1
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.
2
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
3
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
4
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
5
{ }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)
6
Bogoliubov inequality yields
210
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)
7
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
p pqpq
p
>≡<>=<
==
=
−+
++
+ ∑ρρ
ρ h
Same choice for A and B as in Hohenberg theorem:
S(q) static structure factor (density fluctuations)
8
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
9
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.
10
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):
11
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 −−=
12
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
13
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.
14
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=⊥ )(/)(
15
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
21
21
∫ =⊥⊥⊥ 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
16
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
17
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
18
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)
19
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 =
20
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:
22
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
23
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.. )
24
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
25
3-body losses in 1D Bose gas
26
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
27
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 ==
28
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.