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 Connecti
on 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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Thermal Fluctuations
At temperature T
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All occupied states give same condensate fraction
Can take one typical occupied state as
representative of density matrix
As T changes band moves to different energy
Typical state gives different f
All occupied states gives same f to 1/vN
Drop subscript j to simplify notation
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Quantum Fluctuations
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widthh/L
BEC
n(p)
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Temperature dependence At T 0 , ?0(r,s)
must be delocalised over volume f0V and
phase coherent.
For T gt TB occupied states ?j(r,s) must be
either localised or phase incoherent.
What is the nature of the wave functions of
occupied states for 0 lt T lt TB?
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BASIC ASSUMPTION

• ?(r,s) b(s)?0(r,s) ?R(r,s)
• ?0(r,s) is phase coherent ground state
• ?R(r,s) is phase incoherent in r
• b(s) ? 0 as T ? TB for typical occupied state
  
• ?R(r,s) ? 0 as T ? 0

• Gives correct behaviour in limits T ? TB, T ? 0
• True for IBG wave functions.
• Bijl-Feynman wave functions have this property
• 4. Implications agree with wide range of
  experiments

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TR(r,s) is sum of terms containing r Phase
incoherent in r rC 1/?k 5 Å in He4 at 2.17K
Fraction of terms in b(s) is (1-M/N) as N ? ? M
? N T(r,s) is phase incoherent (T ? TB) M ? 0
T(r,s) is phase coherent (T ? 0)
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Consequences
?(r,s) b(s)?0(r,s) ?R(r,s)
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Microscopic basis of two fluid behaviour
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Momentum distribution and liquid flow split into
two independent components of weights wC(T),
wR(T).
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Bijl-Feynman wR determined by number of
excitations

• True to within term N-1/2
• Only if fluctuations in f, ?S and ?N are
  negligible.
• Not in limits T ?0 T ? TB

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Superfluid has extra Quantum pressure
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SR(q) ? S0(q) ? ?0(r,s) and ?R(r,s) ? 0 for
different s
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For s where ?0(r,s) ? 0 7 free volume
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Phase coherent component ?0(r,s)
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Phase incoherent component ?R(r,s)
s such that ?R(r,s) is not connected Localised
phase incoherent regions. Localised quantum
behaviour over length scales rC 5 Å No MQE or
quantised vortices
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Excitations
Normal fluid - momentum of excitations is
uncertain to h/rC Superfluid - momentum can
be defined to within h/L
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Only true in presence of BEC
Landau criterion vC (?/q)min
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Summary
BASIC ASSUMPTION
?(r,s) b(s)?0(r,s) ?R(r,s)
Phase coherent ground state
Phase incoherent

• Has necessary properties in limits T?0, 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
Existing microscopic theory does not provide even
qualitative explanations of the main features of
neutron scattering data
This is the only experimental evidence of the
microscopic nature of Bose condensed helium.
Theory given here explains quantitatively all
these features
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