physics of 2d cold bose gas - zhejiang...
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Physics of 2D Cold Bose Gas
based mainly onbased mainly onZ. Hadzibabic and J. Dalibard
arXiv0912.1490
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• Role of dimensionality in phase transition and the type of order present ( density of states dynamics)order present ( density of states, dynamics)
• Robustness of order in high dimension vs thermal and quantum fluctuation in low dimensions
• 2D is marginal (e.g. quasi‐long range order, algebraic instead of exponential decay of the first order correlation function )†
1( ) ( ) (0)g r r
• Mermin‐Wagner theorem : LRO is impossible in any non‐zero temperature in 1D and 2D system with short range p y ginteraction with a continuous Hamiltonian symmetry . LRO is associated with the spontaneously breaking of a p y gcontinuous symmetry. The LRO is destructed by long wave length thermal fluctuations in low dimensions and the gsymmetry is restored.
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• I Infinite Uniform Cold Bose Gas• Density of states
3D3/2
1/22( ) V mD 3D2D energy independent
2 2( )4
D
2 2( ) / 2D mL
* Chemical potential of classical ideal gas3/2
the quantum concln entrationBmk Tnk T n
Qn n
2 the quantum concln ,2
entrationBB Q
Q
k T nn
It approaches zero when T is decreased and below /kT N
, 1/ BZ e k T 1 1
ppBE temperature it is fixed at 0.Fugacity BThe Bose‐Einstein distribution ( )
1 11 / 1
ne e Z
g y
( ) determinesN ( ) determines il
n N
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1. The ideal gas• The phase space density (dimensionless): the capacity of containing particles
1/2
3D1/2
3 1/33
2, QB
D n nmk T
Thermal de Broglie wave length
B
In 3D D=2.612 for Z=1 . For a definite T, the density has an upper limit (the critical density) BEC takes place Conditionupper limit (the critical density) BEC takes place. Condition for BEC: saturation of single particle excited states at a critical temperature2D
l i b d b f d i
22D n
temperature
Relation between Z and D In absence of condensation2
222 ( ) , ln(1 )
2 1 / 1x
mL d dxN D n ZZ
221 nZ e
22 ( )0 0
, ( )2 1 / 1xe e Z
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ill d h• We will study the 2D system as• The homogeneous ideal gas and interacting g g ggas, in which at sufficiently low temperature the BKT transition leads to quasi‐LRO and qsuperfluidity.
• The finite system (box harmonic trap) of idealThe finite system (box, harmonic trap) of ideal gas and interacting gas. In both cases subtle interplay of BKT transition and Bose‐Einsteininterplay of BKT transition and Bose‐Einstein condensation happens.
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In 2D the relation can be satisfied at any finite temperature, h h d i i li i d BEC d fi i
Di ib i f M S d h fi
hence the density in unlimited. BEC does not occur at finite temperature.• Distribution of Momentum States and the first order correlation function
2 21 k( )
1 ,1 2k kk
kne m
* 2
1 20
1( ) ( ) (0)2
ik rkg r r e n d k
Fourier transform of the momentum distributionNon‐degenerate and degenerate regimes have
different momentum distributions and correlation functions. †
'Re : ik r ik rk k ka e a n e
', '
k k kk k k
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• The non‐degenerate regime 221 nZ e g g
In this regime all states are weakly populated2 21 1, / 1BD n Z n k T
In this regime all states are weakly populated2 22 /4 2 2, / 4k k
k kn Ze n e k
and the correlation function is Gaussian2 2/
1( ) rg r ne
with a correlation lengthh d
1( )g r ne/
. The degenerate regime1, 1, / 1BD Z k T
In this regime the momentum distribution is bimodal depending on the energy of states
, , B
bimodal, depending on the energy of states
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• For the high energy states
2 2
2 2 2 2
/4
/ 4 1, 4 /
1 1 Gaussiank
k
k k
Z n e
The states are weakly populated.
1, 1 GaussiankZ n e
. For the low energy states2 2 2 2/ 4 1 4 /k k
2 2 2 2
))
/ 4 1, 4 /
1kk
k
k
k k
e e
((
2 2 2
4 1 , 2 /
k
Bk c
k Tn k m
Lorentz form. States are strongly populated.
2 2 2 , /k ck c
n k mk k
g y p p
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• The correlation function is bimodal. At short d ( h h ) distance ( high energy states important)
is still Gaussian. The Lorentzian form r up to
1( )g r s st Gauss a . e o e t a ocorresponds to exponential decay for
2r
2/ 1 /21( ) , with a correlation length
4r l n
cg r e l k e
Actually the Lorentz mode is much more strongly populated then the Gaussian modepopulated then the Gaussian mode.
2.Interaction in2D Bose gas at low temperaturesg p. 3D
2 3 23 2 3 34 4( ) ( ) ( )
DDgV E d
3 2 3 3int( ) ( ), ( ) ,
2D
i j s i j sgV r r a r r E n r d r g a
m m
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. Real 2D scattering amplitude is logarithmically d denergy dependent
• Quasi 2D : z‐degree of freedom is frozen, but the Quas : deg ee o eedo s o e , but t escattering is actually 3D: /z z sa m a
We attempt to have the interaction strength for 2D case independent of energy, as in the 3D case:
2 2int 3 2
2
( , ( ) 2D local density, /2 zgE n r d r n r n n a
)
22based on dimensional analysis, , dimensionlessEquating to the 3D interaction energy we obtain
g g gm
Equating to the 3D interaction energy, we obtain 2 / . Actually 8 / (Petrs z s zg a a g a a ov et al)2D healing length = / 1/mgn gn 2D healing length = / 1/mgn gn
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• Strong interaction limit defined:• E int of N particles=E kin of N non‐interacting particles distributed equally over lowest N single pa t c es d st buted equa y o e o est s g eparticle levels: the interaction is strong enough to keep the particles residing on different statesto keep the particles residing on different states
2 22D DOS ( ) / 2 L t b th f th NthD L E2 2
2
2 2
2D DOS: ( ) / 2 Let be the energy of the Nsingle particle state, then / 2 /
thN
N N
D mL ED N E E n m
2 2int/ 2 / equated to /
strong interaction limit: 2kin NE NE Nn m E gNn m
g
This limit is independent
33
of density, as contrasted to 3Dwhere the dimensionless measure is sn a3 s
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3. Suppression of density fluctuations and the l H ilt ilow‐energy Hamiltonian
We assume that the weakly interacting 2d Bose gas at T =0 is y g gdescribed by , where n and θ are classical fields. At finite T assuming that the system is enclosed in a box of
ine
length L such that the wave function description is still valid, both number and phase fluctuation exist. We argue that for repulsive interaction at sufficiently low temperature therepulsive interaction at sufficiently low temperature the number fluctuation is suppressed, because it is tied up with the interaction energy 2 2 2 2( ) ( )g gn r d r L n r
the interaction energy. The number fluctuation is where is the density density correlation
( ) ( )2 2
n r d r L n r 2 2 2 2
2( ) (0) 1n n r n g n
2( ) ( ) (0) /g r n r n n
where is the density‐density correlation function (normalized).
At fixed density n minimizing the interaction is equivalent to
2 ( ) ( ) (0) /g r n r n n
At fixed density n, minimizing the interaction is equivalent to minimizing the number fluctuation.
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In ideal Bose gas g(0)=2 (bunching) and in absence f l ( ) hof correlation g(0)=1 . We can estimate the
minimum cost of density fluctuation by equating it to the interaction energy increase by adding a single particle. Setting g(0)=1, assuming thesingle particle. Setting g(0) 1, assuming the suppression of the fluctuation, we have 2 2 2 2/ 2 / 2E gn L gN L have .
Comparing with the thermal energy of the particle, int / 2 / 2E gn L gN L
we get At sufficiently low temperatures given by Bk T gnAt sufficiently low temperatures given by or the thermal energy is far less than
t d it fl t ti d
Bk T gn
2 /D g
necessary to cause a density fluctuation and
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the density fluctuation is strongly suppressed. In numerical calculations it turns out that the numbernumerical calculations it turns out that the number fluctuation is already suppressed for D>1. In this limit the interaction energy becomes a constant 2 2 / 2gn Lthe interaction energy becomes a constant ,
The kinetic energy arises from the phase θ/ 2gn L
22 2
2H d r
m
This gives a good description of long range physics for albeit phenomenological in nature for , albeit phenomenological in nature,r
superfluid velocity v=
22
superfluid velocity v
1kinetic energy=
m
m n d r
kinetic energy
2 sm n d rm
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An exact formulation is given by the Bogoliubov analysis, (th d t ) (th B li b dˆ + )
aiming at the justification of the density fluctuation suppression and the superfluidity of the state
0 (the condensate) (the Bogoliubov modes+ )
suppression and the superfluidity of the state The classical field Hamiltonian
l bl dTreating as canonical variables, one can derive the G.P. equation. Here we use the order parameter, which
b h d h
* and
seems absent in the 2D case. We can consider the system as finite in size, where the decay of correlation function is ill i ifi d d l i h d*still not significant, and deal with as order
parameters. The applicability of Bogoliubov approach is j ifi d b C i Ph R A 67 053615 2007
*and
justified by Castin Phys. Rev. A 67, 053615, 2007
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( , )( , ) ( , ) ,i r tr t r t e
2
( , ) (1 ( , )), 1 real( , ) (1 2 ( , )),r t n r tr t n r t
Density and phase separated
( , ) (1 2 ( , )),r t n r t
H leads to equations of motion for the phase and density fluctuation. The canonical variables are now and (both real) fluctuation. The canonical variables are now ( )
* *d dFrom the Hamiltonian for the field one obtains 0
,0
k k k kc c d dd
c is the “momentum” d is the “position”
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Bogoliubov modes are simple harmonic oscillator states with eigenfrequencieseigenfrequencies
1 At low k the eigenmodes are phonons with /ck c gn m 1. At low k the eigenmodes are phonons withSuperfluid behavior follows from the order parameter instead of the LRO At high k we have free particle modes
, /k ck c gn m
of the LRO. At high k we have free particle modesCross‐over happens at
2 The relative importance of phase and density fluctuations
2 2 / 2k k m gn 1/k gn 2. The relative importance of phase and density fluctuations can be calculated by using the virial theorem for harmonic oscillators which leads to2 2 2/ 2 / 2m x p m oscillators , which leads to/ 2 / 2m x p m
We conclude: long wave length phonons involve only phase fluctuations while high momentum phonons involve bothfluctuations while high momentum phonons involve both density and phase fluctuations in equal weight.
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3. Density fluctuations are not “soft” Goldstone d d h f d l d dmodes, and therefore do not lead to divergent
effects, while the phase fluctuations do not cost energy for k→0. Density of states at low k leads to divergent effect destructing the LRO.to divergent effect destructing the LRO.
4. Temperature dependence of density fluctuation2 2 2 2 2 2 2 2 2
To calculate we simplify the Bose‐Einstein
2 2 2 2 2 2 2 2 2( , ) (1 2 ( , )) , ( , ) , / 4n r t n r t n n r t n n n
To calculate, we simplify the Bose Einstein distribution by using the equipartition law that in thermal equilibrium each degree of freedom hasthermal equilibrium each degree of freedom has an energy / 2Bk T
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Here we introduced a cut‐off, (otherwise divergence would arise). Physically this is justified that higher modes are rarely populated. We obtain
,
For realistic values of T and g , we conclude that the density fluctuation is suppressed at .2 1n uctuat o s supp essed at
4. Algebraic decay of correlation function
1n
We study the correlation function at low temperature for ( density fluctuation neglected) which depends,r
For low k the B‐E distribution can be simplifiedk Bk T
essentially on the phonon modes 1/k
For low k the B E distribution can be simplifiedk Bk T
i.e. equipartition:
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.( ) ik rkr c e
22
24k
L d k
21/
21/2 ln
r
d k rk
Mermin‐Wagner
Theoremk
More exactly 2D Poisson eq. of a point charge
Theorem
of a point charge
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II The BKT phase transitiona simple picture: the vortices drive the BKT phase transition we begin from a vortex inphase transition we begin from a vortex in the superfluid
V t iti l i tVortex energy critical point Entropy
m
EntropyFree energy critical point 2 4 ln
2B
sk T RF n
2 4sn
2For F is large and positive, the superfluidis stable against proliferation of vortices,
2 4sn
g p ,For F is negative, and the superfluidb t bl
2 4sn
becomes unstable.
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In 2D superfluid density cannot have any value b 4/ d 0 U d h BKT i i2between 4/ and 0. Under the BKT transition vortex pairs play the dominant role. Energy of pair is finite, so th t th f i l ti At fi it T
2
that the free energy is always negative. At any finite Tpairs can be continuously created and annihilated by th l fl t tithermal fluctuations.
S fl id N lSuperfluid Normal
2At the critical temperature 2 / 42
KT s s KTT n n mTm
2
2 jump (Nelson-Kouniv sterersa lit )l zs s KTmmn T
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As T is increased, the density of pairs grows, and the size of pairs also grows As T approaches the BKT temperaturepairs also grows. As T approaches the BKT temperature pairs begin to dissociate, and free vortices proliferates eventually destructing the superfluidity The universaleventually destructing the superfluidity. The universal jump result is elegant, it is independent of the interaction strength But it does predict the value of Tstrength. But it does predict the value of The microscopic theory (Prokofiev, Ruebenbackner , Svistunov) gives C
KTT
Svistunov) gives
This relation holds for weak interaction We can set one
2 ln , 380c c
CD n Cg
This relation holds for weak interaction. We can set one obvious bound to its validity: Since , we can set sn n
4 which leads to 7CD g 4, which leads to 7.CD g
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It also gives that when T is approached from above thecorrelation length is
exp BKTaTl 1a
which diverges logarithmically when T decreases toward
expcBKT
lT T
1a
KTTwhich diverges logarithmically when T decreases towardwhile 2nd order transition in 3D has coherent lengthsd l l h l h l
KT
diverging in a polynomial way. The critical region where l becomes larger than λ as a precursor of the BKT transition istherefore very broad in the case of finite size systems.Diverging correlation length is typical in phase transitionsDiverging correlation length is typical in phase transitions.When the system has a finite size, certain fraction of
d h l b l h h icondensate can appear when l becomes larger than the size.
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3D
Critical behavior of 3D trapped interacting Bose gas Esslinger group ETH Science 315, 1556, 2007Correlation function for T>>Tc: Gaussian, with correlation ~ de Broglie wave lengthcloser to Tc, correlation length > de Broglie wave length, At Tc the correlation diverges, the correlation function becomes algebraic
i i l
/1
1( ) rg r er
1( ) 1/g r rν—critical exponent
c
c
TT T
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• III 2D Bose Gas in a Finite BoxFinite size effects can play a significant role in phase transition, especially in 2D case, in which thermo‐, p y ,dynamical limit can be achieved only in marginal cases
ln /R
1. Ideal gas There are 3 regimesN d i 2 22 /Non‐degenerate regimeDegenerate regime for
2 22 /11, ( ) , for <<rD g r ne L
2/2
2
4, i.e. ln4
D Ll e L D
g gthe finite size has no appreciable effectCondensate regime for significant
24
24l iLD l L
/1( ) r lg r ne
Condensate regime for significant phase coherence exists between any two points in the gas
2ln i.e. D l L
in the gas
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2. Interacting gas The finite size effect and the effect of interaction co‐exist and compete
????
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We study the case with repulsive interactions to see what can be expected in the vicinity of BKT transition The size of thebe expected in the vicinity of BKT transition. The size of the sample is L, and we assume that the point of BKT transition is well before BEC When D approaches BKT
24ln LD Dwell before BEC . When D approaches , BKT
transition occurs, and acquires an algebraically long range order
2ln cD
CD
1( )g r2( ) ( / ) and =1/ 4g r n r n range order .1( ) ( / ) , and =1/ 4s sg r n r n
In sharp contrast with the infinite sample case , the BKT transition is accompanied by a significant fraction of condensate fraction. This fraction is associated with thecondensate fraction. This fraction is associated with the largest eigenvalue of the single particle density matrix0
( ) (0) ( )g r N r 1( ) (0) ( )j j j
jg r N r
The condensate fraction is associated with the state0 0 1/ L
Integrating the LHS over a circle of radius L/π we get
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and we integrate the RHS over a square of side L. For j=0, and non‐zero j states are integrated to 0. 0 0(0) ( ) 1/L L , and non zero j states are integrated to 0.
The result is then0 0( ) ( )
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• Width of the critical region and crossover1/2ll
1/2ln /BKT BKT
l aT T T
1, width of critical region BKTTa
* What comes first: BEC or BKT?TBKTTfor / 10 100, / 2 %0% 5
B
BKT
KT
L T T
* What comes first: BEC or BKT?a subtle question
TCritical region
Large system We can either increase D under constant toward or increase under
2 2ln 380 / ln 4 /cD g L g CD g
constant D so that decreases toward D, the first relevant mechanism
gCD
is BKT. Appreciable condensed fraction appearspp ppwhen BKT is approached. The first case is called the BKT
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driven condensation, and the second case is called the interaction enhanced In the square boxthe interaction enhanced. In the square box, these variants are equivalent.
* Small system2 2 l n ( 4 / )CD D L
2 2ln 4 /cD L Small system c
When D is increased, it reaches the BEC point first, the condensation is conventional.
2 2 ln(4 / ) CD L D
In the intermediate cases it is hard to distinguish the two regimes experimentally, even if the criteria are separated by more than the crossover region.
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• IV 2D Bose Gas in a Harmonic TrapWe find dramatic change of the properties of the system through the change of density of states.
1. The ideal case 2 2( ) / 2; 1V r m r E j
Levels with quantum number j have j+1 fold d j i di ib d h
( ) / 2; 1jV r m r E j
degeneracy: j is distributed among the two dimensions: hence the d
x yj n n
degeneracy.Putting equal to the energy of the ground state, g q gy g
we get the number of atoms on all excited states / k T / Bk T
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• For the sum is replaced by integral, and 1 21 1d
For atoms begin to accumulate on theidN N
2
2 20
1 11 6x
xdxe
For atoms begin to accumulate on the ground state
cN N
These results can be obtained from the ideal case by using the local density approximationy g y pp
( )V r
For the above result is recovered. 20 ( 1), / 6Z In the ideal gas case,
( )
0 ( 0) diverges in BEC transitionn r 0 ( 0) diverges in BEC transitionn r
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2. Interacting case, LDAI fi ld H t F k th th i t tiIn mean field Hartree‐Fock theory the interaction energy when no condensate exists is 2gn (direct + exchange),
2 ( ) /gn gD r The number of atoms on all excited states is
2 ( ) /gn gD r
where D is the solution of
and it depends only on Z and , do not enter: scaling f th t b ith t fi d Z i th i
g and Td Tof the atom number with at fixed Z is the same as in
the ideal case. For any and non‐zero g , N can be bit il l b l h i Z Th d ti
and Tand T
arbitrarily large by properly choosing Z. The condensation does not occur. For ideal gas the saturation on excited states
h N(0)di hil l i i t tioccur when N(0)diverges , while repulsive interaction removes the singularity, and the mean field theory provides
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a solution for any value of N. Interactions when treated at the mean eld level dramaticallyInteractions, when treated at the mean‐eld level, dramatically change the nature of the solution of eqs. For a given trapping frequency ω and temperature T and for any non zero ~g thefrequency ω and temperature T, and for any non‐zero ~g, the atom number N obtained can be made arbitrarily large by choosing properly the fugacity Z The condensation that waschoosing properly the fugacity Z. The condensation that was obtained in the ideal gas case does not occur anymore. This can be understood qualitatively For an ideal gas saturation ofcan be understood qualitatively. For an ideal gas, saturation of the atom number occurs when the central density in the trap becomes infinite In presence of repulsive interactions thisbecomes infinite. In presence of repulsive interactions, thissingular point cannot be reached and the mean‐field treatment provides a solution for any atom numbertreatment provides a solution for any atom number.We give the prediction of the mean‐field approach for the central phase space density D(0) as a function of the totalcentral phase space density D(0) as a function of the total number of atoms in the trap, for various values of ~g.
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Given and N, we solve the implicit equation g
together with . This determines D(R) as a function of and N plotted below
12
0
( )2D R RdR N gfunction of and N, plotted belowg
D(0)Black squares indicate the values qof N at which BKT criterion is met at the center of the trap.
The mean field theory cannot provide an accurate descriptionThe mean field theory cannot provide an accurate description of the transition. Indeed the mean‐field expression2gn(r) for the interaction energy can only be valid at relatively lowthe interaction energy can only be valid at relatively low density, where the density fluctuations are important, so that
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2 22 (0) 2( ) and 2 .g bunching n n When the density increasesand/or the temperature decreases, density fluctuations are p yreduced and one eventually reaches a situation at very low temperature where those fluctuations are frozen out 2 (0) 1g pAnd . At zero temperature, one expects a quasi‐pure condensate in the trap with a density profile given by the
2 2n n
p y p g yThomas‐Fermi law gn(r) = μ‐V (r) (whereas the Hartree‐Fockapproximation would lead to replacing g by 2g in this equation).To capture the reduction of densityfluctuations as the phase space density increases, we now use the numerical results of Prokofiev and Svistunov (classical field Monte‐Carlo)
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Holzmann et al (mean field Hartree‐Fock)
For a given trap at a given temperature the BKT threshold in presence of interaction requires a larger atom number thanpresence of interaction requires a larger atom number than the BEC of ideal gas. For a given atom number the superfluidtransition temperature in the presence of interaction is lowertransition temperature in the presence of interaction is lower than the ideal condensation temperature.
3. What comes firstThis question is more subtle than the box case because of theThis question is more subtle than the box case because of the change of the density of states, and depends on what we keep constant and what we vary in an experimentkeep constant and what we vary in an experiment.i. For non‐zero g increasing D until it reaches , BKT comes first with induced condensation conventional BEC does not
cDfirst, with induced condensation, conventional BEC does not occur in the mean field theory.
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ii. For fixed N, increasing g further increases andinteraction enhanced condensation never occurs. These
BKTCN
interaction enhanced condensation never occurs. Thesetwo processes are not equivalent in the harmonic trap case.N t th t it i t i i t t th t th BKT t itiNote that it is not inconsistent that the BKT transitionoccurs at a lower critical density but higher critical numberthan ideal gas BEC, because in a harmonic trap withA fixed N and T the peak (phase space) density in aA fixed N and T the peak (phase space) density in arepulsively interacting gas is lower than in an ideal gas.Al t th t if k ithi th BKT th d thAlso note that if we work within the BKT theory and thenFormally take the ~g →0 limit, we exactly recover thecriterion for ideal gas condensation, which is usuallyderived from a conceptually completely different viewpoint ofderived from a conceptually completely different viewpoint ofthe saturation of single‐particle excited states
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iii. BEC transition can be considered as a special non‐interacting limit of the more general BKT theory in the case of g g ya harmonically trapped gas. We briefly comment on the case of a realistic harmonic trap, where the spacing of the single‐p p g gparticle energy levels is non‐zero. The results for the critical atom numbers for the BKT and the ideal gas BEC transition gare essentially unaffected by the finite level‐spacing. It therefore remains true that interaction‐enhanced condensation is impossible. However, the ideal gasin this case occurs at a finite phase space density in the BECDtrap center, which can in principle be lower than Dc for some values of ~g. In this case BEC would come first independent of paths. But the value of g would be unrealistically small , this regime is essen ally indis nguishable from the ~g → 0 limit, where the BKT and the BEC transition are no longer distinct.
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V. Probing 2D Atomic Gasesg1. Interaction in a 2D gasPetrov, Holzmann, Shlyapnikov
3 : 0, ( ) ; 2D energy dependence even in low energyD k f k a
Consider instead the 3D scattering in confined geometry (quasi‐2D), the authors obtained 0 ( ) ik
sc z e
(quasi 2D), the authors obtained3.5
0 ( )sc
and establishing the equivalence, they got
When the log term is negligible compared to the leading termWhen the log term is negligible compared to the leading term
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2. In situ density distributionThere is no loss of information due to the integration alongthe line of sight (vs 3D). In the local density approximationg ( ) y ppthe local density can be obtained from that of homogeneousgas where F at this stage is an unknowngas where F at this stage is an unknowndimensionless function. LDA and dimensional analysissuggests ( )V r
2 2 2( ) / 2 , (*), /T BD n r G r r g k T is the length scaleTr
T B
This expression clearly shows a scale invariance for a given interaction strength ~g. Suppose that different density profiles n(r) are recorded for various temperatures T and various atom numbers N (hence different chemical potentials). According to eq. (*) the profiles can all be
i d th G id d th l tt dsuperimposed on the same curve G, provided they are plotted as a function of and translated along the x axis by α. (checked MC)2 2/ Tr r
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For , various asymptotic forms of the function h b l h b
1g ( ; )G g have been given earlier. When interactions can be neglected, the equation of state is ln(1 )D e
In presence of interactions and for small phase space densities, the mean‐field Hartree‐Fock method amounts to replace μ by μ‐2gn into the ideal gas result, which leads to the implicit equation from which /ln(1 ),gDD e
one can extract D as a function of and ~g .In the strongly degenerate limit, where ,
2 / /Bgn k T gD
, 1Bk T D g y g , ,density fluctuations are strongly reduced and oneexpects μ = gn which can be written as D = 2 / g expects μ gn, which can be written as D
In the intermediate regime, in particular close to the BKT transition point one can use the results of the classical
2 / g
transition point, one can use the results of the classicalfield Monte‐Carlo analysis
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The results are represented below
( , )F g ( / )TD r rHomogeneous system system in trap
BKT
LDA: ( ) / BV r k T
Remarkable: F is featureless at the BKT transition! The transition is an infinite order one.
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2. Time of flight expansion Generally speaking, a Time‐of‐Flight (TOF) procedure consists in switching off abruptly theFlight (TOF) procedure consists in switching off abruptly the potential confining the atoms, letting the cloud expand for an adjustable time and then measuring the density profile If theadjustable time, and then measuring the density profile. If the role of interactions is negligible during the expansion, the density profile after a long TOF is proportional to the in trapdensity profile after a long TOF is proportional to the in‐trap momentum distribution. For a two‐dimensional system, two types of TOF can be consideredtypes of TOF can be considered. 2D TOF switching off at t=0 the confining potential in xy plane and keep the axial confinement A Bogoliubov analysis (Kaganand keep the axial confinement. A Bogoliubov analysis (Kaganet al PRA 54 R1753 1966) predicts that the density at time t is given by the scaling lawgiven by the scaling law
This means that the global form of the spatial distribution isThis means that the global form of the spatial distribution is preserved during the TOF. As the expansion proceeds, the
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interaction energy that was initially present in the gas isconverted into kinetic energy in such a way that the densityconverted into kinetic energy in such a way that the density profile at time t is obtained using a scaling transform of the initial oneinitial one.3D TOF Confining potentials are switched off simultaneously.The axial and radial expansions have very different time scalesThe axial and radial expansions have very different time scales
and the expansion has two phases. During the 1 1z r
first phase whose duration is a few (typically 1 ms if1 / 2 first phase, whose duration is a few (typically 1 ms if = 3 kHz), the thickness of the gas along z increases by a factor much larger than 1 but the xy spatial distribution is nearly
z / 2z
much larger than 1, but the xy spatial distribution is nearly not modified. At the end of this first phase, the interactionsbetween atoms have become negligible During thebetween atoms have become negligible. During the subsequent phase the expansion in the xy plane becomes significant but on a much longer time scale It corresponds tosignificant, but on a much longer time scale. It corresponds to the expansion of an ideal gas, whose initial state is equal to
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the state of the system in the xy plane before the beginning of the TOFthe TOF.
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3. Interference between different planesTime of flight experiment harmonic oscillator ground stateTime‐of‐flight experiment harmonic oscillator ground state
2 2/20 1/4
1( ) ,z
z
u z em
Z‐confinementF ti i t l th f ti d f l
/ , /z z z zp m v m t
z
For time interval the wave function expands freely along the z direction for a distance
t
zd
The expansion in xy plane is much longer 1/ /z z zm m t t 1 1
z z z z
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thermal gas , / , TOF far fieldsuperfluid / central region
cr pr r p r
Hadzibabic, Dalibard et al Nature 2006, PRL 2007, ,superfluid / central regionc T Tr r p r
1 To characterize the normal and superfluid regimeNot aiming at studying phase difference between planes, but
1. To characterize the normal and superfluid regime (according to the behavior of )
2 To demonstrate the BKT transition1g
2. To demonstrate the BKT transitionrepresenting a new direction of probing the physics of quantum correlations a property of quantum many‐bodyquantum correlations, a property of quantum many‐body systems.
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The planes are prepared in identical conditions (same temperature and density) 3D TOF From(same temperature and density). 3D TOF. From the interference pattern the behavior of can be extracted Polkovnikov et al suggested to
1g
be extracted. Polkovnikov et al suggested to characterize both normal and superfluid regime using a single experiment.
The state of the planes are described by and a b The state of the planes are described by distance Ketterle(97) : the relative speed at any point after time t is The fringe period is the
a b
/d tzd
point after time t is . The fringe period is the de Broglie wave length associated with the l i i f
/zd t
/D ht d/relative motion of atomsThe spatial density is
/z zD ht md/ m v
p y
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The local contrast (complex) is proportional to * ( ) ( )a bz z if one performs absorption imaging along the y axis, the image involves an integration of the local contrast *
a b g galong the y direction. Averaging the result of this contrast measurements one can define the average contrast C(A) :
a b
g ( )
* * 2 2( ) ( ) ( ') ( ') 'a b a br r r r d rd r
this quantity is independent of the phase difference between the two planes, but is
The correlation functions are the same, but the two planes
p pdetermined by the phase fluctuation within each condensate.The correlation functions are the same, but the two planes fluctuate independently, therefore 2 2
1( )A g r d r
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In the two extreme limits:NormalNormalSuperfluid 1/ 4
If a single isolated vortex is presentin one of the planes, the pattern exhibits a sharp dislocation at the coordinate of the vortex core.
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• Observation of a 2D Bose Gas: From Thermal to Quasi‐condensate to Superfluid W D Phillips group PRL 102condensate to Superfluid W. D. Phillips group PRL 102, 170401, 2009 by measuring the profile of the quasi‐2D system after time of flightsystem after time‐of‐flight
Na atoms (weakly interacting with g~=0.02) in optical d l l d hdipole trap, cooled to ~100nK. The temperature is determined by the trap depth. Absorption imaging l h l d d falong the vertical direction is carried out after turning off the trap and waiting for a TOF period, and the 2D d f l f h l d l d b ddensity profile of the released atomic cloud is obtained. During the experiment the temperature is fixed while h b f i i dthe number of atoms N is varied.
Prokofiev et al did MC calculations for (homogeneous) 2D weakly interacting system and obtained 2 ln( / )BKTn C g
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Prokofiev et al predicted the presence of a non‐superfluid quasi‐condensate (low phase coherence) The present authors applied LDAcondensate (low phase coherence). The present authors applied LDA to the MC results and calculated the total density, quasi‐condensate density and the density of superfluid, as functions of N/Nc, with
2 , the critical number for BEC of a 2D ideal gas. It is seen that at low densities the profile is a Gaussian, and at higher d iti th ( d) li i bi d l iti f t G i
22 / 6 /c BN k T
densities the (red) line is bimodal :a superposition of two Gaussians.After 5 μs of TOF time, the QC and the superfluid components cannot be resolved For 10 μs acannot be resolved. For 10 μs aa tri‐modal profile can be observed. BKTN N
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4. Interfering a single plane with itself: alternative for g_1Philips group PRL 102 170401 2009Philips group PRL 102, 170401, 2009
r RA B
B: point of observation, receiving waves of F=2 from point A, where the atom results from the 1st Raman beams at O acquiring velocity v 0 and moving to A
R
rbeams at O acquiring velocity v_0 and moving to A, and remaining in the state F=2, and from the point O, where the atom remaining in F=1 at the 1st Raman beams and becoming F=2 by the 2nd Raman beams
Obeams and becoming F=2 by the 2nd Raman beams
1phasedifference momentum difference R k R
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The modulated density profile of the above equation gives a direct access to the function g1. It can be observed with an imaging beam along the z direction, so that its measurement does not involve any line‐of‐sight integration. This method can then reveal finer details than the one presented earlier. In particular the present authors could observe a gradual increase of the coherence length l of the cloud, . For small phase space densities the
t i d l i t h l l h th t tl exp / ( )KT KTl aT T T
measurement gives , and l increases to much larger values when the temperature decreases towards the critical temperature TBKT. When T < TBKT a significant interference contrast is observed for all values of R within the size of the central supeflruid region.
l
Upside down!
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• The contrast as a function of R is measured for different densities For small density the correlation ends at 1μmdensities. For small density, the correlation ends at 1μm, the thermal wave length λ for 100nK. For intermediate densities the correlation drops but then remainsdensities the correlation drops, but then remains corresponding to the quasi‐condensate up to 10 μm, and for still larger densities (> ) the superfluidNand for still larger densities (> ), the superfluidcomponent appears, and the correlation exists at 20 μm, corresponding to algebraic decay of coherence
BKTN
corresponding to algebraic decay of coherence.
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• V. Outlook• 1. Higher order correlation functionsgThe matter‐wave interference between two statistically similar, butindependent quasi‐condensates can reveal a wealth of information onthe correlations within each individual 2d gas. So far only a fraction ofthis information has been harnessed, with the study of the averagecontrast of the interference pattern integrated over some area ofcontrast of the interference pattern integrated over some area ofinterest. A convenient tool for extracting more complete informationon g1 as well as higher‐order correlation functions is the full statisticalg gdistribution of interference contrasts. Two limiting cases can easily becharacterized: (i) If the two independent fluids are fully condensed, each image shows a 100% contrast, with the position of the fringesfluctuating randomly from shot to shot. (ii) If each cloud exhibits onlyshort ranged correlations the observed interference results fromshort‐ranged correlations, the observed interference results frommany uncorrelated fringe patterns along the light of sight, and thedistribution of contrasts is an exponential function. For 1d gases, it isp g ,possible to describe quantitatively the transition between these twolimiting cases .
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In the 2d case, the evolution of the contrast distribution through the BKT transition is still an open problemThe 1D interacting Bose liquid can be described by
b i i h L i li i d h (C lill )
through the BKT transition is still an open problem
bosonizing the Luttinger liqiud theory (Cazalilla)Quasi 1D Luttinger liquid theory T=0, quasi LRO , finite T, exp
decay. 1D: no need to integrate along the line of sight. Average over g g g g
a tunable distance along the longitudinal directionWe probe more deeply in the quantum correlations in theWe probe more deeply in the quantum correlations in the
many‐body problem system Shot to shot record enables to measure the statisticalShot‐to‐shot record enables to measure the statistical
distribution of the local contrast, the higher order moments of which is associated with the higher ordermoments of which is associated with the higher order correlation functions of the 1D boson gas
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Demler et alDemler et alNphys 2006,theoryHofferberth,Hofferberth,Demler et alExpt nphys2008
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NB z and yInverted!
Local contrast averaged over distance L
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Luttinger liquid theory gives 1( ; ( , , ))QA L n g
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Temperature and L dependence1. Low temperature/small size / ( ) 1 quantum fluctuation (arising from interaction between atoms) dominates
L T
2 2
2
( g )ideal gas no phase fluctuation ( )QA L L
2strong interaction (Tonks-Girardeau) ( )
inte
QA L L2 2 1/rmediate ( ) ( )new KA L L intermediate ( ) ( )
2. Finite temperature: thermal fluctuation results in ( )new QA L L
T
2/ ( ) 1 thermal fluctuation dominates ( ) ( )QforL T A L L T
Shot‐to‐shot individual measurement of fluctuationsenables to find higher moments and the distribution with the normalized fluctuation
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For the distribution of averaged constrast ( ( )), we obtain( ) : quantum fluctuation dominates
W LL T
( ) : quantum fluctuation dominates
for ideal gas is a delta function for weak interaction has a narrow peak, width 1/
L TW
W K
for finite interaction approaches a Gumbel distribution( ) : both quantum and thermal fluctuations are importantd bl k it ti
WL T a double peak situation
( ) : thermal fluctuation dominates, apprL T W oaches a Poisson distribution as a sum over interference from many Poisson distribution as a sum over interference from many
uncorrelated domains
31nK
60nK
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2. Out‐of‐equilibrium dynamical effectsB k t l [112] h t di d th d i f d h b t t l BBurkov et al. [112] have studied the dynamics of decoherence between two planar Bosegases, assuming that their local phases are initially locked together, and then the twogases are allowed to evolve independently. This can be achieved experimentally by havinga weak potential barrier and hence large tunnel coupling between the two planes fora weak potential barrier and hence large tunnel coupling between the two planes fort < 0, and then suddenly raising the barrier at t = 0. The contrast of the interferencebetween the two gases gives access to the evolution of the phase distribution under theinfluence of thermal fluctuations
3. Transition from 2D to 3D behaviorinfluence of thermal fluctuations.
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4. Tunable Interactions
5. Superfluid Density
2D system: both transport and coherence measurements can be performed , allowing experimental scrutiny of the theory
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Two promising schemes for a direct measurement of the superfluid density (as traditionally defined through transportsuperfluid density (as traditionally defined through transport properties [Leggett]) in an atomic gas have recently been proposed The first scheme [Ho and Zhou] is based onproposed. The first scheme [Ho and Zhou] is based on extracting the superfluid density from the in‐situ density profiles of a rotating 2d gas The second scheme [Cooper andprofiles of a rotating 2d gas. The second scheme [Cooper and Hadzibabic] is based on using a vector potential generated by Raman laser beams to simulate slow rotation of a gas andRaman laser beams to simulate slow rotation of a gas, and allows direct spectroscopic measurement of the superfluiddensitydensity.
Spielman
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Linear gaugepotential
1, 2 , 0, 0 , 1, 2L Lk k 3 states mixed by Raman beams
Nature 462,628, 2009
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Proposal by N. Cooper and Z. Hadzibabic PRL 104, 030401 Azimuthal gauge potential
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Significance of : it is as if the optical field causes the *l
A shift in the min is equivalent to an azimuthal vector potential
g plaboratory frame to behave as a frame of reference that is rotating with angular frequencyg g q y
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