Download - Lecture 3 BEC at finite temperature
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Lecture 3BEC at finite temperature
Thermal and quantum fluctuations in condensate fraction.
Phase coherence and incoherence in the many particle wave function.
Basic assumption and a priori justification
Consequences
Connection between BEC and two fluid behaviour
Connection between condensate and superfluid fraction
Why BEC implies sharp excitations.
Why sf flows without viscosity while nf does not.
How BEC is connected to anomalous thermal expansion as sf is cooled.
Hoe BEC is connecged to anomalous reduction in pair correlations as sf is cooled.
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jj
j FTTf )()(
Thermal Fluctuations
2
),(1
)0( rdsrsdV
nF jjj
Boltzman factorexp(-Ej / T)/Zj
22 )( j
jj fFTf
At temperature T
Δf ~1/√NBasic assumption;
(√f is amplitude of order parameter)
Fj = f ± ~ 1/ √ N
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dEEEgE jj
j )()(
All occupied states give samecondensate fraction
g(E)
η(E)
E
ΔE, Δf ~1/ √ N-1/2
As T changes band moves to different energy
“Typical” state gives different f
Can take one “typical” occupied stateas representative of density matrix
Drop subscript j to simplify notation
All occupied states gives same f to ~1/√N
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Quantum Fluctuations
2
),(1V
dV
F rsr sss dfP )()(
sss dfFPF 22 )()(
V
dV
f rrs s )(1
)( V
dP rsrs2
),()()(
),()(
s
srrS
P
F = f ±~1/√N f(s) ~ f ± 1/√N
ΔF ~1/√N f(s) = F±~1/√N
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width~ħ/L
Weightf
to ~1/√Nfor any
state and any s
Delocalised function of r(non-zero within volume > f V)J. Mayers Phys. Rev. Lett. 84 314 (2000),Phys. Rev.B 64 224521, (2001)
)(rS
2
).exp()()( rrprp S din
Phase correlations in r over distances ~L)(rS0~)( rrS dotherwise
BEC n(p)
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Phase coherent
Condensate
Phase incoherent
No condensate
rC
~1/rC
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Temperature dependence
At T = 0 , Ψ0(r,s) must be delocalised over volume ~ f0V and phase coherent.
For T > TB occupied states Ψj(r,s) must be either localised or phase incoherent.
What is the nature of the wave functions of occupied states for 0 < T < TB?
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BASIC ASSUMPTION
Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)
• Ψ0(r,s) is phase coherent ground state• ΨR(r,s) is phase incoherent in r • b(s) 0 as T TB for typical occupied state
• ΨR(r,s) 0 as T 0
1. Gives correct behaviour in limits T TB, T 0 2. True for IBG wave functions.3 Bijl-Feynman wave functions have this property4. Implications agree with wide range of experiments
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),(),(),( 0 srsrsr
k
nN
nn
k
riksr
1
).exp(),(
Bijl-Feynman wave functionsJ. Mayers, Phys. Rev.B 74 014516, (2006)
),()(),( srsbsr R
b(s) is sum of all terms not containing r = r1
Phase coherent in r.
Fraction of terms in b(s) is (1-M/N) as N M N Θ(r,s) is phase incoherent (T TB)M 0 Θ(r,s) is phase coherent (T 0)
• nk = number of phonon-roton excitations with wave vector k.•M = total number of excitations• sum of NM terms.
k
knM
ΘR(r,s) is sum of terms containing r Phase incoherent in rrC ~1/Δk ~ 5 Å in He4 at 2.17K
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2
0
22
),()(1
),(1
rdsrsdsbV
rdsrsdV
f
Consequences
Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)
• If Δf ~1/N1/2
Nwsb C /1~)(2
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rdsrrdsrwrdsr RC
22
0
2),(),(),(
MacroscopicSystem
X)(sX
dr N/1~
),(0 sr
),( srR
Microscopic basis of two fluid behaviour
NsXrdsrrdsrwrdsr RC /1~)(),(),(),(22
0
2
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Momentum distribution and liquid flow split into two independent components of weights wC(T), wR(T).
rdsrrdsrwrdsr RC
22
0
2),(),(),(
2
).exp(),(1
)( rdrpisrsdV
pn
)()(0 pnwpnw RRC
Parseval’stheorem
wR = 1- wC.
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RRC EwEwE 0
00 VC TES
Thermodynamic properties split into two independentcomponents of weights wc(T), wR(T)
Bijl-Feynman wR determined by number of “excitations”
wc(T) = ρS(T) wR(T) = ρN(T)
• True to within term ~N-1/2
• Only if fluctuations in f, ρS and ρN are negligible.• Not in limits T 0 T TB
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)0()()( fTTf S
)0(
)(
f
Tf
o o T. R. Sosnick,W.M.Snow and P.E. Sokol Phys. Europhys Lett 9 707 (1989).X X H. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000).
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NNST
PPV
EP
0
Superfluid has extra “Quantum pressure”
PN = PB
)0(
)(
T
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)()(0 qSqS RNS
sdrdrqiN
srqS ni
1
2
2
1 ).exp(1
),()(
α < 1 → S less ordered than SR
SR-1
S-1
q
1)(
1)(
qS
qS
R
1 NS
]1)[(1)( 0 TT S
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SR(q) S0(q) → Ψ0(r,s) and ΨR(r,s) 0 for different s
α(T)α0
V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 2009 (1979).
]1)[(1)( 0 TT S
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For s where Ψ0(r,s) 0 ~7% free volume
Why is superfluid more disordered?
Assume for s where ΨR(r,s) 0 negligible free volume
Ground state more disordered
J. Mayers Phys. Rev. Lett. 84 314 (2000)
Quantitative agreement with measurement at atomic size and N/V in liquid 4He
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Phase coherent component Ψ0(r,s)
s such that Ψ0(r,s) is connected
nrdr 2).(
(Macro loops)
Quantised vortices, macroscopic quantum effects
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s such that ΨR(r,s) is not connectedLocalised phase incoherent regions.Localised quantum behaviour over length scales rC ~ 5 ÅNo MQE or quantised vortices
Phase incoherent component ΨR(r,s)
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Phase incoherent Regions of size ~rC
Normal fluid - momentum of excitations is uncertain to ~ ħ/rC
Superfluid - momentum can be defined to within ~ ħ/L
Excitations
)()(1
),(2
ff
f EEAN
S qq
srrqsrsrq ddiNA ff ).exp(),(),()( *
Momentum transfer = ħqEnergy transfer = ħω
|Aif(q)|2 has minimum widthΔq ~ 1/rC
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Anderson and StirlingJ. Phys Cond Matt (1994)
q (Å-1)
ε ( deg K)
0 < T < TB
h/rC
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Landau Theory
Basic assumption is that excitations with well definedenergy and momentum exist.
Landau criterion vC = (ω/q)min
Normal fluid vC = 0
ω
q
Only true in presence of BEC
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Summary
BASIC ASSUMPTION Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)
Phase coherentground state
Phase incoherent
• Has necessary properties in limits T0, T TB
• IBG, Bijl-Feynman wave functions have this form
•Simple explanations of•Why BEC is necessary for non-viscous flow
Why Landau theory needs BEC.
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Summary
Theory given here explains quantitatively all these features
Existing microscopic theory does not provide even qualitativeexplanations of the main features of neutron scattering data
Why the condensate fraction is accurately proportional to the superfluid fraction
Why spatial correlations decrease as superfluid helium is cooled
Why superfluid helium is the only liquid which contains sharp excitations
Why superfluid helium expands when it is cooled
This is the only experimental evidence of the microscopic nature of Bose condensed helium.