Sandro Stringari Lecture 4 7 Mar 05 · Sandro Stringari Lecture 4 7 Mar 05 Fluctuations of the...
Transcript of Sandro Stringari Lecture 4 7 Mar 05 · Sandro Stringari Lecture 4 7 Mar 05 Fluctuations of the...
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Chaire Européenne du College de France (2004/2005)
Sandro Stringari
Lecture 47 Mar 05
Fluctuations of the order parameter
Previous Lecture. Equation for the order parameter.Gross-Pitaevskii theory. Healing length. Time dependent theory. Bogoliubov equations.
This Lecture.- Quantum fluctuations, BEC depletion and kinetic energy- Beyond mean field effects on collective oscillations- Thermal depletion. - Shift of critical temperature due to interactions
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Fluctuations of order parameter
)(ˆ)()(ˆ 0 rrr Ψ+Ψ=Ψ δField operator
Total density of the system:
>ΨΨ<+Ψ>=ΨΨ=< ++ )(ˆ)(ˆ)()(ˆ)(ˆ)( 20 rrrrrrn δδ
0n nδ
Experiments based on imagingtechniques measure total density )()( 0 rnrn ≠
Physical origin of nδ
- Quantum fluctuations (small in very dilute Bose gases)- Thermal fluctuations (vanish at T=0)
Using plane wave representation of fieldoperator the many-body Hamiltonian
takes the form
)(ˆ)(ˆ)(ˆ)(ˆ2
)(ˆ)(ˆ2
ˆ 22
rrrrdrgrrdrm
H ∫∫ ΨΨΨΨ+Ψ∇Ψ−= +++h
h/1ˆ)(ˆ ripp
p
eV
ar ⋅∑=Ψ
∑∑ +−
++
+ +=p
ppqpqpppp
aaaaVgaa
mpH
2121ˆˆˆˆ
2ˆˆ
2
2
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Quantum fluctuations in T=0 uniform Bose gas(brief summary of Bogoliubov theory)
Zero-th order term in H is obtained by keeping only termswith p=0 and using Bogoliubov replacement
Next approximation consists of keeping terms with two operatorswith + normalization condition
Hamiltonian can then be rewritten as:Naaaa
p pp =+∑ ≠++
000 ˆˆˆˆ
pp aa ˆ,+)
NNaa ≈≡=+000 ˆˆ
pp aa ˆ,+) 0≠p
2/gnNEH ==) (mean field energy)
)ˆˆˆˆ(21ˆˆ)
2(ˆ
2
0 pppppppaaaagnaagn
mpconstH −
+−
++≠
++++= ∑
4Hamiltonian is diagonalized by Bogoliubov transformation
+−−+= ppppp bvbua ˆˆˆ *
'' ,],[pppp bb δ=+
transforms particles into quasi-particles. Diagonalization is ensured by choice
2222
2
2)( cp
mpp +
=ε
2/122
21
)(22/,
±
+±=
pmcmpvu
ε maggnmc/4 2
2
hπ=
=
Results for u,v and coincide with predictionsof linearized time dependent Gross-Pitaevskii equation
h/εω =
Bogoliubov transformationreduces H to the form
bbpEH ˆˆ)( +∑+= ε pppgs
contains non trivial ultraviolet divergence. Its evaluation requires renormalization of effective potential)gsE
Physical observables can be expressed in terms of quasi-particle operators. Ground state is vacuum of quasi-particles 0=vacbp
)
Behaviour of :pn∞→
→pp 0
4/1
2/
pn
pmcn
p
p
→
→
is convergent in 3D. Gives number of atomsout of the condensate:
∑≠
=0p
pnNδ
2/13 )(38 na
NN
πδ
=
- Quantum depletion fixed by gas parameter. - Depletion should be small in order to apply Bogoliubobv theory.
Quantum depletionof the condensate
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)(22/ˆˆ
222−
+=>==< +
pmcmpvaan pppp ε
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1° Example: Momentum distribution ( T=0): )ˆˆ)(ˆˆ(ˆˆ ** +
−−−−++ ++= pppppppppp bvbubvbuaa
0≠p
because of two-body interactions0≠pn
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Due to behaviour at large p, Bogoliubov theorypredicts divergence for kinetic energy in 3D.
4/1 p
Differently from total energy (kinetic + potential), whichcan be safely calculated in terms of scattering length, kinergy energy cannot be calculated in Bogoliubovtheory. Bogoliubov theory fails when . Kinetic energy depends on microscopic details of the force.
ap /h≈∞=∑
ppn
mp )(2
2
Behaviour of kinetic energy in a dilute Bose gas
Hard sphere Realistic microscopic potential
a aScattering length
gn21
gn21
mEnergy ( )ag
24 hπ=
gn21
≠gn21Kinetic energy
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Quantum depletion in trapped Bose gas at T=0
32/13 )0(85))((
38)( anarnrdrn
NN
TFTFTFπ
πδ
== ∫
local density approximation
))((1)( 0 rVg
rn extTF −= µ
Effect is small in available configurationsCan become larger with Feshbach resonance
%1≈5/26/13 )(1.0)0(ho
TF aaNan =
Can one extract quantum depletion through measurement of density profile?
N=70 Helium droplet
total density
condensate
Quantum depletion arises from fluctuationsof the condensate. Separation in space between condensate and quantum depleted components is not possiblebecause they fully overlap.
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2° example: density fluctuations
S(q)=1in T=0 idealBose gas
Density-density correlations arerelated to static structure factor >=< −qqqNS ρδρδ ˆˆ)(
By using Bogoliubov approach one can express density operatorIn terms of particle (and hence quasi-particle operators):
)ˆˆ)(()ˆˆ(ˆˆˆ qqqqqqkqkkq bbvuNaaNaa −+
−−++
+ ++=+==∑ρ
Using Bogoliubov results for u and v one finally finds (uniform gas)
∞→→→→
==qasqasmcq
qmqqS
102/
)(2)(
22 hh
εphonon regime
free particle regime22222 )2/()( cpmpp +=εBogoliubov dispersion law
Density fluctuations are strongly quenched in phonon regime
Result for S(q) coincides with predictions given by densityresponse function calculated in time-dependent GP theory
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ideal gas
interacting gas
Response function
)),(),((),('' ωωπωχ −−= qSqSq
measured at MIT(Stamper-Kurn et al. 1999)
in situafterexpansion
- Result for structure factor can be generalized to trappedBEC gases (local density approximation).
- Structure factor has been measured with Bragg spectroscopy experiments (inelastic photon scattering)
Static structure factor
∫= ),()( ωω qSdqNS h
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Beyond mean field effects:density profile and collective oscillations
Quantum correlations in interacting Bose gases modify equation ofstate with respect to mean field prediction)(nµ gnn =)(µ
Inclusion of quantum fluctuations in Bogoliubov theory(+ renormalization of coulpling constant) yieldsfirst correction to equation of state
)3321()( 3nagnnπ
µ +=Lee-Yang-Huang (1957) beyond mean field correction
New equation of state modifies both equilibrium and dynamic properties
0)()( µµ =+ rVn extEquilibrium density profile defined by
meanfield TFdensity
))((1)( 0 rVg
rn extTF −= µ3)(332)()()( arnrnrnrn TFTFTF π
−=
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Beyond mean field effects: collective oscillations
Beyond mean field corrections in density profile are difficult to measure(would require 1% accuracy)
More promising possibility concerns the study of collective oscillations.
Change in equation of state change in frequency of hydrodynamic modes (easily measurable within 1% accuracy)
Corrections in collective frequencies due to beyond mean field affectseasily evaluable using perturbation theory in hydrodynamic equations
0))(21(
0)(
2 =++∇+∂∂
=∇+∂∂
extSS
S
Vnmvvt
m
nvnt
µ)
3321()( 3nagnnπ
µ +=
12
))((2
2
nn
nnt
m δµδ∂∂
∇∇=∂∂Linearized hydrodynamic equations:
In trapped gas beyond mean field effects affect both equilibrium density and chemical potential. One finds:
)(316)( 2/32
2/32 nnangnn TFTF δ
πδδω ∇−=∇∇+ ))((1)( 0 rV
grn extTF −= µ
Left hand side: HD equation in mean field regimeRight hand side: perturbative correction due to beyond mean field
Perturbative correction to collective frequencies is obtained bymultiplying HD eq. by mean field value and integrating by parts
δω*nδ
In uniform gas one recovers Belayev (1958) result for renormalized sound velocity
)161( 32 nagnmcπ
+=
13
Result for beyond mean field shift of collective frequencies:
∫∫ ∇
−=nndr
nnndr
ma TF
δδ
δδ
ωπωδω
*
2/3*2
2
2/3 )(
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where are density variations and frequencies of hydrodynamics equations in mean field (Gross-Pitaevskii) regime.
ωδ ,n
- Surface modes are unaffected by beyondmean field correction (insensitive to equation of state)
- Shift in m=0, l=0 mode (compression) in spherical trap:
and
02 =∇ nδ
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53 Rrn −∝δ hoωω 5= 3)0(
12863 anπ
ωδω
=
Pitaevskii and Stringari, 1999Braaten and Pearson, 1999
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1616968
21)(
)()0(12863
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2
3
0
0
+−
+±=
=
±
±=
=
λλλλ
λπωδω
f
fanm
m
Result is easily generalizedto axially deformed trap:
⊥= ωωλ /z
Application to radial compression mode in cigar geometry
ENS experiment (2002):
Theory predicts
Compared to mean field value
510,2.5,26.1 === Nnmamaho µ55/26/13 102.2)/(1.0)0( −×== hoTF aaNan
⊥= ωω 007.2
⊥ω2
( , and ) 6/5=+f1<<λ ⊥= ωω 2 beyond mean field effect?
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Quantum depletion is small in usual geometriesIt can be increased significantly by changing geometry of configurationExamples: - Low dimensional configurations (Lecture 5).
- Periodic potentials (Lecture 9) - Double well potential
Quantum depletion in double well potential
Hamiltonian in double well potential can be written in boson Hubbard form
)ˆˆˆˆ(2
)ˆˆˆˆˆˆˆˆ(4
abbabbbbaaaaEH JC ++++++ +−+=δ
tunneling between two wells
Eigenstates of are
(ground state) with energy
(excited state) with energy
)ˆˆˆˆ)(2/( abbaH Jsp++ +−= δ
vacba )ˆˆ(1++ +=ϕ 2/0 Jδε −=
2/1 Jδε +=vacba )ˆˆ(0++ −=ϕ
on site energy
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Let us introduce operators and)ˆˆ(ˆ0 baa +=21 )ˆˆ(
21ˆ1 baa −=
Bogoliubov prescription then corresponds to setting NNaa ≈== +000 ˆˆ
By retaining only terms quadratic in and usingrelationship , the Hamiltonian takes the form
+11 ˆ,ˆ aa
Naaaa =+ ++1100 ˆˆˆˆ
Boson Hubbard Hamiltonian can be rewritten in the form)ˆˆ2ˆˆˆˆ(
8ˆˆ 11111111 aaaaaaEaaconstH C
J++++ ++++= δ
++= ββ ˆˆˆ1 vuaH is diagonalized by Bogoliubov transformation
with and , ±±= CJvu2/1
21
24/
+
J
NEε
δ)2/( CJJJ NE+= δδε
Hamiltonian takes diagonal form: ββε ˆˆ ++= JconstH
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Excitation energy approaches :
“single particle” regime ( , Rabi frequency) when
“plasmon” regime ( ) when
JJ δε = NE JC /δ<<
)2/( CJJJ NE+= δδε
2/CJJ ENδε = NE JC /δ>>
Both regimes are described by Bogoliubov theoryprovided quantum depletion is small (they correspondto single particle and phonon regimes in uniform gas)
Quantum depletion is associated with occupation of state 1. One finds
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24/ˆ 2
110 −+
=>==< +
J
CJ NEvaaNε
δδ
Condition of applicability of Bogoliubov theory ( ) isequivalent to condition (compatible with )
NN <<0δ
JC NE δ<< NE JC /δ>>
- Condition can be easily violated by reducing tunneling parameter, through increase of height of barrier.
- In the regime Bogoliubov theory is no longer valid and system jumps into BEC fragmented configuration (Lecture 1)
JC NE δ>>
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Thermal depletion of the condensate (ideal gas)
)()()( 0 rnrnrn T+=Depletion in ideal Bose gas is due to thermal effect:
Calculation in harmonic trap yields ∫ −== ])/(1[)()( 300 CTTNrdrnTN
3/194.0 NkT hoC ωh=with
Differently from uniform gas BEC is visible in coordinate space because it is well separated from thermalcomponent (crucial feature of harmonic trapping)
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ho
BT m
TkRω
≈Width of thermal component :
Width of condensate:ho
ho maR
ωh
=≈ 220
Ratio is large if3/120
2
NTTTk
RR
Cho
BT ≈≈ωh hokT ωh>
5/10 )/15( hoho aNaaR =
- Thermal component scarcely affected by interactions(thermal gas is very dilute)
- Condensate component strongly affected by interactionsIn Thomas-Fermi limit (a>0)
Role of interactions and BEC visibility in coordinate space19
6/1320
2
))0((1anT
TRR
C
T ≈and hence:
thermalBEC
Visibility in coordinate space isreduced with respect to ideal gas, but still since gas parameter is small.
Furthermore shapes of thermalAnd condensate components are different (bimodal distribution)
0RRT >
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Visibility of BEC in momentum space
In superfluid helium information on BEC comes frommeasurement of momentum distribution. Data are availablefrom inelastic neutron scattering at high momentum and energy transfer where impulse approximation (IA) holdsand dynamic structure factor is proportional to n(p):
)()22
)((),(22
pnmp
mqpdpqSIA +
+−= ∫ ωδω h
)(~)()( 0 pnpNpn += δ
BEC quantum depletion at T=0
In practice corrections to IA should be included (final state interactions)
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∫=
−=
=
),,()(
)2
(
)(),(
2
YppndpdpYJ
mq
qmY
YJqmqS
yxyx
hω
ω
scaling variable
At high momenta and energy transfer dynamic structure factorscales according to
Scaling functionmeasured in superfluidhelium (Sokol 1996).Delta peak due BECis smoothed out byfinal state interactionsand instrumental resolution
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Measurement of momentum distribution in trapped Bose gases
In Bragg scattering experiments photons scatter inelastically fromatoms transferring or absorbing large momentum and energy(measured at MIT, Stenger et al. 1999). These measurements probe dynamic structure factor propotional,at high momentum transfer, to momentum distribution.
Width of measured signal is proportional to width of n(p)
Experiment at MIT has provenuncertainty limit RP ∆≈∆ /h
after 3ms
mPq /∆∝∆ν
thermal
BEC
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Visibility of BEC in momentum space: trapped Bose gas
Width of thermal componentis scarcely affected by interactions:
TmkP BT ≈2
IDEAL GAS: width of condensatehom
RP ωh
h≈≈ 2
0
220
3/120
2
NTTTk
PP
Cho
BT ≈≈ωh
Ratio coincides with ratio 20
2 / RRT(in harmonic oscillator momentum and coordinate variables play symmetric role)
ROLE OF INTERACTIONS: width of condensate in momentum space reduced by interactions
00 RP h≈
Ratio strongly enhanced by interactions3/26/1320
2
))0(( NanTT
PP
C
T ≈
Higher visibility of BEC in momentum than in coordinate space
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Effect of interactions on critical temperature
- Many body effect (due to correlations) relevant in uniformsystems (constant density). Even sign of the effect is not trivialRecenty work (Baym et al, 1999; Arnold, Moore 2001, Kashurnikovet al. 2001) predicts positive shift, proportional to 3/1an
- Mean field effect relevant in non uniform systems (constant N)(example: harmonic trapping)
- repulsive interactions, although small, tend to expand the gas, therebyreducing average density and hence the value of critical temperature.
Calculation of mean field effect:Just above critical temperature one can use Hartree Fock-Theory(HF theory can be developed also below , in the presenceof the condensate, not discussed here)
CT
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Hartree-Fock theory describes the gas as a system of statisticallyindependent s.p. excitations governed by the Hamiltonian ( )
gnVm
H extsp 22
22
++∇−=h
Shift of CT
CTT ≥
Factor 2 arises from exchange term(absent in the condensate)
In semi-classical approximation one can write
1]),([exp1
)2( 3 −−= ∫ µεβπ pr
drdpNTh
If )(2
),(2
rVmppr ho+=ε
ideal gas
BEC starts at 0=µ
3/10 94.0 NTkNN hoCBT ωh=⇒=
ideal gas result
If BEC starts at )0(2gn=µ)(2)(2
),(2
rgnrVmppr ho ++=ε
In uniform gas (and hence ) does notdepend on interaction and one recovers ideal gas result
µε −),( prTN
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In non uniform gas one can expand
Yielding, after straightforward integration,
δµµ
δ∂∂
+∂∂
+= TC
TTT
NTTNNN
C0 )0(2gn=δµ
6/10 43.1 N
aa
TT
hoC
C −=δ
Giorgini et al, 1996
Mean field negative shift:small, but measurable effect(Orsay; Gerbier et al 2003):
Many-body effect gives higherorder corrections in harmonictrap.
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This lecture. Fluctuation of the order parameter.Quantum fluctuations and BEC depletion.Thermal depletion. Shift of critical temperature due to interactions
Next 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.