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Chaire Européenne du College de France (2004/2005)

Sandro Stringari

Lecture 514 Mar 05

Bose-Einstein condensation in low dimensions

Previous lecture. Fluctuations of the order parameter.Quantum fluctuations and BEC depletion.Thermal depletion. Shift of critical temperature due to interactions

This Lecture- Theorems on long range order. Algebraic decay in low D. - Mean field and beyond mean field. - Collective oscillations in 1D gas.

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In low D role of thermal and quantumfluctuations is enhanced.

FIRST EXAMPLE: thermal fluctuations in ideal Bose gas

Number of thermal particles (coincides with N if ) ∫= −

)()( µεβ

εε gdNT 0≠µ

density of single-particle states)),(()2(

)( 0 prdrdpg D εεδπ

ε −= ∫∫ h

)(2/),( 20 rVmppr ext+=ε

−1e

semi-classical s-p energy

In the presence of BEC (where ) : 0min 0 == εµ ∫ −

=1)(

βε

εεegdNT ε

ε )(gTkB0→ε

In uniform gas ( ) one finds and integral diverges for D<3 (infrared divergence).

12/)( −∝ DDLg εε0=extVTN

Absence of Bose-Einstein condensation

Harmonic trapping changes energy dependence of )(εg

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SECOND EXAMPLE: quantum fluctuations in Bogoliubov gas at T=0:

In Lecture 4 we have shown that Bogoliubov theory, based onassumption of Bose-Einstein condensation, yields infrared divergence for momentum distribution of uniform gas.1/p behaviour is consequence of the interactions.

ensured in D=1 since integral exhibits infrared divergence

ppn /1)( →

Absence of Bose-Einstein condensation

Normalization condition cannot be )(0

0 pndpNN D∫+

+=

It is possible to generalize results on thermal and quantum fluctuations to more general framework (beyond ideal or mean field picture) with the help of theorems of quantum statistical mechanics

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THEOREMS ON LONG RANGE ORDER (LRO)

thermal fluctuations

Hohenberg-Mermin-Wagner (HMW) theorem:Absence of LRO in 1D and 2D at T>0

Quantum generalization of HMW theorem:Absence of LRO in 1D at T=0

quantumfluctuations

][1, mnmnnm

ECA EE

mAnnCmEE

mCnnAme

Qm

−+

−=

++−∑+βχ

HMW theorem (1966,1967) is based on a series of inequalities starting from Schwartz inequality

where

Is static response relative to operators A and C (<A>=<C>=0).

2ACCCAA

χχχ ≥++

Choosing C=[H,B] and using completeness relation Schwartz inequality becomes

2],[]],[,[ ><>≥< ++ BABHBAA

χ

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{ }AABTkAA +>≥< + χ2,Using fluctuation-dissipation theorem

One finally finds

Bogoliubov inequality{ } 2],[]],[,[, ><>≥><< ++ BATkBHBAA B

- Exact inequality of statistical mechanics (provides rigorous bound for the fluctuations of A in terms of auxiliary variable B)

- Holds at thermal equilibrium and is useless at T=0

Hohenberg theorem applies Bogoliubov inequality to choice ( )qp h=

{ }[ ][ ]

[ ] 00

2

,

,

12,

NaBAmpNBHB

aaAA

aaB

aA

pp

k kqkq

p

==

=

+=

==

=

+

++

++

+ ∑ hρ

>=< +ppp aan particle occupation number

normalized to ∑ =p p Nn

particle operator (annihilates particle with momentum p)

density operator (q-component)

f-sum rule (model independent)

Bogoliubov prescription (gauge symmetry breaking)

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Bogoliubov inequality yields

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2 −≥nn

pmTkn Bp V

Nn 00 = condensate density

diverges in 2D and 1D

No BEC in 2D and 1D

Divergent behavior at low p characterizesBEC systems at T>0

ppn∑

- Similar theorem holds for ferro- and anti-ferromagnetic order (Mermin Wagner theorem, 1966)

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Generalization of HMW theorem to T=0(Pitaevskii and Stringari,1991)

Use uncertainty type inequality { } { } 2],[,, ><>≥><< ++ BABBAA

{ } 2],[2]],[,[, ><>≥><< ++ BATkBHBAA BWith respect to Bogoliubov Inequality

Uncertainty inequality- does not require thermal equilibrium- does not involve H explicitly- sensitive to quantum fluctuations (useful at T=0)

{ } )(22, qNSBB

aaB

aA

qq

p pqpq

p

>≡<>=<

==

=

−+

++

+ ∑ρρ

ρ h

Same choice for A and B as in Hohenberg theorem:

S(q) static structure factor (density fluctuations)

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0)/()12( npSnp ≥+ hUncertainty inequality yields

At T=0 result can be further simplified using inequality

mcqq

mqqS q 2

)(4

)( 0

22 hh→≈≤ χ

quantum fluctuations(Gavoret-Nozieres, 1964)

cfr Hohenberginequality n

npmTkn Bp

020 ≥→

2/1)( mcq →χ

nn

pmcnp 0

0 2≥→

(compressibility sum rule)

thermal fluctuations

No BEC in 1D at T=0

Present proof assumes only finite compressibility

∑p

pnSince diverges in 1D

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What happens to 1-body density matrix when BEC=0 ?

Decay of long range order can be investigated developing formalism of quantum hydrodynamics (QHD)QHD provides link between theory of superfluidityand long range behaviour of 1-body density matrix

Classical equations of irrotational hydrodynamics can be written as:

δρδφ

δφδρ H

tH

t−=

∂∂

=∂∂ , ∫

+∇= )()(

21 2 ρφ edrH φ∇=v

Phase of order parameter is related to velocity potential by .and are conjugate variables.

Can be quantized according to rulewhich transforms classical into quantum hydrodynamics.Procedure limited to the study of macroscopic phenomena at low T

φ)/( hmS =φ ρ

)'()]'(ˆ),(ˆ[ rrirr −−= δρφ h

with and

- Differently from Bogoliubov theory, quantum HD is applicable also to highly correlated fluids. Important examples:- Belyaev decay of phonons in superfluid Helium, - Fluctuations in low D.

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Quantization of velocity potential quantization of the phase

Access to long range behaviour of 1-body density matrix

By writing (only quantum fluctuations of the phase are relevant at large distances, modulus of field operatorIs safely approximated by its classical value)

2/))'(ˆ)(ˆ(0

))'(ˆ)(ˆ(0

)1( 2

)',( >−<−−− >=<= rSrSrSrSi enenrrn

)(ˆ0)(ˆ rSienr =Ψ

can be written as))()0((

0)1( )( sensn χχ −−=

with (phase correlation function)>=<−= )'(ˆ)(ˆ)'( rSrSrrsχ

average value ofgaussian distributionof phase fluctuations

Long range behaviour of determines long range behaviour of 1-body density matrix

)(sχ

Phase correlation function (prediction of quantum HD, holds at large s and low T):

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Next step: evaluation of expectation value by accounting for thermal bath + quantum fluctuations of HD phonons(velocity potential, and hence phase operator, can be writtenin terms of creation and annihilation phonon operators)

>< )'(ˆ)(ˆ rSrS

peNdpcms

sip

pD

D h

h

/2

21

)2()(

⋅

+= ∫ πρ

χ1/ )1( −−= Tkcp

pBeNwhere

number of thermal phononsthermal fluctuations quantum fluctuations

In 3D at large distances and

approaches finite value (long range order)

0)( →sχ ))()0((0

)1( )( sensn χχ −−=

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Quasi-long range behaviour in 1D

In 1D phase correlation function diverges at large distances and consequently 1-body density vanishes (no long range order).

))()0((0

)1( )( sensn χχ −−=

)(sχ

one findsν

ν

−→

∞→

∝

∝1

0

)1(

/1)(

/1)(

ppn

ssn

p

s with 12 n

mchπ

ν =

Exponent of power low decay fixed by sound velocity

(Efetov and Larkin, 1975; Haldene, 1981)

However at T=0 divergence of is logarithmic and 1-body density decaysas a power law (algebraic long range order). Integration at T=0 yields ξπ

χχ sn

mcs ln2

)()0(1h

=−

LNn /1 =

)(sχ

If mean field picture holds (order extends over large distances and GP is applicable).

1<<ν If concepts of long range order and classical field should be abandoned

1≈ν

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How to reach low dimensional configurations

In magnetic traps usual configurations correspond to 3D Thomas-Fermi regime, characterized by condition

If condition is not satisfied, Thomas-Fermi result is no longer valid and motion is frozen in 1 or 2 directions

zyxho

ho aaN ωωωωµ hhhh ,,)15(

21 5/2 >>=

zyx ωωωµ hhh ,,>>

If and the degrees of feedom along zare frozen (atoms occupy lowest harmonic oscillator level) 2D

⊥>> ωµ h zz ωωµ hh <<− 2/

Analogously, if and the degrees of freedom along x,y are frozen and the sytem becomes 1D

zωµ h>> ⊥⊥ <<− ωωµ hh

It is now possible to achieve low D configurations with optical methods:1D optical lattices produce 2D discs. 2D optical lattices produce 1D tubes.

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Evidence of achivement of 1Dregime: expansion of the gas

(MIT: Gorlitz et al. 2001; ENS: Schreck et al. 2001)

During expansion radial and axial radii behave differentlyin deformed traps (see Lecture 2):For large times ( ) radial expansion follows law while axial radius changes slowly : ( )

tRtR ⊥⊥⊥ = ω)(

tZR

tZtR

⊥⊥⊥ = ω

)()(

ZtZ ≈)(

1>>⊥tω

3D cigar1D behaviour1<<tzω

- In 3D initial aspect ratio is and hence , independent of N.

- In 1D one has while Z increases with N

.

⊥⊥ = ωω // zZR

⊥⊥ ≈ aREmergence of N dependence in aspect ratio is indicator of one-dimensional effects

ttZtR zω=⊥ )(/)(

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Mean field predictions for low D configurations

From 3D to 1D in the framework of mean field Gross-Pitaevskii theory.(cylindrical geometry with radial harmonic trapping)

Order parameter with one dimensional density and f obeying dimensionless equation

)(/)( 21 ⊥⊥⊥ =Ψ ρρ fan LNn /1 =

⊥⊥

⊥⊥⊥

=

++

∂∂

−∂∂

−ωµπρ

ρρρ hffan 2

12

2

2

421

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∫ =⊥⊥⊥ 1)(2 2 ρρρπ dffunction f is normalized to

Two important limits

11 >>an )1()( 2⊥⊥ −∝ ρρf 2/1

1)(2 an⊥= ωµ hand

11 <<an )2/exp()( 2⊥⊥ −∝ ρρf and )21( 1an+= ⊥ωµ h

square root

linear

3D cigar

1D mean field

Both configurations have 1D geometrical character

radial TF condition

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Solitons in 1D mean field regime (grey and bright)

- Solitons are stationary solutions of GP equation. - Their existence is ensured by non linearity. - 1D condition is crucial in order to ensure stability against density fluctuations along transverse direction.

- Nature of solitons crucially depends on sign of scattering length:

a>0: grey solitons

a<0: bright solitons

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Grey solitons (a>0)

In uniform ( ) configurations one finds exact non uniformsolutions of time dependent Gross-Pitaevskii equation (Tsuzuki 1971)

constVext =

tielvtz

cu

cvintz µ−

−

+=Ψ tanh),( 122 vcu −=

- is average 1D density- is sound velocity of uniform medium ( ) - is velocity of propagation of soliton - is width of soliton

1n

vmul /h=

c µ== 112 ngmc D

- Soliton is stationary in frame moving with velocity - Solitons propagate with velocity smaller than sound velocity- Width of soliton increases as cv→

v

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Density of grey soliton:

−+=

lvtz

cu

cvntzn 2

2

2

2

2

1 tanh),(

density vanishes at z=vt if 0(dark soliton)

=v

In the presence of axial harmonic trapping solitons oscillate with frequency Grey solitons behave like particles of mass 2m !(Bush and Anglin, 2000)

2/hosol ωω =

Dark solitons in BEC’s have been generated with phase imprintingat Hannover (Burger et al., 1999) and NIST (Denschlag et al., 2000)

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Bright solitons a<0

Uniform condensates with negative scattering length are unstable in 3D and collapse into much denser configurations.

Gross-Pitaevskii equation admits non uniform 1D solitonicsolution, corresponding to localized wave packet (bright soliton):

tielz

tz µ−Ψ=Ψ/cosh

1)0(),(

with and 112/ ngml Dh= 11)2/1( ng D−=µ

Bright soliton moves freely in space like ordinary particles without dispersion. In the presence of smooth harmonic trappingsoliton is expected to oscillate with the frequency of the harmonictrap: (difference with respect to grey solitons!)hosol ωω =

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Solitons in BEC condensates (a<0)

a=0

a<0

ENS, 2002Khaykovich et al

Rice Univ., 2002Strecker et al.

Train of solitons

Results discussed so far are valid if one can use mean field GP equations.Condition is that order extends over large distances. In 1D one has:

Validity of mean field description21

Dannmc

111 211

2 ππν ==

h

νssn /1)()1( → with

Applicability of 1D mean fieldthen requires two conditions:

111 >>Dan in order to ensure order at large distances ( )

11 <<an in order to be 1D

1<<⊥aaNλ 12

2

>>⊥

aaNλ

1<<ν

aaa D /21 ⊥=

effective 1D scatteringlength (radial harmonic trapping)

⊥= ωωλ /z

If system is also axially trapped ( ) the two conditions become⊥= ωωλ /z

and respectively

622 1015/ ×=⊥ aaNλ24.0/ =⊥aaNλ 001.0=νExample: first 1D experiment at MIT (Gorlitz et al.) with magnetic trapping:

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Beyond mean field in 1D

What happens if long range order condition is not satisfied?111 >>Dan

Answer is provided by Lieb-Liniger (1963) who solved the 1D many body problem with the Hamiltonian

2

2

1

2

122

⊥

==aa

mmag

DD

hh

∑∑∑ +===−+−=

N

ij jiN

iDN

jj

zzgdzd

mH

1111 2

22

)(2

δh where

is 1D coupling constant

111 >>Dan

Lieb-Liniger solution provides complete behaviour of equation of state as a a function of dimensionless parameter . Two opposite limits:

Dan 11

111 <<Dan

long range order, 1D mean field(previous slides)

Tonks-Girardeau gas of impenetrable bosons

111 2 angn ⊥== ωµ h

linear

21

22

2n

mhπµ =

quadratic

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Low density does not mean small interaction effects !

Low density limit has simple physical interpretation in 1D: scattering properties are determined by reflection probability R. R tends to unity when energy of colliding particles tends to zero. At T=0 energy tends to zero as the density tends to zero.

GAS OF IMPENETRABLE BOSONS (Tonks-Girardeau limit)

In Tonks-Girardeau limit sound velocity takes simple form . Power law coefficient

of 1-body density approaches 1/2

and momentum distribution diverges at small p like

mnc /1hπ=

ppn p /1)( 0 ∝→

12/ nmc hπν =

Girardeau (1960) has proven that eigenfunctions of gas of impenetrable bosons are exactly mapped into free gaswave functions of independent fermions.(recent work by Girardeau, Olshanii, Shlyapnikov etc.. )

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Girardeau mapping between interacting bosons and ideal Fermi gas

,....),(....),( 21,21 xxxx FB ψψ =

Slater determinant

Ground state.Easy generalization to excited states

In Tonks-Girardeau limit - local properties (density distribution, density correlations etc.) of 1D Bose gas are equal to the ones of 1D ideal Fermi gas.

- non local properties (momentum distribution) are different

For example:- 2-body and 3-body density probabilities vanish at short relative distance, differently from behaviour of 3D dilute Bose gas ( suppression of 3-body recombination)

- while momentum distribution of Fermi gas is step function,in Tonks-Girardeau gas one finds (recent expevidence in 1D periodic optical trap (Paredes et al, 2004)

ppn /1)( ∝

(NIST,seeW. Phillips talk)

DD KgK 333

13 )0(=

Theory byGangard, Shlyapnikov et al. 2002, 2003

Dan 11

2=γ

1D mean field

Tonks-Girardeau

exp with 2Doptical lattices

)0(2g

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3-body losses in 1D Bose gas

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Equation of state of different 1D configurations

Different 1D regimes different equations of state

- 3D cigar, elongated TF configuration,

11 >>an2/1

1)(2 an⊥= ωµ h

- 1D mean field

11 <<an 111 >>Dan 12 an⊥= ωµ haaa D /21 ⊥=

- Tonks-Girardeau

11 <<an 111 <<Dan21

22

2n

mhπµ =aaa D /2

1 ⊥=

Different equations of state - different density profiles (recent experimental evidence for Tonks-Girardeau density profiles, Kinoshita et al. 2004)

- different collective frequencies

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Equation of state and collective frequencies in harmonically trapped 1D configurations

)]1(2[2

22 −+= kkzk γωω Menotti and

Stringari, 2002...21

11 ++= −− kk azznn δγ

γµ 1cn=

))(( 02

2

nn

nnt

δµδ∂∂

∇∇=∂∂

For polytropic equation of state one finds analytic solutions of 1D Hydrodynamic equationin the presence of harmonic trapping

zωω =

22 )2( zωγω +=

- k=1 (center of mass)independent of eq. of state

- k=2 lowest compression mode

k=2

- 3D cigar

- 1D m.f.

- TG gas

zωωγ 2/52/1 ==

zωωγ 31 ==

zωωγ 22 ==

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This Lecture. BEC in low dimensions. Theorems on long range order. Algebraic decay in low D.Mean field and beyond mean field. Collective oscillations in 1D gas.

Next Lecture. Moment of inertia and superfluidity.Irrotational vs rotatational flow. Moment of inertia and scissors mode. Expansion of rotating BEC.