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Bose-Einstein Condensation and Superfluidity
Lecture 1. T=0
•Motivation.
•Bose Einstein condensation (BEC)
•Implications of BEC for properties of ground state many particle WF.
•Feynman model
•Superfluidity and supersolidity.
Lecture 2 T=0
•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.
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
Existing microscopic theory does not explain the
only existing experimental evidence about the
microscopic nature of superfluid helium
What is connection between condensate fraction
and superfluid fraction?
Accepted consensus is that size of condensate
fraction is unrelated to size of superfluid fraction
Superfluid fraction
J. S. Brooks and R. J. Donnelly, J Phys. Chem.
Ref. Data 6 51 (1977).
Normalised condensate fraction
oo
T. R. Sosnick,W.M.Snow and P.E.
Sokol Europhys Lett 9 707 (1989).
x x H. R. Glyde, R.T. Azuah and W.G.
Stirling Phys. Rev. B 62 14337 (2000).
f (T )
f ( 0)
Superfluid helium becomes
more ordered as it is heated
Why?
Line width of excitations
in superfluid helium is
zero as T → 0. Why?
Basis of Lectures
J. Mayers
J. Low. Temp. Phys
109 135
109 153
(1997)
(1997)
J. Mayers
Phys. Rev. Lett.
80, 750
84 314
92 135302
(1998)
(2000)
(2004)
J. Mayers,
Phys. Rev.B
64 224521, (2001)
74 014516, (2006)
Bose-Einstein Condensation
T>TB
0<T<TB
T~0
ħ/L
D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).
BEC in Liquid He4
3.5K
0.35K
T.R. Sosnick, W.M Snow
P.E. Sokol
Phys Rev B 41 11185 (1989)
f =0.07 ±0.01
Kinetic energy of helium atoms.
J. Mayers, F. Albergamo, D. Timms
Physica B 276 (2000) 811
Definition of BEC
N atoms in volume V
Periodic Boundary conditions
Each momentum state occupies volume ħ3/V
n(p)dp = probability of momentum p →p+dp
BEC
Number of atoms in single momentum state (p=0) is proportional to N.
Probability f that randomly chosen atom occupies p=0 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 p=0 state is ~1/N
Quantum mechanical expression for n(p) in ground state
n(p)   dr2 ,..drN  (r1 , r2 ,..rN ) exp( ip.r1 )dr1
r  r1
2
s  r2 ,..rN
n(p)   ds   (r, s) exp( ip.r)dr
2
What are implications of BEC for
properties of Ψ?
ħ/L
|Ψ(r,s)|2 = P(r,s) = probability of configuration r,s of N particles
P(s)   P(r, s)dr
Define
= overall probability of configuration
s = r2, …rN of N-1 particles
 S (r)  (r, s) / P(s)
ψS(r) is many particle wave function normalised over r
  (r) dr  1
2
S
|ψS(r)|2 is conditional probability that particle is at r, given s
1
n(p)   P(s)ds
V

2
S
(r ) exp( ip.r )dr
2
1
f S  nS (0)   S (r )dr
V
momentum distribution
for given s
Condensate fraction for given s
Implications of BEC for ψS(r)
Probability of momentum ħp given s
nS (p) 
2
1

(
r
)
exp(
i
p
.
r
)
d
r
S
V 
nS (p)
ψS(r) non-zero function of r over length scales ~ L
ψS(r) is not phase incoherent in r – trivially true in ground state
Phase of ψS(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 Ψ(r,s,) is independent of r and s
Phase of ψS(r) is independent of r
Not true in Fermi systems
Feynman model for 4He ground state wave function
Ψ(r1,r2, rN) = 0 if |rn-rm| < a
a=hard core diameter of He atom
Ψ(r1,r2, rN) = C otherwise
ψS(r) = 0 if |r-rn| < a
ψS(r) = cS otherwise
ΩS is total volume within which ψS is non-zero
2
2

(
r
)
d
r

1

c
S S
 S
cS =1/√ΩS
2

cS S 
S
1
f S   S (r )dr 

V
V
V
2
Calculation of Condensate fraction in Feynman model
f   P (s) f S ds
Take a=hard 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)
“free volume”
Calculate “free” volume fraction for each randomly generated s with P(s) non-zero
Bin values generated.
J. Mayers PRL 84 314, (2000)
PRB64 224521,(2001)
f ~ 8%
O. Penrose and L. Onsager
Phys Rev 104 576 (1956)
24
atoms
Δf
f
f
192
atoms
fS 
Periodic boundary conditions.
Line is Gaussian with same mean
and standard deviation as
simulation.

S

1
 S (r )dr

V
(r ) dr

1/ N
2
Has same value for all
possible s to within terms
~1/√N
What does “possible” mean?
Gaussian distribution with mean z and variance ~z/√N
Probability of deviation of 10-9 is
~exp(-10-18/10-22)=exp(-10000)!!
N=1022
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).
Feynman model - ψS(r) is non –zero within volume fV.
In general ψS(r) is non –zero within volume >fV.
1
f 
V
  S (r)dr
V
2

V
 S (r) dr  1
2
Assume ψS(r) is non zero within volume Ω
ψS = constant within Ω → maximum value of f = Ω/V
Any variation in phase or amplitude within Ω
gives smaller condensate fraction.
eg ideal Bose gas → f=1 for ψS(r) =constant
For any given f ψS(r) non-zero within vol >fV
PRB 64 224521
(2001)
ψS(r) must be non-zero within volume >fV.
In any Bose condensed system
ψS(r) must be phase coherent in r in the ground state
1
For any possible s ψS(r) must connected
over macroscopic length scales
2
Loops in ψS(r) over macroscopic
length scales
Macrocopic ring of He4 at T=0
In ground state
S (r )
 r .dr  0
Rotation of the container creates
a macroscopic velocity field v(r)
 mv(r ) 
.r 



S (r )  S (r ) i
Galilean
transformation
but
S (r )
 r .dr  2n
if BEC is preserved
 mv(r).dr  nh
At low rotation
velocities v(r)=0
Quantisation of circulation
Superfluidity
ψS(r) in solid
Can still be connected over macro length
scales if enough vacancies are present
BEC
Supersolidity
But how can a solid flow?
Leggett’s argument (PRL
25 1543 1970)
Ω
Ω = angular velocity of ring rotation
R = radius of ring
dR
R
dR<<R
In ground state
 (r )
 r .dr  0
Maintained when
container is slowly rotated
 (r )
 r .dr  
In frame rotating with ring
 
m
(2R) 2 

x is distance around the ring.

2R
0
dS
dx  
dx
v( x) 
h dS
m dx
F=|ψS|2v(x)

2R
0
Simplified model for ψS
h
v( x)dx  
m
ρ1=|ψ1|2
ρ2=|ψ2|2
Mass density conserved
In ring frame if
1v1  2v2
h  1  2
F 
m R 1   2
h  1  2
F
m R 1   2
ρ1=|ψ1|2
ρ2=|ψ2|2
ρ2 → 0 → F=0
ρ1= ρ2= ρ → F=ρRΩ
100% of mass rotates
with the ring.
0% supersolid
No mass rotates with ring
100% supersolid.
Superfluid fraction determined by amplitude in connecting regions.
Can have any value between 0 and 1.
Condensate fraction determined by volume in which ψ is non-zero
ψ1→ 0 → 50% supersolid fraction in model
connectivity suggests f~10% in hcp lattice.
solid
O single crystal high purity He4
X polycrystal high purity He4
□ 10ppm He3 polycrystal
M. A. Adams, R. Down ,O. Kirichek,J Mayers
Phys. Rev. Lett. 98 085301 Feb 2007
Supersolidity not due to BEC
in crystalline solid
liquid
J. Mayers, F. Albergamo, D. Timms
Physica B 276 (2000) 811
Summary
BEC in the ground state implies that;
ψS(r) is a delocalised function of r. – non zero over a volume ~V
1
f

(
r
)
d
r

S
V
V
1 

1



N

for all s
Mass flow is quantised over macroscopic length scales
Superfluidity and Supersolidity