79
Cold Atoms for Condensed Matter Theorists Austen Lamacraft
Cold Atoms for Condensed Matter Theorists
Austen Lamacraft
Contents
1 Introduction 31.1 Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The ideal Bose gas: a reminder . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 The experimental system 52.1 Atomic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Boson or Fermion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 The Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Excited states and polarizabilty . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Trapping and imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Magnetic traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Optical traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Effective interaction between like species . . . . . . . . . . . . . . . . 122.3.2 Interaction between species . . . . . . . . . . . . . . . . . . . . . . . . 152.3.3 The Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Superfluidity and Bose-Einstein condensation 193.1 BEC and off-diagonal long-range order . . . . . . . . . . . . . . . . . . . . . . 193.2 Superfluidity defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Non-classical rotational intertia . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Metastability of superflow and vortices . . . . . . . . . . . . . . . . . . 23
3.3 Experimental status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Bose superfluids 264.1 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.1 Time-independent Gross-Pitaevskii theory . . . . . . . . . . . . . . . . 264.1.2 Time-dependent Gross-Pitaevskii theory . . . . . . . . . . . . . . . . . 28
4.2 Interlude: structure factors and sum rules . . . . . . . . . . . . . . . . . . . . 314.3 The Bogoliubov approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.1 Pair approximation and ground state energy . . . . . . . . . . . . . . . 324.3.2 Structure of the ground state and excitations . . . . . . . . . . . . . . 35
4.4 Atom optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4.1 Fock states and coherent states . . . . . . . . . . . . . . . . . . . . . 37
1
4.4.2 Interference of two condensates . . . . . . . . . . . . . . . . . . . . . 394.4.3 Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.5 Spinor condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5.1 General considerations for multicomponent BEC . . . . . . . . . . . . 444.5.2 The Gross-Pitaevskii description . . . . . . . . . . . . . . . . . . . . . 454.5.3 Metastability of superflow . . . . . . . . . . . . . . . . . . . . . . . . . 464.5.4 Fragmented condensates . . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Fermi superfluids 495.1 Fermionic condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2.1 The pairing hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2.2 The BCS-BEC crossover . . . . . . . . . . . . . . . . . . . . . . . . . 555.2.3 Quasiparticle excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.4 Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 The effect of ‘magnetization’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.1 Sarma state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3.2 Magnetization in the BCS-BEC crossover . . . . . . . . . . . . . . . . 63
6 Hydrodynamics of condensates 666.1 Galilean invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Hydrodynamic description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.3 Quantum Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.1 Hamiltonian and commutation relations . . . . . . . . . . . . . . . . . 686.3.2 Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Strong correlations: low dimensions and lattices 717.1 Bose fluids in one dimension: the Tonks gas . . . . . . . . . . . . . . . . . . . 717.2 Lattice systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2.1 Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.2.2 The Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . 75
2
Chapter 1
Introduction
1.1 Preamble
This set of lectures is supposed to provide a reasonably up-to-date introduction to the con-densed matter physics of atomic gases. Of course, as condensed matter physicists, wehave no particular right to dictate what is interesting in a branch of what is essentially atomicphysics. Nevertheless, there are many elements of the subject that fit into a traditional con-densed matter context. An incomplete but illustrative list of invaluable concepts includesstates of matter, equilibrium phase diagrams, single particle vs. collective behavior, the roleof dimensionality, and long-range order. All the time we should keep an eye to the fact thatour usual (often equilibrium) thinking may not be the most relevant for the experiments thatare done
Because of the pre-eminence of superfluidity and BEC as quantum states of matter at lowtemperatures, the majority of the lectures will involve aspects of these phenomena. Whilethis has certainly been the most headline-grabbing part of the story so far, we should bear inmind that (perhaps more mundane sounding) phases such as the ferromagnet – still a bonafide quantum state – are likely to have their day in the near future.
There is a regime of BEC that is a bit like linear optics: the atom laser is a coherent phe-nomenon which nevertheless doesn’t depend on interactions. But there is some interestingmany-body physics there (even if the bodies are non-interacting), so we will discuss it a bit,especially as we can certainly look forward to continuing cross-fertilization between quantumoptics, atomic physics and condensed matter for the next few years . From this point of view,the condensed matter physics of atomic gases could be called the non-linear quantum opticsof matter waves!
1.2 The ideal Bose gas: a reminder
In 1923 Bose gave a derivation of Planck’s radiation law that treated light quanta as indistin-guishable particles to yield the equilibrium distribution that now bears his name
n(E) =1
eE/kBT − 1. (1.1)
The following year, in applying this distribution to massive bosonic particles, Einstein madethe following observation. Because the number of particles is conserved, we introduce a
3
chemical potential µ and chose it such that the total number is fixed
N =∑k
1e(εk−µ)/kBT − 1
, εk =~2k2
2m(1.2)
Recalling∑
k −−−−→Ωd→∞
Ωd
∫ddk
(2π)d in d dimensions, where Ωd is the d-dimensional volume, we
have for d = 3
n ≡ N
Ω3=
∫d3k
(2π)31
e(εk−µ)/kBT − 1
=∫dk
k2
2π2
1e(~2k2/2m−µ)/kBT
=1~3
(kBTm
2π
)3/2
Li3/2(µ/kBT ) (1.3)
(Lin(z) =∑∞
k=1 zk/kn is the polylogarithm). Note that (kBTm)1/2 is the typical magnitude
of a particle’s momentum. As the density of particles increases, or the temperature falls,µ increases to zero at some critical value of ‘phase space density’ n × np ∼ ~−3 (wherenp ∼ (kBTm)−3/2), corresponding to
kBTc = α~2
mn2/3, α ≡ 2π/ [ζ(3/2)]2/3
Clearly µ has to be negative for the distribution Eq. (1.2) to make sense, so what happens?Let’s think about zero temperature first. Since the particles are bosons, the ground state
consists of every particle sitting in the lowest energy state (k = 0, if we think of a box withperiodic boundary conditions). But such a singular distribution was excluded by the abovereplacement of the sum by the integral. Supposing that at T < Tc we have a finite fractionf(T ) of particles sitting in this state. Then the chemical potential may stay equal to zero andEq. (1.3) gives
n = n
[f(T ) +
(T
Tc
)3/2], so f(T ) =
[1−
(T
Tc
)3/2]. (1.4)
The unusual, highly quantum degenerate state emerging below Tc came to be known as aBose-Einstein condensate (BEC).
4
Chapter 2
The experimental system
In this first lecture, we are going to try and give a lightning introduction to the most importantaspects of the atomic gases that are actually realized in experiments. The goal is to providejustifications for the kinds of models that we will be considering in the following lectures, andto gain some feel for when experimental reality is likely to intrude!
We will be concerned with the properties of gases of neutral alkali atoms. The numberof atoms in typical experiments range from 104 to 107: often N → ∞ will be a good enoughapproximation. The atoms are confined in a trapping potential of magnetic or optical origin,with peak densities at the centre of the trap ranging from 1013 cm−3 to 1015 cm−3.
As we just discussed in the previous chapter, the observation of quantum phenomenalike Bose-Einstein condensation requires a phase-space density of order one, or nλ3
dB ∼ 1.The above densities then correspond to temperatures
T ∼ ~2n2/3
mk∼ 100nK − fewµK
At these temperatures the atoms move at speeds of ∼√kT/m ∼ 1 cm s−1, which should be
compared with around 500 m s−1 for molecules in this room, and ∼ 106 m s−1 for electronsin a metal at zero temperature. Achieving the regime nλdB ∼ 1, through sufficient cooling isthe principle experimental advance that gave birth to this new field of physics.
It should be noted that such low densities of atoms1 are in fact a necessity. We aredealing with systems whose equilibrium state is a solid (that is, a lump of Sodium, Rubidium,etc.). The first stage in the formation of a solid would be the combination of pairs of atomsinto diatomic molecules, but this process is hardly possible without the involvement of a thirdatom to carry away the excess energy. The rate per atom of such three-body processes is10−29 − 10−30 cm6 s−1, leading to a lifetime of several seconds to several minutes2. Theserelatively long timescales suggest that working with equilibrium concepts may be a usefulfirst approximation.
1Compare 1019 cm−3 for the number density of air molecules at ground level, and ∼ 1022 cm−3 for atomicdensities in liquids and solids.
2This three-body rate is reduced by the appearance of Bose-Einstein condensation, further enhancing thesample lifetime.
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2.1 Atomic properties
2.1.1 Boson or Fermion?
Since the alkali elements have odd atomic number Z, we readily see that alkali atoms withodd mass number are bosons, and those with even mass number are fermions.
Alkali atoms have a single valence electron in an nS state, so have electronic spinJ = S = 1/2. Thus bosonic and fermionic alkalis have half integer and integer nuclear spinrespectively. We list the following experimental ‘star players’
Bosons Nuclear spin, I87Rb 3/223Na 3/27Li 3/2
Fermions6Li 140K 4
2.1.2 The Zeeman effect
The Zeeman effect plays a crucial role in the trapping of atoms. Assuming that we deal withatoms in a state of zero orbital angular momentum, the effect of magnetic field is describedby the Hamiltonian
HZ = AI · J + gµBB · J, (2.1)
where I and J are the nuclear and electronic angular momenta respectively, and the first termoriginates from the hyperfine interaction. We can ignore the Zeeman effect of the nuclearspin, as the nuclear magneton is approximately me/mp ∼ 1/2000 of the Bohr magneton.
We will consider this case I = 3/2, exemplified by 87Rb, 23Na, and 7Li. Solving theHamiltonian (2.1) gives energy levels labeled by the conserved quantum numbers F(F = I + J) and mF , the component parallel to the field. In the future, when we speak ofan atomic ”species”, we will normally mean one of these hyperfine-Zeeman states. Thezero-field splitting between the F = 2 and 1 states is 2A, and we can define a crossoverscale Bbf ≡ |A|/µB beyond which the complexities of the hyperfine coupling becomeunimportant. The full field dependence of the energy levels is3
mF F E(B)2 2 A (1 +B/Bhf )1 2, 1 ±A[1 +B/Bhf + (B/Bhf )2]1/2
0 2, 1 ±A[1 + (B/Bhf )2]1/2
−1 2, 1 ±A[1−B/Bhf + (B/Bhf )2]1/2
−2 2 A (1−B/Bhf )
These are plotted in Fig. 2.1. Many experiments are done in the regime B Bhf , so alinear expansion of the above energies suffices.
E(B) = ±[A+
12µBmFB
], (2.2)
3We shunt all the energy levels up by A/4 for convenience
6
Figure 2.1: Magnetic field dependence of atomic states of an atom with J = 1/2, I = 3/2.
with plus (minus) for the upper (lower) multiplet. Since magnetic traps have a local minimumin the field, it is the ”low field seekers” with positive gradient that can be trapped. In thepresent case these are F = 2, mF = 2, 1, 0, and F = 1, mF = −1.
We will see that in general collisions between atoms can convert low field seekers to highfield seekers that are then lost from the trap. Two states are however of special experimentalimportance in being immune to this process. They are the doubly polarized state with F =I + 1/2, mF = F and the maximally stretched state with F = I − 1/2, mF = −F .
2.1.3 Excited states and polarizabilty
A second approach to trapping atoms is to use the potential that they feel in the presence ofthe electric fields created by a laser. The field polarizes the atoms, giving them an electricdipole moment that in turn interacts with the field.
Let us start by considering the effect of a static electric field. Second-order perturbationtheory gives us an expression for the polarizability, defined through the quadratic energy shiftin the presence of an electric field ∆E = −αE2/2
α = 2∑
n
|〈n|d · ε|0〉|2
En − E0, d ≡ −e
∑j
rj . (2.3)
We leave in the unit direction vector of the electric field ε in order to avoid the tensorialstructure of α. It is convenient to write this result in terms of the dimensionless oscillatorstrengths
fn0 =2me (En − E0)
e2~2|〈n|d · ε|0〉|2.
These satisfy the f-sum rule (or Thomas-Reiche-Kuhn sum rule)∑n
fn0 = Z (2.4)
7
Problem 1 Prove this by considering[[H, di
], di
](no sum).
We have
α =e2
me
∑n
fn0
ω2n0
, (2.5)
with ωn0 = (En − E0) /~.The response to external fields is mostly determined by the ”fundamental” transition of
the valence electron nS → nP (a doublet due to spin-orbit coupling). The wavelength of thistransition is in the range 500− 700 nm. We can use these facts to estimate the polarizability.First we assume that the valence electron states are well-approximated by those of a singleelectron moving in a coulomb potential due to the nucleus and core electrons. Thus theiroscillator strengths satisfy Eq. (2.4) with Z = 1. Next we neglect all but the nS → nPtransition, giving
α ∼ 1(∆E)2
.
The result should be understood in atomic units with α measured in units of a30, and energies
in e2/a0 ∼ 27.2 eV.The case of oscillating fields can be treated by considering the ground state to ground
state amplitude in second order perturbation theory in the field E(t) = Eωe−iωt +E−ωe
iωt, withE−ω = E∗ω 4
〈0|0〉t = 1− 12~2
∫ t
0dt1dt2〈0|Td(t1) · Eω d(t2) · E∗ω|0〉e−iω(t1−t2) + ω → −ω
= 1 +i
~2
∑n
1ωn0 + ω
(t− i
ωn0 + ω
[e−i(ωn0+ω)t − 1
])|〈0|d · Eω|n〉|2
+ω → −ω. (2.6)
After time averaging the imaginary linear in t term can be thought of as a shift in the energyof the ground state, leading to a phase factor e−i∆Et/~ ∼ 1− i∆Et/~, equal to
∆E = −1~
∑n
|〈0|d · Eω|n〉|2
ωn0 + ω+ ω → −ω.
This gives us the (real part of the) dynamical polarizability through ∆E = −α′(ω)〈E2(t)〉/2
α′(ω) =∑
n
2 (En − E0) |〈n|d · ε|0〉|2
(En − E0)2 − (~ω)2
=e2
me
∑n
fn0
ω2n0 − ω2
, (2.7)
which generalizes the static results Eq. (2.3) and Eq. (2.5). Note that in contrast to thestatic case, where an attractive force is always felt towards regions of high field intensity,
4Field-theoretic types may prefer to think of the self-energy of the atom at second order in the ”dressing”electric field
8
the dynamical polarizability can be of either sign. In particular, when the polarizability isdominated by a single transition we have
α′(ω) ∼ |〈n|d · ε|0〉|2
~ (ωn0 − ω), (2.8)
and the sign from positive to negative as we go from ω < ωn0 (red detuning) to ω > ωn0 (bluedetuning).
Finally, inclusion of a finite excited state lifetime Γe tends to broaden this behaviour. Theeffect can be included by putting an imaginary part in the denominator of Eq. (7.5) to obtainthe complex polarizability
α(ω) ∼ |〈n|d · ε|0〉|2
~ (ωn0 − ω − iΓe). (2.9)
In particular the imaginary part of this expression can be thought of as giving the rate oftransitions out of the ground state (an ‘imaginary part to the energy’ Im ∆E = −α′′(ω)〈E(t)2〉)
Γg = −2~Im ∆E =
1~α′′(ω)〈E(t)2〉 (2.10)
Excited state lifetimes are of order 10 ns.
Problem 2 Consider a harmonically confined electron. The equation of motion for the dipolemoment d = −er is
d + ω20d =
e2
meE
Show that the expression for the classical polarizability is
α(ω) =e2
me
1ω2
0 − ω2
This correpsonds to Eq. (2.5) with one oscillator strength equal to one at ω0. Verify that thisis case in a quantum mechanical calculation.
2.2 Trapping and imaging
In the previous section we have discussed the two important features of the alkali elementsthat make them such a versatile experimental system. The spin of the valence electronallows magnetic trapping, while the wavelength of the nS → nP transition means that laserscan be used for trapping and cooling. Cooling is discussed in some detail in Ref. [1]. Herewe discuss briefly trapping and imaging, which are probably more important for the theoristseeking to understand experimental papers.
2.2.1 Magnetic traps
In free space B cannot attain a maximum, so magnetic traps work by creating a field mini-mum in which the low field seeking states are confined. Most produce an axially symmetricmagnetic field of the form
|B(r, z, φ)| = B0 +12αr2 +
12βz2. (2.11)
9
Provided that the atoms move slowly, we can make an adiabatic approximation and assumethat the atoms remain in the instantaneous hyperfine-Zeeman state appropriate to their cur-rent position, even though the direction of B(r) may change. This is fine as long as |B(r)|does not become too small, which in practice means various strategies are required to plugsuch ‘holes’ where unwanted transitions between species can occur. Details of the varioustypes of magnetic traps can be found in Ref. [1].
The relationship between (2.11) and the potential experienced by the atom is then verysimple in the linear regime described by Eq. (2.2). There is one subtlety that makes it-self known only occasionally. The effective quantum Hamiltonian of an atom of a particularspecies in the adiabatic approximation does not simply involve a conservative potential dueto the magnetic field, but in general includes a gauge potential whose origin is the Berryphase accumulated by a varying direction of B(r). This potential is expressed in terms ofthe instantaneous hyperfine-Zeeman state |α(r)〉 as
Aα(r) = i〈α(r)|∇|α(r)〉 (2.12)
Problem 3 The magnetic field of a quadrupole trap is
B = Bz z +B′ (xx− yy)
If Bz B′, we can think of the hyperfine states as being eigenstates of mF , the componentin the z-direction. Inversion of Bz would carry atoms adiabatically from mF to −mF , as thedirection of B rotates through π. The rotation axis, however, depends on where they arein the trap, being about an axis parallel to yx + xy. Show that the effect of this rotationis to multiply the wavefunction by the position-dependent phase factor e−2imF φ, where φis the azimuthal angle. This technique has been used to create vortices in Bose-Einsteincondensates [2].
2.2.2 Optical traps
Optical traps work by focusing a laser to create a field maximum. If the laser is red-detunedthe result is a potential minimum for the atoms, as discussed in Section 5.3. This type oftrap is useful when we are interested in interactions that depend on species or Feshbachresonances, which are tuned by magnetic field. In such circumstances we don’t want theZeeman energy confusing things.
In this context, it is useful to consider if the optical potential really is independent ofspecies. Any species-dependence requires spin-orbit coupling to be taken into account inthe excited state. In the nP state spin-orbit causing a splitting into nP1/2 and nP3/2 cor-responding to total electron angular momentum of 1/2 and 3/2 respectively. Since thesestates contain Kramers doublets due to time-reversal symmetry, the dipole matrix elementsbetween these states and the ground state are independent of the initial spin state of theelectron for linearly polarized light. Thus any effect will arise from the hyperfine splitting,which alters the detuning from a particular transition. If the detuning is much larger thanthis splitting, as it usually is, the effect is negligible. For circularly polarized light the matrixelements can differ, and the greatest effect is achieved by tuning between nP1/2 and nP3/2.
10
Figure 2.2: Absorption image of around 7 × 105 atoms just above, at, and below the Bose-Einstein condensation temperature. The gas is allowed to expand for 6 ms. Reproducedfrom Ref. [3].
Optical traps are usually made using far-detuned light with Γe/(ωn0 − ω) ∼ 10−7 − 10−6.Although the trapping potential is inversely proportional to the detuning, the ground statelifetime Eq. (2.10) goes like the inverse square. Note that the absorption of even one photonwould be a disaster for a sample, heating it far from degeneracy.
2.2.3 Imaging
As we saw, the scale of the dependence of optical properties on frequency is set by theexcited state lifetime Γ and is typically a few MHz. This is much less than the zero fieldhyperfine splitting on the GHz scale. Thus distinguishing optically between different F valuespresents no problem. For the sublevels within a multiplet, the dependence of transitions uponpolarization can be exploited to further enhance resolution. Usually it is possible to imagethe individual species in a gas separately.
The images that one most frequently sees in experimental papers are simple absorptionimages, in which light is absorbed by the gas, creating real transitions and heating the gas ,see Fig. 2.2 . Such measurements are therefore of a one-shot character in that they destroythe sample.
The second kind of imaging is dispersive (phase-contrast) imaging that relies on diffrac-tion. Many images may be taken with this technique without much heating. The creation ofsuch images can be regarded as instantaneous.
2.3 Interactions
For the most part, interactions between alkali atoms in the parameter ranges of experimentalinterest are highly amenable to theoretical analysis. The effective range of the potential isalways small compared to other length scales, and normally we are also in the dilute gas limit
11
Figure 2.3: Sketch of the interatomic potentials for two alkali atoms with valence electrons insinglet and triplet states.
na3s 1 where as is the s- wave scattering length. These happy circumstances go some
way to explaining the popularity of these systems with theorists.
2.3.1 Effective interaction between like species
Working within the standard Born-Oppenheimer approximation we consider the interatomicpotential describing the interaction between two alkali atoms. In general this potential isstrongly dependent on the spin state of the valence electrons. The singlet state has a fardeeper minimum, of the order ∼ 5000K, than the triplet state. This is because the twoelectrons in the singlet state can share an orbital and form a covalent bond. By contrast, theminimum in the triplet potential results from an interplay between the −C6/r
6 van der Waalsattraction at large distances, and a hardcore repulsion at short distances, see Fig. 2.3.
Two atoms in the same hyperfine state clearly have a symmetric electron spin wavefunc-tion, so the triplet potential is the appropriate one5. The van der Waals potential defines alength r0 ≡
(2mrC6/~2
)1/4, where mr is the reduced mass. This scale, the typical extentof the last bound state in the potential, is of order 50 Angstroms, much smaller than the deBroglie wavelength. This allows us to ignore all but s-wave scattering.
On general grounds, we expect all features in the scattering amplitude to be at the highenergy scale ∼ ~2/2mrr
20. Then the low energy scattering of interest is described simply
by the s-wave scattering length, defined through the form of the wavefunction in the relativedisplacement of two atoms
ψ(r) = const.sin [k (r − as)]
r. (2.13)
5In the next section we will consider interactions between different species, which in general will involve bothsinglet and triplet potentials. As explained at the start of this chapter, however, the strongly bound molecularstates are slow to form so still have little effect, and singlet and triplet scattering lengths can be roughly equal.
12
The scattering length will prove to be a basic parameter of all theories of the alkali gases. Forall but the heaviest atoms theoretical calculation of scattering lengths is extremely difficult.They can, however, be reliably measured using photoassociative spectroscopy, see Ref. [1].It is of the same order as the scale r0 introduced above.
as normally enters into our theoretical considerations through the notion of the pseudopo-tential. The idea is that provided that all other scales in the problem are much larger than as
– in particular this requires the smallness of the gas parameter na3s – the expectation value
of the interaction energy has the form6
〈Hint〉 =12
4πas~2
m
∑i6=j
∫ ∏k
drk δ(rij)|Ψ(r)|2. (2.14)
δ (rij) denotes a delta-function smeared on a scale much larger than as but much smallerthan all other scales. The integral in Eq. (2.14) is just the (spatially averaged) probabilitydensity for two particles to approach each other, and the interaction energy is just propor-tional to this quantity summed over all pairs. It is natural to expect that a pairwise form holdsin the dilute limit and the quantity summed over in then the energy of a pair in vacuo. To beabsolutely clear about the origin of Eq. (2.14), I present an argument leading to it in moredetail7.
• At low densities interaction energy (meaning the expectation value of the interactionhamiltonian) should have a pairwise form.
• Wavefunctions we will consider correspond to energy per particle Etyp much less than~2/2ma2
s. It’s reasonable that such wavefunctions satisfy the ‘boundary condition’
Ψ(r) ∼ A(1− as/rij),
at ~/√
2mEtyp rij as.
• In the case of a single pair, such a wavefunction has energy
4πas~2
m|A|2, (2.15)
which is found by solving the Schrodinger equation in a spherical box of size R as,then finding the normalization constant for this wavefunction. (don’t forget the reducedmass!). Eq. (2.15) is in fact theO(as/R
3) part of the energy, with the leading term beingπ2~2/4mR2. Since it represents the leading as dependence, however, it is reasonableto call this the ’interaction energy’. Note, however, that the energy can range frombeing all kinetic for a hard sphere potential, to all potential for a very ‘soft’ potential. Inany case, it arises from a scale ∼ as and should be affected by long distance changesin the wavefunction only through A.
• Finally, |A|2 corresponds to the value of the integral in Eq. (2.14).
6This is a slight abuse of notation: the origin of this effective potential is both kinetic and potential in general:see below
7Alternative versions may be found in Ref. [4] and Ref. [5]. The interesting feature of Leggett’s argument isthat it seems to apply even when na3
s 1 as long as nr30 1, e.g. near a Feshbach resonance.
13
Several other ways of writing the pseudopotential are in common usage. One is to dis-pense with the slightly cumbersome form of Eq. (2.14) and write the interaction Hamiltonianas a delta-function potential.
U(r) =4πas~2
mδ(r), (2.16)
or in second quantized notation
Hint =12
4πas~2
m
∫drφ†(r)φ†(r)φ(r)φ(r). (2.17)
Obviously this requires careful handling in light of the above. Another way of approachingthese difficulties is to ask how we can define a δ-function potential U0δ(r) in three dimen-sions. The scattering amplitude in general satisfies the integral equation
F (k,k′) = −U(k,k′)− 12
∫dq
(2π)3U(k′,q)F (k,q)ε(q)− ε(k)− i0
(2.18)
where ε(k) = k2/2m. The δ-function potential can then be taken to be the limit of the (non-translationally invariant!) separable potential
U(k,k′) = U0g(k)g(k′)
where g(k) is equal to unity at small k, but falls to zero at some cut-off scale. Then the integralequation is solved trivially for the low energy scattering amplitude F (k,k′) → −4π~2as/m
m
4πas~2=
1U0
−∫
dq
(2π)3g(k)2εk
.
which is compatible with Eq. (2.16), apart from the (divergent) second term. This is just themomentum space version of our real space difficulties.
Finally, Eq. (2.16) is sometimes written
U(r) =4πas~2
mδ(r)∂rr.
This serves to remove the 1/r piece from the boundary condition Ψ(r) ∼ A(1 − as/rij)when the expectation value is taken.
All of these complexities aside, by far the most important thing we will do with the pseu-dopotential is use it to find the interaction energy for trial wavefunctions that have no addi-tional correlations built in between particles (the ‘Gross-Pitaevskii’ approximation). Then thesummand in Eq. (2.14) is n/N , and we find the energy per particle
E/N =2πnas~2
m.
The remarkable thing is that, although we used the diluteness condition to derive it, thepseudopotential can be used to compute the next order in the ground state energy of asystem of bosons as an expansion in
(na3
s
)1/2
E/N =2πnas~2
m
(1 + α
(na3
s
)1/2 + . . .), (2.19)
where α = 128/15√π.
14
2.3.2 Interaction between species
The natural generalization of the pseudopotential Eq. (2.17) to several species is
Hint =12
∑αβγδ
4πaαβγδ~2
m
∫drφ†α(r)φ†β(r)φγ(r)φδ(r). (2.20)
Bose (Fermi) statistics allows us to take aαβγδ = (−)aβαγδ = (−)aαβδγ8. WhereB Bhf and
the hyperfine splitting is large enough to rule out scattering to other values of F , rotationalinvariance allow us to write, for the states within a given hyperfine multiplet
aαβγδ =12
[a1δαγδβδ + a2Fαγ · Fβδ ± α↔ β] , (2.21)
where F is the total spin operator within the multiplet. Even when scattering between mul-tiplets is possible, the total angular momentum and projection mF = mF1 +mF2 of the twoparticles is conserved.
We are now in a position to explain the importance of the low-field seeking doubly polar-ized state with F = I + 1/2, mF = F and the maximally stretched state with F = I − 1/2,mF = −F , introduced in Section 2.1.2. Atoms in the doubly polarized state can only scat-ter into the same state, as no other states have larger mF . Two atoms in the maximallystretched state could scatter so that one ends up with mF = −F − 1, but these states lie inthe F = I + 1/2 multiplet. Since we are concerned with positive splittings A on the scale100 mK - 1 K, collisions might not be expected to depopulate the maximally stretched state.Transitions to other states within the F = I − 1/2 multiplet do however occur due to themagnetic dipole interactions, see Section 2.3.4, but these are typically much less frequent.
2.3.3 The Feshbach resonance
If the centre-of-mass energy of two scattering atoms is close to the energy of a bound state,the scattering amplitude can be strongly modified. This phenomenon is called a Feshbachresonance. In recent years its exploitation by experimentalists has made the strength ofthe interaction between atoms a continuously tunable experimental parameter, somethingunthinkable in conventional condensed matter systems.
The idea is illustrated in Fig. 2.4. The curves that we drew in Fig. 2.3 are really justrepresentative of the many that we could draw for the interatomic potentials correspondingto different hyperfine states of the two atoms. As we just explained, the relatively largehyperfine splitting makes it impossible for either of the atoms to scatter at low energy intoa higher multiplet (the ‘closed channel’ of the diagram). Their wavefunctions will, however,in general hybridize with bound or nearly bound states in these closed channels due thepresence of exchange interactions.
The simplest model for this kind of scattering is the two-channel model, that accounts forone nearby bound state in one closed channel
H =∑p,s
εpa†s,pas,p +
∑q
(εq2
+ ε0
)b†qbq +
g√V
∑p,q
bqa†1,q+pa
†−1,−p + h.c (2.22)
8The definition of these quantities is again from the scattering state, where the incoming and outgoing wavesare now in the internal states γ1δ2 ± δ1γ2 and α1β2 ± β1α2 respectively
15
Figure 2.4: A Feshbach resonance is caused by hybridization of a closed channel boundstate with the open channel.
Here bq annhilates a molecule (bound state of two atoms in the closed channel), and as,p
annihilates an atom in the open channel. The energy of the closed channel bound state isε0. We have introduced the two species s = ±1 so that we can discuss either bosons orfermions: the atoms are distinguishable in either case 9.
Problem 4 Find the scattering amplitude for two atoms in the model Eq. (2.22).Solution For two atoms the wavefunction in the centre-of-mass frame has the form
|ψ〉 =
[βb†0 +
∑p
αpa†1,pa
†−1,−p
]|0〉.
Substituting into Eq. (2.22) gives the equations
2εpαp +g√Vβ = Eαp
ε0β +g√V
∑p
αp = Eβ.
Eliminating β
2εpαp +g2
V (E − ε0)
∑p′
αp′ = Eαp. (2.23)
9The same model can be applied to bosons of one species – the result for the scattering amplitude belowshould be doubled – while fermions of the same species have no s-wave scattering. This exemplifies the generalrelation for the scattering amplitude of identical bosons or fermions f(θ) ± f(π − θ) in the s-wave case, whenf(θ) = const
16
We look for a scattering state of the form
αp(E) = δp,p0 +4π~2
m
f(E)2εp − E − i0
,
where p0 is the wavevector of the incoming wave with 2εp0 = E. Substituting into Eq. (2.23)gives
f(E) +g2
(E − ε0)
[m
4π~2+
∫dp
(2π)3f(E)
2εp − E − i0
]= 0
(c.f. Eq. (2.18)). In order to tame the singular behaviour of the integral, we need to shift thedetuning parameter ε0 by the infinite constant
ε0 → ε0 + g2
∫dp
(2π)31
2εp, (2.24)
to give
f(E) +g2
(E − ε0)
[m
4π~2+ f(E)
∫dp
(2π)3
(1
2εp − E − i0− 1
2εp
)]= 0.
Taking real and imaginary parts now yields
Re f(E) +g2
(E − ε0)
[m
4π~2− m3/2
√E
4πIm f(E)
]= 0
Im f(E) +g2
(E − ε0)m3/2
√E
4πRe f(E) = 0.
The final result for the scattering amplitude is
f(E) = − ~γ√m
1E − ε0 + iγ
√E
(2.25)
with γ = g2m3/2/4π.
The pole in f(E) is lies at real energies for ε0 < γ2/4, passing through zero when ε0 = 0.When the pole lies at negative values of energy, its position corresponds to the bound stateenergy (modified by coupling). When the pole is no longer at negative energy we refer to avirtual state10. The scattering length f(0) = −a is
a = − ~γ√m
1ε0,
and displays the divergence characteristic of a Feshbach resonance as we pass from positiveto negative detuning, signaling the occurrence of a bound state11. In a sense the model weintroduced can now be discarded. The form of the scattering amplitude Eq. (2.25) is in factthe most general one allowed at low energies [6]: the model was just a convenient physicalrealization where this two-parameter asymptotic description turns out to be exact. The γ
10Although the pole is at real positive energies up to for 0 < ε0 < γ2/4, there is no singularity in the scatteringamplitude as the pole is not on the physical sheet, see Ref. [6].
11A background scattering length is normally added to this to include the effect of non-resonant scattering.
17
parameter characterizes the width of the resonance. If we were only interested in energiesvery low compared to γ2, a single parameter description in terms of the scattering lengthonly would suffice.
Feshbach resonances have been found in a variety of alkali atoms. Those in the fermions6Li and 40K have been exploited to great effect recently to probe the BCS-BEC crossover thatwe will discuss later. The consensus view is that these resonances are broad in the abovesense, so that a single parameter description is possible. Tuning through the resonanceis achieved by varying an applied field, as the atom and molecule generally have differentmagnetic moments.
The divergence of the gas parameter na3s implied by the approach to a Feshbach res-
onance suggests that sample lifetime will be dramatically reduced, as three-body collisionsleading to the formation of diatomic molecules become more frequent. One should bear inmind, however, that such processes are a function of statistics. In a fermionic system a Fes-hbach resonance for scattering between two species can occur in the s-wave channel, butof any three particles scattering in this way, two will be of the same species. The formationof a molecule of size r0 is then suppressed by some power of r0q, for q a typical wavevector.This power turns out to be about 3.33, so that even when as > 3000 Angstroms, the moleculelifetime can be > 100 ms.
2.3.4 Dipolar interactions
Finally, there is a dipole-dipole interaction between the valence electron spins of two atoms
Umd =µ0 (2µB)2
4πr3[S1 · S2 − 3 (S1 · r) (S2 · r)] . (2.26)
This interaction only conserves total angular momentum (orbital plus spin), so can lead toa decay from the doubly polarized or maximally stretched states. It is possible to show,however, that the rate for such processes is slow and in general does not limit the lifetime inexperiments on alkali atoms [1].
Since the creation of a Bose-Einstein condensate of Chromium atoms last year, the mag-netic dipole interaction has returned to prominence. Chromium has a dipole moment of 6µB,six times larger than the alkalis, so the dipole interaction is 36 times stronger. Recent theo-retical work has focussed on understanding the consequent properties of the condensate.
18
Chapter 3
Superfluidity and Bose-Einsteincondensation
In this second introductory chapter, we will introduce the concepts of Bose-Einstein con-densation (BEC) and superfluidity in a general way, before we move on to consider specificmodels in later chapters. We’ll also take a look at the experimental status of these distinctphenomena.
3.1 BEC and off-diagonal long-range order
BEC, according to Einstein’s original idea, means that a finite fraction of the total number ofparticles in our system occupy one single-particle state below some critical temperature. Forthe usual case of periodic boundary conditions and translational invariance, this is the zeromomentum state, with energy zero. The distribution n(k) of the number of particles in eachmomentum state is then
n(k) = Ncδk + · · · , (3.1)
where Nc is O(N), and f ≡ Nc/N is the condensate fraction. For non-interacting bosonsf = 0 at the condensation temperature and f = 1 at T = 0. For a uniform 3D system wehave
f(T ) = N
[1−
(T
Tc
)3/2].
This basic idea can be elaborated in a number of ways. For the case of atomic gases,held in a trapping potential, it is necessary to have a definition that does not depend upontranslational invariance. The most commonly used one is based on the behaviour of theone-body density matrix
ρ1(r, r′) ≡ 〈〈φ†(r)φ(r′)〉〉. (3.2)
〈〈· · · 〉〉 is the average over the many-body density matrix of the system 1. Since the distri-bution n(k) = 〈〈φ†kφk〉〉, we identify this as the Fourier transform of ρ1(r, r′) ≡ g1(|r − r′|) in
1The first-quantized version of Eq. (3.2) is ρ1(r, r′) ≡ N
Pn pn
Rdr2 · · · drNΨ∗
n(r, r2, . . . , rN )Ψn(r′, r2, . . . , rN ),for a statistical mixture of orthogonal states Ψn occupied with probabilities pn
19
a translationally invariant, isotropic system. The presence of a δ-function in Eq. (3.1) showsthat
g(r) → Nc
V, |r| → ∞,
The non-vanishing of the right hand side is referred to as off-diagonal long range order, orODLRO2. More generally, this property implies that in the spectral resolution of the densitymatrix
ρ1(r, r′) =∑
i
nαχ∗i (r)χi(r′, )
there is at least one eigenvalue of order N . This may be seen by using a trial eigenfunctionχ0(r) = 1/V1/2. The useful thing about this last criterion is that it is very general, and doesn’trequire translational invariance. For trapped gases it is therefore useful to define BEC asthe presence of such an eigenvalue. The corresponding eigenfunction χ0(r) is called thecondensate wavefunction (though it is the solution of no Hamiltonian).
As long as BEC is simple, meaning that there is only one thermodynamically large eigen-value (the multicomponent case will be discussed in Section 4.5), the order parameter ofthe BEC can be introduced as Ψ(r) =
√N0χ0(r), where N0 and χ0(r) are the eigenvalue
and eigenfunction respectively. Writing χ0(r) as |χ0(r) = |χ(r)|eiϕ(r) we define the superfluidvelocity
vs(r) =~m∇ϕ(r), (3.3)
We will justify this choice microscopically when we discuss the Gross-Pitaevskii equation.Although it looks just like the corresponding formula from elementary quantum mechanics, itrefers to a macroscopic quantity, one whose quantum fluctuations are much smaller than itsaverage. It immediately follows that the superfluid velocity is irrotational : ∇∧ vs(r) = 0, andits circulation (line integral) satisfies the quantization condition∮
vs(r) · dl =nh
m, n ∈ Z. (3.4)
Finally, note that none of the definitions we made here refer exclusively to equilibrium oreven time- independent quantities – all can be considered to be functions of time without anyconceptual difficulty.
3.2 Superfluidity defined
Superfluidity is not really a single phenomenon but rather a complex of related phenomena,see Ref. [7] for a very complete discussion. Here we focus on two of the conceptuallysimplest properties of those systems usually deemed superfluid.
3.2.1 Non-classical rotational intertia
Suppose we have a container in the form of a cylindrical annulus containing some ‘matter’consisting of N particles of mass m (Fig. 3.1). If we rotate this at some angular frequency ω,we expect that the free energy of this system has ω-dependence of the form
2If all the particles were in a finite momentum state, the RHS would tend to a plane wave, for instance
20
Figure 3.1: Thought experiment used to define the superfluid fraction. An annulus of fluid isrotated slowly
F (ω) = F0 +12Iω2
where I = NmR2 is the moment of inertia, and we neglect the mass of the container and theorder d/R effects of finite container thickness. The phenomenon of non-classical rotationalinertia (NCRI) corresponds to an additional ω-dependent contribution ∆F (ω), which at smallω has the form
∆F (ω) = −12
(ρs/ρ) Iω2,
defining the superfluid density ρs or superfluid fraction ρs/ρ. In other words, the equilibriumstate of the system is one in which a fraction of the mass is not rotating with the container3.
It may be surprising that we are proposing to characterize a situation that seems intrin-sically dynamical using equilibrium concepts. It is easy to show, however, that conditions ofconstant ω are rather special. Consider the time-dependent Hamiltonian
H(t) =N∑
i=1
p2i
2m+ Ucont(r′i(t)) +
12
N∑i,j=1
V (|ri − rj |, )
where Ucont(r) is the potential due to the container, and
r′i(t) = (xi cosωt+ yi sinωt, yi cosωt− xi sinωt, zi)
is the position of the ith particle in the rotating frame. The time evolution of wavefunctions in3To be careful that we are talking about the equilibrium state, one could imagine tuning the system into the
superfluid state while the container is in motion, thus excluding the possibility that the system merely takes avery long time to catch up with the rotating walls.
21
the rotating frame is governed by the time-independent Hamiltonian4
Hrot = H(0)− ω · L, (3.5)
where L is the operator of total orbital angular momentum
L =∑
i
ri ∧ pi.
Thus a formal procedure to calculate the superfluid fraction would involve obtaining the equi-librium density matrix in the rotating frame using Eq. (3.5), then using it to calculate theaverage of H(0) − TS to give the free energy in the lab frame. As a trivial example, a rigidbody has H = L2/2I, so that Hrot is minimized for L = Iω, giving 〈H〉 = Iω2/2, as expected.Taking quantum mechanics into account means that L is quantized in units of ~, so that thenon-classical part of the energy in this example is of order ~2/2I. Applied to our annulus,we see that for our defintion of NCRI to make sense we must have NmR2ω/~ → ∞. Atthe same time we require, for reasons that will become clear, mR2ω/~ → 0. With these twoconditions and d/R→ 0 met we are free to take both the thermodynamic limit and ω → 0.
Thus we see that the superfluid density is defined through a response of an equilibriumsystem to an infinitesimal perturbation. Specializing now to zero temperature, we see that,since 〈Hrot〉ω = −ω〈L〉ω/2 for small ω (by perturbation theory in ω, for example)5
ρs/ρ = limω→0
2Iω2
[〈Hrot〉ω − 〈Hrot〉0 +
12Iω2
]. (3.6)
We now note that the quantity in square brackets is the expectation value of the Hamiltonian
Hω =N∑
i=1
[pi −mω ∧ ri]2
2m+ . . .
The ‘vector potential’6 mω∧ri can be removed by a gauge transformation which returns us tothe original Hamiltonian H(0) but changes the boundary conditions from periodic to ‘twisted’
Ψ(θ1, . . . , θi + 2π, . . . , θN , rj , zj) = eiϕΨ(θ1, . . . , θi, . . . , θN , rj , zj),
with ϕ = 2πR2mω/~. The definition Eq. (3.6) is thus seen to be equivalent to
ρs/ρ = limϕ→0
4π2I
N2~2
∂2E0(ϕ)∂ϕ2
, (3.7)
(the strange notation is to remind us that ϕ 1/N by the earlier discussion of the thermo-dynamic limit) where E0(ϕ) is the ground state energy . The definition in terms of ‘rigidity’ totwisted boundary conditions is very appealing, and constitutes a pleasing abstraction of theoriginal thought experiment.
4Note that this procedure would break down e.g. for the case of the magnetic dipole interaction if the magneticfield does not rotate with the container. We assume such effects to be negligible.
5More generally, ∂〈Hrot〉ω/∂ω = −〈L〉ω is a consequence of the Hellman-Feynmann theorem.6This formulation makes the analogy between the superfluid response and Meissner effect in a superconduc-
tor clear.
22
Problem 5 Satisfy yourself that a noninteracting Bose gas at zero temperature has ρs/ρ = 1.What about a non-interacting Fermi gas?
Problem 6 (for enthusiasts, but related) The definition Eq. (3.7) seems to coincide with the‘Drude weight’ in Kohn’s theory of the insulating state. In that theory, a metal has a finiteDrude weight. But a metal is not a superconductor (charged superfluid). What’s going on?
3.2.2 Metastability of superflow and vortices
A characteristic of superfluidity, which is probably more familiar than the resistance to rotationat low angular velocity just discussed, is the property of persistent circulation. This refers tothe ability of a superfluid to keep rotating after its container has stopped, without slowingdue to friction. The state with no rotation is evidently lowest in energy, so the implicationis that there are metastable configurations of the fluid with finite angular velocity. Suchconfigurations can exist up to some critical angular velocity ωc (or velocity vc).
Using ∂〈Hrot〉ω/∂ω = −〈L〉ω, we can formally introduce the Legendre transform
E(L) = 〈Hrot〉ω(L) + Lω(L),
where ω(L) is the function inverse to 〈L〉ω, assuming it exists. From Eq. (3.5), it’s clear thatE(L) = 〈H(0)〉 when the container is rotating at angular velocity ω(L). It seems reasonablethat in order to be metastable when the rotation stops, E(L) must have regions of negativecurvature E′′(L) < 0, see Fig. 3.2. But since E′′(L) = ω′(L), and ω(L) is the inverse ofsome function with 〈L〉0 = 0, this can’t be true, and the assumption that the inverse existswas wrong. Barring the unrealistic scenario that 〈L〉ω decreases over some region where ωincreases, we conclude: metastability implies jumps in 〈L〉ω7.
Unlike the definition of the superfluid fraction, there is no general formalism that tellsus whether metastable superflow is possible, so this is about as far as we can get withoutdiscussing a concrete physical system. If we are dealing with a Bose-Einstein condensate,with a macroscopic number of atoms in the same state, things are immediately a lot clearer.Since the atoms behave as one, their quantized angular momentum n~ becomes a quantizedmacroscopic quantity Nn~ (if we ignore all deviations from axial symmetry and the effect ofinteractions). This explains the resistance of the condensate to rotation at small ω discussedearlier: the n = 0 and 1 states have 〈Hrot〉 = 0 and N~2/2mR2−ωN~ respectively, so n = 1is not favored until ω > ω1 ≡ ~/2mR2. It’s possible to argue that the inclusion of interactionsonly quantitatively changes this conclusion.
Metastability, on the other hand, implies that if the angular momentum is made to deviatefrom Nn~, there is a energy barrier (Fig. 3.2). We will see that the origin of this barrieris the (repulsive) interaction between particles, which penalizes the order parameter Ψ(r)going to zero in some part of the annulus. The resulting metastable configurations satisfythe quantization condition Eq. (3.4), where the phase of the order parameter winds through
7In the interests of full disclosure, I have to say that I don’t really like this argument. The problem is that itsneaks in the physically attractive idea that E(L) tells us about the possible rotating states at ω = 0, even thoughit is just a formal transform introduced at finite ω. Probably the only honest thing one can say is that metastabilityis associated with a first order transition as ω is changed, and that the jump in 〈L〉ω is an associated discontinuity
23
1 2 3 4 5
0.5
1
1.5
2
2.5
Figure 3.2: E(L/N~), showing metastable minima.
2π n times, corresponding to the angular momentum quantum number just discussed. Notethat the mere existence of the order parameter is almost enough to explain persistent flow,and requires only some reasonable assumptions about its stable configurations together withthe definition of the superfluid velocity.
When we introduced the order parameter, we stressed that we required BEC to be sim-ple (one component). We will see later that in multicomponent condensates the issue ofmetastability is considerably more complicated.
If the asymmetry of the container is too large we might not get any metastable configu-rations (see Problem 7 in the next chapter). This is the situation for so called ‘weak links’or Josephson junctions, which does not stop such situations displaying superfluidity in thesense of the previous section.
What happens if we don’t have an annular container but an (approximately) cylindricalone? If Ψ(r) is finite everywhere, then n = 0 in the quantization condition. n 6= 0 requires, byStokes’ theorem, that the irrotationality condition ∇ ∧ vs(r) breaks down somewhere insideany surface bounded by the contour we integrate around. For such configurations to havefinite energy, Ψ(r) mush vanish at this point. The resulting line defect is a called a vortex.The simplest vortex configuration, for a vortex along the r = 0 line in cylindrical coordinates
vφs (φ, r, z) =
n~m
1r. (3.8)
The metastable states resulting from halting a rotation with Iω & N~ in this simply connectedgeometry are generically (multi-)vortex configurations.
3.3 Experimental status
The experimental situation regarding the demonstration of the phenomena of BEC and su-perfluidity in atomic gases is in some ways the reverse of that in the study of other quantumfluids like liquid Helium. There, the belief that BEC is the cause of the observed superfluidbehaviour was part of the theoretical explanation of superfluidity, not an independently veri-
24
Figure 3.3: Vortices in an atomic gas of 23Na. Reproduced from Ref. [8].
fied experimental fact. In contrast, BEC in atomic gases was observed in 1995, but the firstexperiments confirming superfluidity had to wait until 1999.
As we mentioned in the previous chapter, absorption images of the trapped gas are oneof the most common experimental probes. If the gas is allowed to expand freely for sometime T , the resulting density profile corresponds to the distribution n(k) in momentum spaceintroduced earlier (as long as vT the trapped cloud). A central peak in absorption in suchimages (corresponding to the logarithm of n(k) column integrated along the line of sight)therefore provides a direct measurement of condensation (see Fig. 2.2).
The other major experimental confirmation of BEC relates to the observation of certaininterference phenomena that we will discuss in Section 4.4. As for superfluidity, experimentsin which a laser was used to ‘stir’ the gas revealed dramatic arrays of vortices in subsequentimaging, which speak for themselves, see e.g. Fig. 3.3.
In order that the vorticity matches, in a coarse-grained fashion, that of a rigid body ∇ ∧vs(r) = 2ωz, we must have a area density of quantized vortices nv = 2mω/h. In the trappedcase, this is only an approximate statement.
Note that in a trap, while the quantization condition Eq. (3.4) remains exact, we expectthat L(ω) is not in general quantized, since the angular momentum density in general de-pends on the magnitude of the order parameter (see next chapter). The jumps in L(ω) thatresult from metastability should still be present, of course.
25
Chapter 4
Bose superfluids
In this chapter, we develop the microscopic theory of the condensed phase of a Bose gas.
4.1 The Gross-Pitaevskii equation
4.1.1 Time-independent Gross-Pitaevskii theory
The first, and most versatile, approach to the problem is to use the Gross-Pitaevskii approx-imation. This is a variational approach that starts from the following ansatz for the groundstate
Ψ(ri) =∏
i
χ0(ri) (4.1)
Such a wavefunction of course displays BEC with the density matrix having an eigenfunc-tion χ0(r) with eigenvalue N . The expectation value of the energy in this state, using thepseudopotential Eq. (2.14), is
〈H〉 = N
∫dr
[~2
2m|∇χ0|2 + Uext(r)|χ0(r)|2
]+
12N(N − 1)U0
∫dr|χ0(r)|4, (4.2)
where the interaction constant is U0 = 4π~as/m. For large N , we can neglect the differencebetweenN andN+1. Minimizing with respect to χ0(r), and introducing a Lagrange multiplierto maintain the normalization of χ0(r) gives the equation[
− ~2
2m∇2 − µ+ Uext(r) +NU0|χ0(r)|2
]χ0(r) = 0.
The multiplier µ = ∂〈H〉/∂N , so is identified with the chemical potential. Rewriting in termsof the order parameter Ψ(r) gives the Gross-Pitaevskii equation[
− ~2
2m∇2 − µ+ Uext(r) + U0|Ψ(r)|2
]Ψ(r) = 0. (4.3)
A fundamental effect of the nonlinearity of the GP equation is that there exists a length scaleset by the typical value of |Ψ(r)|2 ∼ n and the interaction strength
ξ ≡(
2mnU0
~2
)−1/2
= (8πnas)−1/2 . (4.4)
26
This healing length determines the scale over which Ψ(r) is disturbed by the introduction ofa localized potential of scale ξ. It is a fundamental length scale in the system. Note that inthe dilute limit when na3
s 1, ξ the interparticle separation. The fact that Ψ(r) varies onsuch long scales compared to the distance between particles is another physical justificationfor the present mean-field approach1. A typical value of ξ may be around 4000 Angstoms.
In a uniform system with Ψ(r) =√n, the GP energy density 〈H〉/V = n2U0/2 provides
us with a formula for the sound velocity via the hydrodynamic relation
c2s =n
m
∂2(E/V)∂n2
=nU0
m(4.5)
Note that mcs = ~/√
2ξ.With the ansatz Eq. (4.1) for the wavefunction, we can obtain various observables without
difficulty. The particle density is just
ρ(r) = ρ1(r, r) = |Ψ(r)|2.
Note that this refers to the number density, not the mass density as in the previous chapter.The current density is
j(r) =−i~2m
(∇r −∇′
r
)ρ1(r, r′)|r′→r =
~m|Ψ(r)|2∇ϕ(r).
Dividing one by the other yields the superfluid velocity defined in Eq. (3.3), though that rela-tion is in fact the more general one. The total z-component of angular momentum is
Lz = −i~∫dr|Ψ(r)|2∂φϕ(r)
For the (ideal) annular container considered in the previous chapter, we would have
Ψ`(φ, r, z) = Ψ0(r)ei`φ
Where Ψ0(r) goes to zero on the inner and outer edges of the container. The normalizationof Ψ(r) means that Lz = N`~
Problem 7 [See Ref. [4] Section VI.D.2] Using the GP approximation, we can give amore informed discussion of the way in which repulsive interactions allow the existence ofmetastable rotational states. As in Chapter 3, we consider a cylindrical annulus, and the twolowest angular momentum states ` = 0, 1. We now wish to include, however, the effect of asmall deviation from cylindrical symmetry, whose effect is to mix these two states. If a†0 anda†1 create atoms in the ` = 0, 1 states, a model version of the rotating frame Hamiltonian Hrot
that includes the kinetic energy, the asymmetry effect, and interactions is
Hrot = −~ (ω − ω1)[a†1a1 − a†0a0
]−V0
[a†0a1 + h.c.
]+U0
2V
[a†0a
†0a0a0 + a†1a
†1a1a1 + 4a†1a
†0a0a1
](4.6)
1In real systems ξ n−1/3 is actually a severe exaggeration, as their ratio is only ∼ (na3s)
−1/6, and thegas parameter is maybe 10−4. Recall from Section 2.3.1, however, that the expansion parameter justifying thepresent approximation is (nas)
1/2, which is still small.
27
where ω1 = ~/2mR2 is the critical angular velocity at which the ` = 1 state has the lowerenergy. If we introduce the GP wavefunction[
cosχ
2eiϕ/2a†0 + sin
χ
2e−iϕ/2a†1
]N|0〉,
show that
• The order parameter has a node for χ = π/2. If V0 is due to a localized potential, thisnode will coincide with the position of that potential.
• The GP variational energy is (up to a constant, and ignoring terms lower order in N )
E(χ)/N = ~ (ω − ω1) cosχ− V0 sinχ+nU0
2sin2 χ,
while the angular momentum is
L(χ)/N~ =12
(1− cosχ)
• A metastable minimum exists for 2U0 > V0 (assuming U0 and V0 are both much lessthan ~ω1). That is, for small enough deviations from perfect symmetry, metastableconfigurations are possible, and have their origin in the repulsive interactions. Thepoint χ = π/2 that corresponds to an order parameter with a node is then a maximumof the energy.
• Repeating the argument with a state of angular momentum ` greater than one, showthat even when V0 goes to zero, metastable configurations are only possible whennU0 > `2~ω1. This corresponds to a critical velocity of `~/mR =
√2nU0/m =
√2cs,
and coincides (parametrically, at least), with the famous Landau criterion.
4.1.2 Time-dependent Gross-Pitaevskii theory
For time dependent problems it is tempting to immediately write down[− ~2
2m∇2 + Uext(r) + U0|Ψ(r, t)|2
]Ψ(r, t) = i~
∂Ψ(r, t)∂t
. (4.7)
Note that this equation conserves the normalization N =∫dr|Ψ(r)|2 – all particles remain in
the condensate. Eq. (4.7) may be derived by generalizing the ansatz Eq. (4.1)
Ψ(ri, t) =∏
i
χ0(ri, t). (4.8)
Substitution into the time-dependent Schrodinger equation yields
∑i
− ~2
2m∇2
i + Uext(ri) + U0
∑k 6=i
δ(rk − ri)
χo(ri, t)∏j 6=i
χ0(rj) = i~∑
i
∂χ0(ri, t)∂t
∏j 6=i
χ0(rj).
(4.9)
28
In order to get a closed equation for χ0(r) we can replace the∑
j 6=i δ(rj − ri) with the ex-pectation value of the density N |χ0(ri)|2 evaluated with Eq. (4.8). With this replacement,Eq. (4.9) is satisfied if Eq. (4.7) is, as long as we normalize Ψ(r) to N .
The simplicity of this derivation is of course deceptive. The sleight of hand comes atthe last stage. This is somewhat clearer if we pass to an orthogonal basis of single-particlestates of which χ0(r) (at some reference time) is a member. We write the boson field operatorin terms of these states
φ(r) =∑α
χn(r)aα.
Then the interaction Hamiltonian has the form
Hint =U0
2
∑αβγδ
Mαβγδa†αa
†βaγaδ, (4.10)
where the matrix elements Mαβγδ are
Mαβγδ =∫drχ∗α(r)χ∗β(r)χγ(r)χδ(r).
Now applied to the ansatz Eq. (4.8), we can see that it is the term in Eq. (4.10) with α =β = γ = δ = 0 that gives 1
2N (N − 1)U0
∫dr|χ0(r)|4 times the original wavefunction, and is
therefore just this that is kept in the GP approximation. One might worry that we are throwingaway all sorts of complexity at this stage, but in fact the only neglected terms correspond toα, β 6= γ = δ = 0. You should satisfy yourself that these have the form
Sχα(r1)χβ(r2)∏
j 6=1,2
χ0(rj), (4.11)
where S denotes the operation of symmetrizaton. The inclusion of such effects is thus quitetractable, but it will have to wait unitl the next (Bogoliubov) stage of approximation. In theequilibrium state, their effect is to give the quantum depletion of the condensate fraction,leading to N0 < N , even at zero temperature. The effect is of order
√na3
s, so it is reasonablethat the present approximation is justified when this is small [9].
The Gross-Pitaevskii theory therefore gives a very straightforward and appealing route tothe computation of observables in time-dependent situations. A great deal of intuition maybe obtained from examining the dynamics of small deviations δΨ(r, t) of Ψ(r, t) from somereference solution Ψ0(r, t), satisfying[
− ~2
2m∇2 + Uext(r)
]δΨ(r, t) + 2U0|Ψ0(r, t)|2δΨ(r, t) + U0Ψ2
0(r, t)δΨ∗(r, t) = i~
∂Ψ(r, t)∂t
.
Note that the nonlinear term couples δΨ(r) and δΨ∗(r). The presence of the chemical po-tential in Eq. (4.3) means that the solution of Eq. (4.7) corresponding to to a solution of thetime-independent problem is Ψ0(r, t) = Ψ0(r)e
−iµt/~. Introducing the harmonic solution
δΨ(r) = e−iµt/~ [u(r)e−iωt + v∗(r)eiωt
],
one obtains easily the Bogoliubov-de Gennes equations (BdG equations)
~ωu(r) =[− ~2
2m∇2 + Uext(r)− µ+ 2U0|Ψ0(r, t)|2
]u(r) + U0Ψ2
0(r)v(r)
−~ωv(r) =[− ~2
2m∇2 + Uext(r)− µ+ 2U0|Ψ0(r, t)|2
]v(r) + U0Ψ∗2
0 (r)u(r). (4.12)
29
In free space µ = nU0, and plane wave solutions of Eq. (4.12) have the dispersion relation~ω(k) = E(k)
E(k) ≡[ε(k)
(ε(k) + 2mc2s
)]1/2, (4.13)
where we use the expression Eq. (4.5) for the speed of sound found earlier. This is thefamous Bogoliubov spectrum that we will encounter again shortly. At k ~/mcs (kξ 1)it has the linear form E(k) = csk, crossing over to the free particle spectrum at highermomentum.
A natural situation to examine is the effect of a weak time-dependent perturbing potentialUext(r, t). Consider the plane wave perturbation
Uext(r, t) = V0 cos (q · r− ωt)
This describes the effect of a pair of laser beams (Bragg spectroscopy) with differentwavevectors q = q1 − q2 and frequency difference ω, generally much smaller than the de-tuning from the fundamental transition2. This allows us to enter a regime of ω, q that probesthe collective behaviour of the system ~q/m ∼ cs ∼ 1 cm s−1, ~ω ∼ h × 1kHz. The BdGequations can be used to compute the resulting density response
δn(r, t) = |Ψ0(r, t) + δΨ(r, t)|2 − |Ψ0(r, t)|2 ∼ Ψ∗0(r, t)δΨ(r, t) + Ψ0(r, t)δΨ
∗(r, t), (4.14)
giving
δn(q, ω) = −nV0ε(q)
E(q)2 − ~2(ω + i0)2≡ V0
2D(q, ω). (4.15)
As usual, the imaginary part of this response function describes the absorption of energyfrom the perturbing field. In more quantum mechanical terms, quanta are created whentheir energy E(q) and momentum matches the change in energy and momentum of photonsscattering from one beam to the other. The golden rule gives the rate for this process as
Γ(q, ω) =2π~V 2
0
4
∑α,i
[δ (~ω − Eα) |〈α|eiq·ri |0〉|2 + δ (~ω + Eα) |〈α|e−iq·ri |0〉|2
], (4.16)
so that the rate of energy absorption is ~|ω|Γ(q, ω). Note that in this situation energy can beabsorbed for positive or negative ω as photons are scattered from the more energetic beamto the less energetic one. Comparing with the response function Eq. (4.15)
1π
(Vol) ImD(q, ω) = −∑α,i
[δ (~ω − Eα) |〈α|eiq·ri |0〉|2 − δ (~ω + Eα) |〈α|e−iq·ri |0〉|2
],
(4.17)which implies that all of the absorption comes from a single transition with3
∑i
|〈α|eiq·ri |0〉|2 =Nε(q)E(q)
. (4.18)
2To put some figures to it, the typical recoil energy ~2k2op/2m on absorption of an optical photon is on the kHz
scale3c.f. Problem 2. The only difference is the colossal ratio (∼ 1011) of energy scales in the two contexts!
30
Noting that∑
i eiq·ri is just a Fourier component ρq of the density ρ(r) =
∑i δ(r−ri), we can
integrate over positive ω in Eq. (4.17) to obtain the line strength of the resonance
− 1π
(Vol)∫ ∞
0dω ImD(q, ω) = 〈0|ρqρ−q|0〉
where we used the completeness relation to get rid of∑
α. This Fluctuation- Dissipationrelation relates the response function that we found to the density fluctuations in the groundstate of our system. Our explicit form Eq. (4.15) implies
〈0|ρqρ−q|0〉 =Nε(q)E(q)
→ N~q2mcs
N, p ~/ξ. (4.19)
The only problem is that our original ground state ansatz Eq. (4.1) disagrees with this result.Because there are no correlations between particles in that state, its density fluctuations arenormal, even at low p
〈GP |ρqρ−q|GP 〉 = N,
while the result Eq. (4.19) only reaches this value at large p4.Nevertheless, we will see that Eq. (4.19) is correct. The time-dependent GP approxima-
tion, though adequate for obtaining the dynamics of expectation values of such one-bodyquantities such as the density, fails to describe the quantum fluctuations implied by thisdynamics through general relations like Eq. (4.17). The remedy to this situation is the Bo-goliubov approximation.
Before closing, we should mention the effect of finite temperature. As long as the conden-sate fraction is close to unity, meaning that both quantum depletion (see next section) andthermal excitation out of the condensate are small, the GP approximation and be justified inboth its static and time-dependent forms. Obviously, the relation just explored between re-sponse and fluctuations has to be modified when the initial state of system may be differentfrom the ground state, as we assumed in Eq. (4.16).
4.2 Interlude: structure factors and sum rules
Let’s take a moment to relate the analysis of the previous section to some more generalformalism. The expression on the right hand side of Eq. (4.17) is related to the dynamicalstructure factor S(q, ω), defined as
S(q, ω) =∑α
δ
(ω − Eα − E0
~
)|〈α|ρq|0〉|2. (4.20)
Using completeness of the eigenstates, we then have∫ ∞
0dω S(q, ω) ≡ S(q) = 〈0|ρqρ−q|0〉,
4We should point out that the finiteness of density fluctuations at q → 0 is not inconsistent with the valueexactly at q = 0 being zero, as it will be for a wavefunction describing a fixed number of particles.
31
where S(q, ω) is called the static structure factor. S(q, ω) satisfies the following two sumrules (see, for instance, Ref. [10])∫ ∞
0~ωS(q, ω) =
N~2q2
2m
limq→0
∫ ∞
0
S(q, ω)~ω
=N
2mc2s, (4.21)
known as the f-sum and compressibility sum rules, respectively. They are the analogues ofEq. (2.4) and Eq. (2.7). In the present case Eq. (4.18) implies
S(q, ω) =Nε(q)E(q)
δ
(ω − E(q)
~
), (4.22)
and you should check that this satisfies both sum rules.
Problem 8 Use the sum rules Eq. (4.21) and the Cauchy-Schwartz inequality |〈A|B〉| ≤|A||B|, interpreting the integrals as inner products, to derive Onsager’s inequality
S(q) ≤ N~q2mcs
(4.23)
Note that Eq. (4.19) shows that in the present case the inequality is saturated. This is thecase (as should be fairly obvious from the proof) whenever S(q, ω) consists of a single mode.
4.3 The Bogoliubov approximation
4.3.1 Pair approximation and ground state energy
With the shortcomings of the GP approximation now manifest, we now turn to their resolution.We have already seen the germ of the idea when we discussed the derivation of the time-dependent GP equation. We saw there that the action of the interaction Hamiltonian onthe GP ground state generates pairwise occupation of single particle orbitals orthogonalto χ0. For now we will stick with the translationally invariant case (and periodic boundaryconditions), so that χ0 = 1/
√V and the single particle orbitals are plane waves. We can thus
imagine an improved ground state ansatz of the form
|pair〉 ≡∑nk
cnk∏k
Λnkk |GP 〉, (4.24)
where Λk = a†ka†−ka0a0 creates a (+k,−k) pair out of the ground state and we sum overall
all assignments nk of the number of such pairs to each k. Provided the mean numberof particles N0 in the zero momentum state remains close to N , the state that results fromthis one after an application of the Hamiltonian is approximately of the same form (withdifferent coefficients, in general) if the interaction is weak, as the c’s are in general small inthe interaction. Putting it another way, any order of perturbation theory in the interaction,
32
starting from the GP state, is clearly going to be dominated by repeated excitation of pairsout of this state5.
The crucial thing about the use of variational states of the form Eq. (4.24) is that
〈pair|H|pair〉 = 〈pair|Hpair|pair〉, (4.25)
where the Hpair is
Hpair = Hkin +U0
2VN (N − 1) +
U0
2V
′∑p
α†pα0 + α†0αp + 2npn0
+U0
2V
′∑p6=q
npnq + α†pαq. (4.26)
Here α†p = a†pa†−p creates a (p,−p) pair, np = a†pap is the occupation number of an orbital,
and∑′ indicates that the zero momentum state is to be excluded. Note that it is probably not
justified to keep the terms on the second line, however. The last term changes a (q,−q) pairto a (p,−p) pair, when we have ignored the possibility of creating a triple with p+q+r = 0. Itturns out that these contributions can be dropped for weak interactions (though strictly doingso destroys the bounding property of Hpair, see below).
The pair Hamiltonian (without the second line) is usually obtained directly from the origi-nal Hamiltonian, by similarly arguing that BEC means that a0 is of ”order
√N ”. It’s important
to realize that these two points of view – starting from a variational state of the form Eq. (4.24)or jumping straight to the pair Hamiltonian Eq. (4.26) – are equivalent in terms of the physi-cal intuition they embody. Furthermore, the strength of having a reduced Hamiltonian is thatone can obtain excited states as well as the ground states, imagining that they have a similarform. Mathematically, the nice thing about the relation Eq. (4.25) is that we know we aregoing to end up with an upper bound for the ground state energy.
Can we solve Hpair? If we were able to treat a0 and n0 as O(N) c-numbers (and neglectthe last term), then we would be left with a quadratic Hamiltonian, and the answer wouldclearly be yes. The one glitch, however, is that 〈a0〉 = 0 on any state with a fixed numberof particles. The usual approach is to consider states that are superpositions of differentparticle number, introducing a chemical potential to fix the average particle number to N .The resulting state, after projection, should be a good approximation to the true ground stateof Hpair at large N , since the fluctuations in particle number in the unprojected state are only∼√N .
An alternative approach is to introduce the operator bp = a†0 (n0 + 1)−1/2 ap and its con-jugate. These satisfy the canonical relations
[bp, b
†p
]= 1 for p 6= 0. An operator like α†pα0
is then (n0 (n0 + 1))1/2 b†pb†−p. Since bp conserves total particle number c-number substitu-
tions in the resulting Hamiltonian are free from the above problem. Dropping the second lineof Eq. (4.26), we have, following the replacement of n0 with 〈n0〉 = N (assuming that the
5It’s necessary to point out that the method of this section should be understood as applying strictly to thecase of weak interatomic potentials U(r), where the Born approximation as ∼
`m/4π~2
´ RdrU(r) is valid. This
is not the same as the diluteness condition na3s 1, as strong potentials can give small scattering lengths. The
derivation of Eq. (2.19), say, with as the true scattering length, is considerably more complicated, see Ref. [5].
33
depletion N −N0 can be neglected)
Hpair =U0
2VN (N − 1) +
∑p
ε(p)b†pbp +U0N
2V
′∑p
b†pb†−p + bpb−p + 2np, (4.27)
which is the same as Bogoliubov’s non-conserving Hamiltonian. It is diagonalized by thetransformation 6
βp = bp coshκp − b†−p sinhκp
tanh 2κp =nU0
ε(p) + nU0. (4.28)
The transformed Hamiltonian is
H =12nU0N +
′∑p
12
[E(p)− ε(q)− nU0] + E(p)β†pβp, (4.29)
where E(p) is the Bogoliubov dispersion relation introduced in Eq. (4.13). The ground statecorresponds to
βp|0〉 = 0, (4.30)
and the ground state energy
E0 =12nU0N +
′∑p
12
[E(p)− ε(q)− nU0] .
The integral is divergent in the ultraviolet, but this can be fixed by writing
E0 =12nU0N
[1− 1
V∑p
U0
2ε(p)
]+
′∑p
12
[E(p)− ε(q)− nU0 +
(nU0)2
2ε(p)
].
In this form, the term we have added and subtracted is recognized as the next order in theBorn approximation for the scattering length as = a0 + a1 + · · · . The second term can nowbe evaluated to give
E0 =12nN
4π~2
m(a0 + a1) +
12nN
4π~2
ma0
12815√π
(na3
0
)1/2. (4.31)
Where
a0 = (m/4π~2)U0, a1 = −(m/4π~2)U2
0
V∑p
12ε(p)
.
This closely resembles the result quoted in Eq. (2.19) for the first two terms of the groundstate energy of a system of bosons as an expansion in
(na3
s
)1/2. The Bogoliubov approxi-mation, as just described, is not able to reproduce that result in all its glory, for reasons we’llexplain below.
6Note that the transformation will not exist at low p if µ is negative, meaning that the Hamiltonian has noground state. The stems from the the neglect of the second line of Eq. (4.26). While systems with negativescattering length are believed to be thermodynamically unstable, their Hamiltonians should still make sense!
34
4.3.2 Structure of the ground state and excitations
We now turn our attention to the nature of the ground state of Eq. (4.27). It is not difficult tosee that the ‘Bogoliubov vacuum’ conditions Eq. (4.30) are satisfied by
|B〉 ≡′∏p
e(cp/2)b†pb†−p |N〉0 |N〉0 =1√N !
(a†0
)N|0〉, (4.32)
if cp = tanhκp (we have not normalized). The factor of 1/2 arises from accounting for p and−p contributions when βp is applied.
A slightly different wavefunction, that coincides with Eq. (4.32) for N 1, is more easilygiven physical interpretation. Using b†p = a†pa0/
√N , one can show that in this limit
|B;N〉 =1√N !
[a†0a
†0 +
′∑p
cpa†pa
†−p
]N/2
|0〉 (4.33)
(assuming N even) is an equivalent choice. This is almost, but not quite, the number projec-tion of the non-conserving Bogoliubov ground state
|B; ζ〉 ≡′∏p
eζ(cp/2)a†pa†−p |ζ〉 |ζ〉 = e√
Nζa†0 |0〉, |ζ| = 1. (4.34)
I emphasize that for N 1 all three of Eq. (4.32-4.34) give the same result (Eq. (4.31))when the energy is evaluated. Eq. (4.33) has the pleasant property of being a member ofthe class Eq. (4.24) in which the amplitudes for multi-pair states exactly factorize into thosefor the constituent single-pair states. In first quantized language we have
|B,N〉 = S∏i<j
ϕ(ri − rj), (4.35)
(c.f. Eq. (4.11) with
ϕ(r) =∑p
cp eip·r
cp ∼ −nU0/2ε(p) at large p, reproducing the scattering wavefunction at lowest order. Sinceit doesn’t work to all orders, there is clearly no way that even the leading order term in(na3
s)1/2 in E0 can come out right, as it depends on the scattering wavefunction being written
in terms of the true scattering length. It is possible to obtain the result Eq. (2.19), however, byusing a variational wavefunction of the form Eq. (4.33), but retaining the full pair HamiltonianHpair [5, 11]. Furthermore, in the thermodynamic limit, the exact solution of Hpair is belivedto be of this form.
The excitations described by Eq. (4.29) have the same spectrum that we found before.Indeed, it should be clear that coshκp and sinhκp are determined by the same Bogoliubov-deGennes equations that determined u(r) and v(r) (compare the quadratic form in Eq. (4.27)with the matrix structure of Eq. (4.12)). In a sense the Bogoliubov theory can be thought of asa quantization of those oscillations. This allows the Bogoliubov approximation to be extendedto the inhomogeneous case quite straightforwardly (see e.g. Ref. [4] for a brief discussion),though in light of the small depletion discussed below this normally isn’t necessary.
35
Figure 4.1: Momentum transfer per particle for a BEC (open circles) and expanded cloud(closed circles), for a given value of the wavevector q = q1−q2 in Bragg spectroscopy. Notehow the peak is suppressed and shifted to higher energies in the BEC. Broadening is due tothe inhomogeneity of the condensate. Reproduced from Ref. [12].
An intuitive picture of the excitations can be obtained by considering the action of thedensity ρq on the ground state. Keeping only the parts of the operator that move particles inor out of the condensate gives
ρq|B〉 ∼√N
[a†qa0 + a†0a−q
]|B〉 =
√N
[b†q + b−q
]|B〉
=√N [coshκp − sinhκp]β†p|B〉, (4.36)
(c.f. Eq. (4.14)) so that an excitation coincides with a density fluctuation. This allows us toimmediately obtain the structure factor
S(q) = 〈B|ρqρ−q|B〉 = N [coshκp − sinhκp]2 =Nε(p)E(p)
→ Np
2mcs, q ~/ξ, (4.37)
matching the result Eq. (4.19) and showing that the structure factor saturates the inequalityEq. (4.23). We therefore see that all of the weight in S(q, ω) lies in the Bogoliubov modes
S(q, ω) = S(q)δ(ω − E(q)/~).
The result Eq. (4.37) can be interpreted as resulting from interference between terms withdifferent number of excited pairs at q, −q. That is, both
|N − 2n〉0|n〉q|n〉−q, and |N − 2n− 2〉0|n+ 1〉q|n+ 1〉−q
terms of Eq. (4.32) contribute to the |N−2n−1〉0|n+1〉q|n〉−q component when ρq is applied,with amplitudes coshκq and sinhκq respectively.
I want to emphasize that response functions in the Bogoliubov approximation, calculatedvia the dynamical structure factor, coincide with those obtained in the time-dependent Gross-Pitaevskii theory. Thus experiments on scattering (see Fig.4.1) that are consistent with these
36
results do not really bear on the validity of the Bogoliubov theory. The qualitatively newfeature of the Bogoliubov ground state is that the zero momentum state is depleted. We findthe momentum distribution
n(p) =|cp|2
1− |cp|2→ mcs
2p, p ξ−1.
The radial density distribution 4πp2n(p) is peaked around ~/ξ. Summing over p gives thefraction of atoms not in the condensate
1N
∑p
n(p) =8
3√π
√na3
s, (4.38)
where we used the Born approximation for the scattering length as = 4π~2U0m . Under typical
experimental conditions the depletion does not much exceed 0.01, which justifies the use ofthe GP approximation. In the presence of an optical lattice, however, we shall see that thecondensate can be depleted to zero, causing a quantum phase transition out of the superfluidstate.
What is beyond the Bogoliubov approximation? Just two brief comments
• The finite temperature depletion of the condensate, and the self-consistent effect it hason the Bogoliubov approximation, can be included in a relatively straightforward way.This is known as the Popov approximation.
• Since the Bogoliubov excitations are of course not exact eigenstates, an improvedcalculation will lead to their interaction. A notable process is Beliaev damping, corre-sponding to the decay of one Bogoliubov excitation into two. This process comes fromterms like a†pa
†p′ap+p′a0 that we discarded in making the pair ansatz. Such processes
are however not significant at low momenta.
Problem 9 How does the superfluid fraction discussed in Section 3.2.1 change in the Bo-goliubov approximation?
4.4 Atom optics
Some of the most dramatic effects of Bose-Einstein condensation are quantum interferenceeffects that the BEC is able to ‘amplify’ to a macroscopic level. Even more surprising, how-ever, is the tendency of separate condensates to respond coherently even when they startout with no coherence. We explore these features in this section.
4.4.1 Fock states and coherent states
When we discussed the Gross-Pitaevskii approximation, we used a trial wavefunction for Natoms of the form
|N〉 =∏
i
χ0(ri),
37
or in second quantized notation
|N〉 =1√N !
(a†0
)N|0〉,
where a†0 creates a particle in the state χ0(r). In general we will call states of this form Fockstates. In some discussions of BEC, one encounters the coherent states
|α〉 ≡ e−|α|2/2 exp(αa†0)|0〉.
This is a superposition of Fock states, that is, a superposition of states with definite particlenumber. While this might be a reasonable state for photons, which can be created anddestroyed, it certainly is never the state of a system of atoms. The coherent state has,however, the nice property that
〈α|(a†0)m(a0)
n|α〉 = (α∗)mαn. (4.39)
In other words, a0 acquires an expectation value α, which can be complex. Thus these arethe states that one is forced to introduce in the traditional non-conserving formulations of theBogoliubov approximation, see Eq. (4.34), where a0 is treated as a c- number. The propertyEq. (4.39) gives us the density matrix
ρ1(r, r′) = 〈α|φ†(r)φ(r′)|α〉 = χ∗0(r)χ(r′)|α|2,
so that for |α|2 = N we have BEC as we defined it before. The difference is that now φ(r)itself has an expectation value
〈α|φ(r)|α〉 = χ0(r)α,
which leads to the idea that the phase of α is really the phase of the condensate wave-function. Since we introduced χ0(r) through the spectral resolution of ρ1(r, r′), its phase isarbitrary in our formulation, Another way of putting it is through the relation
|N〉 ∝∫ 2π
0
dϕ
2πe−iϕN |eiϕ〉, (4.40)
i.e. coherent states and Fock states are conjugate in more or less the same was as momen-tum and position states in ordinary single particle quantum mechanics. Since the number isfixed for a system of isolated particles, the phase is completely unknown.
The situation is quite different, however, when we consider that atoms may occupy twopossible states χ0(r) and χ1(r), not necessarily orthogonal, which could be different orbitalstates, hyperfine states, etc.. Then the Gross-Pitaevskii state(
a†0 + αa†1
)N|0〉 (4.41)
is clearly a physically sensible state, and quite distinct from the product of two Fock states
|n,N − n〉 ≡ (a†0)n(a†1)
N−n|0〉. (4.42)
We do, however, have a relation just like Eq. (4.39)
|n,N − n〉 ∝∫dϕ
2πe−iϕn
(a†0 + |α|eiϕa†1
)N|0〉. (4.43)
We see that Eq. (4.41) describes a state in which atoms in states 1 and 2 have a definiterelative phase, while Eq. (4.42) can be thought of as a state in which that phase fluctuates.As we will now show, this is not just a mathematical nicety.
38
4.4.2 Interference of two condensates
With this background, let us consider the case where the states χ0 and χ1 represent twospatially separated condensates. Schematically, this is the situation in classic interferenceexperiment demonstrating coherence of condensates [13]. Such a system can be preparedin a state like Eq. (4.41) of definite relative phase or in the Fock state Eq. (4.42), dependingon whether the gas is divided in two below the condensation temperature or above it.
Suppose that the two states are initially sufficiently separate that they can be thought ofas orthogonal. Then the state of definite relative phase ϕ is
|N0, N1, ϕ〉 ≡1√N !
[√N0
Ne−iϕ/2a†0 +
√N1
Neiϕ/2a†1
]N
|0〉, (4.44)
where N0, are the expectation values of particle number in each state N = N0 + N1. Weallow the system to evolve for some time T , so that the two ‘clouds’ begin to overlap (typ-ically achieved by allowing free expansion). Ignoring interactions, the many-particle stateis just Eq. (4.44) with the wavefunctons χ0,1 evolving freely. We compute the subsequentexpectation value of the density
ρ(r) = φ†(r)φ(r), φ(r) = χ0(r)a0 + χ1(r)a1
〈ρ(r, T )〉ϕ = N0|χ0(r, T )|2 + N1|χ1(r, T )|2 + 2√N0N1Re eiϕ χ∗0(r, T )χ1(r, T ). (4.45)
If the clouds are now overlapping, the last term in Eq. (4.45) comes into play. Its origin is inquantum interference between the two coherent subsystems, and it depends on the relativephase, demonstrating the real macroscopic effects of this quantity.
As an illustration, consider the evolution of two Gaussian wavepackets with width R0 atT = 0, separated by a distance d R0
χ0,1(r) =1
(πRT )3/2exp
[−
(r± d/2)2(1 + i~t/mR2
0))
2R2T
], (4.46)
with
R2T = R2
0 +(
~tmR0
)2
.
The final term of Eq. (4.45) is then
ρint(r, T ) =2√N0N1
R3T
A(r, T ) cos[ϕ+
~r · dmR2
0R2T
T
]. (4.47)
The interference term therefore consists of regularly spaced fringes, with a separation atlong times of 2π~T/md.
Now we consider doing the same thing with two condensates of fixed particle number,which bear no phase relation to one another. The system is described by the Fock state
|N0, N1〉 ≡1√
N0!N1!(a†0)
N0(a†1)N1 |0〉.
Computing the density in the same way yields
〈ρ(r, T )〉F = N0|χ0(r, T )|2 +N1|χ1(r, T )|2, (4.48)
39
i.e. the same as before, but without the interference term. This is consistent with the principleexpressed in Eq. (4.43), that we can think of the Fock state as a kind of ‘phase-averaged’state.
This is not the end of the story, however. When we look at an absorption image ofthe gas, we are not looking at an expectation value of ρ(r) but rather the measured value ofsome observable (s) ρ(r). Unlike the situation in most condensed matter experiments, wherewe suppose that a kind of averaging occurs in space or time, there is no particular reasonwhy the expectation value should tell us everything there is to know about such a ‘one-shot’measurement. Consider now the correlation function of the density at two different points
〈ρ(r)ρ(r′)〉F = 〈ρ(r)〉F 〈ρ(r′)〉F+N0(N1 + 1)χ∗0(r)χ
∗1(r
′)χ0(r′)χ1(r) +N1(N0 + 1)χ∗1(r)χ∗0(r
′)χ1(r′)χ0(r).(4.49)
We see that the second line contains interference fringes, with the same spacing as before.The correlation function gives the relative probability of finding an atom at r′ if there is oneat r. It seems that in each measurement of the density, fringes are present but with a phasethat varies between measurements, even if the samples are identically prepared7. Indeed, itis clear that, for N0, N1 1
〈ρ(r)ρ(r′)〉F =∫ 2π
0
dϕ
2π〈ρ(r)〉ϕ〈ρ(r′)〉ϕ, (4.50)
The rather surprising implication is that predictions for measured quantities for a system in aFock state are the same as in a relative phase state, but with a subsequent averaging overthe phase.
Problem 10 Prove this in general by showing that the density matrix corresponding to astatistical mixture of phase states |N0, N1, ϕ〉 with random phase coincides with that of amixture of Fock states with binomial distribution of atoms into states 0, 1. At large N thisdistribution becomes sharply peaked at occupations N0, N1.
Turning this observation around suggests that after we have observed fringes, the systemis in a state of definite relative phase. It is useful to consider how this happens, but in thesimpler situation depicted in Fig. 4.2. Atoms from two condensates are released to passthrough a beam splitter. Assuming that there is some way to determine when k atoms havepassed the splitter, without measuring whether they came from condensate 0 or 1, we canask for the resulting probability that k+ arrive at the counter, with the other k− ≡ k − k+
passing along the other arm of the beam splitter. The simplicity of this situation lies in thefact that there are only two possible final positions for the atoms. Assuming that these finalstates |±〉 are created by 1√
2[a0 ± a1]a
†± (i.e. the beam splitter is 50 : 50 with no relative
phase shift), the observation of k+ counts leaves the condensate, initially in the Fock state7We note that Eq. (4.49) is not real, so that strictly 〈[ρ(r), ρ(r′)]〉 6= 0 and the density measured at two points
does not correspond to two commuting variables. This is, however, only an O(N) effect, compared to the O(N2)magnitude of the correlator. For a large number of particles, then, an absorption image can be assumed to bean instantaneous measurement of the function ρ(r) (projected onto a plane, of course).
40
Figure 4.2: Beam splitter configuration with two condensates
|N/2, N/2〉, in the state [14]
[a0 +a1]k+ [a0−a1]
k− |N/2, N/2〉 ∝∫ π
−π
dϕ
2π[cos(ϕ/2)]k+ [sin(ϕ/2)]k− |(N−k)/2, (N−k)/2, ϕ〉,
(4.51)where we have used the relationship between the Fock and phase states to resolve theresulting state into phase states. We see that the effect of the measurement is to createuncertainty in the relative number of atoms remaining in the two condensates. From ourdiscussion of the conjugate relationship between particle number and phase, this meansthat the relative phase can become more precisely known. At large k±, we can write theintegral in Eq. (4.51) in the stationary phase approximation as∫ π
−π
dϕ
2π
[e−k(ϕ−ϕ0)2 + (−)k− e−k(ϕ+ϕ0)2
]|(N − k)/2, (N − k)/2, ϕ〉, (4.52)
where ϕ0 is the solution in [0, π] of k+ = k cos2 (ϕ0/2). This corresponds to the value of thephase that we infer (up to a sign) from a measurement of the fraction k+/k. Eq. (4.52) showsthat the residual uncertainty in the phase is 1/
√2k. The measurement of the phase is shot
noise-limited : the fraction k+/k estimates the magnitude of a wavefunction of definite phase,but the discreteness of the individual measurements provides a limit to how accurately wecan know it. An even more fundamental point is that a necessary condition for the phaseto be a sharp observable is that we are dealing with a large number of particles in the firstplace.
An experiment that very closely resembles this situation has recently been performed.In Ref. [15], the beam-splitter consists of the crossed laser set-up used to perform Braggspectroscopy, as described in Section 4.1.2. The input states correspond to two trappedcondensates, as in Fig. 4.2, and the two output states correspond to momentum zero and prespectively. The number of atoms kp ending up in p can be monitored through the number ofphotons transferred from one beam to another. The rate of this process is given by Eq. (4.16)
Γ(p, ω) =2π~V 2
0
4[S(p, ω)− S(−p,−ω)] . (4.53)
41
The relation to our previous discussion is that we take k → ∞ with k+/k → 0 such that theexpected number of counts k+ is Γτ , with τ the time for which the Bragg pulse is applied.
4.4.3 Superradiance
When we discussed Bragg spectroscopy in Section 4.1.2, the optical potential due to thecrossed lasers was treated classically as a ‘running wave’
Uext(r, t) = V0 cos (p · r− ωt) .
This is acceptable, because under conditions of Bragg spectroscopy, there are a large num-ber of photons in each laser mode. There are other phenomena, however, that require thephotons to be treated more quantum mechanically. This can be done by considering thefollowing interaction, describing the scattering of photons (described by operators ck, c†k) byatoms
Hint =∑
k,l,m,n
Cklmnc†la†nckamδl+n−k−m. (4.54)
Note that this expression is second order in the photon operators, and so is second order inthe electric field, so we expect a correspondence with our earlier discussion of AC Stark shiftin Section 5.3. The coupling constants Cklmn can be found by comparison with those results.Considering a single photon mode to begin with, the number of photons with wavevector k is
nck =
ε0|E|2V2~ck
,
whereas an electric field field of this intensity gives an AC Stark shift
∆E =d2E2
4~∆,
where d is the dipole matrix element for the transition, and ∆ is the detuning ω−ωn0 (assumedmuch larger than the excited state lifetime). The l = k term of Eq. (4.54) then gives Ckkmmn
ck
for the shift of each atom, and we have
Ckkmm =ckd2
2ε0V∆.
For the case of crossed lasers at wavevectors k, l, the same consideration leads to
Cklm(m+k−l) =ckd2 cosφkl
2ε0V∆, (4.55)
where φ is the angle between the axes of polarization of the two beams8. Now, how are weto understand a formula like Eq. (4.53) on this basis? Consider transitions out of the initialstate
|i〉 = |g;nck, n
ck−q〉 (4.56)
8It is certainly possible that l − k is on the scale of typical atomic momentum, which would lead to somedependence on the momentum transfer. We neglect this for now, as we did implicitly in Section 5.3.
42
At lowest order in Hint, we change the momentum of one atom by q in all possible waysthrough scattering of one photon between states k and k − q. The square of the matrixelement of Hint between |i〉 and a given final state |f〉 is then
|〈f |Hint|i〉|2 = |Ck,k−q|2〈g|ρq|f〉〈f |ρ†q|g〉(nc
k−q + 1)nc
k
where ρ†q =∑
n a†n+qan is a Fourier component of the density operator. We then write down
the Golden Rule expression for the transition rate as
Γ+ =2π~
∑f
|〈f |Hint|i〉|2δ (Ef − Eg − ~ω)
=2π~
∑f
|Ck,k−q|2〈g|ρq|f〉〈f |ρ†q|g〉(nc
k−q + 1)nc
kδ (Ef − Eg − ~ω)
=2π~|Ck,k−q|2S(q, ω)
(nc
k−q + 1)nc
k. (4.57)
This is the rate at which +k is transferred. To get the total rate we must subtract the rate forthe −k transitions to give the total rate
Γ = Γ+ − Γ− =2π~|Ck,k−q|2
[S(q, ω)
(nc
k−q + 1)nc
k − S(−q,−ω) (nck + 1)nc
k−q
]. (4.58)
In the limit of large mode occupations nc 1, the difference between nc and nc + 1 is ne-glected and the resulting formula, using the above expressions for C, reduces to Eq. (4.53).The result Eq. (4.58) is much more general and includes, for example, the more familiar caseof Rayleigh scattering, in which the photon mode k− q is empty, nc
k−q = 0, so that only theforward rate survives. It is via this process that light is scattered into all other modes, whilethe second laser is required to preselect the momentum transfer q by creating an opticalgrating. Note also that the discussion works as well for photon modes in Fock or coherentstates.
Now we have a new way of looking at this arrangement. Instead of considering theresponse of the condensate to a perturbation due to the light fields, we can think insteadof the scattering of light by the condensate. An interesting case occurs when both the zeromomentum state and some finite momentum state q of the condensate are occupied. Thiscorresponds to the wavefunction, in the GP approximation
(a†0)N0(a†q)Nq |0〉,
and static structure factorS(q′) = N0 (Nq + 1) δq′−q.
Ignoring all interaction effects, the dynamic structure factor is
S(q′, ω) = N0 (Nq + 1) δq′−qδ(ω − εq). (4.59)
We can then use Eq. (4.58) to give the total rate for scattering of light from a mode k into aninitially empty mode k− q, leading to an increase in the number of atoms in mode q
Nq = Gq (Nq + 1)− Γ2,qNq. (4.60)
43
The gain coefficient Gq depends on N0, nck, and geometric factors, but the crucial thing is
the ‘bosonic stimulation’ term (Nq + 1). The loss term in Eq. (4.60) represents the linewidthof the two-photon process we are considering. The instability present for Gq > Γ2,q corre-sponds to the onset of superradiance: atoms begin to accumulate in the state q, and light isemitted coherently into the mode k−q. An alternative view of this phenomenon comes fromconsidering scattering from a grating formed by the coherent state[√
N0
Na†0 +
√Nq
Na†q
]N
|0〉.
From our earlier discussion, this corresponds to a oscillating density 〈ρ(r)〉. Furthermore,we know that the scattering of light by this density is going to give the same answer asEq. (4.59), based on the general result that the measurements on the Fock state coincidewith those for the phase state (with an average over relative phase). In more concrete terms:though 〈ρ(r)〉 has no q component in the Fock state, the density-density correlator does,and this gives the structure factor Eq. (4.59). A detailed discussion of experiments on thescattering of light by condensates, and the associated theory, may be found in Ref. [12].
In this section, then, we have explored the relationship between the Fock and coherentstates, and the remarkable tendency of bosons to end up in the latter, starting from theformer. We have also learnt about the reciprocal nature of atom-light scattering, where,depending on the occupancies of the various modes, we can think about light scatteringfrom a grating formed by the atoms, or vice versa. Both cases have a conjugate viewpoint interms of bosonic stimulation of scattering by occupation of the final state.
4.5 Spinor condensates
So far our discussion of the effect of the hyperfine degree of freedom was limited to the oneand two atom level. In Chapter 2 we introduced the Hyperfine-Zeeman states of a singleatom, and briefly discussed the scattering of these states off one another. When we turnto the many-boson problem, the existence of different atomic species leads to a number ofqualitatively new and surprising phenomena.
4.5.1 General considerations for multicomponent BEC
It is straightforward to enlarge our definition of BEC to include different species of atoms.Our definition of the one-particle density matrix now bears indices labeling these species
ρ1(rα, r′α′) ≡ 〈〈φ†α(r)φα′(r′)〉〉
=∑
i
niχ∗i (r, α)χi(r′, α′). (4.61)
BEC can then be defined exactly as before, using the spectral resolution of the densityoperator. We shall see that non-simple BEC, where more than one eigenvalue of order N , isnow more common.
If we do have simple BEC the condensate wavefunction and order parameter are intro-duced exactly as in the one component case, as the eigenfunction χ0(rα) and
√N0χ0(rα)
44
respectively. Now we meet a significant complication. χ0(rα) is in general a non-trivialfunction of α. Thus it is not immediately obvious how to generalize the decomposition intoamplitude and phase that was necessary to define the superfluid velocity. In Section 3.1 weswept this under the carpet, effectively by assuming that the condensate wavefunction was
χ0(rα) = χ0(r)ηα(r),
where ηα(r) is some normalized vector in the hyperfine space, and then decomposing χ0(r)in the naive way. The problem is that this prescription is not at all well-defined, as we couldhave equally well chosen a different η′α(r) = eiϑ(r)ηα(r), differing from the first by a position-dependent phase factor. The solution is to define the superfluid velocity using the ‘covariantderivative’
D = ∇− iη†(r)∇η(r). (4.62)
In this way the definition
vs =~m
Dϕ, ϕ = Argχ0(r)
is independent of how we apportion the phase between χ0(r) and ηα(r). A vanishing vs
is in general inconsistent with the single-valuedness of ϕ. Thus vs is non-zero even in theequilibrium state of the non-rotating system. Furthermore, vs is no longer irrotational, as canbe seen from the Mermin-Ho relation
∇∧ vs =~mmF εijkBi∇Bj ∧∇Bk, (4.63)
where B is the unit vector in the direction of the magnetic field.
Problem 11 Verify Eq. (4.63) for the simplest case of the spin-1/2 spinor
χ1/2(θ, φ) =(
eiφ/2 cos θ/2e−iφ/2 sin θ/2
),
corresponding to the unit vector B = (sin θ cosφ, sin θ sinφ, cos θ).
4.5.2 The Gross-Pitaevskii description
Now we turn to the effect of interactions, and develop the Gross-Pitaevskii approximation. Itis simplest to forget the above complexities associated with non-constant adiabatic bases,at least to begin with. Let us therefore consider bosons with F = 1 in an optical trap, sothat all hyperfine species are present, and have equal single-particle energy. Accordingto the discussion of Section 2.3.2, it is clear that the Gross-Pitaevskii variational energycorresponding to the wavefunction
Ψ(ri, αi) =∏
i
χ0(riαi) (4.64)
is
〈H〉 =∫dr
[~2
2m|∇Ψ|2 + Uext(r)|Ψ(r)|2
]+
12
∫drU1|Ψ(r)|4 + U2[Ψ†FΨ]2, (4.65)
45
where |Ψ|2 = Ψ†Ψ, U1,2 = 4π~2a1,2/m, and as before we have neglected the differencebetween N and N − 1, although we will see that we must examine this point more carefullylater.
The following two cases immediately present themselves
• Antiferromagnetic interactions, U2 > 0. In this case the energy is minimized for 〈F〉 = 0,the ground state energy is EA = nNU0/2. This is the case as long as χ† ∝ (χx, χy, χz),for χx,y,z all real. The symmetry group of such configurations is SO(3)/U(1) i.e. rota-tions about the axis of Ψ are excluded. This state is sometimes called the ‘polar state’by analogy with superfluid 3He.
• Ferromagnetic interactions, U2 < 0. In this case we take the spin to be as big aspossible, with each particle having |〈F〉| = 1. Thus the ground state energy is EF =nN(U1 + U2)/2. The spinor with 〈F〉 = (0, 0, 1) has the form
1√2
1−i0
, (4.66)
and all other possible ferromagnetic configurations can be generated from the onethrough appropriate SO(3) rotations. Note that because the spinor is complex, allrotations are effective. Thus a rotation of Eq. (4.66) about the z-axis yields
1√2
e−iϕ
−ie−iϕ
0
.
Although the overall phase of the condensate wavefunction has no meaning, the factthat a phase changing in space is equivalent to a rotation through differing angles atdifferent points can have real physical consequences, as we shall see.
4.5.3 Metastability of superflow
The internal states of the condensate have rather intriguing repercussions for the metastabil-ity of superflow. We return to our old friend the annular container to illustrate these. Supposethat we have only two internal hyperfine states | ↑〉 | ↓〉 (F = 1/2 clearly doesn’t correspondto a boson, but let’s not worry about that for the moment), and consider the family of orderparameter configurations
cosθ
2eiφ| ↑〉+ sin
θ
2| ↓〉 0 ≤ θ ≤ π. (4.67)
(φ is the angular variable in cylindrical coordinates). This interpolates between a state withorbital angular momentum ` = 1 and 0, just like the state considered in Problem 7 for thesingle component case. The crucial difference, however, is that all members of the abovefamily have the same |Ψ|2 (in particular we can move from ` = 1 to 0 without passing througha state with a node) and therefore the same interaction energy, if interactions are invariantunder rotations in the internal space. Since the kinetic energy decreases monotonically as θgoes from 0 to π, there is no metastability.
46
Consider now the ferromagnetic state of spin-1 bosons discussed in the previous section.We want to attempt the same trick: rotating from mF = +F = 1 to mF = −F = −1 usinga rotation axis in the x-y planes that rotates by 2π as go around the annulus, we try to undothe winding of the phase. Thus we take the family of condensate wavefunctions
χθ(φ) =1√2
cosφ− i sinφ cos θ− sinφ− i cosφ cos θ
−i sin θ
. (4.68)
We see that
χ0(φ) =1√2
e−iφ
−ie−iφ
0
,
and furthermore that the spin rotates in the desired way
〈F(φ)〉θ =
− sinφ sin θ− cosφ sin θ
cos θ
.
But what state do we end up in? Surprisingly, the answer is
χπ(φ) =1√2
eiφ
−ieiφ0
.
We have not got rid of the winding in the phase at all, but rather just switched its sense frompositive to negative! It is clear, then, that a winding number of two is equivalent to no windingin this sense, and odd winding number configurations are metastable.
To summarize, when mF = 0, including in the antiferromagnetic case for F = 1, thereare metastable configurations of arbitrary winding number. For mF = 1/2, there are none,and when mF = 1 there is just one: all even winding numbers are topologically equivalent tozero winding number, and all odd winding numbers are equivalent to one. The general rulethat these three exemplify is: the winding number of the phase can change by 2mF withoutenergetic penalty. Of course, there is still the issue of the dynamics of this unwinding process– it may take a long time – but there is no energetic barrier.
Experimentally, we are still waiting for the observation of the ferromagnetic state, but thesame physics is behind the topological vortex formation discussed in Problem 3. The resultcan also be interpreted in the light of Eq. (4.63).
[A Java applet for the belt trick shown in the lecture can be found at http://www.math.toronto.edu/∼drorbn/Students/Song/]
4.5.4 Fragmented condensates
There is a slight wrinkle to the above discussion of the antiferromagnetic case. If we assumethat all the atoms are in a single spatial orbital – the zero momentum state – but do notconstrain the spin wavefunction of this system at all, we arrive at the spin Hamiltonian
Hspin =U1
2VN (N − 1) +
U2
2V(S2 − 2N
), (4.69)
47
where S is the operator of total spin. It’s clear that the ground state of this problem corre-sponds to the N -spin singlet state, with energy
U1
2VN (N − 1)−N
U2
V.
This is smaller than the ground state of the Gross-Pitaevskii state that we considered earlier,because of the (non-thermodynamic) second term. As is fairly clear, the mean number ofparticles in each of the three hyperfine states of the system is N/3, with fluctuations alsoof order N . We have a fragmented condensate, in which more than one eigenvalue of thedensity matrix is thermodynamically large.
The consensus view of this situation is that:
• The original GP state is a better description of the ground state in a magnetic field, withthe fluctuations decreasing very rapidly
• The two states are in any case very hard to distinguish on the basis of an interferencemeasurement such as those discussed in Section 4.4.
48
Chapter 5
Fermi superfluids
5.1 Fermionic condensates
In Chapter 3, we saw that superfluidity was a natural consequence of Bose-Einstein conden-sation. The existence of the condensate order parameter and the definition of the superfluidvelocity in terms of it make superfluidity almost inevitable. Thus it is natural to ask whethersuperfluidity is a phenomenon which is entirely confined to systems of bosons. It is worthbearing in mind, however, that so far we have been treating composite objects – alkali atoms– as indivisible interacting quantum particles. But an atom is made of electrons, protons, andneutrons (if we stop at this level of description), which are all fermions. So the answer to ourquestion is clearly negative.
To make the discussion simpler, consider a gas of two different types of fermions withattractive interactions between the atoms of different types. Since the two types are distin-guishable, there is s- wave scattering, with an associated scattering length as, as we havebefore. If the potential between the two types of fermion is made sufficiently attractive, abound state may form. The resulting molecule can be thought of as a boson, in the sensethat if we swap the positions of the two atoms with the corresponding coordinates of anothertwo atoms bound in another molecule, the wavefunction is unchanged. If we are concernedonly with energy scales low compared to the binding energy of the molecule, so that thewavefunction of the relative coordinate can be almost always be taken to be the bound statewavefunction (except when we are considering collisions between molecules), we can forgetabout the internal structure altogether, and the system can bose condense. Putting it moreformally, the system displays ODLRO, but in the two-particle density matrix. By analogy withour discussion in Section 3.1, and using the label s =↑, ↓ to denote the two types of fermions
ρ2(r1, r2; r′1, r′2) ≡ 〈〈ψ†↑(r1)ψ
†↓(r2)ψ↑(r
′1)ψ↓(r
′2)〉〉
→ Nc
VF ∗(r1 − r2)F (r′1 − r′2), |r1 − r′1|, |r2 − r′2| → ∞, (5.1)
where F (r) is the wavefunction of the molecule, and Nc, as before, is the number of particles(this time diatomic molecules) in the condensate. Thus we might conjecture that a superfluidcan result only if we have a bound state, and then only if we are at temperatures below thebinding energy (and of course, the degeneracy temperature), see Fig. 5.1. The surprisingthing is that this expectation is not borne out. The long-range order described by Eq. (5.1)can be present even when the interaction between two isolated particles is not strong enough
49
Figure 5.1: Schematic phase diagram of a system of two species of fermions (equal innumber). The dashed line represents a naive expectation, without accounting for the Cooperphenomenon
to create a bound state, as a consequence of the restrictions imposed by the Pauli principlein a many-fermion sysetem. This is known as the Cooper phenomenon after Leon Cooper,who later became the C in the famous BCS theory of superconductivity, in which this effectplays a prominent role.
As a historical note, the idea that superconductivity (the analog of superfluidity in acharged system) in metals is caused by the condensation of electron pairs existed beforeBCS, and in this context a bound fermion pair is sometimes called a Schafrorth pair. Theproblem for these early authors was that they could not figure out any way that two electronscould bind.
Returning to our original discussion, we can start out with weakly attractive interactionsand describe the resulting condensate using the BCS theory, but then what happens if wecontinue to increase the strength of the attraction between fermions? A two-particle boundstate eventually will become possible, and the picture we outlined above, of the condensationof tightly bound molecules, will apply. The suggestion is that, at least as far as the groundstate goes, the evolution from the BCS to the BEC descriptions is smooth. The experimentalrealization of this BCS-BEC crossover in 2004 was perhaps the greatest triumph since thecreation of bosonic condensates in 1995 (see Ref. [16] for a non-technical introduction to the40K experiments at JILA). Understanding this phenomenon more closely will be one of maingoals in this chapter
5.2 The BCS theory
5.2.1 The pairing hypothesis
We start from the obvious Hamiltonian describing two species of Fermions of equal mass,which we will denote with a label s =↑, ↓, deferring a discussion of the actual experimental
50
system until later.
H =∑p,s
ξpa†p,sap,s +
U0
V∑
p,p′,q
a†p+q↑a†−p↓a−p′↓ap′+q↑. (5.2)
We take the interaction between the two species to be attractive U0 < 0. Starting from theground state of the non-interacting problem
|FS〉 =∏
|p|<pF
a†p↑a†p↓|0〉 (5.3)
describing a fermi sea with all states below the Fermi energy filled, we note that the applica-tion of the interaction terms generates terms of the form
a†p+q↑a†−p↓a−p′↓ap′+q↑|FS〉. |p|, |p + q| > pF , |p′|, |p′ + q| < pF .
Note the difference from the Bose case: because the state p = 0 plays no special role– like every other state below the Fermi surface it is occupied with one fermion of eachspecies – we do not just create pair excitations with zero centre of mass momentum q = 0.Nevertheless, the BCS theory starts from the assumption that such finite momentum pairsdo not contribute significantly to the ground state, and makes pair ansatz of essentially thesame form that was used in the Bogoliubov theory1
|pair〉 ≡∑
Pp nP
p =N/2
cnPp
∏p
Λnp
p′ |0〉, (5.4)
where Λk = a†p↑a†−p↓ creates a (+p,−p) pair and the numbers np,p′ are either 0 or 1. Note
that in writing Eq. (5.4) the number of each species is assumed to be exactly N/2 – the moregeneral case will be discussed later. As in the Bogoliubov case, such a form allows us towork with a pair Hamiltonian2
Hpair =∑p,s
ξpa†p,sap,s +
U0
V∑p,p′
a†p↑a†−p↓a−p′↓ap′↑. (5.5)
That is, just Eq. (5.2) with all but the q = 0 part discarded. Now, can we solve Eq. (5.5)?It is illuminating to introduce the operators b†p = a†p↑a
†−p↓ and its conjugate, that create and
destroy a (+p,−p) pair. Because our pair ansatz Eq. (5.4) only includes amplitudes for agiven (+p,−p) pair of momenta having either none or two fermions, the pair Hamiltoniancan be written in terms of the pair operators b†p, bp as
Hpair = 2∑p
ξpb†pbp +
U0
V
′∑p,p′
b†pbp′ . (5.6)
1In the book Ref. [17], Schreiffer explains that the BCS wavefunction was directly motivated by such treat-ments.
2One difference from the Bogoliubov case is that 〈pair|H|pair〉 = 〈pair|Hpair|pair〉 + EHartree, where theHartree energy is EHartree/V = U0n↑n↓, and ns is the expectation value of the density of species s. Sincethe overall Hamiltonian commutes with the total number of each component, our variational calculations are notaffected by dropping this term.
51
This may now look like a quadratic problem, but the pair operators bp, while commuting withthose at another momentum
[bp, bp′ ] = [b†p, b†p′ ] = [b†p, bp′ ] = 0 p 6= p′, (5.7)
obey the hardcore constraint(b†p)2 = 0, (5.8)
which is a result of b† being composed of a pair of fermions obeying the exclusion principle.Nevertheless, given that a pair that ‘hops’ into a level at p could come from any other levelp′, it seems reasonable to try, as a variational state, one in which the amplitudes for theoccupancy of each level are uncorrelated
|N〉 ≡
[∑p
cpb†p
]N/2
|0〉, (5.9)
corresponding to Eq. (5.4) but with a set of coefficients cnPp that factorizes. Finding the
variational energy of Eq. (5.9) is still a tricky problem. For instance, what is the expectationvalue of the kinetic energy?
K.E = 2∑p
ξp〈b†pbp〉 ≡ 2∑p
ξp〈nPp 〉, (5.10)
Finding the average number of pairs 〈nPp 〉 in Eq. (5.9) is however not obvious. We can make it
so by following the route taken by BCS, and considering instead the normalized wavefunction
|BCS〉 =∏p
[vpb
†p + up
]|0〉 |up|2 + |vp|2 = 1. (5.11)
This is a superposition of states with different total number of particles. One can see quiteeasily, however, that the projection onto a fixed number N of particles corresponds exactlyto Eq. (5.9) if cp = vp/up. Since Eq. (5.11) is a product of factors corresponding to eachmomentum separately, 〈nP
p 〉 is easily found to be v2p. The total variational energy of this state
is〈H〉BCS = 2
∑p
ξpv2p +
U0
V∑p,p′
u∗pvpup′v∗p′ . (5.12)
What about our use of a non-conserving wavefunction? The expectation value of any oper-ator that itself conserves the number of particles can evidently be written
〈O〉BCS =∑N
PN 〈O〉N ,
where 〈· · · 〉N denotes the expectation with respect to the N -particle projection Eq. (5.9). Theprobabilities PN are
PN =∑
PnPp =N/2
∏p
[nP
p v2p +
(1− nP
p
)u2p
],
which is strongly peaked around 〈N〉 = 2∑
p v2p = 2
∑p〈nP
p 〉, with a variance that is O(N).Thus at large N
〈O〉BCS → 〈O〉〈N〉. (5.13)
52
In the thermodynamic limit, we might as well work with the non-conserving form Eq. (5.11).There is an interesting alternative interpretation of the pair Hamiltonian Eq. (5.6). Acting
within the pair subspace, the three operators bp, b†p, and b†pbp − 1/2 behave as spin-1/2operators (Anderson spins) [
b†p, bp
]= 2
(b†pbp − 1/2
)[b†p,
(b†pbp − 1/2
)]= −b†p, (5.14)
meaning that Eq. (5.6) can be written as a spin chain
Hpair = 2∑p
ξpSzp +
U0
V∑p,p′
S+p S
−p′ . (5.15)
If we parameterize (up, vp) as (cos(θ/2)eiϕ/2, sin(θ/2)e−iϕ/2) then the variational energyEq. (5.12) has the form (except for a constant)
〈H〉BCS = −∑p
ξp cos θp +U0
4V∑p,p′
sin θp sin θp′ cos(ϕp − ϕp′
). (5.16)
The interpretation of Eq. (5.16) is the following. The first term tends to align the spins withthe z- axis in the - direction for ξp < 0 and in the + direction for ξp > 0. On the other hand,the second term, originating from the potential energy between the constituents of a pair,wants the spins to lie in the x-y plane3.
It remains to actually minimize the energy Eq. (5.12) to determine the u’s and v’s, orequivalently, the configurations of the spins. For U0 > 0 (repulsive interactions), the spins allpoint in the ±z direction, forming a ‘domain wall’ where ξp changes sign at the fermi surface,see Fig. 5.2. The relationship between the spin picture and the average number of pairs is
〈nPp 〉 = v2
p = [1− cos θp] /2,
so we see that this corresponds simply to a sharp fermi step. For U0 < 0, the systemcan lower its energy by taking sin θp 6= 0. The lowering of the interaction energy morethan compensates the increase in kinetic energy that comes from smearing the step, seeEq. (5.10). Clearly all of the angles ϕp, describing the angle in the x-y plane, should beequal. Taking the extremum of Eq. (5.16) with respect to the angles θp gives the condition
ξp sin θp − |∆| cos θp = 0,
where it is convenient to introduce the gap parameter
∆ = −U0
2V∑p
eiϕ sin θp = −U0
V∑p
upv∗p. (5.17)
Thus we have
cos θp =ξpEp
, sin θp =|∆|Ep
, Ep =√ξ(p)2 + |∆|2. (5.18)
3Of course the spins are really quantum mechanical spin-1/2 operators. The direction corresponds to thedirection of 〈S〉 in the BCS state
53
Figure 5.2: Anderson spin configurations and the associated distribution functions for thefree fermi gas (top) and the BCS state (bottom).
The meaning of these solutions is very simple. They correspond to the alignment of the spinvector with the direction of the effective ‘magnetic field’
(Re ∆.Im ∆, ξp) (5.19)
To be self-consistent, the solution must further satisfy
∆ = −U0
2V∑p
∆Ep
. (5.20)
Eq. (5.20) is a fundamental relation of the BCS theory. It is clear that for U0 > 0 there areno non-trivial solutions (∆ = 0 always), while for any U0 < 0 there is always a solution atfinite ∆ (One can also show that it corresponds to a minimum of energy. In the repulsivecase, the solution at ∆ = 0, which always exists, corresponds to a maximum). Passing tothe continuum limit we have
∆ = −U0
2
∫dp
(2π)3∆Ep
(5.21)
This integral is divergent in the ultraviolet. We turn to the question of how to regularize it inthe next section. More significant, however, is the dependence of the right hand side on ∆for small ∆. This is
∼ −U0
2ν(µ)∆ lnΛ/∆,
where Λ is the UV cut-off (we will shortly identify it with the Fermi energy). This shows thatno matter how small the attraction U0 < 0, there will always be a solution of Eq. (5.21) withfinite ∆. This is the essence of the Cooper phenomenon. It should be compared with thesituation in which there are not a macroscopically large number of particles present. In thatcase µ = 0, so that ξp = ε(p) > 0. Then the right hand side of Eq. (5.21) has no divergence
54
at low energies because the density of states ν(E) vanishes4. The finite value of ∆ is thusseen to be a consequence of the fermi sea.
What are the implications for the order of the system? In the BCS state
〈BCS|a†p↑a†−p↓|BCS〉 = u∗pvp.
Obviously this is a consequence of the non-conserving form of the wavefunction. But basedon our previous argument we have
〈N |a†p↑a†−p↓ap′↑a−p′↓|N〉 = u∗pvpup′v
∗p′ ,
which is equivalent to our statment of ODLRO, Eq. (5.1), with
Fp = upv∗p.
Thus we have demonstrated the presence of a fermi condensate for an attractive interaction,no matter how weak5. To be more honest, what we have really shown is that the BCS statealways has a lower energy than the free fermi problem. Since this certainly is the groundstate at U0 = 0, and the BCS is smoothly connected to it, it seems clear that the BCS statecan be trusted at least for small coupling. At larger coupling some other state of matter couldwin out, but given our argument that we should eventually arrive in a BEC state of molecules,which also has ODLRO of the form Eq. (5.1), this seems unlikely.
Problem 12 As well as the average occupancy of a given momentum state we can considerthe correlations between the occupancy of different p states
Css′(p,p′) ≡ 〈np,snp′,s′〉 − 〈np,s〉〈np′,s′〉 (5.22)
Show that for the BCS state
C↑↓(p1,p2) = δp1,−p2u2p1v2p1
= δp1,−p2
|∆|2
4E2p1
C↑↑(p1,p2) = δp1,p2
|∆|2
4E2p1
Interpret these two expressions.
5.2.2 The BCS-BEC crossover
Taking our argument one step further, we can argue that not only is the order specified byEq. (5.1) the same whether two fermions can form a bound state or not, but the wavefunction
4As should be clear, this picture in fact depends on dimensionality. In d ≤ 2, the form of the density of statesat low p means that a bound state will always form. It is only in three dimensions that a critical strength of U0 isrequired. In contrast, our discussion of the Cooper phenomena for µ > 0 is independent of dimensionality.
5The statement that pairing always occurs is strictly true only for exactly equal numbers of the two species,as we will see in Section 5.3 below
55
in the extreme molecular limit is of precisely the same form as the BCS wavefunction. Thiscan be seen from the first quantized form of the number conserving wavefunction Eq. (5.9)
|N〉 = AN/2∏i<j
ϕ(ri↑ − rj↓), (5.23)
where ϕp = vp/up. At weak coupling the extent of the ‘pair wavefunction’ ϕ(r) is largecompared to the separation between pairs. In this limit the antisymmetrization operation,required by the exclusion principle, plays a dominant role, as we have seen. When the pairwavefunction ϕ(r) has a much smaller extent than the typical separation between pairs, wecan expect that the anitsymmetrization operation in Eq. (5.23) is not too important, as twofermions of the same type rarely overlap. In this limit, any given momentum state has alow average occupancy, and the hardcore constraint Eq. (5.8) does not play a significantrole. Then Eq. (5.6) can really be thought of as a Hamiltonian for isolated pairs, with thecorresponding binding energy (see also the discussion at the end of the previous section).The resulting wavefunction is then essentially a Gross-Pitaevskii state of molecules, whichwas the picture that led us to suggest Eq. (5.1) in the first place. This suggests that we canuse Eq. (5.11) as a variational wavefunction all the way through the BCS-BEC crossover.
First we have to address the issue of regularizing Eq. (5.21). Suppose we had workedwith a finite-range interaction with fourier transform U0(p). It is easy to see that the onlyway that the equations of the previous section are modified is through the gap parameter ∆becoming momentum dependent
∆p = − 12V
∑p′
U0(p− p′) sin θp′ = −∑p′
U0(p− p′)up′v∗p′ , (5.24)
so that the self-consistent equation is now
∆p = − 12V
∑p′
U0(p− p′)∆p′
Ep′. (5.25)
The solution of Eq. (5.25) is in general very difficult, but it can be greatly simplified if weassume that the range of the potential U0(r) is the shortest length scale in the problem. Thisis of course the case for the systems in which we are interested. With this assumption wecan write Eq. (5.25) as
∆p = −U0(p)V
∑p′
[∆0
2Ep′− ∆0
2εp′
]− 1V
∑p′
U(p− p′)∆p′
2εp′. (5.26)
We have added a subtracted a term to the right hand side. The first term of the resultingexpression converges on a scale set by the momentum corresponding to the larger of ∆p
or µ, which by assumption is much less than the scale on which U0(p) varies, and justifiesreplacing U0(p− p′) with U0(p), and ∆p′ with ∆0. We now make the following ansatz for ∆p
∆p = − m
4π~2as∆0F (0,p),
56
in terms of the scattering amplitude F (p,p′) and scattering length as corresponding to thepotential U0(r). From the integral equation Eq. (2.18) satisfied by the scattering amplitudewe see immediately that
− m
4π~2as∆0 =
1V
∑p
[∆0
2Ep′− ∆0
2εp′
]. (5.27)
In the weak-coupling limit, the gap ∆ (we drop the subscript 0 from now on, as we never needto discuss the high momentum behaviour of ∆p again) is expected to be much smaller thanthe Fermi energy, and the chemical potential is just equal to the Fermi energy EF = p2
F /2m.The integral in Eq. (5.27) can then be done explicitly to give the gap
∆BCS =8e2EF exp
[− π
2|kFas|
](5.28)
Outside of the weak-coupling limit, we have to account for a change in the chemical potential,in order to keep a fixed density. This is apparent from the equation
N = 2∑p
〈nPp 〉 = 2
∑p
v2p =
∑p
[1− ξp
Ep
](5.29)
The two equations Eq. (5.27) and Eq. (5.29) are conveniently cast in the dimensionless form
π
2kFas=
∫ ∞
0dx
[1− x2√
(x2 − µ)2 + ∆20
]23
=∫ ∞
0x2
[1− x2 − µ√
(x2 − µ)2 + ∆20
](5.30)
where µ and ∆0 are measured in units of EF = p2F /2m, and the unit of length is p−1
F . Inthese units the total density of particles of both types is 1/3π2. The behavior of the gap andchemical potential is shown in Fig. 5.3. Recall from Section 2.3.3 that the point 1/kFas = 0,where the scattering length diverges, corresponds to the formation of a bound state. Thisis an interesting part of the phase diagram (sometimes called the unitary point), becausehere (if the temperature is zero) there is only one energy scale (the Fermi energy) and onlyone length scale (the fermi wavelength). All quantities such as ∆ and µ are simply someuniversal fraction of the Fermi energy. In particular the equation of state of the system is
E/V = α35EFn ∝ n5/3. (5.31)
The numerical factors are to emphasize the resemblance of the unitary gas to the free fermigas, where α = 1. The mean field theory above gives α = 0.59, while a recent Monte Carlocalculation found α = 0.44±0.01 [18]. There is of course no reason to believe the quantitativepredictions of the mean-field theory in the region where interactions are so strong.
Problem 13 Show that the in the BCS limit (1/kFa large and negative, ∆ EF , and µ =EF ), the variational enenergy Eq. (5.12) of the BCS state relative to the free fermi gas is
EBCS(∆)− EBCS(0) = −12ν(EF )|∆|2 (5.32)
where ν(EF ) = mpF
2π2 is the fermi surface density of states. Note that this arises from the sumof a large increase in the kinetic energy, and a large decrease in the pairing energy.
57
-2 -1 1 2
-1.5
-1
-0.5
0.5
1
1.5
1/kFa
∆0/EFµ/EF
Figure 5.3: Variation of gap and chemical potential. Also plotted is the result for the gap onthe BEC side
√16/(kFas)/π. Note the exponential dependence of the gap on the BCS side,
consistent with the analytic result Eq. (5.28)
Problem 14 When µ becomes large and negative for 1/kFa > 0, the angles θp in Eq. (5.16)are all close to 0, since cos θp = ξp/Ep. Expand Eq. (5.16) in small deviations from θp = 0and interpret the result.
Problem 15 In Section 2.3.3 we introduced the simplest two-channel model that displays aFeshbach resonance, Eq. (2.22)
H =∑p,s
εpa†s,pas,p +
∑q
(εq2
+ ε0
)b†qbq +
g√V
∑p,q
bqa†↑,q+pa
†↓,−p + h.c. (5.33)
Construct a mean-field theory to describe condensation in this model by replacing the oper-ator b†0 creating a bosonic molecular state a zero momentum with a c-number b∗0. Show thatthe self- consistent equation Eq. (5.27) is replaced by
ε0g2
=1V
∑p
[1
2Ep− 1
2εp
], (5.34)
and ∆ = gb0 = g√nbV, where nb is the density of b molecules. [Hint: you will need to shift
the detuning ε0 by an infinite constant as in Eq. (2.24) of Problem 4]Show that the number equation is then
n = nb +1V
∑p
(1− ξp
Ep
). (5.35)
Argue that in the limit of a broad resonance, where the parameter γ = g2m3/2/4π →∞, themean-field is equivalent to that considered previously. Note that in this limit the occupancyof the molecular state goes to zero. This demonstrates in the many-body context that broadresonances are adequately described by the single channel model.
58
5.2.3 Quasiparticle excitations
Like the Bogoliubov theory, this BCS theory also lets us discuss excitations out of the groundstate. We didn’t solve the BCS hamiltonian by a Bogoliubov transformation, as is often done,but we can introduce the Bogoliubov-type excitations after the fact. Recalling the BCS state
|BCS〉 =∏p
[vpa
†p↑a
†−p↓ + up
]|0〉, (5.36)
it’s easy to see that the operators
αp↑ = upap↑ − vpa†−p↓
αp↓ = upap↓ + vpa†−p↑, (5.37)
satisfy the canonical fermion anticommutation relations and annihilate the BCS stateαp,s|BCS〉 = 0. Consider the state
|p, s〉 = α†p,s|BCS〉 = a†p,s
∏p′ 6=p
[vpa
†p↑a
†−p↓ + up
]|0〉,
corresponding to the momentum state p certainly containing one particle with (pseudo-)spins, and the (−p,−s) state certainly being empty. The result is an eigenstate of momentumand spin, but is it an energy eigenstate, and thus a sharply defined excitation? Note that ifwe chose s =↑ so that the (p, ↑) state is certainly occupied it means that a†p↑a
†−p↓|p, ↑〉 = 0,
so that the corresponding term no longer appears in the interaction term when it is appliedto this state. The level is said to be ‘blocked’. Thus it certainly is an eigenstate of the pairproblem, if |BCS〉 is. What is its energy? We have to take into account the kinetic energy aswell as the loss of attractive interaction energy, see Eq. (5.16)
Es(p) = ξp[
(p,s) occupied︷ ︸︸ ︷(1− 〈nP
p 〉) (−p,−s) empty︷ ︸︸ ︷
−〈nPp 〉 ] +
‘blocking′︷ ︸︸ ︷∆ sin θp = Ep (5.38)
Note that these quasiparticle excitations always have a gap ∆s given by
∆s = minpEp =
∆, µ > 0,√
∆2 + µ2, µ < 0.(5.39)
As µ turns from positive to negative as we pass from BCS to BEC (see Fig. 5.3), the densityof states of the quasiparticle excitations turns from having a square root singularity ν(ε) ∼(ε−∆s)
−1/2 to vanishing like a square root ν(ε) ∼ (ε−∆s)1/2.
5.2.4 Effect of Temperature
When we discussed the quasiparticle excitations, we didn’t account for the effect that theyhave on the self-consistent equation. This is fine when there are few excitations in thesystem, but when many levels are blocked, we have to take this effect into account, leadingto a reduction in the gap parameter. The obvious example of this effect is a system at
59
finite temperature, where the quasiparticles, being fermionic, have the fermi-dirac distributionfunction
ns(p) =1
eβEs(p) + 1, β = 1/kBT, (5.40)
where we allow the quasiparticle energies, and hence distributions, to differ for the twospecies. Since the occupancy of a given state can be zero or one only ns(p) is also theprobability for that state to be occupied. Thus the probability of a (p, s;−p,−s) state beingblocked is ns(p) [1− n−s(−p)] + n−s(−p) [1− ns(p)]. What if we have both a (p, s) and a(−p,−s) quasiparticle present?
α†p↑α†−p↓|BCS〉 =
[upa
†p↑a
†−p↓ − vp
] ∏p′ 6=p
[vpa
†p↑a
†−p↓ + up
]|0〉, (5.41)
which is orthogonal to the BCS state. In the Anderson spin language this corresponds to asingle flipped spin, and has energy given by twice the ‘magnetic field’ each pair experiences2E(p), see Eq. (5.19). The spin interpretation allows us to see easily that this state con-tributes −upvp to the self consistent equation, and occurs with probability n↑(p)n↓(−p). Theoverall result is thus
∆ =U0
V∑p
∆2Ep′
(
no qp︷ ︸︸ ︷[1− n↑(p)][1− n↓(−p)]−
two qp︷ ︸︸ ︷n↑(p)n↓(−p))
=U0
V∑p
∆2Ep′
[1− n↑(p)− n↓(−p)]. (5.42)
For the case Es(p) = E(p), and in the BCS limit where µ ∼ EF , ∆ EF , this is convenientlypresented in the form
1U0ν(EF )
= −∫
∆dE tanh
(E
2T
)∆√
E2 −∆2,
where ν(EF ) is the fermi surface density of states, and we have left out the upper cut-off.This expression needs to be regularized as in the zero temperature case considered before.The resulting ∆(T ) varies from ∆(0) = ∆BCS given by Eq. (5.28), to zero at
kBTc =γ
π∆BCS. (5.43)
The key point is that the variation is smooth, suggesting a second order transition to a nor-mal (non- superfluid) state above Tc. It looks like we have gone some way to verifying theschematic phase diagram that we sketched in Fig. 5.1, with a small but finite transition tem-perature on the ‘weak’ side (now identified with 1/kFa < 0). Unfortunately there is a problem.If we repeat the calculation throughout the crossover, the temperature at which a non-zero∆ develops continues to increase indefinitely on the BEC side [19]
kBTc ∼ Eb/2 [lnEb/EF ]3/2 , 1/kFa 0 (5.44)
where Eb = 1/ma2 is the binding energy of a pair. This temperature should be thought ofas a dissociation temperature below which pairs can form (the logarithmic factor is entropicin origin). The temperature at which these pairs condense will however be lower, tending tothe ideal bose TBEC in the 1/kFa → ∞ limit. A more sophisticated treatment is required tofind a smooth interpolating Tc(1/kFa), see Fig. 5.4.
60
Figure 5.4: Tc in the mean-field theory (solid line), compared with the result of a treatmentthat smoothly interpolates to TBEC (dashed). Reproduced from Ref. [19]
5.3 The effect of ‘magnetization’
We turn now to the effect of having differing densities of the two species of fermions. Thisproblem has a long history in the theory of superconductivity, where it corresponds to themagnetization of the system of electron spins in an external field. There, the Meissnereffect and other complications originating from the orbital effect of the magnetic field onthe electrons often dominate the behaviour of the system. A neutral gas is free of suchcomplications, and moreover achieving finite ‘magnetization’ is straightforward
To study this situation we introduce a conjugate variable h that couples to the imbal-ance in number between the two species, just as the chemical potential couples to the totalnumber6
Hµ,h = H − µ [N↑ +N↓]− h [N↑ −N↓] .
The resulting free energy F (µ, h) gives the thermodynamic relations
− ∂F
∂µ= N ≡ N↑ +N↓, −∂F
∂h= M ≡ N↑ −N↓, (5.45)
but in fact it is possible to write down the complete set of equations that we need withoutcomputing F (µ, h). Since the quasiparticles carry spin, it is natural that the energy Es(p) is
Es(p) = Ep − sh, (5.46)
6In studies of the BCS-BEC crossover in 6Li, a Feshbach resonance between the |F, mF 〉 = |1/2,±1/2〉states has been used, whereas for 40K, the two states are |9/2,−9/2〉, and |9/2,−7/2〉. Thus the gases reallyare magnetized at finite species imbalance. We stress, though, that the magnetic field h is simply a conjugatevariable. The real magnetic field is varied to tune through the resonance, see Section 2.3.3.
61
and this energy appears in the equilibrium quasiparticle distribution function Eq. (5.40). Thuswe have
− m
4π~2as=
1V
∑p
[1
2Ep′[1− n↑(p)− n↓(−p)]− 1
2εp
]N =
∑p
2v2p +
(u2p − v2
p
)[n↑(p) + n↓(−p)]
M =∑p
n↑(p)− n↓(p). (5.47)
This is a complete set of equations to study the problem of finite M , although their analysismay be numerically quite difficult. The interpretation of the equation forN , which we have notseen before, is actually straightforward. With no quasiparticles we get a contribution 2v2
p, theoccupancy of the paired state as before. With one quasiparticle we get a contribution of one(one state of the pair (p,−p) occupied), and with two a contribution of u2
p, see Eq. (5.41).
5.3.1 Sarma state
If h is positive, only ↑ quasiparticles may exist, and then only if h > ∆s. In the BCS limit atT = 0 the analysis is simple enough to be done analytically. If h > ∆ quasiparticle states areoccupied for |ξp| >
√h2 −∆2. By taking the difference between the self-consistent equation
at h = 0 (solved by ∆BCS) and finite h, we get the condition
0 = ln
[h+
√h2 −∆
h−√h2 −∆
∆BCS
∆
], (5.48)
with solution∆Sarma
∆BCS=
√2h
∆BCS− 1. (5.49)
Thus for ∆BCS/2 < h < ∆BCS there are three solutions of the self-consistent equation: ∆ = 0(always a solution, though we have often divided through by ∆), ∆BCS, and ∆Sarma. It isuseful to think about what this means for the variational energy of the system as function of ∆,see Fig. 5.5, and one quikly concludes that the appearance of a new solution at h = ∆BCS/2correpsonds to the maximum at ∆ = 0 turning to a minimum, with a maximum moving awayto finite ∆Sarma. Note that for ∆ > h, the potential coincides with its zero field value. Thuswe can use the result Eq. (5.32) for the condensation energy to determine when the ∆ = 0minimum becomes the global minimum of the potential, assuming this occurs for h < ∆BCS.The free energy density of the magnetized normal system is
E(0, h)/V = −ν(EF )h2, h EF
so that for h > hc = ∆BCS/√
2, the system makes a first order transition to a normal fermigas, with magnetization density
mN = −∂E(0, h)/V∂h
∣∣∣hc
= 2ν(EF )hc
We have already argued in Section 5.2.4 that the transition to the superfluid state at finitetemperature is second order at h = 0. Together with the result just found at zero temperature,
62
Figure 5.5: Schematic plots of ground state energy versus ∆ for different h.
we are led to the conclusion that at some point on the superfluid-normal phase boundary inthe (T, h) plane, the transition changes from being first order to second order: a tricriticalpoint, see Fig. 5.6
Since we are interested in an experimental situation where m is fixed and not h, it is im-portant to realize that values ofm in the region 0 < m < mN cannot be accommodated withina single phase, see Fig, 5.7. At finite temperature, the superfluid state can be magnetized,as thermally excited quasiparticles will be principally of ↑ type, but beneath the tricritical pointthere will still be a jump from some mS to some larger mN at a critical hc. The result is phaseseparation, leading to two coexisting but spatially separated regions, one of superfluid withm = mS , the other a magnetized normal fluid. It is straightforward to show that the fraction pof superfluid is
p =mN −m
mN −mS, mS < m < mN
5.3.2 Magnetization in the BCS-BEC crossover
Now we discuss in a completely qualitative way what happens as we pass through the BCS-BEC crossover [21]. The main difference from the analysis of the BCS limit is that it ispossible to have h > ∆s, so that there is a finite concentration of quasiparticles in the su-perfluid, before the normal state becomes favoured. Physically, we expect that on the farBEC side we can have a coexisting mixture of N↓/2 bosonic molecules with the remainingM unpaired ‘majority’ fermions, see Fig. 5.8.
At finite temperature, the phase diagram will in general have two tricritical points, one oneach side of the crossover, which meet near the unitary point at some temperature of orderof the Fermi energy.
63
Figure 5.6: The tricritical point in the (T, h) plane. Reproduced from Ref. [20]
Figure 5.7: Magnetization versus field at zero (black line) and finite temperature (red line),and the resulting region of phase separation (PS) in the (T,m) plane.
64
a) b)
Figure 5.8: a) Schematic phase diagram of the zero temperature BCS-BEC crossover withmagnetization. The red dot marks a tricritical point that separates the trivial second-orderphase transition when the number of pairs N↓/2 goes to zero from the phase separatedregion. Point A is where the normal state becomes fully polarized, and point B is where thefree energy of the superfluid state with gap ∆s(kFaS) is equal to that of the normal state inmagnetic field h = ∆s. b) The same, but at finite temperature. Now the second tricriticalpoint on the BCS side is visible.
65
Chapter 6
Hydrodynamics of condensates
6.1 Galilean invariance
Let us go back to the time-dependent Gross-Pitaevskii theory describing the dynamics of acondensate [
− ~2
2m∇2 + Uext(r) + U0|Ψ(r, t)|2
]Ψ(r, t) = i~
∂Ψ(r, t)∂t
, (6.1)
and ask what happens when we pass to a reference frame moving with relative velocity −vto the original frame. One can verify directly that if Ψ(r, t) is a solution of Eq. (6.1) then
Ψ(r− vt, t) exp[i
~
(mv · r− 1
2mv2t
)], (6.2)
is also a solution. Eq. (6.2) gives the transformation property of the condensate wavefunctionunder Galilean transformations. Upon making the decomposition Ψ =
√neiϕ, we see that
this implies a transformation law for the phase of the condensate
ϕ→ ϕ+1~
(mv · r− 1
2mv2t
). (6.3)
Recalling the identifications1
vs =~m∇ϕ µ = −~ϕ,
we have the transformation laws for the superfluid velocity and chemical potential
vs → vs + v µ→ µ+12mv2, (6.4)
which seems sensible. The general character of these transformation laws leads us to sup-pose that they are more general that the equation of motion Eq. (6.1) that we started from.Indeed, we could have made exactly the same arguments for the equation of motion of theBose field φ(r, t), valid for arbitrary density-density interactions.
1in the lab frame
66
6.2 Hydrodynamic description
We can make the hydrodynamic character of the time-dependent GP equation more explicit.If we rewrite ∂|Ψ|2/∂t as the difference of Eq. (6.1) and its complex conjugate we get thecontinuity equation
∂n
∂t+∇ · [nvs] = 0, (6.5)
while Ψ∗Ψ− Ψ∗Ψ, which involves the sum of the equation and its conjugate, yields
~ϕ+12mv2
s −~2
2m√n∇2√n+ U0n+ Uext = 0. (6.6)
If n varies sufficiently slow in space – in practice this means slower than the healing lengthξ ≡
(2mnU0
~2
)−1/2– we can drop the ∇2√n term. Taking the gradient of the resulting equation
gives
mvs +∇(
12mv2
s + µ(n) + Uext
)= 0, (6.7)
where we wrote µ(n) = nU0. For the static case we have
µ(n(r)) + Uext(r) = µ0, (6.8)
which defines the Thomas-Fermi approximation, widely used to compute density profiles oftrapped gases.
The equation Eq. (6.7) coincides with the Euler equation for the flow of a non-viscousfluid, usually written as
ρ [∂t + v · ∇]v +∇P = 0,
where we used ρ = mn, the mass density, and v∧∇∧v = ∇(v2
)/2−(v · ∇)v. The pressure
gradient is introduced through ∇P = n∇µ, which follows from the Gibbs-Duhem relation.One should not forget, however, that the velocity field is in the present case irrotational. Notethat in the one-dimensional case that we will consider in Chapter 7 there is no vorticity, sothis difference disappears.
It is illuminating to consider the linearization of the equations Eq. (6.5) and Eq. (6.7). Inmany ways the analysis is clearer than the linearization of the TDGP theory considered inSection 4.1.2. Assuming that n(r, t) = n0 + δn(r, t), and that vs is of the same order as δn,one can easily derive the wave equation
δn− c2s∇2δn = 0,
with c2s = n0U0/m giving the sound velocity that we found before. Obviously this descriptionis not Galilean invariant. The requirement that vs be small amounts to a choice of referenceframe. What happens if you include the ∇
√n term in Eq. (6.6)?
As in the previous section, Eq. (6.5) and Eq. (6.7) are far more general than the weakinteraction limit of the TDGP theory, and even apply to situations in low-dimension wherethere is no condensate. For instance, Eq. (6.7) could have been obtained by taking the timeand spatial derivatives of the transformation law Eq. (6.3).
67
6.3 Quantum Hydrodynamics
So far, our hydrodynamic description has been purely classical. Now we show how, withoutrecourse to microscopic theory, this description can be quantized to reveal some genericfeatures of interacting condensates.
6.3.1 Hamiltonian and commutation relations
Our starting point is the hydrodynamic equations (continuity and Euler) that we found previ-ously
∂n
∂t+∇ · [nvs] = 0,
mvs +∇(
12mv2
s + µ(n))
= 0, ∇2√n√n/ξ2. (6.9)
(we have dropped the external potential). This set of equations can be obtained from theHamiltonian
H =∫dr
[12vρv + V (ρ)
], (6.10)
together with the commutation relation[ρ(r), ϕ(r′)
]= iδ
(r− r′
)(6.11)
where v = ∇ϕ (I am setting ~ = m = 1 from now on, so that the mass density ρ equals thenumber density n), and µ = dV/dρ. The Hamiltonian Eq. (6.10) is just what one would expectfor a fluid, only with an irrotational velocity v = ∇ϕ. Let’s check the continuity equation
ρ = i [H, ρ] =i
2
∫dr vρ [v, ρ] + [v, ρ] ρv
= −12∇ · (vρ+ ρv) = −∇ · ρv, (6.12)
and you should also check that the Euler equation is reproduced. The commutation relationEq. (6.11) plays a fundamental role in the following development, and follows from the fun-damental quantum commutator. It can be obtained from the microscopic expressions for thedensity and current
ρ(r) =∑
i
δ(r− ri)
j(r) =12
∑i
piδ (r− ri) + δ (r− ri)pi pi = −i∇i, (6.13)
from which one readily obtains[j(r), ρ(r′)
]= −iρ(r)∇rδ(r− r′).
If we use j = (ρv + vρ) /2, then the commutation relation for ϕ(r) follows immediately.
68
6.3.2 Mode expansion
It is convenient to re-write the commutation relation in terms of the Fourier modes of thefields. Introducing the canonical boson operators βk, β†k, with
[βk, β
†k′
]= δk,k′ , the commu-
tation relations are consistent with
ρ(r) = ρ0 +√ρ0
∑k
e−κkβkeik·r + h.c.
ϕ(r) =√
14ρ0
∑k
−ieκkβkeik·r + h.c., (6.14)
where κk is for now a free parameter that will be fixed by some dynamical input (i.e. theHamiltonian). We now write the Hamiltonian Eq. (6.10) as
H =∫dr
12ρ0v2 +
12U0(ρ− ρ0)2. (6.15)
At this point we have approximated the full Hamiltonian by a quadratic one. This assumesthat v and ρ − ρ0 are small, and is equivalent to the linearization of the equations of motiondiscussed at the end of Section 6.2. In a narrow sense we could think of the interactionterm as the usual quadratic approximation from the Gross-Pitaevskii theory, where U0 is amicroscopic interaction parameter (or pseudopotential, at the next level of sophistication).The Hamiltonian Eq. (6.15) is more general, however, and if the scale of variation of ρ ismuch larger than the interparticle separation, U0 can be thought of as V ′′(ρ0), with V (ρ) thepotential energy density of the fluid. In general this has a more complicated relationship tomicroscopic parameters. We will see how this works when we discuss the Tonks gas.
Substituting the mode expansion Eq. (6.14) into the Hamiltonian Eq. (6.15) yields aquadratic form in the operators βk, β
†k, which in general contains β−kβk terms and their
complex conjugates. A judicious choice of κk sets such terms to zero and yields (aside froman infinite constant)
H =∑
kρ1/d0
cs|k|β†kβk,
e−2κk =~|k|2mcs
, c2s =nU0
m(6.16)
Of course, this is recognizable as nothing more than the Bogoliubov transformation in an-other guise (we have restored the units of mass for familiarity’s sake). In line with the abovediscussion, we have we have included only low wavevectors in Eq. (6.16).
6.3.3 Correlation functions
To see the power of this approach, let’s consider the density matrix of the bosons. Using thedensity-phase representation, we write the boson operator as b(r) =
√ρ(r)eiϕ(r). Assuming
that the phase-phase correlations are sufficiently small, we can expand the exponents toobtain for the density matrix
ρ(r, r′
)= 〈b†(r)b(r′)〉 ∼ ρ0
(1− 1
2〈(ϕ(r)− ϕ(r′)
)2〉), |r− r′| → ∞ (6.17)
69
It turns out that in d > 1 this expansion is a reasonable thing to do, because
12〈(ϕ(r)− ϕ(r′)
)2〉 → −cdmcs
ρ0|r− r′|d−1|r− r′| → ∞,
(the constant cd depends on dimension) giving for the momentum distribution
n(p) = noδp,0 +mcs2|p|
, |p| → 0.
We see that the effect of interactions is to give rise to a ground state in which some particleshave been removed from the zero-momentum state. The effect is called the quantum de-pletion of the condensate, and is more commonly discussed in the context of Bogoliubov’stheory of the weakly interacting gas. The advantage of that approach is that the full wavevec-tor dependence of quantities can be calculated, not just the k ξ−1 asymptotes. In this wayone can show that the total depletion is
1N
∑p
n(p) =8
3√π
√na3
s, (6.18)
where we used the Born approximation for the scattering length as = 4π~2U0m . Under typical
experimental conditions the depletion does not much exceed 0.01, which justifies the use ofthe GP approximation.
The nice thing about the present discussion, however, is that it is not restricted to weakinteractions. As we’ll see in the next chapter, it applies even when the condensate is totallydepleted.
70
Chapter 7
Strong correlations: low dimensionsand lattices
The realm of strong interactions is interesting in its own right, providing a challenge to many-body theorists. But strongly correlated lattice systems also offer the promise of providingextremely clean realizations of the lattice models of traditional condensed matter physics,which may in turn lead us to new insights into real solids.
A natural question is: how generic is Bose-Einstein condensation for a system of boseparticles? Can anything else happen as we move to zero temperature? In this chapter wedescribe two situations in which the quantum depletion examined previously can be total,leading to the destruction of the condensate.
7.1 Bose fluids in one dimension: the Tonks gas
So far, we have worked in the limit of weak interactions, where energy scales such as thechemical potential are simply proportional to U0 (µ = nU0). What happens as interactionsbecome strong? There is no general answer to this very difficult question, but one situationwhere progress is often possible is in one dimension 1. In fact, the case of δ-function inter-actions that we have been considering can be exactly solved in one dimension to yield thewavefunction for the ground and excited states. We will briefly discuss the character of thissolution. Firstly, notice that we can form the dimensionless parameter
γ ≡ mU0
~2n,
because the density n has the units of inverse length in one dimension, while the strength ofthe δ-function has units [energy]× [length]. Our earlier result for the energy density at smallU0 goes through as before and can be written
E/Ω1 = U0n2/2 = ~2n3γ/2m.
In general then, the energy density will be of the form E/Ω1 = n3~2e(γ)/2m, where thefunction e(γ) is shown in Fig. 7.1. Notice that e(γ →∞) = π2/3. How can we understand thisresult? Surprisingly, the wavefunction has a simple form in this limit, which can be obtained
1In a sense, the interactions are always strong in one dimension, see Problem 16.
71
Figure 7.1: The curve shows the function e(γ) obtained in Ref. [22] for the δ-function Bosegas in 1D, in terms of which the energy is E = Nn2~2e(γ)/2m.
by noticing that for infinitely repulsive interactions the many-body wavefunction has to vanishwhenever the coordinates of two particles coincide. When none coincide, the wavefunctionsatisfies the free Schrodinger equation because of the δ-function nature of the interaction.There is an obvious class of wavefunctions that satisfy both of these properties, namely theSlater determinants. The drawback that these function are completely antisymmetric, ratherthan symmetric as dictated by bose statistics, is readily solved by taking the modulus. Thusany eigenstate of the γ →∞ problem can be written
ΨB(x1, · · · , xN ) = |ΨF (x1, · · · , xN )|, (7.1)
where ΨF (x1, · · · , xN ) is an eigenstate of a system of non-interacting fermions. Such a one-dimensional system of impenetrable bosons is known as a Tonks-Giradeau gas. Let’s checkthis idea by calculating the ground state energy. If the fermi gas has fermi wavevector kF ,the total energy density is
E/Ω1 =∫ kF
−kF
dk
2π~2k2
2m=
~2k3F
6πm=n3~2
2mπ2
3,
using n = kF /π for the density. The chemical potential is ∂E/∂N = ~2k2F /2m ≡ EF and the
hydrodynamic speed of sound is
c2s =n
m
∂2(E/Ω)∂n2
, cs =~kF
m≡ vF . (7.2)
Another very useful feature of the fermion mapping is that it allows us to calculate any ob-servable that depends on the local density operator ρ(r) = b†(r)b(r), as the modulus inEq. (7.1) doesn’t interfere with such a calculation.
72
Problem 16 (Momentum distribution in a 1D Bose fluid) Using the harmonic Hamilto-nian Eq. (6.15)
• Show〈0|ρqρ−q|0〉 =
N~|q|2mcs
, |q| → 0, (7.3)
consistent with Onsager’s inequality.
• Show that in 1D the behaviour of the phase correlation function is such that the fullexponential has to be retained in Eq. (6.17) to give
n(p) ∝ 1
|p|1−1η
, |p| → 0, η ≡ 2π~nmcs
In particular, this result tells us that there is no condensate in a one-dimensional sys-tem.
Problem 17 (Structure factor and momentum distribution for the Tonks gas) Let’scompute some properties of the Tonks gas using the fermionic mapping Eq. (7.1).
• First show that the dynamical structure factor defined in Eq. (4.20) is
S(q, ω) =
Nm
2~qpF
∣∣∣ qpFm − ~q2
2m
∣∣∣ < ω <∣∣∣ qpF
m + ~q2
2m
∣∣∣0 otherwise
(7.4)
• Check that this is consistent with the sum rules Eq. (4.21) as well as the inequalityEq. (4.23), with cs = vF ≡ pF /m. What is the value of η, and the resulting n(p), for theTonks gas? Note that deriving this result for the momentum distribution is hard to dostarting from the wavefunction. In particular, it does not coincide with the momentumdistribution of a fermi gas.
7.2 Lattice systems
7.2.1 Optical lattices
Another way in which the rather weak interactions between atoms in a dilute gas can bemade strong is by quenching the kinetic energy by confining them to an optical lattice. Suchoptical potentials are created by the interaction between the oscillating electric field of a laserand the electric dipole moment it induces in an atom. Thus the strength of resulting potentialis proportional to the square of the field
Vopt. = −α(ω)|Eω(r)|2,
where α(ω) is the polarizability of the atom. In particular, when the polarizability is dominatedby a single atomic transition of angular frequency ωn0 we have
α′(ω) ∼ |〈n|d · ε|0〉|2
~ (ωn0 − ω), (7.5)
73
Figure 7.2: Counterpropagating lasers form a) a two-dimensional and b) a three-dimensionallattice. Reproduced from the recent review Ref. [23].
(ε is the direction of E) and the sign changes from positive (attractive potential) to negative(repulsive) as we go from ω < ωn0 (red detuning) to ω > ωn0 (blue detuning).
By superimposing counterpropagating lasers one can form an optical standing wave withperiod λ/2 that generates a periodic potential. In this way one can create a one-, two- orthree-dimensional lattice, see Fig. 7.2. The quantum states of a particle propagating in aperiodic particle are the Bloch waves, characterized by (pseudo-)momentum p and bandindex n
Ψn(p) = eip·rϕn(r), ϕ(r + ai) = ϕ(r)
where ai are the lattice vectors. If we are concerned only with low energies in the lowestband, the following tight-binding model is an adequate description of the kinetic and latticeparts of the Hamiltonian
Htb = −t∑〈ij〉
b†ibj . (7.6)
(where 〈ij〉 denotes nearest neightbours – we are thinking of the one-dimensional case).This can be diagonalized to give the dispersion relation ε(k) = −2t cos k with bandwidth4t. With the experimental parameters such as lattice period and depth in the hands of theexperimentalist, t can be made small so that we enter the regime of small effective mass andstrong interactions.
74
7.2.2 The Bose-Hubbard model
Adding the simplest on-site interactions to the tight-binding Hamiltonian Eq. (7.6) gives theBose Hubbard model
HBH = −t∑〈ij〉
b†ibj + (U/2)∑
i
ni(ni − 1). (7.7)
Given the short-range character of interatomic interactions this is in fact a very good de-scription of the result of confining bosonic atoms to an optical lattice. One could discussthe behaviour of this model starting from small U/t using the Bogoliubov theory (if we hadcovered it). This would tell us that, as in the case of bosons in free space, the condensatefraction is depleted with increasing interaction strength: in this case the relevant parameteris U/t.
A simpler approach is to start in the limit of strong interactions with U/t→∞. In this casewe can neglect the hopping term in Eq. (7.7), so that the Hamiltonian becomes a sum of on-site Hamiltonians. After including the chemical potential, the free energy at zero temperatureis minimized by a state
∏i
(b†i
)n|0〉, with integer n particles on each site and
n = [µ/U + 1/2], (7.8)
where the square brackets [· · · ] denote the nearest integer. What happens when t/U 6= 0?Let’s fix µ at a value corresponding to n bosons per site, so that µ/U = n − 1/2 + α,with −1/2 < α < 1/2. Then there is an energy (1/2∓ α)U to add or remove a par-ticle (add a hole). An added particle (or hole) can hop freely, giving a contribution tothe energy of order −t (recall the dispersion relation of the model Eq. (7.6)). Thus ift . min ((1/2 + α)U, (1/2− α)U), we expect that no extra particles will be added or re-moved from the ground state in some finite region of t/U = 0, provided we stay away fromthe degeneracy points where α = ±1/2. For larger values of t, particles enter the groundstate and (presumably) condense to form a BEC.
We can make this argument more precise by noting that the minimum hopping energy ofa particle or hole is −2td in d-dimensions. Thus we expect the asymptotic phase boundaries
tc(µ) ∼ µ
2d(1 +O(t/U)) −∞ < µ < U/2
with corrections that are higher order in t/U . This result is readily generalized to the stateswith larger numbers of particles where the hopping energy to add a particle when there arek particles per site already, or a hole when there are k + 1, is −2td(k + 1), leading to
tc(µ) ∼
µ2d (1 +O(t/U)) −∞ < µ < U/2|µ−kU |2d(k+1) (1 +O(t/U)) (k − 1/2)U < µ < (k + 1/2)U, k ≥ 1.
(7.9)
In this way we arrive at the schematic phase diagram in Fig. 7.3. The states with fixed numberthat prevail at low hopping are characterized by vanishing compressibility −∂〈n〉/∂µ andin condensed matter physics are known as Mott insulators, owing their incompressibility tointerparticle interactions. The transition between this state and the condensate is sometimescalled the superfluid-insulator transition. Its observation in a gas of 87Rb in 2002 [25] is whatreally made the community wake up to the potential of cold gases for doing fundamentalcondensed matter physics.
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Figure 7.3: Schematic phase diagram of the Bose Hubbard model, including the estimateEq. (7.9). The regions marked ρ = 0, 1, 2, etc. are the Mott phases with different fillings.Reproduced from Ref. [24].
Problem 18 (More phases of the Bose-Hubbard model) In the lectures we discussed thephase diagram of the Bose Hubbard model Eq. (7.7) as a function of chemical potential andU/t.
• Qualitatively, what happens when we add an additional term describing repulsion be-tween neighbouring sites?
H ′BH = −t
∑〈ij〉
b†ibj + (U/2)∑
i
ni(ni − 1) + V∑〈ij〉
ninj , (7.10)
Hint: as in the lectures, start by considering the case of zero hopping. See Ref. [26] formore details.
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