1 Chaire Européenne du College de France (2004/2005) Sandro Stringari Lecture 9 18 Apr 05 BEC in periodic potentials Previous Lecture. Ultracold Fermi gases Ideal Fermi gas in harmonic trap. Role of interactions. BCS-BEC crossover. Unitarity and universality. Effects of superfluidity. This Lecture Momentum distribution and interference. Bloch oscillations. Josephson oscillations. Superfluid vs insulator phase. Lectures and seminars available at: http://www.phys.ens.fr/cours/Sandro/index.html
Transcript
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Chaire Européenne du College de France (2004/2005)
Sandro Stringari
Lecture 918 Apr 05
BEC in periodic potentialsPrevious Lecture. Ultracold Fermi gasesIdeal Fermi gas in harmonic trap. Role of interactions. BCS-BEC crossover. Unitarity and universality. Effects of superfluidity.
This LectureMomentum distribution and interference. Bloch oscillations. Josephson oscillations. Superfluid vs insulator phase.
Lectures and seminars available at: http://www.phys.ens.fr/cours/Sandro/index.html
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Ultracold gases in periodic potentials (optical lattices)
Periodic potentials are produced by two counter propagating laser beamsgiving rise to standing wave of the form
Time averaged effective fieldtakes the form
dipole olarizability
(natural extension to 2D and 3D periodic potentials)
..cos),( ccqzEetrE ti += − ω
><−= ),()()2/1()( 2 trErVopt ωα
qzErVopt22 cos)()( ωα−= ≡)(ωα
Ideal crystal-like systems: no impurities, possibility of tuning depth of the potential
New physics in the presence of periodic potentials. Some examples:
- Without interaction (spin polarised Fermi gas or dilute Bose gas):Interference in momentum distribution, Bloch oscillations
- With interactions: Josephson oscillations, dynamic instabilities,superfluid-Mott insulator transition.
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Ideal Bose gas in 1D optical potential
qzsEzV ropt2sin)( =
Order parameter can be written aswhere sum extends to wells. Each well is occupied by atoms. If s>>1 the potential can beexpanded harmonically around each minimum, yielding
)()( ldzfzl
−=Ψ ∑0N
0/ NN
4/122
2/14/1 ;)2/exp(1)(sdzzf
πσσ
σπ=−=
d
- recoil energy, periodicity d fixed by laser wavelength
- dimensionless parameter, fixed by laser powerand by atomic polarizability (detuning)
222 2/ mdEr πh=
s
4In this lecture:k = momentump = quasi-momentum
Momentum distribution
∫ ∑∑ −− =−=Ψ
Ψ=
ildkl
ikzl
ekfldzfdzek
kkn
)()(21)(
)()(
0/
2
h
hπ
)2/exp()(
)2/(sin)2/(sin)()(
2222/14/1
2/1
0
2
22
0
hh
h
h
σπσ kkf
kdkdNkfkn w
−=
=
Momentum distribution is characterized by series of peaks locatedat , and width
n(k) is modulated by the momentum distribution of the Wannier function f.
ZdNw // hh =dnk /2 hπ=number of wells
..2,1,0 ±±=n
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Momentum distribution
size of the atomic cloud
periodicity of the lattice
width of wf in each well
Bragg momentum
σd
dNZ w=
dqB /πh=
...2,1)/4 2222 =− ndn σπexp(relative weightof each peak
-3 -2 -1 0 1 2 3
k/qB
d/1Z/1
σ/1
Pedri et al. 2001
exp
- After release of the trap atomsoccupying lateral peaks expandfast according to law
- Since expansion is welldescribed by ideal gas model
rE<µ
dmnttzk /2)( hπ=
)()(),(..1,0
ldzfrrzMlll −Ψ=Ψ ∑
±±=⊥⊥Ansatz for order parameter:
( ) ∫∑∫∑∫
∫∑∫
Ψ+Ψ+
Ψ
+∂=
⊥⊥⊥
⊥
244
2222
),(2
)(2
lhol
ll
ll
optz
ldrVdrdrdzfg
drVffm
dzE h
∫ =12dzf
rE<µ
- Interactions and harmonic trapping do not changesignificantly the mechanism of the expansion of lateral peaks provided
-They instead determine occupation number of atomsin each well and hence shape of density distribution.
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Role of interactions and harmonictrapping in 1D periodic potentials
Energy of the system in mean field (Gross-Pitaevskii) theory is given by(neglecting small overlap between condensates of different wells (s>>1))
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Minimization of energy with respect to yieldslΨThomas-Fermi profile:
−−=Ψ ⊥⊥⊥
222222
21
21
~)( rmdlmgdr zl ωωµ
with effective coupling constant
and renormalized chemical potential ( )
∫ ==~ 4 gddzfgdg
15 = ho Nωµ h πσ2/~ ada =
πσ2
5/2~
21
hoaa
Shape of density profile, after averaging over distance d separatingtwo consecutive wells, preservestypical inverted parabola form
−−=
⊥
⊥⊥ 2
2
2
2
1~),(Zz
Rr
gzrn µ
Size of condensate increases according to(enhancement of repulsive effectproduced optical trapping)
==⊥
5/1~
)0()(
)0()(
⊥ gg
ZsZ
RsR
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Effect of periodic potential on density profiles
- in situ density profiles exhibit increase in size (effect is small withoptical lattices of moderate intensity (s=3-5))
- modulations of in situ density are too narrowto be measurable ( )
- density profiles after expansion show pronounced and measurablepeaks due to interference effects in momentum distribution.
md µ4.0≈
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Energy bands and Bloch oscillations
Ground state solution can be generalized to solutions carryingquasi-momentum p. Look for stationary solutions (Bloch solutions):
)()( / zuez pipz
ph=Ψ with periodic function ( )pu )()( dzuzu pp +=
Gross-Pitaevskii equation for order parameter yields:
)()()()]()([)(2
222
zupzuzVzugzupidzd
m ppoptpp µ=++
−−
h
h
dpd // hh ππ ≤≤−
10
8
−1 −0.5 0 0.5 10
2
4
6
NEpEp )0()()(0
−=ε
Value of p restricted to first Brillouin zone
for each value of p several solutions (band structure)
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From solution of GP equation one can define energy per particle (Bloch spectrum): N0
EpEp )0()()( −=ε
Without lattice
p-dependence of Bloch spectrum can be calculated by solving GP equation. Analytic solution for lowest band is available in ideal gas forlarge intensity s (tight binding limit):
mpp 2/)( 20 =ε
In general at low quasi-momentaone can expand: *2
0 2/)( mpp =ε
h2/sin2)( 20 pdp Jδε =
is tunneling energyrelated to effective mass 2*
2
dmJh
=δ
*m
( ) lextkJ Vmrd ϕϕδ ∫ +∇−−= 22 2/2 hr
increases exponentially at large s
mm /*
s
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Bloch oscillations (periodic motion ofwhole system along the Bloch band)
Bloch oscillations can be produced via:
- acceleration of the lattice, through time dependent detuning of twocounter propagating laser beams
- switch on of weak uniform force (e.g. gravity)
)2/(sin)( 2 tqzsEzV ropt ω∆−=
Two competing conditions:
- Acceleration should be slow (adiabaticity condition) in order to avoidtransitions to higher energy bands.
- Acceleration should be fast to avoid dynamic instability (see later).
Acceleration of the lattice:12
ω+δω, k+δk
tqaqv LL 22 ==∆ω velocity of the lattice at time t Lv
ω, -k
)(pvvv gLm +=
Current carried by the gas in Lab frame: where is the current relative to the lattice. By defining(measurable by imaging expanding atomiccloud after release of the lattice), one finds:
ω, -kRelationship between velocity of the lattice and quasi-momentum p isfixed by adiabaticity condition, yielding
Lmvp −=
In the figure is Braggvelocity, fixed by periodicity of theperiodic potential (Morsh et al 2001)
mdvB /πh=
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Behaviour of momentum distributionduring the acceleration of the lattice
))(())((),( /)(/)()(
/)( ldtzfeetzetz ldtipl
ztimvtp
ztimvLAB
LL −=Ψ=Ψ ∑ hhh
quasi-momentum
)()( / ldzfez ipldlp −=Ψ ∑ h
Consider simplest case of ideal gas filling wells, each occupied with equalnumber of atoms. By writing one can easilycalculate momentum distribution starting from:
)2/exp()(
)2/(sin)2/(sin)()(
2222/14/1
2/1
0
2
22
0
hh
h
h
σπσ kkf
kdkdNmvkfkn w
L
−=
−=
momentum
Acceleration of the lattice doesnot modify position of peaks. it affects their modulation
-2 -1 0 1 2
-2 -1 0 1 2
k/qB
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Bloch oscillations produced by external force
Initial condition: - atomic gas trapped by harmonic + gravity+ periodic confinement- at t=0 one switches off harmonic confinement. - at t>0 system feels periodic potential +gravity.
)(),( / ldzfetz imgldtlLAB −=Ψ ∑ h Atoms filling different wells
evolve with different phase !
)2/exp()(
]2/)[(sin]2/)([sin)()(
2222/14/1
2/1
0
2
22
0
hh
h
h
σπσ kkf
dmgtkdmgtkNkfkn w
−=
−−
=
External force affects position of peaksit does not change their modulation
Bloch frequency h/mgdBloch =ω
-2 -1 0 1 2
-2 -1 0 1 2
k/qB
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Bloch oscillations: Bose vs Fermi gases
Spin polarized Fermi gas is ideal non interacting system (s-wave scattering suppressed by Pauli blocking)
Bosons spin-polarized Fermions
Advantages narrow n(k) stability
Disadvantages dynamic instability broadened n(k)
Example: Fermi gas in 3D harmonic trap +1D optical lattice.Harmonic trap is suddenlyswitched off(Modugno et al, 2004)
images takenafter expansion
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Josephson oscillations in periodic potential
Generalization of Josephson equations in double well potential:
canonically conjugate variables)(
)(
kH
t
Hkt
J
J
h
h
∂∂
=Φ∂∂
Φ∂∂
−=∂∂ Φ−−= cos
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21 222 kNkEH JCJ δ
Φ,khbaba SSNNk −=Φ−= ;2/)(
For an array of periodic wells equations can be derived fromJosephson Hamiltonian
Small oscillations around equilibrium(equilibrium: ) 0' == ll SN
)2(42
)2(
'1
''1
0
'
110'
−+
−+
+−+−=∂∂
+−−=∂∂
lllJ
lC
l
lllJl
NNNN
NESt
SSSNNt
δ
δ
h
h eq. of continity
eq. for the phase
quantum pressure effect
By looking for periodic solutions
one finds dispersion relation in tight binding limit (Javanainen 1999)
with)()()( 22 ppENp εεε +=
]/))((exp[)(),( ' htplpditNtS ll ε−∝
h2sin2)( 2
0pdp Jδε =000 C
lowest Bloch spectrumin non interacting model
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Properties of small amplitudeoscillations in periodic potential
h2sin2)( 2
0pdp Jδε =)]()[()( 000
2 pENpp C εεε +=
- In non interacting limit ( ) one recovers single particle dipersionlaw (Bloch band)
- Typical Bolgoliubov structure withreplaced by .
- In long wave length limit withand dipersion law takes phonon like form with
- Using relation and identitywhere n is average density, one finds result
0=CE
)2
2(2
)(2
22
2
mpmc
mpp +=ε
mp 2/2 )(0 pε
cpp =)(ε
0/2 NEC ∂∂= µ nnNN ∂∂=∂∂ // 00 µµ
)(0 pε
*20 2/)( mpp =ε 2*2 / dmJ h=δ
*0
2 2/ mENc C=
nn
mc
∂∂
=µ
*
1for sound velocity
From energy functional
(n is smooth density averaged over several sites)
one derives HD equations
in the absenceof harmonic trap
( ) 0~
0)()()( *
=+∇+∂∂
=∂+∂+∂+∂∂
ho
zzyyxx
Vngtvm
nvmmnvnv
tn
rr
Solutions of HD equations are obtained from results without opticaltrap (Lecture 2), by simple replacement
In harmonic traps HD frequencies do not depend on coupling constantand results are immediately obtained by replacing(major role of effective mass)
*,~ mmgg →→
zz mm ωω */→
nn
mc
∂∂
=µ
*
1
∫
++
++= hozyx nVngvmmvmvmndrE 22
*
222 ~
21
222
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Hydrodynamic theory of superfluids in the presence of periodic + harmonic potential
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Predictions for the frequencies of the lowest modes(Kraemer et al., 2002)
Dipole mode:
Axial quadrupole mode (elongated trap) zQ
zD
mmmm
ωω
ωω
25
*
*
=
=
(Cataliotti et al, 2001)
2/5Fort et al.2002
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Energetic vs dynamic instability in optical lattices
When the velocity of the condensate is large or the lattice moves with large velocities, the system exhibit instabilities.Instabilities can be either energetic (negative energy) or dynamical (complex frequency) (see Lecture 6)
The problem is easily studied if one restricts investigation to phonon modes in periodic potential
Consider fluid moving at uniform velocity along the direction of the lattice (stationary solution). Long wave length oscillations around stationary solutionare described by HD equationswhere
is smooth averaged densityand is energy density:
0),(
0),(
=∇+∂∂
=⋅∇+∂∂
vntvm
vnjtn
µ
0v
nevemj ∂∂=∂∂= /,/)/1( µ
),( vne ),( vnedrE ∫=n
In the absence of periodic potential:
In the presence of periodic potentialboth density and velocitydependence are affected(n is here smoothed density) with
22
21
21),( nmvgnvne +=
h2sin2~
21),( 22 pdnngvne Jδ+→
mvp =
22
Looking for small amplitude oscillations:withone finds dispersion relation:
),(,),( 00 tzvvvtznnn δδ +=+=)(exp, tqzivn ωδδ −∝
qpn
qpn ∂∂
∂±
∂∂
∂∂
=εεεω
2
2
2
2
2 Dopplereffect
In the absence of periodic potential whereNo dynamic instability. Energetic instability if (usual Landau criterion).
qvc )( 0±=ω mgnc /=
cv >0
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Dynamic instability in tight binding limit
In periodic potential one finds:
h
h
h
dmvqdm
qdmvmng 0
*0
* sincos~
±=ω is velocityof the fluid
0v
Dynamic instability if i.e if
instability occurs only in the presence of interaction
0cos 0 <h
dmvd
mvp20hπ
>=
Dynamic instability in BEC gases always occurs during Bloch oscillation.Fast crossing through the instability region is needed in order to avoidthe consequences of instability
Dynamic instability in the presenceof moving periodic potential
0.0 0.5 1.0
0.00
0.01
0.02
s = 0.2
loss
rate
[ms-1
]
quasimomentum [q/qB]
(Fallani et al., 2004)
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ω+δω, k+δk ω, -k
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Superfluid Mott insulator transition
If number of particles per site is of order of unityFormalism of Josephson Hamiltonian is no longer adequate(usually the case for 3D optical lattice)
Convenient approach is based on Boson-Hubbard Hamiltonianderivable from many-body Hamiltonian
)(ˆ)(ˆ)(ˆ)(ˆ2
)(ˆ)(2
)(ˆ 22
rrrrdrgrrVm
rdrH ext ∫∫ ΨΨΨΨ+Ψ
+∇−Ψ= +++ h
and writing field operator on a basis of single site operators
Ignoring higher order terms, the Hamiltonian takes the form:
kkkaˆ ϕ∑=Ψ
)ˆˆˆˆ(2
)1ˆ(ˆ4 , kllklk
Jkkk
C aaaannEH ++><
+−−= ∑∑ δ pairs of first neighbours
( ) lextkJ Vmrd ϕϕδ ∫ +∇−−= 22 2/2 hr
∫= 42 kC rdgE ϕr
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Boson-Hubbard model
- Phase diagram of Boson-Hubbard Hamiltonian predictssuperfluid- Mott insulator transition for integer values of theaverage occupation number per site.
- Superfluid phase corresponds to non vanishing of average(order parameter).
- For occupation number =1 many-body theory theory predictsquantum phase transition at critical value(Fisher et al. 1989) .
- For larger values of insulator phase (no long range order)
- For smaller values superfluid phase (long range order).
8.34/ =JCE δ
0ˆˆ >≠>=<Ψ< ∑ kkkaϕ
JCE δ/
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Quantum phase transition from superfluid toMott insulator in trapped Bose gases
- Extension of theory to harmonic trapping: Jacksch et al. 1998
- In Bose gases superfluid phase can be tested by measuring inteferencepatterns in expanding condensate. Intereference is result of orderparameter and reflects coherent behaviour in momentum space
- Disappearence of fringes at high laser intensities revealsoccurrence of transition into Mott insulator phase
superfluid superfluidMott
(Bloch et al.Nature 2002)
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This Lecture:Momentum distribution and interference. Bloch oscillations. Josephson oscillations. Superfluid vs insulator phase.
References relative to the various lectures of the course soon available at the addresshttp://www.phys.ens.fr/cours/Sandro/index.html
