BoseEinstein Condensation and Superfluidity

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Bose-Einstein Condensation and Superfluidity

• Lecture 1. T0
• Motivation.
• Bose Einstein condensation (BEC)
• Implications of BEC for properties of ground
  state many particle WF.
• Feynman model
• Superfluidity and supersolidity.

• Lecture 2 T0
• Why BEC implies macroscopic single particle
  quantum effects
• Derivation of macroscopic single particle
  Schrödinger equation

• Lecture 3 Finite T
• Basic assumption
• A priori justification.
• Physical consequences
• Two fluid behaviour
• Connection between condensate and superfluid
  fraction
• Why sharp excitations why sf flows without
  viscosity while nf does not.
• Microscopic origin of anomalous thermal
  expansion as sf is cooled.
• Microscopic origin of anomalous reduction in
  pair correlations as sf is cooled.

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Motivation
A vast amount of neutron data has been
collected from superfluid helium in the past 40
years.
This data contains unique features, not observed
in any other fluid.
These features are not explained even
qualitatively by existing microscopic theory
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What is connection between condensate fraction
and superfluid fraction?
Accepted consensus is that size of condensate
fraction is unrelated to size of superfluid
fraction
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Superfluid helium becomes more ordered as it is
heated Why?
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Basis of Lectures
J. Mayers J. Low. Temp. Phys 109 135
(1997)
109 153 (1997) J. Mayers
Phys. Rev. Lett. 80, 750 (1998)

84 314 (2000) 92
135302 (2004) J. Mayers, Phys. Rev.B
64 224521, (2001)
74
014516, (2006)
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Bose-Einstein Condensation
TgtTB
0ltTltTB
T0
D. S. Durfee and W. Ketterle Optics Express 2,
299-313 (1998).
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BEC in Liquid He4
f 0.07 0.01
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Definition of BEC
N atoms in volume V Periodic Boundary
conditions Each momentum state occupies volume
h3/V
n(p)dp probability of momentum p ?pdp
BEC Number of atoms in single momentum state
(p0) is proportional to N. Probability f that
randomly chosen atom occupies p0 state is
independent of system size.
No BEC Number of atoms in any p state is
independent of system size Probability that
randomly chosen atom occupies p0 state is 1/N
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Quantum mechanical expression for n(p) in ground
state
What are implications of BEC for properties of ??
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?(r,s)2 P(r,s) probability of configuration
r,s of N particles
?S(r)2 is conditional probability that particle
is at r, given s
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Implications of BEC for ?S(r)
?S(r) non-zero function of r over length scales
L
?S(r) is not phase incoherent in r trivially
true in ground state
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Phase of ?S(r) is the same for all r in the
ground state of any Bose system.

• Fundamental result of quantum mechanics
• Ground state wave function of any Bose system
  has no nodes (Feynman).
• Hence can be chosen as real and positive

Phase of ?(r,s,) is independent of r and s
Phase of ?S(r) is independent of r
Not true in Fermi systems
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Feynman model for 4He ground state wave function
?(r1,r2, rN) 0 if rn-rm lt a ahard core
diameter of He atom ?(r1,r2, rN) C otherwise
OS is total volume within which ?S is non-zero
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Calculation of Condensate fraction in Feynman
model
Take ahard core diameter of He atom N / V
number density of He II as T ? 0
Generate random configurations s (P(s) constant
for non-overlapping spheres, zero otherwise)
Calculate free volume fraction for each
randomly generated s with P(s) non-zero
Bin values generated.
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f 8 O. Penrose and L. Onsager Phys Rev 104
576 (1956)
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What does possible mean?
Probability of deviation of 10-9 is
exp(-10-18/10-22)exp(-10000)!!
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Pressure dependence of f in Feynman model
Experimental points taken from T. R.
Sosnick,W.M.Snow and P.E. Sokol Europhys Lett 9
707 (1989).
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Feynman model - ?S(r) is non zero within volume
fV.
Assume ?S(r) is non zero within volume O ?S
constant within O ? maximum value of f O/V
Any variation in phase or amplitude within O
gives smaller condensate fraction. eg ideal Bose
gas ? f1 for ?S(r) constant
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?S(r) must be non-zero within volume gtfV.
In any Bose condensed system
?S(r) must be phase coherent in r in the ground
state
Loops in ?S(r) over macroscopic (cm) length
scales
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Macrocopic ring of He4 at T0
At low rotation velocities v(r)0
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?S(r) in solid
Can still be connected over macro length scales
if enough vacancies are present
But how can a solid flow?
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Leggetts argument (PRL 25 1543 1970)
O
O angular velocity of ring rotation R radius
of ring dRltltR
dR
R
Maintained when container is slowly rotated
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?1 ?2 ? ? F?RO No mass rotates with ring 100
supersolid.
?2 ? 0 ? F0 100 of mass rotates with the
ring. 0 supersolid
Superfluid fraction determined by amplitude in
connecting regions. Can have any value between 0
and 1. Condensate fraction determined by volume
in which ? is non-zero ?1? 0 ? 50 supersolid
fraction in model
connectivity suggests f10 in hcp lattice.
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M. A. Adams, R. Down ,O. Kirichek,J Mayers Phys.
Rev. Lett. 98 085301 Feb 2007
Supersolidity not due to BEC in crystalline solid
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Summary
BEC in the ground state implies that
?S(r) is a delocalised function of r. non zero
over a volume V
Mass flow is quantised over macroscopic length
scales