N↑(a†↓)N↓√
N↑!N↓!|0〉. (5.4)
In any of these states 〈Sx〉 = 〈Sy〉 = 0, and 〈Sz〉 = N↑ −N↓ ≈ 0. How, then, did Siggia and
Ruckenstein obtain a macroscopic magnetization in the x-y plane, while we have just shown
that there cannot be any net magnetization in that plane?
Resolution of the paradox
For the purposes of this paradox it suffices to consider the state with N↑ = N↓ = N/2.
That state is given by:
|Ψ〉 =(a†↑)
N/2(a†↓)N/2
(N/2)!|0〉. (5.5)
This state apparently has 〈Sx〉 = 〈Sy〉 = 0. This means that in an ensemble of measurements,
the average value of Sx or Sy will be equal to zero. If we average the results of Eq. (5.3)
over all values of χ between −π and π, we find that 〈Sx〉 = 〈Sy〉 = 0. Therefore, there
is no contradiction between the two results. Furthermore, in the state given in Eq. (5.5),
〈S2x + S2
y〉1/2 = 〈S2 − S2z 〉1/2 ≈ N/2. That means that if we measure the component of the
70
magnetization in the x-y plane, we will always find a macroscopic value, also in agreement
with the predictions of the spontaneous symmetry breaking argument. In fact, this problem
is exactly that of measuring the relative phase between two condensates ‘that have never seen
one another’, in disguise [see ?]. Every definite value of the relative phase χ corresponds to a
direction in the x-y plane along which the macroscopic magetization points. A measurement
of the magnetization of the system will yield a macroscopic value in some direction, in the
same way that a definite relative phase is found whenever one tries to measure it. And
in the same way that one can think of the measurement as building up the relative phase,
one can also think of the measurement as building up this macroscopic magnetization. It is
straightforward to see that the same result can be obtained for N↑ = N↓, as long as they
are both macroscopic. In order to have the magnetization purely in the x-y plane, we need
that the difference N↑−N↓ be nonmacroscopic. Otherwise, we would also find a macroscopic
component of the magnetization in the z direction as well. However, for the thermal initial
distribution assumed above, 〈S2z 〉1/2 ∼ √
N N , and the z component of the magnetization
is negligible.
We now digress for a moment to discuss a somewhat related point. Let us look at the state
(5.5). The way it is expressed above looks quite normal, i.e. there is nothing about it that
strikes a reader with a condensed matter background. However, as we have discussed in the
previous paragraph (see also Appendix A), this state can also be expressed as a superposition
of states, each of which has a macroscopic magnetization along a different direction:
|Ψ〉 =
(πN
2
)1/4 ∫ π
−π
dχ
2π√N !
(eiχ/2
√2a†↑ +
e−iχ/2
√2a†↓
)N
|0〉. (5.6)
The states inside the integral of Eq. (5.6) are macroscopically distinct. The condensate
is therefore in a superposition of macroscopically different quantum states. Thus the state
|Ψ〉 qualifies as a Schroedinger’s cat. The problem can be stressed further by considering
the following situation: let us take two experimentalists, one in Boston and the other in
Boulder. If each one of them creates a condensate (of the same atomic species) with a
well-defined number of atoms, the composite system can alternatively be described by a
71
superposition of (infinitely many) states, each of which has a well-defined relative phase
between the two condensates. The latter state can also be thought of as a Schroedinger’s
cat. The question then is: Which expression [(5.5) or (5.6)] should we use to determine
whether the state is a Schroedinger’s cat or not? The answer, somewhat surprisingly, is
that whether a given state is a Schroedinger’s cat or not depends on the basis one uses to
describe the system. There is no systematic procedure that can be used to give a universal
measure of the “catness” of a quantum state. The above argument aslo shows that it is
very easy to create a Schroedinger’s cat state. In fact, many such states are created every
day, naturally or in the lab alike. There are two reasons why they “go unnoticed”. First of
all, it is because they appear as Schroedinger’s cat states in unnatural bases. For example,
it is very unnatural to describe condensates in different cities in the basis of states with
definite relative phase. Secondly, they do not do anything that defies our classical intuition,
as quantum mechanics usually does. In other words, there is no way to check that they are
really quantum mechanical objects. The reason why physicists still spend years trying to
create a Schroedinger’s cat can be considered by a devout believer in quantum mechanics
to be a matter of adventure and challenge, since to that believer any quantum state of
a macroscopic system is a Schroedinger’s cat in some basis. Experimentalists are trying
to create a state that takes the form of a Schroedinger’s cat in the most natural basis for
describing the system, and to be able to show unclassical results with them (e.g. interference
between two macroscopically different states). From that point of view, they are essentially
just trying to beat decoherence by isolating the system from external uncontrolled noise. As
far as quantum mechanics is concerned, there is nothing fundamentally new in the search
for Schroedinger’s cats. In short, Schroedinger’s cats are all over the place, except that they
are sitting in places where we are not surprised to find them and doing things we are not
surprised to see them doing.
72
5.2.2 Paradox 2
Now we turn to another paradox whose resolution will lead us to the main subject of this
Chapter. We have argued that in the initial state of the system 〈S2z 〉1/2 N , and therefore
we concluded that there will be a spontaneous macroscopic magnetization in the x-y plane.
However, at least as far as the effective nuclear-spin Hamiltonian is concerned, our choice of
the z axis was arbitrary. We could have started by considering the magnetization in the x
direction and concluded that there will be a spontaneous macroscopic magnetization in the
y-z plane. With a similar argument for the y axis, we find the magnetization to be in the x-z
plane. These three conditions obviously cannot be satisfied simultaneously. The question
then is: What will actually happen? Will there be a macroscopic magnetization? And if so,
in what direction will it point?
Resolution of the paradox
The resolution of the paradox lies in the interesting property of a condensate of spin
1/2 atoms that it must be an eigenstate of the total spin operator with eigenvalue N/2,
i.e. S2 = N2(N
2+ 1), provided that all the atoms occupy the same orbital wave function. To
prove the above statement, we consider the complete basis of states defined by the occupation
numbers of the single-particle states |k, ↑〉 and |k, ↓〉, where k refers to the orbital part of
the wave function (e.g. the single-particle ground state of the trapping potential). The
many-body quantum states can be expressed as |N↑, N − N↑〉, where the two quantum
numbers correspond to the occupation numbers of the states |k, ↑〉 and |k, ↓〉. The many-
body quantum state is uniquely defined by the number N↑, since the many-body wave
function must be symmetric under the exchange of any two atoms. Now we look at the state
|N, 0〉. This state has N atoms in the ↑ state and no atoms in the ↓ state. Therefore, it
has S2 = N2(N
2+ 1) and Sz = N/2. Any of the other states can be obtained by repeated
application of the spin lowering operator S− ≡ a†↓a↑. The operator S− commutes with the
operator S2, and therefore all the states |N↑, N − N↑〉 must be eigenstates of S2 with the
73
same eigenvalue, namely N2(N
2+ 1).3 Noting the above property is the main step towards
resolving the paradox. It implies that the ground state of a weakly-interacting Bose gas is
one with macroscopic magnetization. Therefore, a gas initially with S N/2 cannot be
cooled down to its ground state while conserving its total spin.
If we start with an initial state above Tc with a definite value of S (that is smaller than
N/2) and impose conservation of total spin, we find that the lowest energy state is one with
N/2 + S atoms in the lowest orbital wave function and N/2− S atoms in the wave function
corresponding to the first-excited single-particle state. If we put more than N/2 + S atoms
in the lowest wave function, there will not be enough atoms left to reduce the total spin to
the desired value. Since in the system described above S N/2, we find that the lowest
energy state that conserves total spin is a fragmented condensate with almost half of the
atoms in each of the two lowest wave functions. The state can be expressed as [?]:
|Ψ〉 = Z(a†0,↑)(S+Sz)/2(a†0,↓)
(S−Sz)/2(a†0,↑a
†1,↓ − a†0,↓a
†1,↑)(N−S)/2
|0〉 (5.7)
=∑N0,↑
cN0,↑ |N0,↑, N/2 + S −N0,↑, N/2 + Sz −N0,↑, N0,↑ − S − Sz〉, (5.8)
where Z is a normalization constant,
cN0,↑ = Z√
(N/2 − S)!(−1)N0,↑
√N0,↑!(N/2 + S −N0,↑)!
(N0,↑ − S − Sz)!(N/2 + Sz −N0,↑)!, (5.9)
and the four quantum numbers inside the ket correspond to N0,↑, N0,↓, N1,↑ and N1,↓. The
operator (a†0,↑)(S+Sz)/2(a†0,↓)
(S−Sz)/2 creates atoms in the lowest orbital wave function with
total spin S and z component Sz, whereas the operator(a†0,↑a
†1,↓ − a†0,↓a
†1,↑)(N−S)/2
creates
(N − S)/2 pairs of atoms, where each pair has no total spin. Therefore, these pairs do not
contribute to the total spin of the condensate. Their purpose is to satisfy the total number
constraint.4
3The number of states in the |N↑, N −N↑〉 basis is N + 1, matching the number of states in the S = N/2
spin-state basis, as it must.4Although the pair-creation operator creates a pair of entangled atoms, it is not possible to extract
entangled pairs of atoms out of the condensate (without further assumptions). The reason for that is that
74
In the above state (Eq. 5.7), we consider two cases of special interest. The first one is
the case where S = Sz = 0, i.e. the singlet state (assuming N is even). Intuitively one can
expect that this case has a vanishingly small probability of occuring in the thermodynamic
limit. In fact, from Eq. (5.17) below, we can see that this probability is ∼ N−3/2. However,
this case is interesting because of its simple appearance and highly entangled nature. To see
that we take S = Sz = 0 in Eq. (5.9) and find that
cN0,↑ = Z
√N
2(−1)N0,↑
=(−1)N0,↑√N/2 + 1
, (5.10)
which means that all the states |N0,↑, N/2−N0,↑, N/2−N0,↑, N0,↑〉 occur in the superposition
with equal probability. We shall shortly re-express the singlet state in another form that will
be useful in the analysis in Section 5.7. The other case of special interest is that with total
spin much larger than 1 but much smaller than N/2, because this is the case to be expected
in the real experiment, as will also become clear from Eq. (5.17) below. In that case the
coefficient cN0,↑ can be approximated by:
cN0,↑ ≈ (−1)N0,↑
(2π)1/4√
∆Nexp
−(N0,↑ −Nmax)
2
4(∆N)2
, (5.11)
where
Nmax =S + Sz
2S
N
2, (5.12)
∆N =
√S2 − S2
z
2S3
N
4. (5.13)
The above result is quite interesting. One might not expect much dependence of Nmax on
Sz, as long as S ∼ √N N . However Nmax depends rather strongly on Sz, and in fact,
it goes from zero, when Sz = −S, to N/2, when Sz = S. If we take S ∼ Sz ∼ √N , we
the condensate contains a large number of pairs, and if we pick any two atoms, it is most likely that they
will belong to different pairs, and therefore they will not be entangled with each other. We shall come back
to that point in Section 5.4.
75
find that ∆N ∼ N3/4 N . We shall use these results in Section 5.6 when we derive the
Gross-Pitaevskii (GP) equations for this system. The single-particle density matrix in the
basis |0, ↑〉, |0, ↓〉, |1, ↑〉, |1, ↓〉 is approximately given by:
ρ ≡ 〈a†nam〉 →
N/4 0 0 0
0 N/4 0 0
0 0 N/4 0
0 0 0 N/4
,when N → ∞, (5.14)
so that all four single-particle states are equally populated. This is also true for the singlet
state (S = Sz = 0).
We now re-express the singlet state as a superposition of coherent-like states5:
|N, S = Sz = 0〉 =
√N
2
∫dΩ
4π(N/2)!
(a†0(Ω)
)N/2 (a†1(−Ω)
)N/2
|0〉, (5.15)
where the operators a†0(Ω) and a†1(Ω) create atoms in the orbital ground and first-excited
states, respectively, with S · d = +1/2, and d is the vector specified by the angle Ω on
the unit sphere. The above state contains N/2 atoms in the orbital ground-state wave
function and N/2 atoms in the orbital first-excited state wave function, and it transforms
into itself under an arbitrary rotation, as it should. Each state inside the integral describes
two condensates, each of which is in a coherent state, with their macroscopic spins pointing
in opposite directions. Classically, the two macroscopic spins would cancel exactly. However,
quantum fluctuations in the transverse direction do not cancel, and therefore any one of the
states inside the integral does not satisfy the fixed spin constraint. We shall see in Section
5.7 that in certain experiments, just as in the case of spin 1 atoms, which was treated in
Chapter 3, a condensate in the singlet state behaves as if it were in one of these coherent-like
states.
5We refer to these states as “coherent-like” instead of “coherent” because we defined coherent states in
Chapter 3 as those states where all the particles occupy the same single-particle quantum state, whereas
here we have two macroscopically-occupied single-particle states. The coherent-like states can in fact be
expressed as superpositions of coherent states (see Appendix A and Section 5.7).
76
In order to determine the distribution of values of S at T = 0, we have to determine
that distribution in the initial state of the system. If we start the quenching above 2.6Tc,
the atoms can be treated as distinguishable (for purposes of calculating their total spin),
since they occupy different orbital wave functions, i.e. the occupation of each orbital wave
function is smaller than 1.6 Therefore, the possible spin states and their probabilities are
equal to those of a system comprised of N independent spins. The (reduced) Hilbert space
describing the spin states has dimension 2N , reflecting the fact that each spin is described
by the basis |Sz = +1/2〉 and |Sz = −1/2〉. In the absence of external fields and assuming
the spins are noninteracting (so that all the different spin states have the same energy), we
find that the probability of having a certain value of Sz (of the total system) is given by:
P (Sz) =1
2N
N !
(N2− Sz)!(
N2
+ Sz)!. (5.16)
And the probability of having a certain value of S is given by [?]:
P (S) = (2S + 1) ·(P (Sz = S) − P (Sz = S + 1)
)
=2S + 1
2N
N !
(N2− S)!(N
2+ S)!
(1 −
N2− S
N2
+ S + 1
). (5.17)
In the large N limit, these expressions can be approximated by:7
P (Sz) ≈√
2
πNe−2S2
z/N (5.18)
P (S) ≈ 27/2
√πN3
S2e−2S2/N . (5.19)
Most of the probability is concentrated in the range of values S ∼ √N , and therefore
the probability of reaching a condensate with most of the atoms in the same orbital wave
function, i.e. a condensate with N1/N0 1, is negligible. In particular, the probability that
the system will reach its ground state, which is equal to the probability of having S = N/2
in the initial state, is given by:
6The occupation of the low-lying states reaches 2 when T = 2Tc and 10 when T = 1.3Tc. These small
occupation numbers slightly above Tc justify our argument that above Tc, S ∼ √N .
7These expressions can also be derived from semiclassical arguments.
77
P (cooling to GS) = P (S = N/2) =N + 1
2N, (5.20)
assuming that the energy does not depend on the spin state. This probability is vanishingly
small for large values of N . The mistake that we made in the original argument is that we
noticed the fact that it is highly unlikely to have a value of Sz ∼ N , but we did not notice
that it is equally unlikely to have a value of S ∼ N . This led to the (wrong) conclusion that√S2
x + S2y(=
√S2 − S2
z ) must be macroscopic. We have thus resolved the two paradoxes,
and in doing so, stumbled across a new interesting phenomenon, namely the formation of a
fragmented condensate as a result of the symmetry of the initial state of the Bose gas.
We now go back to the question of the direction of the macroscopic magnetization in the
original problem above. The initial state of the system above Tc is described by a density
matrix corresponding to a thermal mixture of pure states. For a given value of S, the
probability of having a certain value of Sz is proportional to the number of different states
that have total spin S and z component of the total spin Sz. However, that number is
independent of Sz, and it can be read off Eq. (5.17).8 Therefore, for a given value of S, the
probability of having any value of Sz is a constant, i.e. independent of the value Sz. This
leads us to the conclusion that if the system happens to be in an S ∼ N/2 state, it is equally
likely to be in any state of Sz. Therefore, the macroscopic magnetization will point in any
direction on the unit sphere (in 3 dimensions) with equal probability, if this macroscopic
magnetization is obtained. We shall see in Section 5.5, however, that if we include certain
spin-changing terms in the Hamiltonian, we recover the original Siggia-Ruckenstein result of
a macroscopic nuclear magnetization in the x-y plane.9
8The number of different state with S and Sz is equal to P (S)2N/(2S + 1).9In fact, we find the Leggett-Sols result that the condensate is described by a mixture of states with
definite N↑ and N↓ without any definite relative phase between them.
78
5.2.3 Conclusion
Before going into the detailed discussion of the fragmented state suggested above, we summa-
rize the main idea of how that state is formed. The first point to note is that the ground state
of a weakly-interacting Bose gas of spin 1/2 atoms [with an SU(2) symmetric Hamiltonian]
is a state with macroscopic magnetization. The ground state is therefore ferromagnetic.10
If the spins of the different atoms are oriented randomly above Tc (so that the total spin
is much smaller than N/2) and the system is cooled down to T = 0, it cannot reach its
ground state unless the total spin changes. If we constrain the total spin to be conserved,
the lowest energy state that can be reached is the fragmented state. In that state the lowest
two orbital wave functions are macroscopically occupied, i.e. two condensates are formed.
Each condensate has macroscopic magnetization, but they point in opposite directions so as
to cancel each other. If we include spin-changing terms in the Hamiltonian, the system will
be able to reach its ground state, as we shall explain in Section 5.5.
An interesting question regarding the formation of the fragmented state is that of the
critical temperature Tc. We suspect that there are two transition temperatures, Tc1 and
Tc2. As the temperature of the gas is lowered below Tc1, the lowest wave function becomes
macroscopically occupied, while the occupation of all the other wave functions remains ∼ 1.
As the temperature is lowered further and goes below Tc2, the second lowest wave function
becomes macroscopically occupied. In order to determine the two transition temperatures,
one would need to have an exact model for the cooling mechanism. For example, we shall
see in Section 5.5 that the effect of evaporative cooling is qualitatively different from cooling
using a heat bath. It would be interesting to calculate Tc1 and Tc2 for some model of the
cooling mechanisms.
10Note that the ferromagnetism in this system is not caused by any kind of interaction, but rather by the
Bose symmetry of the many-body wave function (see Section 5.3).
79
5.3 The Role of Interactions in Forming the Fragmented
State
It is well known that certain forms of interatomic interactions can cause a Bose-Einstein
condensate to form a fragmented state [see ?]. Somewhere in the details of the above argu-
ment, one might be misled to think that interatomic interactions play a role in causing the
condensate to form the fragmented state presented in this Chapter as well. However, we did
not use interactions in any energetic arguments so far, except for possibly using them as a
means of thermalization. Our arguments would go through even if the atoms were noninter-
acting and are cooled down purely by coupling to an external heat bath. The fragmentation
in this system results from the initial symmetry of the many-body wave function. If we do
not include symmetry-changing terms in the Hamiltonian, the symmetry will persist as we
lower the temperature, and the fragmented state will form.
The relation of this fragmented state to other fragmented states11 is therefore analogous
to the relation of the BEC transition to other phase transitions. One of the unique aspects
about the BEC transition is that it is not caused by any interatomic interactions, and it
can occur in noninteracting systems, whereas all other known phase transitions result from
the competition between interatomic interactions and some other mechanism. Similarly,
all other fragmented states in BEC depend on the form of the interatomic interactions.
This fragmented state, on the other hand, does not. The BEC transition is caused by the
constraint on the many-body wave function of bosons to be symmetric, and this fragmented
state is caused by the constraint of conserving the symmetry of the initial many-body wave
function.
Although interactions are not responsible for the formation of the fragmented state, they
do play a role in determining the two macroscopically-occupied states. We shall address this
11Here we have in mind condensates with attractive interactions, condensates in a double-well potential
in the Fock regime and antiferromagnetic spinor condensates [see ?].
80
question in detail in Section 5.6.
Another interesting topic to discuss the role of interactions in the present problem is the
ground state (without any constraint on the total spin). As we mentioned in Section 5.2.3,
the ground state is ferromagnetic. This is the only noninteracting system known to us that
exhibits ferromagnetism. For example, in a system of spins fixed at their lattice sites, it is
necessary to have a certain form of interactions to obtain ferromagnetism. In a system of
free, interacting spin 1/2 fermions, if the interaction energy scale is larger than the kinetic
energy scale, it is also possible to see ferromagnetic behaviour. In this system, however,
all we need is a Hamiltonian with nondegenerate orbital states. For example, even if the
Hamiltonian contains only a kinetic energy term and nothing else, the ground state of the
system is ferromagnetic. That is another unique feature of Bose gases of spin 1/2 particles.
5.4 Extracting Entangled Atom Pairs
Let us take another look at the state (5.7) and set the total spin of the condensate equal to
zero:
|Ψ〉 = Z(a†0,↑a
†1,↓ − a†0,↓a
†1,↑)N/2
|0〉.
The above state is formed by creating N/2 pairs of atoms, where each pair has zero total
spin. Now one might suggest that if we take a pair of atoms out of the condensate, one atom
from each occupied wave function, the spins of the two atoms will be entangled to give zero
total spin. Another argument can go as follows: Let us start by creating one entangled pair
of atoms in the vacuum, and let us denote the atoms by the numbers 1 and 2. Next we create
a second pair of atoms (denoted by the numbers 3 and 4). Now we take out one atom from
each of the two occupied wave functions. If the two atoms happen to be 1 and 2, they will
be entangled to form the singlet state. The same applies to picking atoms 3 and 4. But what
if we happen to pick atoms 1 and 3? Using the symmetry of the many-body wave function
we can argue that since atom 3 was taken out of the same wave function as atom 2, their
81
spin state must be symmetric and therefore identical.12 Therefore, atoms 1 and 3 are also
entangled to form the singlet state. This suggests that if we pick any two atoms, with the
requirement that each atom comes from a different spatial wave function, the atoms will be
an entangled pair. This would be a major achievement for quantum information purposes.
Unfortunately, the above argument had a small flaw. We innocently assumed that when
atoms 3 and 4 were added, they did not affect atoms 1 and 2. This can be naively inferred
from the appearance of the many-body state:
|Ψ〉 = Z(a†0,↑a
†1,↓ − a†0,↓a
†1,↑)(
a†0,↑a†1,↓ − a†0,↓a
†1,↑)|0〉.
However, this assumption is incorrect. When the second pair is added, it destroys the
entanglement in the first pair. In order to see that, we look at the many-body wave function
in first-quantized language:
Ψ(1, 2, 3, 4) =1√72
(ψ0,↑(1)ψ1,↓(2) + ψ0,↑(2)ψ1,↓(1) − ψ0,↓(1)ψ1,↑(2) − ψ0,↓(2)ψ1,↑(1))
(ψ0,↑(3)ψ1,↓(4) + ψ0,↑(4)ψ1,↓(3) − ψ0,↓(3)ψ1,↑(4) − ψ0,↓(4)ψ1,↑(3))
+(ψ0,↑(1)ψ1,↓(3) + ψ0,↑(3)ψ1,↓(1) − ψ0,↓(1)ψ1,↑(3) − ψ0,↓(3)ψ1,↑(1))
(ψ0,↑(2)ψ1,↓(4) + ψ0,↑(4)ψ1,↓(2) − ψ0,↓(2)ψ1,↑(4) − ψ0,↓(4)ψ1,↑(2))
+(ψ0,↑(1)ψ1,↓(4) + ψ0,↑(4)ψ1,↓(1) − ψ0,↓(1)ψ1,↑(4) − ψ0,↓(4)ψ1,↑(1))
(ψ0,↑(2)ψ1,↓(3) + ψ0,↑(3)ψ1,↓(2) − ψ0,↓(2)ψ1,↑(3) − ψ0,↓(3)ψ1,↑(2)).(5.21)
Due to the symmetrization of the many-body wave function, there are now terms where, for
example, both atoms 1 and 2 have spin ↑. Therefore, these two atoms are not necessarily
entangled any more. The above argument can be generalized toN particles straightforwardly.
The many-body wave function of the system is then given by:
12That can be seen from the fact that by measuring the spin of atom 1, the spin of atom 2 is determined,
and to have a symmetric state between atom 2 (whose state is now well defined) and atom 3, the two spin
states must be identical.
82
Ψ(1, 2, ..., N) =1√
N !(N/2 + 1)!
∑∏(ψ0,↑(j1)ψ1,↓(j2) + ψ0,↑(j2)ψ1,↓(j1)
−ψ0,↓(j1)ψ1,↑(j2) − ψ0,↓(j2)ψ1,↑(j1)), (5.22)
where the sum and product cover all the possible pairings of the particles. The probability
that two particles taken from the two orbital wave functions are in a spin singlet state can
be calculated relatively straightforwardly and is given by:
P (Singlet) =N + 2
4(N − 1). (5.23)
At first sight it might look counter-intuitive that no matter how many atoms are added
and the entanglement between a certain pair of atoms decreases, the probability of being
entangled in the singlet state stays above 1/4. The explanation of this lower limit is simple.
If we take two spin 1/2 particles on two different galaxies in unknown states, they still have
a 1/4 probability of being in the singlet state. The reason behind that phenomenon is that
the Hilbert space of this system is four dimensional, and in the total spin basis, the singlet
state is one of the four basis states. As a result, a density matrix that is proportional to the
unit matrix has probability 1/4 of being in that state. Therefore, the probability of being in
the singlet state is a misleading measure of entanglement. There have been several attempts
in the literature to define a universal measure of entanglement, but to date that goal has
not yet been achieved [For an attempt to quantify entanglement, see ?]. Here we define the
probability of a pair of atoms to be in different orbital states and entangled as follows:
P (Entangled) ≡ 4
3(P (Singlet) − 1) =
1
N − 1. (5.24)
This definition satisfies the two necessary conditions that P = 1 when N = 2 and P → 0
when N → ∞.
5.5 Possible Spin-Relaxation Mechanisms
Our argument that led to the prediction of the fragmented state depended strongly on the
assumption that the different interactions involved in the cooling process conserve total spin.
83
That is the result of the fact that the total spin of the ground state is macroscopically different
from that of the initial state. If we include spin-nonconserving terms in the Hamiltonian, the
system will be able to eventually reach its ground state. In fact, in the systems that we are
considering, the longitudinal relaxation time scale T1 is typically of the order of the life time
of the trapped cloud, and therefore spin relaxation in the z direction can be neglected. The
transverse relaxation time scale T2, on the other hand, is shorter than the time of running the
experiment (in current experiments), and that would give rise to relaxation in the x-y plane.
If T2 is short enough, we recover the spontaneous-symmetry-breaking result of a macroscopic
magnetization in the x-y plane. In this Section we consider the effect of including certain
spin-relaxation mechanisms.
(1) Uniform magnetic field: The effect of a uniform magnetic field, as well as any differ-
ence in chemical potential between the Sz = ±1/2 states, is simply to cause any finite spin
to precess around the direction of the field. Any such effect can be eliminated by going to
a rotating frame. Therefore, a uniform magnetic field cannot change the magnitude of the
total spin of the system.
(2) Magnetic field gradient: The effect of a magnetic field gradient is qualitatively different
and more complicated than that of a uniform magnetic field, because it does not conserve
total spin.13 Therefore, one can argue that as the system is cooled down and the lowest
wave function becomes macroscopically occupied, the total spin can increase and eventually
reach the value N/2 (and by conservation arguments, Sz remains at its initial value ≈ 0).
However, if we assume that this really happens and at some point in time we have a uniform
condensate with S = N/2, the field gradient will now cause the spins of different parts of the
condensate to precess at different rates, so that the total spin is reduced and the condensate
is destroyed.
The correct approach to this problem is to determine the ground state of the system in
13We limit our treatment here to fields pointing in the z direction. The generalization to a varying direction
is, in principle, straightforward.
84
the presence of a field gradient. We use the constraint that Sz ≈ 0, since Sz is still conserved
as long as the external field points in the z direction everywhere, and the time scale of
the experiment is much shorter than the longitudinal relaxation time T1. The problem
now reduces to that of two coexisting condensates, which has been treated by ?. With the
constraint that Sz ≈ 0, the ground state of the condensate will be nonuniform. There will
be a finite magnetization parallel to the external field in high-field regions and antiparallel
to it in low-field regions (assuming the g-factor of the magnetic moment is positive). The
value of this component of the magnetization depends on the magnitude of the gradient. If
the gradient is weak enough that the condensates overlap significantly, we find that there
will be a macroscopic magnetization in the x-y plane. This magnetization occurs because
there are no variations in the phase of the wave function of each condensate. As a result, if
the x-y component of the magnetization of an atom makes an angle θ with the x axis, the
same will be true for all the atoms in the condensate. In the limit of zero field gradient,
〈Sz〉 = 0 everywhere, and we recover the Siggia-Ruckenstein (or the Leggett-Sols) result
exactly. However, the weaker the field gradient, the longer the transverse relaxation time
T2, which is the time scale for relaxing S to its (macroscopic) equilibrium value. In fact, if
B′ → 0, we find that T2 → ∞. Here we give a rough estimate of the time scale for spin
relaxation due a magnetic field gradient: We assume that the condensate spin is constant
in time, since it is formed in the ground state of the trap with the gradient. If we also
assume that the atoms in the normal cloud are fixed in space, we find that any finite spin
that may build up in the normal cloud to compensate for the spin buildup in the condensate
will decay (due to different precession rates of the different atoms) over a time scale of the
order of /gpµpB′L, where gp is the g-factor of the proton, µp is the magnetic moment of
the proton, B′ is the magnetic field gradient and L is the length scale of the system. If we
use the values gp = 5.6, µp = 5 × 10−27J/T, B′ ∼ 10T/m, L ∼ 10−4m, we find a relaxation
time T2 ∼ 10−5s. This time scale is much shorter than the period of oscillation inside the
trap (∼ 10−2s), and therefore our assumption that the atoms in the normal cloud are fixed
85
in space is justified. The relaxation time scale that we have just calculated is very short
compared to the time scale of running the experiment, and it would prevent the formation
of the fragmented state. In the above calculation, however, we used numbers from spin-
polarized hydrogen experiments, where magnetic field gradients are very large. If optical
and magnetic fields are combined, it is possible to reduce the magnetic field gradient while
still trapping the gas. In that case, one can try to reduce B′, and therefore increase T2. If
T2 is increased to substantially larger than the time of quenching the gas down to T = 0,
the effect of magnetic field gradients can be neglected.
The effect of the magnetic field gradient can be summarized as follows: In the weak
gradient limit, if the cooling is very fast, we get the four-fold fragmented state (and spin
relaxation proceeds from there), whereas if the cooling is very slow, the total spin will change
adiabatically from a small value (∼ √N) to a macroscopic one (∼ N). On the other hand,
in the strong gradient limit the two components separate completely, and we obtain two
condensates, one with S ≈ Sz ≈ N/4 and the other with S ≈ −Sz ≈ N/4, which gives a
total spin of the system S ≈ Sz ≈ 0.
(3) Spin-dependent interatomic interactions: In general, in a condensate with two internal
states, there are three scattering lengths in the problem, namely a11, a22 and a12. Unless
these three quantities are equal, the component of the total spin in the x-y plane (defined by
mapping the two states onto the | ↑〉, | ↓〉 basis) is not conserved. The two-atom Hamiltonian
can be expressed as:
Hint =
g11
1 + 2S1z
2
1 + 2S2z
2+ g22
1 − 2S1z
2
1 − 2S2z
2
+g121 + 2S1z
2
1 − 2S2z
2+ g12
1 − 2S1z
2
1 + 2S2z
2
δ(r1 − r2)
=c0 + c1(S1z + S2z) + c2S1zS2z
δ(r1 − r2), (5.25)
where
86
c0
c1
c2
=
1/4 1/4 1/2
1/2 −1/2 0
1 1 −2
g11
g22
g12
. (5.26)
If we are considering a system of pseudospin 1/2, the appearance of the above Hamiltonian
might not worry us, since we simply added the four terms corresponding to the different
possible combinations of internal states. If we are considering a system of real spin 1/2,
however, the appearance of the S1z, S2z and S1zS2z terms look rather suspicious. There
should be symmetry between the different directions, and instead of the S1zS2z term, one
would expect to have an S1 · S2 term. If we start the derivation by considering the case of
real spin 1/2 with its symmetries, and assume that we only have contact interactions, we
find that
Hint =c+ c′S1 · S2
δ(r1 − r2), (5.27)
where c and c′ are some constant coefficients. However, if the two colliding atoms overlap,
their relative orbital wave function must be symmetric. It then follows that their spin
state must also be symmetric, and the operator S1 · S2 can be replaced by its value in the
symmetric triplet state, namely 1/4. Therefore, spin operators should not appear in the
interaction Hamiltonian at all. In fact, if we take the rotationally invariant case, we require
that the values of the three scattering lengths must be equal (and equal to the scattering
length in the triplet channel). That gives g11 = g22 = g12 and therefore c1 = c2 = 0, in
agreement with the above reasoning. In that case there are no spin-changing interatomic
interactions. In the general case (i.e. no rotational symmetry), the terms containing S1z, S2z
and S1zS2z can change the total spin of the system, and as the system is cooled down, they
can cause the spin to relax to a macroscopic value in the x-y plane. For a gas of 87Rb atoms,
the time scale for pseudospin relaxation T2 is ∼ 0.01s (taking c1 ∼ c2 ∼ 10−18s−1m3 and the
density ρ ∼ 1020m−3). In spin-polarized hydrogen the situation is more complicated.14 The14That is, in part, because we have not been able to find the values of the different scattering lengths in
the literature.
87
main contribution to the interatomic potential comes from the Coulomb potential, which is
almost the same for both states of the nuclear spin. There are small effects that depend on
the nuclear spins, such as dipole-dipole interactions between the nuclei. If the asymmetry
caused by these effects is small, the interactions can be treated as spin independent, which
is desirable for our purposes.
(4) Cooling walls/Sympathetic cooling: In certain systems, such as 4He, the surrounding
walls are important for providing the cooling mechanism. At the same time, the surrounding
walls can change the total spin of the system, especially since in the present problem changing
the total spin is essential to lowering the energy. However, since in the alkali atomic gases the
cooling walls can be replaced by a coolant (sympathetic cooling), if one chooses a coolant
with no spin-dependent interactions, it is possible to perform the desired spin-conserving
cooling. Another advantage of using sympathetic cooling is the fact that it relaxes the
need for thermalization within the cooled gas. Thermalization can now occur solely due
to the interaction between the cooled gas and the coolant. The different time scales in
the problem are then independent and one can, in principle, increase the speed of cooling
without necessarily decreasing T2. In trapped atomic gases, the cooling is usually done using
evaporative cooling, which we discuss next.
(5) Evaporative cooling: In present experiments on condensation of spin-polarized hydro-
gen and the alkali atomic gases, the last stage of cooling to BEC is evaporative cooling. In
this technique the system is cooled down by allowing the high-energy atoms to escape, leav-
ing the atoms in the trap with lower average energy and therefore lower temperature. The
energy splitting of the two relevant states (labelled by ↑ and ↓) in spin-polarized hydrogen
is of the order of a few mK. This is several orders of magnitude higher than Tc. Therefore,
one might expect that one of the two states will be lost first in evaporative cooling, long
before we reach Tc. However, this energy scale does not affect cooling, because there is no
machanism to release this internal energy. In other words, the mechanism of evaporative
cooling depends only on the kinetic energy of the atoms and is insensitive to their internal
88
(e.g. hyperfine) energy. An atom with low internal energy and high kinetic energy is more
likely to be lost than one with high internal energy and low kinetic energy, even if the internal
energy scale is orders of magnitude larger than the kinetic energy scale. Therefore, the ratio
between the number of ↑ and ↓ atoms is expected to be conserved in evaporative cooling.
By choosing certain initial values N↑ and N↓ at the beginning of the experiment, the ratio
between them remains constant until the end of the experiment.
A possible scenario in evaporative cooling is the following: As the system is cooled down
below Tc, a large number of atoms start to populate the lowest wave function, while the
other atoms essentially remain in different orbital states. Therefore, the condensate acquires
a magnetization equal to n/2, where n is the number of atoms occupying the same orbital
wave function, and the normal cloud (comprised of the rest of the atoms) acquires the same
value of magnetization in the opposite direction, such that the total magnetization remains
equal to its initial value. As evaporative cooling continues, it is more likely to lose atoms
from the normal cloud than from the condensate. In this process the magnetization of the
condensate keeps growing, and any (or at least part of the) compensating magnetization will
be lost. Again, this scenario breaks conservation of total spin (in an indirect way, of course).
Better understanding of the effect of this scenario requires further detailed analysis.
In conclusion, there are, in general, several mechanisms at play during the cooling of the
gas. Magnetic field gradients, anisotropy in interatomic interactions and spin-changing inter-
actions with the heat bath all contribute to the transverse relaxation rate 1/T2, and therefore
they push in the direction of creating a single condensate with macroscopic magnetization.
Evaporative cooling also favours the formation of a single condensate with macroscopic mag-
netization. Spin-conserving interactions with the heat bath, on the other hand, determine the
quenching time Tquench, and therefore they push in the direction of cooling without changing
the total spin. That, in effect, favours the creation of two condensates whose macroscopic
spins add up to a much smaller value (∼ √N). The end result of any experiment will be
the creation of a single condensate with macroscopic magnetization. However, the compe-
89
tition between the above mechanisms determines the intermediate state of the system. If
the spin-conserving cooling mechanism is slow enough, the magnetization of the system will
grow as the condensate grows. On the other hand, if the spin-conserving cooling mechanism
is fast enough, the system will pass through an intermediate state in which two condensates
(in two different orbital wave functions) are created, and the other mechanisms then cause
the system to eventually relax to a single-condensate state.
5.6 The Gross-Pitaevskii Equations
In Section 5.2 we concluded that in order to satisfy the condition of conserved total spin, a
gas of spin 1/2 bosons will condense into two spatially different (and, in fact, orthogonal)
wave functions. It is now natural to ask what these wave functions will look like. In the
noninteracting case, the answer is rather trivial, namely the single-particle ground- and first-
excited states. If we include interatomic interactions, however, the problem is not that simple
any more. In order to avoid the two main spin-changing mechanisms mentioned in Section
5.5, we shall assume that both the external potenial and interatomic interactions are spin
independent. The Hamiltonian describing the system is:
H =
∫dr
∑α=↑,↓
ψ†α(r)
(−
2
2m∇2 + U(r)
)ψα(r) +
g
2
∑α,β=↑,↓
ψ†α(r)ψ†
β(r)ψβ(r)ψα(r). (5.28)
The many-body quantum state (Eq. 5.7) can be expressed as:
|Ψ〉 =∑N0,↑
cN0,↑
(a†0,↑
)N0,↑ (a†0,↓
)N0,↓ (a†1,↑
)N1,↑ (a†1,↓
)N1,↓
√N0,↑!N0,↓!N1,↑!N1,↓!
|0〉, (5.29)
where cN0,↑ is given by Eq. (5.9) and
a†i,↑↓ =
∫drφi(r)ψ
†↑↓(r), (5.30)
and φi(r) are the wave functions corresponding to the two different orbital states. Now we
calculate the expectation value of the Hamiltonian with the ansatz (5.29):
90
〈H〉 =∑N0,↑
|cN0,↑|2∫
dr∑i=0,1
Niφ∗i (r)
(−
2
2m∇2 + U(r)
)φi(r)
+g
2
( ∑i,j=0,1
NiNjφ∗i (r)φ
∗j(r)φj(r)φi(r) + 2(N0,↑N1,↑ +N0,↓N1,↓)φ∗
0(r)φ∗1(r)φ1(r)φ0(r)
)
+∑N0,↑
(c∗N0,↑−1cN0,↑
√N0,↑(N0,↓ + 1)(N1,↑ + 1)N1,↓
+c∗N0,↑+1cN0,↑
√(N0,↑ + 1)N0,↓N1,↑(N1,↓ + 1)
)φ∗
0(r)φ∗1(r)φ1(r)φ0(r)
≈∫dr
∑i=0,1
Niφ∗i (r)
(−
2
2m∇2 + U(r)
)φi(r)
+g
2
( ∑i,j=0,1
NiNjφ∗i (r)φ
∗j(r)φj(r)φi(r) + 2〈N0,↑N1,↑ +N0,↓N1,↓〉φ∗
0(r)φ∗1(r)φ1(r)φ0(r)
)
−2⟨√
N0,↑N0,↓N1,↑N1,↓⟩φ∗
0(r)φ∗1(r)φ1(r)φ0(r). (5.31)
In the last step we used the approximation cN0,↑+1 ≈ −cN0,↑ , and we neglected terms of
order N in the interaction Hamiltonian. The wave functions φ0(r) and φ1(r) must obey the
orthonormality constraints:∫drφ∗
0(r)φ0(r) =
∫drφ∗
1(r)φ1(r) = 1, (5.32)∫drφ∗
0(r)φ1(r) = 0. (5.33)
In fact, the orthogonality constraint can be decomposed into two constraints as follows:
Re
∫drφ∗
0(r)φ1(r) =
∫drφ∗
0(r)φ1(r) + φ0(r)φ∗1(r)
2= 0, (5.34)
Im
∫drφ∗
0(r)φ1(r) =
∫drφ∗
0(r)φ1(r) − φ0(r)φ∗1(r)
2= 0. (5.35)
We can now minimize the energy functional with respect to the wave functions φ0,1(r) under
the orthonormality constraints, and we find the following set of GP equations:
N0
(−
2
2m∇2 + U(r)
)φ0(r) + g
(N2
0φ∗0(r)φ0(r) + (N0N1 + 〈N0,↑N1,↑〉 + 〈N0,↓N1,↓〉
−⟨√
N0,↑N0,↓N1,↑N1,↓⟩φ∗
1(r)φ1(r))φ0(r) = N0µ0φ0(r) + Λφ1(r), (5.36)
N1
(−
2
2m∇2 + U(r)
)φ1(r) + g
(N2
1φ∗1(r)φ1(r) + (N0N1 + 〈N0,↑N1,↑〉 + 〈N0,↓N1,↓〉
−⟨√
N0,↑N0,↓N1,↑N1,↓⟩φ∗
0(r)φ0(r))φ1(r) = N1µ1φ1(r) + Λ∗φ0(r),(5.37)
91
where the average values can be calculated using the amplitude cN0,↑ given in Eq. (5.9).
First we consider the case S = Sz = 0. From Eq. (5.9) we have cN0,↑ = (−1)N0,↑/√N/2 + 1
and therefore
N0 = N1 =N
2, (5.38)
〈N0,↑N1,↑〉 = 〈N0,↓N1,↓〉 =⟨√
N0,↑N1,↑N0,↓N1,↓⟩
=N2
24. (5.39)
Using the above values, we arrive at the GP equations:
(−
2
2m∇2 + U(r) + g
N
2φ∗
0(r)φ0(r) +7
12gNφ∗
1(r)φ1(r)
)φ0(r) = µ0φ0(r) + λφ1(r), (5.40)(
− 2
2m∇2 + U(r) + g
N
2φ∗
1(r)φ1(r) +7
12gNφ∗
0(r)φ0(r)
)φ1(r) = µ1φ1(r) + λ∗φ0(r). (5.41)
The above set of equations is the generalization of the GP equation of the spinless case. µ0
and µ1 are the chemical potentials of the two wave functions. Note that the GP equations
must be complemented by the orthonormality conditions (Eqs. 5.32 and 5.33).
In the Thomas-Fermi (TF) limit, we can neglect the kinetic energy term and the GP
equations reduce to:
(U(r) + gρ0(r) +
7
6gρ1(r) − µ0
)φ0(r) = λφ1(r), (5.42)(
U(r) + gρ1(r) +7
6gρ0(r) − µ1
)φ1(r) = λφ0(r), (5.43)
where ρi(r) is the number density in the i-th orbital wave function, i.e. ρi(r) = N2|φi(r)|2.
We have replaced λ∗ by λ, since the ground state wave functions φ0 and φ1 are expected to
be real. Let us for a moment take µ0 = µ1 and λ = 0. Eqs. (5.42) and (5.43) then describe
the well-studied problem of two coexisting condensates with equal numbers of atoms [see ?].
Clearly the interaction term between the two wave functions is stronger than that within each
wave function.15 Therefore, in order to minimize the energy, the overlap between the two
15In fact, this phenomenon is well known in other related contexts, one of which is the argument that
repulsion in configuration space leads to attraction in momentum space. That consequently leads to the
92
wave functions must be minimized (Here we are assuming that g is positive). In the TF limit,
the two components separate completely on opposite sides of the trap. The overlap between
the two wave functions is equal to zero, so that our approximations (µ0 = µ1 = µ and λ = 0)
are satisfied automatically. Assuming the separation between the two condensates occurs in
the x direction, the density distribution of the two condensates is given by:
ρ0,1 =µ− U(r)
gθ(±x), (5.44)
where θ(x) is the Heaviside step function, and the upper and lower signs correspond to the
two different condensates, in random order. If we take S ∼ Sz ∼ √N , the GP equations are
qualitatively unaffected (in the TF limit). The only difference is that the coefficient in front
of the interaction term between the two wave functions φ0(r) and φ1(r) is changed. Since
N0 ≈ N1 ≈ N
2, (5.45)
〈N0,↑N1,↑〉 ≈ 〈N0,↓N1,↓〉 ≈⟨√
N0,↑N1,↑N0,↓N1,↓⟩≈ Nmax
(N
2−Nmax
)
≈ S2 − S2z
4S2
N2
4, (5.46)
the GP equations now read:
(−
2
2m∇2 + U(r) + g
N
2φ∗
0(r)φ0(r) + ηgN
2φ∗
1(r)φ1(r)
)φ0(r) = µ0φ0(r) + λφ1(r), (5.47)(
− 2
2m∇2 + U(r) + g
N
2φ∗
1(r)φ1(r) + ηgN
2φ∗
0(r)φ0(r)
)φ1(r) = µ1φ1(r) + λ∗φ0(r), (5.48)
where
η =5S2 − S2
z
4S2. (5.49)
The coefficient η now takes a different value between 1 and 5/4 in every run of the experiment.
When Sz = ±S, we find that η = 1, and the interaction term does not favour separation
result that, in the spinless case, a single condensate is energetically more favourable than a fragmented one
when the single-particle ground state is degenerate. The extra energy in the fragmented state is due to the
Fock term in the expectation value of the interaction Hamiltonian.
93
between the two condensates any more.16 This case can be understood by noting that when
Sz = ±S, Nmax = 0 or Nmax = N/2, so that the atoms in the two condensates are in
different internal states, and the Fock term disappears, which gives η = 1. The appearance
of Sz in the GP equations looks somewhat odd and suspicious, especially that one of the
main ideas in this Chapter is to assume perfect rotational symmetry. However, in deriving
these equations we have assumed that initially the system had well-defined values of S and
Sz. If we now take the appropriate statistical distributions, we get the GP equations with
the factor η taking a random value between 1 and 5/4, without any further information
about the origin of that random value. Sz then plays no special role in the equations, and
the same results could be obtained using Sx or Sy instead. If we take S 1, we find that
〈S2z 〉 = S2/3, and therefore η = 7/6 as in the case S = Sz = 0. In the TF limit, all the
different values of η give the same density distribution, since in that limit as long as η is
greater than 1, there will be complete separation between the two wave functions.
5.7 Detection of the Fragmented State
Let us assume that we have prepared the fragmented state suggested in Section 5.2 (i.e.
two spatially distinct condensates with total spin of the whole system ∼ 0). Now we ask
the question of whether there is a way to verify experimentally that we do in fact have the
fragmented state. Another question of interest is the evolution of the quantum state of the
condensate due to the measurement, assuming that initially it is in the fragmented state. As
a first step, let us imagine a device that measures the total spin of the condensate, regardless
of the spatial coordinates of the different atoms. We have found in Section 5.2 that if we
impose conservation of total spin, the system is described by a mixture of states with a
distribution of values of S and Sz. The proposed measurement simply tells us which of these
states the system is in. Therefore, in every run of the experiment, one state of the mixture
16Needless to say, it does not favour miscibility either.
94
will be chosen (or simply measured), and the system ends up in a pure state corresponding to
the measured values. Otherwise, the two condensates remain unchanged and, in fact, remain
entangled to conserve the small value of the total spin. Although this measurement would
distinguish easily between a fragmented condensate and the ground state of the condensate,
we do not know whether such a measurement would be simple to carry out experimentally.
In the alkali atomic gases, one usually measures density distributions using optical imaging
techniques, which we analyze next.
Now we analyze the density distribution that is obtained when the condensate is imaged,
for example, using absorption imaging. In the true many-body ground state, only one
orbital wave function is macroscopically occupied. Therefore, the density distribution will
simply reflect the square of the absolute value of that wave funtion, and there will not
be any interference effects. Since in the fragmented state two orbital wave functions are
macroscopically occupied, one can expect to see some interference effects in the density of
each of the two components (↑ and ↓). That would be a clear signature of the fact that the
system is not in its ground state. We shall see below that seeing interference fringes does not
necessarily mean that we have the fragmented state. The results that we shall find can be
understood in terms of coherent states. As in Chapter 3, we find that one cannot distinguish
between the fragmented state and a certain mixture of coherent states. Nevertheless, the fact
that a macroscopic number of atoms do not occupy the lowest orbital orbital wave function
(neglecting depletion) proves that the physics discussed in this Chapter is playing a role in
the problem.
First we treat a simpler problem whose use we shall justify shortly. Let us take a conden-
sate in the state |N1, N2〉, where N1 and N2 are the numbers of atoms in the single-particle
orbital states denoted by the indices 1 and 2, with no internal degrees of freedom. The
quantum state can be expressed as:
95
|Ψ〉 =
(a†1)N1
(a†2)N2
√N1!N2!
|0〉
=
(2πN1N2
N
)1/4 ∫ π
−π
dχ
2π√N !ei(N2−N1)χ/2
(√γeiχ/2a†1 +
√1 − γe−iχ/2a†2
)N
|0〉, (5.50)
where γ = N1/N , and N = N1 + N2. Imaging the condensate is equivalent to measuring
the relative phase between the two independent subcondensates (see Section 4.2). In the
interference pattern, the condensate will mimic a state of the form:
|Ψ〉 =1√N !
(√γeiχ/2a†1 +
√1 − γe−iχ/2a†2
)N
|0〉, (5.51)
with a randomly chosen value for the relative phase χ. The density is then given by:
ρ(r) =∣∣∣√γφ1(r) +
√1 − γe−iχφ2(r)
∣∣∣2= ρ1(r) + ρ2(r) + 2
√γ(1 − γ)Re
(φ1(r)φ2(r)e
−iχ). (5.52)
The last term in Eq. (5.52) describes interference effects. To go further with the analysis
of the exact shape of the interference pattern, we need to have the exact form of the two
occupied wave functions. For definiteness and simplicity, we take them to be the single-
particle ground state and one of the two first-excited states of a one dimensional box with
periodic boundary conditions. We shall see in Section 5.8 that this example is realistic and
quite interesting. The two wave functions are given by φ1 = 1/√L and φ2 = e2πix/L/
√L,
where L is the size of the box. The density now reduces to:
ρ(r) =N
L+ 2
√N1N2
Lcos(
2πx
L− χ). (5.53)
The position of the central maximum of the density distribution is determined randomly in
every run of the experiment (xmax = χL/2π). The visibility of the interference pattern is
given by:
V ≡ ρmax − ρmin
ρmax + ρmin
=2√N1N2
N. (5.54)
The visibility goes from 0%, when N1 = 0 or N2 = 0, to 100%, when N1 = N2 = N/2, as we
obtained in Section 4.2.
96
Now we explain why that problem is relevant here. Let us go back to the many-body
wave function describing the fragmented state (Eq. 5.7):
|Ψ〉 =∑N0,↑
cN0,↑ |N0,↑, N/2 + S −N0,↑, N/2 + Sz −N0,↑, N0,↑ − S − Sz〉. (5.55)
Each term inside the sum in Eq. (5.55) contains a well-defined number of atoms in each of
the four populated single-particle states. If we make a measurement on the ↑ component
of the condensate, we have to use a description that discards any information about the ↓component. That is done by taking the reduced density matrix that describes only the ↑component. In other words, we have to take the trace of the full density matrix over the
states of the ↓ component. For given values of S and Sz, there is only one degree of freedom
in the above quantum state (Eq. 5.55). We shall use the quantum number N0,↑ to describe
that degree of freedom, since the other three quantum numbers are determined exactly by
N0,↑, N , S and Sz. Tracing over one of the quantum numbers in this case is equivalent to
introducing N0,↑-conserving decoherence to the system. That has the effect of eliminating
the off-diagonal elements in the density matrix, leaving the diagonal elements unchanged.
The reduced density matrix (after tracing out the ↓ component) is then given by:
ρ(N0,↑, N ′0,↑) = 〈|cN0,↑|2〉δN0,↑,N ′
0,↑ , (5.56)
which for the case S = Sz = 0 gives:
ρ(N0,↑, N ′0,↑) =
1
N/2 + 1δN0,↑,N ′
0,↑ . (5.57)
For the case 1 S N/2, it also gives:
ρ(N0,↑, N ′0,↑) =
⟨1√
2π∆Nexp−(N0,↑ −Nmax)
2
2(∆N)2
⟩δN0,↑,N ′
0,↑
≈ 1
N/2 + 1δN0,↑,N ′
0,↑ . (5.58)
We obtained Eq. (5.58) by considering some value of S 1. All the different values of
Sz appear in the density matrix with equal probability. Since Nmax is a linear function
97
of Sz, we expect all the different values of N0,↑ to appear with approximately the same
probability. This approximation is probably invalid near the edges of the possible values
of N0,↑, and there can also be some finite fluctuations around it. However, for the puposes
of the following calculation, it should give a good description of the density matrix. Now,
in order to analyze the density distribution in real space of the ↑ component, we have to
describe the system using a mixture of states each of which has a definite N0,↑ (and the
corresponding N1,↑). Each state in that mixture has the form (the second number inside the
ket corresponds to N1,↑):
|Ψreduced〉 = |N0,↑, N/2 + Sz −N0,↑〉
≈ |N0,↑, N/2 −N0,↑〉, (5.59)
and appears with probability given by the (diagonal) density matrix above (Eq. 5.58).
Therefore, in every run of the experiment a different value is obtained for N0,↑. For a given
value of N0,↑, an interference pattern is obtained with a random value for xmax, just as in
any interference experiment, but with a definite value for the visibility given by Eq. (5.54).
The probability distribution of the visibility is given by:
dP
dV=
dP
dN0,↑× 1
|dV/dN0,↑|≈ 2
N× 1
2N
∣∣∣√N/2−N0,↑N0,↑
−√
N0,↑N/2−N0,↑
∣∣∣=
V√1 − V 2
. (5.60)
The probability distribution has more weight in the large-V region than the small-V region.
Therefore, there is more probability for seeing an interference pattern with high visibility
than seeing one with low visibility. Now we ask the following question: What do we obtain
if, after measuring the density distribution of the ↑ component, we measure the density
distribution of the ↓ component? To answer that question, we have to take the initial state
of the condensate (Eq. 5.55) and study the evolution of that state due to the annihilation of
98
the already-detected atoms. Although it is possible to carry out that procedure, it involves
somewhat messy algebra. Therefore, we take a different approach to answer that question.
Now we rederive the above results using a faster and more intuitive method. In particular,
we consider the singlet state, and we express it in the form of Eq. (5.15):
|N, S = Sz = 0〉 =
√N
2
∫dΩ
4π(N/2)!
(a†0(Ω)
)N/2 (a†1(−Ω)
)N/2
|0〉. (5.61)
The different states inside the integral, corresponding to the different values of Ω, give rise to
different interference patterns. Even if we pick a definite value of Ω, there is still one relative
phase (that is not part of Ω) that is determined randomly upon measuring the interference
pattern, since we still have two condensates with N/2 atoms each. For given values of the
solid angle Ω and the relative phase χ, we find the following coherent state:
|Ψ(Ω, χ)〉 =1√N !
(eiχ/2a†0(Ω) + e−iχ/2a†1(−Ω)
)N
|0〉
=1√N !
(eiχ/2 cos
θ
2a†0,↑ + eiχ/2 sin
θ
2a†0,↓ + e−iχ/2 sin
θ
2a†1,↑ − e−iχ/2 cos
θ
2a†1,↓
)N
|0〉
∝(∫
dx
(cos
θ
2+ ei( 2πx
L−χ) sin
θ
2
)ψ†↑(x)
+
∫dx
(sin
θ
2+ ei( 2πx
L−χ−π) cos
θ
2
)ψ†↓(x)
)N
|0〉, (5.62)
where θ is the azimuthal angle (of Ω). It is the extra phase factor of π in the last equation
that is responsible for the fact that if the ↑ component has a maximum at a certain point
xmax, the ↓ component will have its minimum at that point. The visibility of the interference
pattern of either component for a certain value of Ω is given by:
V = sin θ. (5.63)
xmax is determined by the randomly chosen relative phase between the condensates and
is independent of Ω. The process of measuring the density distribution can therefore be
thought of as a measurement of the angle θ and the relative phase χ. In every run of the
experiment, one of the different values of θ is chosen randomly. As a result, a different value
99
of the visibility is obtained in every run of the experiment. If we calculate the probability
distribution of the visibility we find that
dP
dV=dP
dθ× 1
|dV/dθ|=
sin θ
| cos θ|=
V√1 − V 2
, (5.64)
in agreement with the above treatment. This treatment (using the basis of definite Ω states)
can, in principle, be generalized to states of nonzero total spin. However, the algebra becomes
rather complicated due to extra phase factors that are required in the basis states. Without
going through the algebra, one can intuitively see that similar results will be obtained for
S N .
We can now answer the question of the evolution of the quantum state of the condensate.
Since the interference measurement reveals the values of θ and χ, the state of the condensate,
which initially was a superposition of all possible values of θ, φ and χ, becomes more and more
localized in the θ and χ directions as atoms are detected and these quantities are revealed
more and more accurately. However, the density distribution of the ↑ and ↓ components does
not reveal any information about the angle φ, which describes the relative phase between
the ↑ and ↓ components. Therefore, the state of the condensate remains a superposition
of all possible values of φ. If the spin density distribution is measured along three or more
different axes, one effectively measures θ, φ and χ. Therefore, the state of the condensate
evolves closer and closer to a coherent state where all three quantities are well defined.
The resemblance between the above treatment and that in Chapters 3 and 4 suggests that
one might not be able to distinguish between the fragmented state and a uniform mixture of
coherent-like states. We have just demonstrated this fact for interference measurements. In
fact, in such experiments one cannot distinguish between the fragmented state and a mixture
of coherent states where the relative phase between the two subcondensates χ is well-defined,
provided one takes a mixture of all the possible values of the relative phase. If we go back
100
to the example of measuring the total spin of the condensate, we find that the coherent and
coherent-like states have total spin of the order ∼ √N . Therefore, in order to determine
with certainty that we do in fact have the fragmented state, we need to measure the total
spin of the condensate with relative accuracy better than N−1/2. One can also argue that
decoherence will destroy the fragmented state and make the coherent states more suitable
to describe the system.17 We stress again, though, that even if we cannot tell with certainty
that we have the fragmented state, it is sufficient to show that two orbital wave functions are
macroscopically occupied (or at least, that the occupied wave function is not the one with
the lowest energy). That proves that something is preventing the atoms from condensing
into the lowest wave function, and that by itself is an interesting phenomenon.
5.8 Condensation in a Toroidal Geometry
Let us add a new twist to the problem. We consider a Bose gas of spin 1/2 atoms trapped
in a toroidal geometry. We assume that the length scales in the system are designed such
that the radial degrees of freedom are frozen out, and the relevant orbital states are defined
by a single quantum number, namely the angular momentum around the axis of the torus.
We also assume that the interaction energy scale is much smaller than the kinetic energy
scales, both along and perpendicular to the angular degree of freedom. The ground state in
this geometry corresponds to a uniform phase around the torus:
φ0(θ) =1√2π, (5.65)
and the first-excited state is two-fold degenerate:
17It is not obvious to us whether atom loss will contribute to decoherence in the same way it does in
condensates of spin 1 atoms, as we have shown in Chapter 3. One complication here is that if an escaping
atom is detected, one still cannot tell which orbital wave function it came from. However, it would be
interesting to explore the idea that atom loss is enhanced for certain values of the relative phase between two
condensates and suppressed for other values. This phenomenon can still serve as a decoherence mechanism.
101
φ±1(θ) =1√2πe±iθ. (5.66)
We now try to use the result of Section 5.2 that when the gas is cooled down to T = 0, two
condensates will form: N/2 + S atoms will occupy the ground state, and N/2 − S atoms
will occupy one of the two first-excited states. However, the resulting many-body quantum
state has a net orbital angular momentum of ±(N/2 − S), depending on which of the two
wave functions is occupied. This would violate conservation of orbital angular momentum,
if the trap is symmetric under rotations about the z axis. This difficulty can be avoided
by occupying three orbital wave functions instead of two. If we put N/2 + S atoms in the
ground state φ0(θ), (N/2−S)/2 atoms in the state φ1(θ) and (N/2−S)/2 atoms in the state
φ−1(θ), the total orbital angular momentum would add up to zero, which we are assuming
to be the initial value. The spins of the atoms in the two excited states can still add up to
the value N/2 − S, which is required to satisfy the constraint of conservation of total spin.
If, due to statistical fluctuations, the system has a finite value of orbital angular momentum
L (∼ √N), the occupation of the states would be N/2 +S atoms in φ0(θ), (N/2−S +L)/2
atoms in φ1(θ) and (N/2 − S − L)/2 atoms in φ−1(θ).
Now let us assume that the trap is not completely axially symmetric, but the states
φ±1(θ) are still degenerate. Orbital angular momentum is no longer a conserved quantity.
If the atoms are noninteracting, the ground state has approximately half the atoms in the
φ0(θ) state, and the other half can be distributed in any way between the states φ1(θ) and
φ−1(θ). Thus the many-body ground state is approximately (N/2)-fold degenerate. If we
include weak interactions, however, we find an energy term of the form:
Eint =ε
2
(N(N − 1) +N0(N1 +N−1) +N1N−1
). (5.67)
The last term clearly breaks the degeneracy, and it is minimized by making either N1 = 0 or
N−1 = 0. The ground state then has half the atoms in the φ0(θ) state and the other half in
one of the two states φ±1(θ). That state has a finite orbital angular momentum (≈ N/2).
The interesting feature about this state is that it spontaneously gains orbital angular mo-
102
mentum. Even without rotating the trap or having initial angular momentum, the orbital
angular momentum of the system increases from zero to N/2 in order to maintain the sym-
metry of the many-body wave function. It is worth stressing again that this phenomenon
is not related to any spin-orbit coupling, since we are not assuming any such terms in the
Hamiltonian. This spontaneously rotating state can be detected easily by imaging the con-
denaste as explained in Section 5.7 and by ?. In every run of the experiment, an interference
pattern is observed with a random value chosen for the offset xmax and the visibility V .
In the strongly interacting limit, where the interaction energy is larger than the kinetic
energy, the results are less interesting. From the results of Section 5.6, we find that instead
of φ0(θ) and φ1(θ), the two occupied states will be nonoverlapping ones localized on opposite
sides of the trap. Therefore, the interesting results of this Section, such as spontaneous
rotation of the gas and the random-visibility interference patterns, would not occur.
5.9 Conclusions
We have studied the problem of condensing a system of effectively spin 1/2 atoms. We
presented a flawed argument based on the concept of spontaneous symmetry breaking in the
Bose-Einstein condensation transition. We discussed two paradoxes related to that argument
and showed that, under certain conditions, the system forms a fragmented state where two
orbital wave functions are macroscopically occupied. We derived a set of Gross-Pitaevskii
equations that describe the two macroscopically-occupied wave functions. We analyzed a
possible method to detect the fragmented state. It turns out that measuring the spin density
of the fragmented state gives interference patterns where both the offset, i.e. the positions
of the maxima, and the visibility are determined randomly in every run of the experiment.
Finally, we studied the possible realization of the problem in a toroidal trap and found that
this system exhibits the interesting behaviour of spontaneously acquiring macroscopic orbital
angular momentum.
103
Chapter 6
External Josephson Effect in
Bose-Einstein Condensates with a
Spin Degree of Freedom
In this Chapter we study the external Josephson effect, i.e. the Josephson effect between two
spatially separated Bose-Einstein condensates (BEC). In Section 6.1 we study the spinless
case, where the condensates have no internal degrees of freedom. We describe the model
system and the Hamiltonian, and then after deriving the equations of motion, we study the
different dynamical regimes and map the dynamics of the system onto that of a classical
particle in a well. We analyze the stability of the different equilibrium points of the motion.
In Section 6.2 we analyze the case of spin 1/2, i.e. when the atoms can occupy two internal
states besides the orbital degree of freedom. We derive simple equations of motion for
this system closely analogous to the Bloch equations. We find density and spin modes of
oscillation and stable equilibrium points of the motion that are unstable in the spinless case.
Finally, in Section 6.3 we consider the oscillation modes in the spin 1 case.
104
6.1 The Spinless Case
The external Josephson effect with no internal degrees of freedom has already been addressed
extensively in the literature [see e.g. ?????????]. Josephson-like phenomena have been
observed experimentally by ? and by ?. In this Section we give an overview of the model
used to describe this problem, and we present some of the basic results. Note that none of
the dynamical behaviours presented in this Section is a result of BEC, and they occur in
internal-state oscillations in noncondensed clouds, as demonstrated by ?.
Let us take a condensate in a symmetric double-well potential where the barrier is larger
than the chemical potential, and therefore, in order to go from one side to the other, the
atoms must tunnel under the barrier. This setup can be achieved by taking a condensate in
a single well and slowly raising a potential barrier in the middle, thereby splitting it into two
parts. We now make a two mode approximation to describe the system, i.e. we assume that
at any point in time the state of the system can be described completely by the occupation
of two single-atom states.1 Let |R〉 and |L〉 be the two single-atom states corresponding
to the occupied wave functions in the right and left wells. The single-atom states are, in
principle, time dependent and will be approximately given, in the adiabatic approximation,
by the Gross-Pitaevskii ground state, which in turn is determined by the number of particles
in each single-atom state. The Hamiltonian is given by:
H = −ω0
2(a†LaR + a†RaL) +
ε
2(a†La
†LaLaL + a†Ra
†RaRaR). (6.1)
The first term is the Josephson term describing the tunneling of atoms between the two
sides of the junction, and the second is the interaction term, which is comprised of the sum
of the interatomic interaction energies on the two sides. ω0 and ε will be taken as constant
coefficients, although we expect them both to change substantially for large values of the
imbalance in the number of atoms on the two sides of the junction. We shall assume that ε is
1Typically the density ρ and the s-wave scattering length as satisfy the relation ρa3s ∼ 10−5, which means
that the above approximation works extremely well, at least in the static limit.
105
positive, because we want to avoid two possible complications, namely the collapse of the gas
and the possible accumulation of atoms on one side of the junction. When each of the two
states defined above is macroscopically occupied, we can use the semiclassical approximation,
i.e. we assume that the fluctuations around the mean values of the physical quantities are
small (see below). With that assumption we can replace the creation and annihilation
operators by their expectation values (aL ≈ ΨL and aR ≈ ΨR, where ΨR,L are c-numbers).
We now define the two variables that we shall be using to describe the system, namely n and
χ [see ?]. These variable are defined implicitly by the formula: ΨR,L =√N ± n exp (±iχ/2),
up to an overall phase factor, where N is one half of the total number of atoms. n is one
half of the difference in the number of atoms between the two wells, and χ is the relative
phase between the condensates in the two wells. It can be shown that the variables n and χ
are canonically conjugate to each other, even when treated as quantum operators [see ?, see
also Appendix A]. We can now write a semiclassical Hamiltonian by replacing the operators
in Eq. (6.1) by their expectation values:2
H = −ω0
√N2 − n2 cosχ +
ε
2n2. (6.2)
To be consistent with the semiclassical description, we require that the standard deviations
of the quantum operators χ and n satisfy the conditions σ(χ) 1 and σ(n) N during
the motion of the system. To estimate these deviations we look at the ground state of the
system. Expanding the Hamiltonian (Eq. 6.2) around n = 0, χ = 0, we find that
H ≈ −ω0N +ω0N
2χ2 +
ω0/N + ε
2n2. (6.3)
This Hamiltonian is exactly that of a harmonic oscillator whose ground state has widths
σ(n) ∼ σ−1(χ) ∼ (ε/ω0N + 1/N2)−1/4. The second inequality is always satisfied for positive
values of ε. The first is satisfied only in the Josephson and Rabi regimes, to which we shall
restrict our analysis from now on.
2For different derivations of these expressions for the Josephson and interaction terms, including their
coefficients, see ???.
106
Since the energy, which is given by Eq. (6.2), is a conserved quantity, we can study the
general behaviour of the system by making a contour plot of the energy as a function of χ
and n [Fig. 6.1]. The equienergetic lines describe the possible paths that the system can
follow in phase space. We identify four different dynamical regimes as the parameters ω0 and
ε are varied, separated by two sharp transitions and one crossover [see Figs. 6.1(a)-6.1(c)].
Starting from the weakly interacting limit, i.e. the so-called Rabi regime [Fig. 6.1(a)], the
transition to the intermediate regime is marked by the appearance of closed orbits oscillating
around nonzero values of n and centered at χ = π [Fig. 6.1(b)]. The second transition occurs
when open orbits appear where χ extends over all values [Fig. 6.1(c)]. This is a transition
from the intermediate to the Josephson regimes. These transitions happen at the values
ω0 = εN and 2ω0 = εN , respectively, for our model. Finally when ω0N ∼ ε there is a
crossover to the Fock regime, where quantum phase fluctuations cannot be neglected. Deep
in the Rabi regime (when ω0 εN), the tunneling energy dominates. In the Josephson
regime (ω0 εN ω0N2), both terms in the Hamiltonian are important and finally, in the
Fock regime (ω0N ε), the interaction term dominates [see ?].
Under current experimental conditions, ω0 can be varied anywhere from 0 to 100s−1.
On the other hand, ε can go from 0.01 to 0.1s−1 [see ?], and N is usually between 104 and
107. With this range of parameters, the Fock and Josephson regimes are easily accessible,
whereas the Rabi regime is more difficult to achieve. It is important that the frequency of
any oscillation between the wells be smaller than the lowest intrawell excitation frequencies,
so that during the motion, these degrees of freedom are not excited. In practice this means
that√ω0εN has to be smaller than the frequency of the lowest intrawell collective mode
(as will become clear below). The experimental observations can be made by measuring the
density (and therefore n) in the usual way, either destructively or by phase-contrast imaging.
Now we turn to the equations of motion of n and χ. Since these two variables are
canonically conjugate to each other, we can use the semiclassical Hamilton’s equations (see
Appendix A for derivation):
107
n =∂H
∂χ= ω0
√N2 − n2 sinχ (6.4)
χ = −∂H∂n
= −ω0n√
N2 − n2cosχ− εn. (6.5)
If we now differentiate Eq. (6.4) with respect to time and use Eqs. (6.4) and (6.5) to
eliminate n and χ, we find that
n = − (ω2
0 − εH(0))n− ε2
2n3, (6.6)
where H(0) = −ω0
√N2 − n2(0) cosχ(0) + ε
2n2(0). Eq. (6.6) is formally identical to the
equation of motion of a particle with unit mass in the quadratic-plus-quartic effective po-
tential
Veff(n) =1
2
(ω2
0 − εH(0))n2 +
ε2
8n4, (6.7)
with effective total energy
Eeff = Veff (n) +1
2n2. (6.8)
Note that Eeff and Veff cannot be chosen independently, since they both depend on the
initial conditions. The variation of H(0) and of n allows us to find three different types of
motion [Figs. 6.2(a)-6.2(c)]. In the first type, the coefficient of the quadratic term is positive,
and n oscillates around zero, which is the minimum of Veff [Fig. 6.2(a)]. This corresponds
to either oscillations around the origin [Figs. 6.1(a)-6.1(c)] or to small oscillations around
the π state [Fig. 6.1(a)]. The second case occurs when the coefficient is negative and Eeff is
positive, which also leads to oscillations of n around zero, although that point is no longer a
minimum of Veff [Fig. 6.2(b)]. It corresponds to large oscillations around the origin or the π
state [(Figs. 6.1(b) and 6.1(c)]. The third one corresponds to both Eeff and the coefficient
being negative [Fig. 6.2(c)] and leads to self-trapped behaviour [oscillations around n = 0;
Figs. 6.1(b) and 6.1(c)]. There is a well-known analogy between the Josephson effect in BEC
and a momentum-shortened pendulum in a gravitational field whose behaviour is also fully
reproduced by this particle-in-a-well model [For the pendulum analogy, see e.g. ??].3
3It is worth mentioning though that the explanation of π states and self-trapped π states is rather artificialand unintuitive in the pendulum analogy.
108
We now go back to Fig. 6.1 and identify three types of equilibrium points: the ground
state (n = 0, χ = 0), π states (n = 0, χ = π) and self-trapped π states (n = 0, χ = π).
For the typical experimental parameters ω0 ∼ 10s−1, ε ∼ 0.01s−1 and N ∼ 106, the system
is in the Josephson regime. From Fig. 6.1(c) we see that π states are unstable, whereas
self-trapped π states exist and are stable. However, let us estimate n for the self-trapped
π state. That can be done by setting Eq. (6.6) equal to zero and evaluating n, which
gives n =√N2 − (ω0/ε)2 = N − O(1). Most of the atoms are therefore on one side of the
junction and only a few atoms, at most, are on the other side, rendering the semiclassical
approximation invalid. Therefore, for these typical parameters, one can say that self-trapped
π states do not exist. Finally, we estimate the frequency of small oscillations around the
ground state. We solve Eq. (6.6) neglecting second and higher order terms in n and χ, and
we find the frequency√ω2
0 + ω0εN ∼ 102s−1.
By this we conclude our discussion of the spinless case. We now generalize the model
and the analysis to cases where the condensates have an internal degree of freedom, and we
investigate whether this extra degree of freedom adds any new physics to the problem.
6.2 Spin 1/2 Case
We now generalize the treatment of the external Josephson effect to the case when there are
two species of atoms in the trap, e.g. the |F = 2, mF = 1〉 and |F = 1, mF = −1〉 states of
87Rb [?]. The system can be realized experimentally by, for example, taking a single-species
condensate of 87Rb in a single well and slowly raising a potential barrier in the middle,
splitting it into two parts. Following this laser pulses may be applied to each side selectively
in order to choose a particular superposition of internal states of the atoms on each side.
An important point about optical imaging techniques is that they allow the determination
of the behaviour of each hyperfine species separately.
To describe the system theoretically, we make a four-mode approximation, with |1, R〉,
109
|2, R〉, |1, L〉 and |2, L〉 being the four single-atom states corresponding to the four modes,
where the labels R and L refer to the right and left wells, respectively, and the labels 1 and 2
refer to the two hyperfine states. The Hamiltonian that we shall use to describe this system
is the simplest generalization of that in Eq. (6.2):
H = HJ +Hint, (6.9)
where
HJ = −ω0
∑i=1,2
√N2
i − n2i cosχi, (6.10)
and
Hint =1
2
(ε11n
21 + ε22n
22 + 2ε12n1n2
). (6.11)
The interaction term conserves the total number of atoms in each hyperfine state separately
(We are here ignoring loss processes that occur in the real system), and there is no external
laser coupling between the two states. Therefore, the relative phase between the two species
does not appear in the Hamiltonian and is not a useful variable for describing the system.
Although the difference in the chemical potential between the two species can be very large
and cause the relative phase between the two species to vary rapidly, this does not affect
the Josephson dynamics. For the same reason, the dephasing between the two hyperfine
components is irrelevant. We take ω0 to be the same for both hyperfine states for simplicity.
The expression for the interaction coefficients εij in terms of the density ρi of species i and
the interaction parameter gij = 4π2aij/m is:
εij =gij
NiNj
∫drρiρj, (6.12)
where aij is the s-wave scattering length between atoms of type i and j.
In order to justify the use of the Hamiltonian in Eq. (6.9), we make all the assumptions
that we made in Section 6.1. A few additional assumptions are required in this case. The
two species must be miscible. In other words, there can be no component separation. If
110
this were not the case, we might not have the same tunneling matrix element ω0 for both
species, and the interaction energy would not have the form that we assume. This means
that a condition must be imposed on the interaction parameters, namely that ε11ε22 > ε212
[see ?].4
The pairs of variables ni and χi are canonically conjugate. The equations of motion can
therefore be simply derived as:
ni =∂H
∂χi= ω0
√N2
i − n2i sinχi, (6.13)
χi = −∂H∂ni
= −ω0ni√
N2i − n2
i
cosχi −∑
j
εijnj. (6.14)
In Sections 6.2.1 and 6.2.2, we shall rewrite the equations of motion in terms of new variables
in order to provide some insight into the dynamics of the system.
6.2.1 Isotropic Case
We first consider the isotropic case, where ε11 = ε12 = ε22 = ε. At first sight the equality
of the interaction parameters seems to violate the miscibility condition, which says that
ε11ε22 > ε212. However, this condition does not take into account the kinetic energy, which
favours miscibility. Therefore, isotropy does not pose any such problems.
We note that the Hamiltonian (Eq. 6.2) is invariant under arbitrary SU(2) transfor-
mations applied simultaneously to the spins in both wells. That is, if we transform the
two-component spinor order parameters ΨL and ΨR with the same unitary operator, the dy-
namics should remain unchanged.5 This suggests that we re-express the equations of motion
in terms of quantities that are invariant under such transformations. We therefore define the
following dot products of spinors:
4It is known that the |F = 2, mF = 1〉 and |F = 1, mF = −1〉 hyperfine states of 87Rb are miscible [see
e.g. ?].5This conclusion, of course, depends on the isotropy of the interaction Hamiltonian Hint.
111
n+ ≡ |ΨR|2 − |ΨL|22
=∑
i
ni,
α+ ≡ Ψ∗LΨR − Ψ∗
RΨL
2i=∑
i
√N2
i − n2i sinχi, (6.15)
β+ ≡ Ψ∗LΨR + Ψ∗
RΨL
2=∑
i
√N2
i − n2i cosχi.
The subscript (+) will be used to distinguish this set of variables from another one with
subscript (−) to be defined below. Using the equations of motion for ni and χi, we find thatn+
α+
β+
=
0 ω0 0
−ω0 0 −εn+
0 εn+ 0
n+
α+
β+
. (6.16)
If we now define the three-component vectors r+ ≡ (n+, α+, β+) and B(t) ≡ (εn+, 0,−ω0),
we can rewrite the equations of motion succinctly as:
r+ = B(t) × r+. (6.17)
Note though that B and r+ are not independent since they are both functions of n+.
Straightforward manipulation of Eq. (6.16), or directly of Eqs. (6.13) and (6.14), leads
to:
n+ = − (ω2
0 − εH(0))n+ − ε2
2n3
+, (6.18)
where H(0) = −ω0β+(0) + ε2n2
+(0). This equation is quite general, since it is valid not
only for two hyperfine states but for any number of them, as long as they interact only
through a n2+ ≡ (
∑i ni)
2 term. In fact, this equation is identical to Eq. (6.6) that we found
for n in the spinless case. In other words, if the interaction term is isotropic, the density
mode (i.e. oscillations of the total number) is insensitive to the internal degrees of freedom,
and its dynamics can be mapped onto that of a classical particle in a well. The analysis
of the dynamics of n given in Section 6.1 applies here as well, with the straightforward
generalizations of Veff and Eeff .
112
Specifying the dynamics of n+ does not describe the motion completely. For example,
even in the spinless case it is known that the self-trapped regime includes two different
behaviours of the relative phases, the so-called “running” and “oscillating” phases. To
further understand the dynamics of the two hyperfine state Josephson effect, we introduce
the additional variables
n− ≡ n1 − n2
α− ≡√N2
1 − n21 sinχ1 −
√N2
2 − n22 sinχ2 (6.19)
β− ≡√N2
1 − n21 cosχ1 −
√N2
2 − n22 cosχ2,
and their equations of motion are:n−
α−
β−
=
0 ω0 0
−ω0 0 −εn+
0 εn+ 0
n−
α−
β−
. (6.20)
Since the matrix is the same as in Eq. (6.16), we define the three-component vector r− ≡(n−, α−, β−) and rewrite the equations of motion as:
r− = B(t) × r−. (6.21)
Now, however, B and r− are independent. Therefore, these equations are formally identical
to the Bloch equations (without any relaxation terms), familiar from the context of NMR
and quantum optics. Note that in going from the original four variables to six, we are
enlarging the configuration space, which means that not all points described by the new set
of variables are physically allowed. Therefore, care must be taken in choosing the initial
conditions of the motion.
We can obtain some physical insight into the variables n± by noting that n+ is one half
of the difference in total number between the right and left wells, and n− is one half of the
difference in z component of spin between the two wells (|Ψ1R|2−|Ψ2
R|2)/2−(|Ψ1L|2−|Ψ2
L|2)/2.
This means that the former describes the density mode whereas the latter, in that limit,
113
describes the spin mode. We can now analyze Eqs. (6.17) and (6.21) in a few limiting cases
to gain some insight into the behaviour of the system. Under appropriate conditions it is
possible to have small oscillations in n+ and no motion in n−, or vice-versa. We consider two
cases: the Rabi limit, where εN ω0, and the Josephson regime, where εN ω0. In the
Rabi case, neglecting higher order terms, the oscillation frequency of n+ (and therefore of the
density mode) can be calculated from Eq. (6.7) to be√ω2
0 + ω0εβ+. Also, using Eq. (6.21),
we can neglect the component of B along the n− axis, so that r− (which describes the spin
mode) rotates around the β− axis with frequency ω0. In the Josephson case, we consider two
types of situations: small oscillations of n+ around zero and around nonzero values. For zero
values and zero relative phases between the two sides, we find density and spin modes with
frequencies√ω2
0 + ω0εN and ω0, respectively. For zero values, a π phase in species 1 and a
zero phase in species 2, the density and spin modes have frequencies√ω2
0 + ω0ε(N2 −N1)
and ω0, respectively. For nonzero values (i.e. when n+ is “self-trapped” around a value
n0+), n+ oscillates with frequency ε|n0
+| and n− oscillates with frequency√ω2
0 + ε2(n0+)2. In
Section 6.2.3 we shall treat special cases of the above results using a different method that
applies in the limit of small oscillations.
6.2.2 Anisotropic Case
As we would expect, the equations of motion of r+ and r− in this case become much more
complicated. However, for completeness, we include them here. We find that the equations
of motion become coupled:
r+ = B1(t) × r+ + B2(t) × r−, (6.22)
r− = B1(t) × r− + B2(t) × r+, (6.23)
where B1 = (εn+ + εBn−, 0,−ω0), B2 = (εAn− + εBn+, 0, 0), ε ≡ 14(ε11 + ε22 + 2ε12), εA ≡
14(ε11 + ε22 − 2ε12) and εB ≡ 1
4(ε11 − ε22).
114
6.2.3 Discussion of the Equilibrium Points of the Motion
We now study the existence and stability of the equilibrium points of the motion. To do
this we can use the equations derived in Sections 6.2.1 and 6.2.2. However, we shall work
directly with Eqs. (6.13) and (6.14), since it turns out that they are more intuitive for our
purposes. The detailed calculations are done in Appendix B, and the main results for the
isotropic case are summarized in Table 1.
Table 6.1: Equilibrium points and oscillation modes around them in the isotropic case.
χ1, χ2 n1, n2 Existence condition Type of mode Frequency Stability condition
density√
ω20 + ω0εN0,0 0,0 always exists
spin ω0
always stable
mixed√
ω20 + ω0ε(N2 − N1) N1 − N2 − ω0/ε < 0
0,0 always existsspin ω0 always stable
π,0mixed ε|n0
+|= 0 N1 − N2 − ω0/ε > 0spin
√ω2
0 + ε2(n0+)2
always stable
density√
ω20 − ω0εN N1 + N2 − ω0/ε < 0
0,0 always existsspin ω0 always stable
π, π
density ε|n0+|= 0 N1 + N2 − ω0/ε > 0
spin√
ω20 + ε2(n0
+)2always stable
As can be expected, the lowest energy state is characterised by χ1,2 = n1,2 = 0. The
density and spin modes of oscillation around that equilibrium point can also be understood
in a simple way as in- and out-of-phase oscillations of two coupled Josephson currents. The
condition of stability of the π states (χ1 = π and χ2 = 0) can be satisfied by controlling the
number of atoms in the two components N1,2. This means that the stability of π states in
spinor condensates is robust, regardless of the ratio ω0/ε. As far as self-trapped equilibrium
points are concerned, we find two stable ones. In the case of the χ1 = π, χ2 = 0 state, the
two components are self-trapped on opposite sides of the junction, whereas in the case of
χ1 = χ2 = π the two components are self-trapped on the same side. However, as we shall see
below, for typical experimental parameters, these states are outside the region of validity of
115
the Gross-Pitaevskii (mean-field) description.
6.2.4 Experimental Considerations
The typical frequencies of small oscillations can be calculated using the following parameters:
N ∼ 106, ε ∼ 0.01s−1, εA,B ∼ 10−4s−1 and ω0 ∼ 10s−1. For these values most of the
frequencies lie between 10s−1 and 100s−1, whenever stable oscillations exist.
For a general initial state near the trivial equilibrium point, i.e. χ1,2 = 0 and n1,2 = 0,
the oscillations in n1,2 and χ1,2 will be a superposition of both density and spin modes. For
the typical parameters that we are using, the frequency of the density mode is one order of
magnitude larger than that of the spin mode, and therefore it should be simple to distinguish
between them experimentally. It should also be possible to prepare an initial state in which
only one of the two modes is significantly excited.
Although π states are unstable in the spinless case for typical parameters (which lie in
the Josephson regime), we have shown that they can be stabilized in spinor condensates. To
prepare them experimentally, we must have N1 < N2 as explained in Appendix B, where the
π phase difference is in species 1. A frequency measurement of the density mode could be
used to detect that in fact a π phase exists in species 1. Alternatively, one could observe the
destabilization of the state suddenly appearing in the form of density oscillations due to the
reduction of N2. Finally, a third possibility would be the direct imaging of the interference
pattern between the left and right condensates of species 1 during its expansion, after the
trapping potentials have been switched off.
We have shown in Section 6.1 that in the spinless case, self-trapped π states are outside
the region of validity of the Gross-Pitaevskii description, because one of the two sides of the
junction is not macroscopically occupied. In the spinor case, we see from Figs. B.1(d) and
B.1(f) (in Appendix B) that it is plausible that n0i is very close to Ni, and we no longer
expect our model to work in that region.
116
6.3 Spin 1 Case
Recently, π states were observed experimentally in 3He by ?. ? and ? attributted their
stability, in part, to the spin degree of freedom. 3He has some similarities with a condensate
of spin 1 atoms in the antiferromagnetic ground state. Therefore, in this Section we consider
the Josephson effect in a BEC of spin 1 atoms in order to investigate whether new phenomena
arise due to the exact spinor nature of these systems. We consider the Hamiltonian, mean-
field ground state and oscillation modes of a Josephson junction containing atoms with total
spin 1. We also study the stability of certain π states. The ground state of a single spinor
condensate has been analyzed in the literature [see e.g. ???, see also Chapter 2]. Here we
extend the analysis to the case where the condensate is comprised of two spatially separated
parts linked by a weak junction. Some of the results in this Section are similar to those
obtained by ? and are related to the bulk excitation spectrum of a condensate of spin 1
atoms. As in the previous Sections, we assume that the trapping potentials are identical for
all three hyperfine states (which can be achieved using optical dipole traps). Under these
conditions it is known that all three hyperfine states of the multiplet are miscible.
One might try to proceed as in Section 6.2 by deriving a set of equations for invariant
quantities, such as n+, α+, β+ and so on. However, it turns out that while this is possible, it
does not lead to simple equations of motion as in the two-component system, and therefore
this approach does not seem to provide a clear insight into the dynamics.
We shall now study the small oscillations around some of the equilibrium points in both
the ferro- and antiferromagnetic cases. We shall use the Hamiltonian:
H = HJ + Hint, (6.24)
where
HJ = −ω0
2
∑α=1,0,−1
a†α,Laα,R +H.c., (6.25)
and [see ?]
117
Hint =∑
i=R,L
ε04
(a†1,ia†1,ia1,ia1,i + a†0,ia
†0,ia0,ia0,i + a†−1,ia
†−1,ia−1,ia−1,i
+2a†1,ia†0,ia0,ia1,i + 2a†0,ia
†−1,ia−1,ia0,i + 2a†1,ia
†−1,ia−1,ia1,i)
+ε24
(a†1,ia†1,ia1,ia1,i + a†−1,ia
†−1,ia−1,ia−1,i + 2a†1,ia
†0,ia0,ia1,i
+2a†0,ia†−1,ia−1,ia0,i − 2a†1,ia
†−1,ia−1,ia1,i + 2a†1,ia
†−1,ia0,ia0,i
+2a†0,ia†0,ia1,ia−1,i − a†1,ia1,i − a†−1,ia−1,i). (6.26)
For 23Na and 87Rb, ε2 is a few percent of ε0. We now derive the equations of motion, and in
the mean-field approximation, since we are assuming a macroscopic occupation, we linearize
by keeping only terms at least of order N in Hint.
For the ferromagnetic case, if we assume that only the Sz = 1 has macroscopic occupation,
we obtain the equations:
(µ+ i
d
dt
)δφ1,L
δφ0,L
δφ−1,L
= −ω0
2
δφ1,R
δφ0,R
δφ−1,R
+
N
2
ε0 + ε2 0 0
0 ε0 + ε2 0
0 0 ε0 − ε2
2δφ1,L + δφ∗1,L
δφ0,L
δφ−1,L
,
(6.27)
and a similar set for δφi,R. µ is the chemical potential given by:
µ = ∓ω0
2+ε0 + ε2
2N, (6.28)
where the upper and lower signs correspond to a 0 and π phase between the two conden-
sates, respectively. Solving the equations of motion gives the following results: For the
ground state (the relative phase between φ1,R and φ1,L equal to zero), we find a density
mode with frequency√ω2
0 + ω0(ε0 + ε2)N , a spin mode with frequency ω0 and a quadrupole
mode with frequency ω0 + |ε2|N . For the π state, we find the same modes with frequencies√ω2
0 − ω0(ε0 + ε2)N , −ω0 and −ω0 + |ε2|N . The density mode can clearly become unstable
for ω0 < (ε0 + ε2)N , which is the case for the typical parameters quoted in Sections 6.1 and
6.2. The two modes with negative frequencies are dynamically stable but thermodynamically
unstable, i.e. this equilibrium point becomes unstable in the presence of dissipation.
118
For the antiferromagnetic case, if we assume that only the Sz = 0 state is macroscopically
occupied, we obtain the equations:
(µ+ i
d
dt
)δφ1,L
δφ0,L
δφ−1,L
= −ω0
2
δφ1,R
δφ0,R
δφ−1,R
+
N
2
ε0 + ε2 0 0
0 ε0 0
0 0 ε0 + ε2
δφ1,L
2δφ0,L + δφ∗0,L
δφ−1,L
+Nε22
δφ∗
−1,L
0
δφ∗1,L
, (6.29)
and a similar set for δφi,R. The chemical potential µ is given by Eq. (6.28) with ε0 replacing
(ε0 + ε2). Solving the equations of motion gives the following results: For the ground state
(the relative phase between φ0,R and φ0,L equal to zero), we have the following three modes:
a density mode with frequency√ω2
0 + ω0ε0N and two degenerate spin modes with frequency√ω2
0 + ω0ε2N . For the π state, we find the same modes with frequencies√ω2
0 − ω0ε0N and√ω2
0 − ω0ε2N . The density mode becomes unstable for ω0 < ε0N , and the spin modes
become unstable for ω0 < ε2N . With the parameters that we are using, at the very least the
density mode is unstable. Therefore, we conclude that the spinor nature of the condensate
by itself is not sufficient to stabilize π states. Further assumptions, e.g. effects of the walls
of the container, must be made to explain the stability of π states in condensates of spin 1
atoms.
6.4 Conclusions
We have considered the Josephson effect between two spatially separated condensates with a
hyperfine degree of freedom. We started by giving an overview of the case where the atoms
have no internal degrees of freedom. We analyzed the different dynamical regimes and found
a mapping of the Josephson dynamics onto that of a classical particle in a well. In the
two-component case, we derived a simple set of equations which, in the isotropic limit, are
119
formally identical to the Bloch equations and which provide insight into the dynamics of the
two-component condensate in a double-well setup. We also demonstrated the existence in
this system of new density and spin oscillation modes. In particular, we found π states that
are stable under experimentally accessible conditions due to the interactions between the
two species. Finally, we analyzed the spin 1 case in the same geometry both for the ferro-
and antiferromagnetic cases and found the low-lying oscillation modes. Our results indicate
that, without further assumptions, π states are unstable in condensates of spin 1 atoms.
120
-2 -1 0 1 2
a)
χ/π
-1
-0.5
0
0.5
1
n/N
-2 -1 0 1 2
b)
χ/π
-1
-0.5
0
0.5
1
n/N
-2 -1 0 1 2
c)
χ/π
-1
-0.5
0
0.5
1
n/N
Figure 6.1: Orbits in phase space of the spinless Josephson effect: (a) Rabi regime ω0/ε =1.2, (b) intermediate regime ω0/ε = 0.6, (c) Josephson regime ω0/ε = 0.4.
121
V
n
effa)
V
n
eff
b)
V
n
eff
c)
Figure 6.2: Veff(n) for different initial conditions: a) when the coefficient of the quadraticterm in Eq. (6.7), namely 1
2ω2
0 − εH(0), is positive; b) when the coefficient is negative andEeff > 0; c) when both the coefficient and Eeff are negative. The horizontal dashed linecorresponds to Eeff .
122
Chapter 7
Conclusions
We have studied a variety of problems in the theory of Bose-Einstein condensates related
to internal degrees of freedom. In particular, we focussed on two cases. The first is that of
spin 1 atoms, where the Hamiltonian possesses the rotational SO(3) symmetry in spin space.
The second case is that of two-internal-state atoms, which is formally identical to that of
spin 1/2 atoms and, under certain conditions, gives rise to condensates that possess SU(2)
symmetry. In both cases we have found several interesting phenomena that are related to
the spin degree of freedom.
We have considered a number of problems related to condensates of spin 1 atoms. We
analyzed the ground state of this system. It turns out that the case of antiferromagnetic
interatomic interactions is more interesting than that of ferromagnetic interactions, because
in the former we find competition between the SO(3) symmetric interactions, which favour
the formation of a spin-singlet state, and symmetry-breaking external fields, which favour
the formation of coherent states. We also analyzed the ground state under the constraint of
a fixed total spin and found that by changing the total spin and the external magnetic field,
we obtain a wide variety of spin density structures. Then we studied the problem of dis-
tinguishing between coherent states and the singlet state using optical imaging techniques.
Our results demonstrate that this type of measurement is essentially a measurement in the
coherent state basis. In other words, if the initial state is a superposition of several different
123
coherent states, the measurement picks out one of them. Therefore, the measurement causes
the system to evolve into a coherent state, even if it were in a singlet state before the mea-
surement. As a result, this type of measurement cannot be used to distinguish between the
singlet state and a uniform distribution of coherent states pointing in all possible directions.
We have analyzed several spatial interference problems in Bose-Einstein condensates. We
derived the well-known result stating that if two spinless condensates (of the same atomic
species, of course) overlap in space, interference fringes are observed in the overlap region.
The method that we used allowed us to generalize the treatment to the case when the
two condensates have a spin degree of freedom. That generalization answers two different
questions for the ferromagnetic and antiferromagnetic cases. In the ferromagnetic case, we
estimated the reduction in the visibility in the experiment by ? due to magnetic field in-
homogeneities and found that this reduction is too small to be seen in current experiments.
In the antiferromagnetic case, we continued our search for a method to distinguish between
coherent states and the singlet state. It turns out that the two different states are indis-
tinguishable in interference experiments. One of the interesting phenomena in interference
between spinor condensates is that in such experiments both the visibility and the offset
of the interference fringes vary from shot to shot. This is unlike single-particle interference
phenomena, where both the visibility and the offset are determined by the setup. It is also
unlike interference between spinless condensates, where the offset varies from shot to shot,
whereas the visibility is always 100%, up to some small effects.
We have considered the problem of condensing a gas of effectively spin 1/2 Bose atoms,
which can be realized, for example, in spin-polarized hydrogen. Due to the Bose symmetry
of the many-body wave function, the ground state of this system is ferromagnetic. Under
certain conditions we find that upon cooling the system does not reach its ground state,
but rather it forms a fragmented state whose formation is not caused by any interactions,
but simply by the symmetry of the initial many-body wave function. Another, somewhat
more physical, interpretation of this phenomenon can be given in terms of spin as follows:
124
The ground state of the system is a singly-condensed state. However, its total spin is
macroscopically different from that of the initial state, whereas the fragmented state has
the same total spin as the initial state. Therefore, if the cooling is done in a time that is
shorter than the spin relaxation time, the fragmented state will form, and it will subsequently
“decay” into the ground state in a time equal to the transverse spin-relaxation time (taking
into account all the different contributions). We derived a set of Gross-Pitaevskii equations
that describe the two occupied wave functions in the fragmented state. We then analyzed
the spin structures that are obtained upon imaging the condensate, since this method can
be used to detect the fragmented state. We find that in every run of the experiment, a
generally nonuniform spin density is obtained. This result is somewhat similar to those we
found when studying interference between condensates of spin 1 atoms. We considered the
consequences of forming the fragmented state in a toroidal trap, and we found that, under
certain conditions, half of the condensate will spontaneously start rotating around the torus,
even if the trap is not rotating at all.
Finally, we have treated the Josephson effect in Bose-Einstein condensates. We considered
three different cases, namely those of spinless, spin 1/2 and spin 1 atoms. In the spinless
case, we mapped the dynamics of the system onto that of a classical particle in a well.
In the spin 1/2 case, we derived a set of equations of motion to describe the system that
are formally identical to the Bloch equations, which have been analyzed extensively in the
contexts of nuclear magnetic resonance and quantum optics. We also found that π states,
which are unstable in the spinless case in the so-called Josophson regime, are stabilized in
the spin 1/2 case due to the interaction between atoms in the two hyperfine components.
In the spin 1 case, we analyzed oscillations around the ground state and certain π states.
Oscillation modes and frequencies around the ground state are related to quasiparticles and
their energies in the bulk of a spinor condensates, which is confirmed by the expressions that
we find for the frequencies. We find that, without imposing any further constraints on the
system, π states are unstable in condensates of spin 1 atoms.
125
Appendix A
Quasiorthogonality and
Overcompleteness of Coherent States
In this Appendix we demonstrate two useful properties of coherent states in Bose-Einstein
condensates, namely quasiorthogonality and overcompleteness. We use the term coherent
states to describe states of the system where all the atoms occupy the same single-particle
state. We consider two types of coherent states (for two different types of condensates):
phase states and coherent states of antiferromagnetic spinor condensate.
A.1 Phase States
We take a condensate in the two-mode approximation, i.e. we assume that only two single-
particle states are occupied. The two states will be denoted by the indices 1 and 2. The
most intuitive basis to describe the state of the condensate is the number-state basis:
|N1, N2〉 ≡(a†1)N1
(a†2)N2
√N1!N2!
|0〉, (A.1)
with the constraint that N1 + N2 = N , the total number of atoms in the condensate. Any
state of the condensate can be constructed by the appropriate superposition of states of the
form |N1, N2〉. We now want to show that phase states can be used to serve, for several
126
purposes, as a basis. Phase states are defined as:
|N,χ〉 ≡ 1√N !
(√γeiχ/2a†1 +
√1 − γe−iχ/2a†2
)N
|0〉, (A.2)
where γ is any number between 0 and 1 (The same value of γ has to be used for all the different
phase states though). The phase states defined in Eq. (A.2) cannot be a basis, simply because
there are infinitely many of them, whereas the Hilbert space isN+1 dimensional. However, if
we prove that the different phase states are quasiorthogonal and overcomplete, phase states
qualify as a basis for certain calculations (as is done in Chapter 4). Quasiorthogonality
means that if we pick any two different phase states, they will be almost orthogonal. To
demonstrate that property of phase states, we take the inner product of two states:
〈N,χ1|N,χ2〉 =1
N !〈0|
(√γe−iχ1/2a1 +
√1 − γeiχ1/2a2
)N
(√γeiχ2/2a†1 +
√1 − γe−iχ2/2a†2
)N
|0〉
=(γei(χ2−χ1)/2 + (1 − γ)e−i(χ2−χ1)/2
)N
=(
γ(1 + i(χ2 − χ1)/2 − (χ2 − χ1)2/8 + ...)
+(1 − γ)(1 − i(χ2 − χ1)/2 − (χ2 − χ1)2/8 + ...
)N
≈(1 + i(2γ − 1)(χ2 − χ1)/2 − (2γ − 1)2(χ2 − χ1)
2/8
−(γ − γ2)(χ2 − χ1)2/2
)N
≈ (1 + i(2γ − 1)(χ2 − χ1)/2)N (1 − (γ − γ2)(χ2 − χ1)
2/2)N
≈ e−N(γ−γ2)(χ2−χ1)2/2eiN(2γ−1)(χ2−χ1)/2. (A.3)
Therefore, if the difference between χ1 and χ2 is larger than N−1/2, the inner product de-
creases to negligible values. In other words, if we take any two general phase states, they will
be almost orthogonal. The fact that the inner product is finite for small values of χ2−χ1 will
play a crucial role in determining what is the most suitable value of γ for a given problem
(as explained in Chapter 4), or perhaps that phase states are not suitable at all for treating
certain problems.
127
The second property that we shall demonstrate is overcompleteness, i.e. the fact that
any general state of the condensate can be expressed as a superposition of phase states. For
that purpose it is sufficient to demonstrate that each one of the basis states in Eq. (A.1)
can be expressed as a superposition of phase states. We begin by stating the result:(a†1)N1
(a†2)N2
√N1!N2!
|0〉 = const.×∫ π
−π
dχ
2π√N !ei(N2−N1)χ/2
(√γeiχ/2a†1 +
√1 − γe−iχ/2a†2
)N
|0〉.
(A.4)
To demonstrate how we derived the above expression, we start with the right-hand side and
go backwards as follows:∫ π
−π
dχei(N2−N1)χ/2(√
γeiχ/2a†1 +√
1 − γe−iχ/2a†2)N
|0〉
=
∫ π
−π
dχei(N−2N1)χ/2∑
k
ckei(2k−N)χ/2
(a†1)k (
a†2)N−k
|0〉
=∑
k
(∫ π
−π
dχei(k−N1)χ
)ck
(a†1)k (
a†2)N−k
|0〉
=∑
k
2πδk,N1ck
(a†1)k (
a†2)N−k
|0〉
= 2πcN1
(a†1)N1
(a†2)N2 |0〉, (A.5)
where ck = N !γk/2(1 − γ)(N−k)/2/(k!(N − k)!). Since any of the basis states |N1, N2〉 can
be expressed as a superposition of phase states, any general state of the condensate can
be expressed as a superposition of phase states. The two properties that we have just
demonstrated for phase states are utilized in Chapter 4 to study interference between two
condensates.
We now turn to a somewhat different question, namely that of the dynamics. In order to
study the dynamics of the two-mode system (as is done in Chapter 6), we need the commuta-
tion relations between the different dynamical variables. We now define these variables from
the two bases discussed above, namely the number- and phase-state bases. The phase states
defined above (with a certain value of γ) are peaked around a certain number state, when
expressed in that basis. However, if we want to treat the general case where the imbalance
128
in the number of atoms in the two modes can take any allowed value, a definition of phase
states that is spread-out over all number states would be more suitable [see ?]. For purposes
of studying the dynamics, we therefore use the basis states
|n〉 ≡ |N2
+ n,N
2− n〉, (A.6)
|χ〉 ≡ 1√N + 1
N/2∑n=−N/2
einχ|n〉. (A.7)
We can now define the quantum operators n and χ as follows:
n|n〉 = n|n〉, (A.8)
χ|χ〉 = χ|χ〉. (A.9)
If we now express the phase state |χ〉 in the form of Eq. (A.7) and apply the phase operator
χ to it, we find that
χ|χ〉 = χ1√N + 1
N/2∑n=−N/2
einχ|n〉
≈ 1√N + 1
N/2∑n=−N/2
(−i ddneinχ
)|n〉
= −i ddn
|χ〉, (A.10)
where in making the approximation above we have treated n as a continuous variable and
neglected edge effects. We can now evaluate the commutator of χ and n:
[χ, n] ≈[−i ddn, n
]= −i. (A.11)
We can now use the commutator above to study the dynamics of the two-mode system.1 In
particular, in the semiclassical approximation used in Chapter 6, since the commutator is a
c-number, we find that
1The sign that we obtain here is different from that of ?. Note, however, that this sign difference does
not affect any physical results. In fact, the opposite sign was used in Ref. [?].
129
id〈χ〉dt
=⟨[χ, H
]⟩≈ [χ, n]
d〈H〉d〈n〉
= −id〈H〉d〈n〉 , (A.12)
id〈n〉dt
≈ [n, χ]d〈H〉d〈χ〉
= id〈H〉d〈χ〉 . (A.13)
In the above derivation, we have used the fact that all the other relevant commutators in
deriving the equations of motion are equal to zero.
A.2 Spinor Condensates
The system we consider here is a condensate of spin 1 atoms, and we assume that all the
atoms occupy the same orbital wave function. The state of the condensate is then determined
by the occupation of the three single-particle states corresponding to Sz=1, 0 and −1. The
most intuitive basis to describe the state of the condensate is:
|N1, N0, N−1〉 ≡(a†1)N1
(a†0)N0
(a†−1
)N−1
√N1!N0!N−1!
|0〉, (A.14)
with the constraint that N1 + N0 + N−1 = N . ? showed that there is an alternative basis
described by the quantum numbers N, S and Sz, where S is the total spin of the condensate
and Sz is the z component of S, without any additional quantum numbers. The coherent
states that we shall use here are defined as:
|N, θ, φ〉 ≡(a†(θ, φ)
)N
√N !
|0〉, (A.15)
where
a†(θ, φ) = −sin θ√2e−iφa†1 + cos θa†0 +
sin θ√2eiφa†−1. (A.16)
130
Overcompleteness means that any state of the condensate containing N particles can be
expressed as:
|Ψ〉 =
∫dΩ c(θ, φ)|N, θ, φ〉, (A.17)
where c(θ, φ) is some function of the angles θ and φ in spherical polar coordinates. For
example, the complete basis defined by the total spin and the z component of the total spin
of the condensate [see ?] can be expressed as:
|N, S, Sz〉 = const.×∫dΩYS,Sz(θ, φ)|N, θ, φ〉. (A.18)
The states (A.18) transform like the spherical harmonics YS,Sz under an arbitrary rotation,
and they possess a nonzero norm. Thus completeness (or rather overcompleteness) is proved.
Quasiorthogonality follows from:
〈N, θ1, φ1|N, θ2, φ2〉 = (cos θ)N , (A.19)
where θ is the angle between two vectors defined by (θ1, φ1) and (θ2, φ2). This result can
be obtained by observing the fact that the inner product is invariant under simultaneous
rotations of the directions (θ1, φ1) and (θ2, φ2). If we rotate them in such a way that the
former points in the z direction, the inner product reduces to:
〈N, θ1, φ1|N, θ2, φ2〉 =1
N !〈0| (a0)
N
(−sin θ√
2e−iφa†1 + cos θa†0 +
sin θ√2eiφa†−1
)N
|0〉
= (cos θ)N . (A.20)
For large N the inner product is nonvanishing at two points: when the two directions are
parallel or antiparallel. Therefore, for integration purposes it can be approximated by:
〈N, θ1, φ1|N, θ2, φ2〉 ≈ const.
sin θ1×(δ(θ1 − θ2, φ1 − φ2) + δ(θ1 + θ2 − π, φ1 − φ2 ± π)
), (A.21)
where the denominator (sin θ1) is needed to cancel a similar factor when expressing d cos θ
in the integral as sin θdθ. The plus sign in front of the second δ-function in Eq. (A.21)
131
would be replaced by a minus sign if N were odd. The constant outside the bracket can be
evaluated by performing the integral
∫dΩ2〈N, θ1 = 0, φ1 = 0|N, θ2, φ2〉 =
∫ π
0
d cos θ2
∫ 2π
0
dφ2(cos θ2)N
=4π
N + 1. (A.22)
Since that integral has two equal contributions from the two δ-functions, we finally find that
〈N, θ1, φ1|N, θ2, φ2〉 ≈ 2π
N sin θ1
(δ(θ1 − θ2, φ1 − φ2) + δ(θ1 + θ2 − π, φ1 − φ2 ± π)
). (A.23)
The state obtained by rotating (θ, φ) to its antiparallel direction is the same state (except
for a possible change of sign). Therefore we shall, with no loss of generality, restrict the
basis states and the integrals to the upper hemisphere, i.e. θ1, θ2 range from 0 to π/2. One
has to be careful that Eq. (A.21) applies only when the coefficient c(θ, φ) varies slowly in θ
and φ. It cannot be applied to states with large Stotal (e.g. the ferromagnetic ground state),
since the function YS,Sz in Eq. (A.18) changes more and more rapidly with increasing S. On
the other hand, S = Sz = 0 for the singlet state and the coefficient c(θ, φ) in Eq. (A.17) is
independent of θ and φ:
c(θ, φ) =
√N
2π. (A.24)
132
Appendix B
Calculation of the Equilibrium Points
and Oscillation Frequencies for the
Two-Component Josephson Effect
At an equilibrium point, n1,2 = χ1,2 = 0. Using Eq. (6.13) this implies that the phases χ1
and χ2 are either zero or π. From Eq. (6.14) we find that
n01 = −n0
2
(ε22ε12
+ω0
ε12√N2
2 − (n02)
2ζ2
), (B.1)
n02 = −n0
1
(ε11ε12
+ω0
ε12√N2
1 − (n01)
2ζ1
), (B.2)
where n0i and χ0
i are the coordinates of the equilibrium point. We have defined ζ1 ≡ cosχ01
and ζ2 ≡ cosχ02 to abbreviate the formulae. These equations define two functions, n0
1(n02)
and n02(n
01), which we can plot on the (n0
1, n02) plane (Fig. B.1). Since both ζ1 and ζ2 can be
1 or −1 (corresponding to χ0i = 0 or π), we have three distinct cases. For all of them, the
trivial point n01 = n0
2 = 0 is always a solution. Therefore, in all cases we have at least one
solution.
Case 1: ζ1 = ζ2 = 1 (χ1 = χ2 = 0)
As is clear from Figs. B.1(a) and B.1(b), the condition for the existence of three solutions
imposes the following condition on the slopes of the curves at the origin:
133
(ε22ε12
+ω0
ε12N2
)(ε11ε12
+ω0
ε12N1
)< 1. (B.3)
Case 2: ζ1 = −1, ζ2 = 1 (χ1 = π, χ2 = 0)
For three solutions to exist [see Fig. B.1(c) and B.1(d)], we require this time that(ε11ε12
− ω0
ε12N1
)(ε22ε12
+ω0
ε12N2
)> 1. (B.4)
Case 3: ζ1 = ζ2 = −1 (χ1 = χ2 = π)
This time we may have one [Fig. B.1(e)], three [Fig. B.1(f)] or five [Fig. B.1(g)] solutions.
To have three we need that(ε22ε12
− ω0
ε12N2
)(ε11ε12
− ω0
ε12N1
)< 1. (B.5)
If this condition is not met, we will have either one or five solutions, depending on whether
the factors on the left-hand side are both negative or both positive, respectively.
In the rest of this Appendix, we shall analyze the behaviour of the system close to the
various equilibrium points. Note that the global ground state is the trivial solution n1,2 = 0
and χ1,2 = 0. All the other equilibrium points are thermodynamically unstable, although
possibly dynamically stable.1
B.1 Isotropic Case
If all the εij’s are equal, some of the conditions above cannot be satisfied. For Case 1
condition (B.3) cannot be satisfied, and therefore only the equilibrium point n01 = n0
2 = 0
is allowed. In Case 2 both single and triple solutions are allowed. Condition (B.4) for the
existence of three equilibrium points becomes N1 −N2 − ω0/ε > 0. In Case 3 the conditions
for the existence of one or three equilibrium points can be satisfied. The condition for three
is N1 +N2 − ω0/ε > 0. However, we cannot have five equilibrium points since, if both terms
are positive in condition (B.5) and each of them is smaller than 1, their product will also be
smaller than 1.1For a discussion of dynamic vs. thermodynamic stability, see e.g. ?.
134
To study the behaviour of the system in the neighbourhood of an equilibrium point, we
shall work with the second-order differential equations for n1 and n2. To obtain these we
differentiate Eq. (6.13) with respect to time and eliminate χ1,2 and n1,2 using Eqs. (6.13)
and (6.14). We now introduce the variables δ1 and δ2, defined by
n1 = n01 + δ1, (B.6)
n2 = n02 + δ2. (B.7)
The linearized equations of motion for the isotropic case are:δ1δ2
= −Ω2
δ1δ2
, (B.8)
where
Ω2 =(ω2
0 + ε2(n0+)2
)1 + ω0ε
√N2
1 − (n01)
2ζ1√N2
1 − (n01)
2ζ1√N2
2 − (n02)
2ζ2√N2
2 − (n02)
2ζ2
. (B.9)
Case 1: ζ1 = ζ2 = 1
As mentioned before, the only stable point is at n01 = n0
2 = 0. In the basis (δ1, δ2),
we find the modes (N1, N2) and (1,−1) (Note that the matrix of the linearized equations
of motion is not Hermitian, and therefore the two eigenvectors are not guaranteed to be
orthogonal, even if the corresponding frequencies are different). The first corresponds to a
‘density’ mode with frequency√ω2
0 + ω0εN and the second to a ‘spin’ mode with frequency
ω0. Note though that, even in the density mode, the total spin on each side of the junction
changes as a function of time (unless N1 = N2).
Case 2: ζ1 = −1, ζ2 = 1
Near the equilibrium point n01 = n0
2 = 0, we proceed as above and find the eigenfrequen-
cies√ω2
0 + ω0ε(N2 −N1) and ω0 with corresponding eigenvectors (N1,−N2) and (1,−1).
135
The system is dynamically stable as long as the frequencies are real, which leads to the con-
dition N1 −N2 − ω0/ε < 0. Since it is easy to change N1 and N2 experimentally, this state
can always be made stable, regardless of the values of ω0 and ε. It is therefore much easier
to obtain a π state in a two-component condensate than in a spinless condensate. However,
there is an additional complication: If N1 = N2, the eigenvectors of small oscillations be-
come parallel, and the amplitude of the oscillations for initial displacements in the in-phase
direction will tend to diverge as (N2 −N1)/N → 0 after some time. Experimentally it is not
difficult to prepare a condensate with different values for N1 and N2 and avoid this pitfall.
Near the equilibrium point n01, n
02 = 0, the frequencies are ε|n0
+| for the density mode and√ω2
0 + ε2(n0+)2 for the spin mode. Both frequencies are real, and therefore this equilibrium
point is stable (however, see Section 6.2.4).
Case 3: ζ1 = ζ2 = −1
For n01 = n0
2 = 0, we find the eigenfrequencies√ω2
0 − ω0εN and ω0 with corresponding
eigenvectors (N1, N2) and (1,−1). The system is dynamically stable as long as N < ω0/ε.
For n01, n
02 = 0 the frequencies of the two eigenmodes are given by the same expressions as in
Case 2 (ε|n0+| and
√ω2
0 + ε2(n0+)2), and therefore they are both stable as long as the points
exist.
Note that all the frequencies found in Section 6.2.1 are in agreement with those derived
here by studying the small oscillation behaviour directly from Eqs. (6.13) and (6.14).
B.2 Anisotropic Case
In the anisotropic case, we can still use Eq. (B.8) but with Ω2 given by
136
Ω2 =
ω2
0 + (ε11n01 + ε12n
02)
2 0
0 ω20 + (ε22n
02 + ε12n
01)
2
+ω0
ε11
√N2
1 − (n01)
2ζ1 ε12√N2
1 − (n01)
2ζ1
ε12√N2
2 − (n02)
2ζ2 ε22√N2
2 − (n02)
2ζ2
. (B.10)
Case 1: ζ1 = ζ2 = 1
When n01 = n0
2 = 0, the eigenvalues are
ω2 = ω20 + ω0
(ε11N1ζ1 + ε22N2ζ2
2±√
(ε11N1ζ1 − ε22N2ζ2)2
4+ ε212N1N2ζ1ζ2
), (B.11)
with ζ1 = ζ2 = 1. For simplicity we shall address only the nearly isotropic case. It is
experimentally relevant since this is the case for both 23Na and 87Rb, where the experimental
values for the ε11, ε22 and ε12 are similar. To do this we use the variables ε, εA,B defined in
Section 6.2.2 since εA and εB quantify the degree of anisotropy. We therefore treat them as
small parameters. Expanding the square root in Eq. (B.11) and keeping terms to first order
in those variables, we obtain the two eigenvalues
ω2 = ω20 + ω0ε(N1ζ1 +N2ζ2) + ω0εA
(N1ζ1 −N2ζ2)2
N1ζ1 +N2ζ2+ 2ω0εB(N1ζ1 −N2ζ2), (B.12)
ω2 = ω20 + 4ω0εA
N1N2ζ1ζ2N1ζ1 +N2ζ2
. (B.13)
We have assumed that N1 + N2 ∼ N1 − N2 ∼ N . Since εA > 0 (which is implied by the
miscibility condition) both modes are stable (εB ε). The instability that would arise at
sufficiently large and negative values of εA has the same origin as the immiscibility condition.
However, we do not consider this region in this paper since immiscibility would have severe
consequences (see Section 6.2.1). It is easy to see from Eq. (B.3) that the case n01, n
02 = 0 is
also ruled out due to the miscibility condition.
Case 2: ζ1 = −1, ζ2 = 1
137
At the origin (n01 = n0
2 = 0), the frequencies are again given by Eqs. (B.12) and (B.13)
but with ζ1 = −1, ζ2 = 1. Let us divide the region into two parts: N1 − N2 > 0 and
N1 −N2 < 0. In the first region, the motion is always unstable: for large values of N1 −N2
the first frequency is imaginary and, for small values, the resonance of Section 6.2.3 will tend
to destabilize the equilibrium point. In the second region, if N1 −N2 > −4εAN1N2/ω0, then
again it is unstable. Otherwise it is stable (provided it is outside the region of resonance).
For n01, n
02 = 0, the conditions for stability become rather complex and offer little insight.
However, it is reasonable to believe that, by suitably choosing the parameters, most of the
equilibrium points can be made stable (here also, see caution in Section 6.2.4).
Case 3: ζ1 = ζ2 = −1
The frequencies at the point n01 = n0
2 = 0 are those given by Eqs. (B.12) and (B.13)
with ζ1 = ζ2 = −1. As in the isotropic case, the point n01 = n0
2 = 0 is unstable for
typical experimental conditions, namely when εN > ω0. The corrections to this criterion
are of order (εA, εB)/ε. Finally, when n01, n
02 = 0, as in the previous case, the stability can
generally be achieved for all equilibrium points for appropriate values of the parameters,
barring immiscibility problems.
138
a) b)
c) d)
e) f)
g)
Figure B.1: Equations (B.1) and (B.2) plotted on the (n1, n2) plane. The intersection ofthe two curves is the graphical solution for the equilibrium points for χ1 = χ2 = 0 (a),(b),χ1 = π and χ2 = 0 (c),(d) and χ1 = χ2 = π (e)-(g).
139
Vita
Sahel Ashhab was born on October 17, 1974, in Amman, Jordan. He received his B.Sc.degree in Physics from the University of Jordan in 1996 and his M.S. degree in Physics fromthe University of Illinois at Urbana-Champaign in 1998.
140
