Bose-Einstein Condensation and Superfluidity
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Transcript of Bose-Einstein Condensation and Superfluidity
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 3 Finite T•Basic assumption•A priori justification.•Physical consequences
Two fluid behaviourConnection between condensate and superfluid fractionWhy 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.
Lecture 2 T=0 •Why BEC implies macroscopic single particle quantum effects•Derivation of macroscopic single particle Schrödinger equation
Motivation
A vast amount of neutron data has been collectedfrom superfluid helium in the past 40 years.
This data contains unique features, not observed in any other fluid.
These features are not explained even qualitativelyby 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
)0(
)(
f
Tf
Superfluid fractionJ. S. Brooks and R. J. Donnelly, J Phys. Chem. Ref. Data 6 51 (1977).
Normalised condensate fraction
o o 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).
Superfluid helium becomesmore ordered as it is heatedWhy?
Line width of excitationsin superfluid helium is zero as T → 0. Why?
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)
Bose-Einstein CondensationT>TB 0<T<TB T~0
D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).
ħ/L
Kinetic energy of helium atoms. J. Mayers, F. Albergamo, D. Timms Physica B 276 (2000) 811
T.R. Sosnick, W.M SnowP.E. Sokol Phys Rev B 41 11185 (1989)
3.5K 0.35K
BEC in Liquid He4
f =0.07 ±0.01
N atoms in volume VPeriodic Boundary conditionsEach momentum state occupies volume ħ3/V
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
n(p)dp = probability of momentum p →p+dp
Definition of BEC
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.
2
11212 ).exp(),..,(,..)( rrprrrrrp diddn NN
Quantum mechanical expression for n(p) in ground state
What are implications of BEC forproperties of Ψ?
1rr Nrrs ,..2
2
).exp(),()( rrpsrsp didnħ/L
|Ψ(r,s)|2 = P(r,s) = probability of configuration r,s of N particles
rsrs dPP ),()( = overall probability of configuration s = r2, …rN of N-1 particles
|ψS(r)|2 is conditional probability that particle is at r, given s
Define )(/),()( ssrrS P
1)(2 rrS dψS(r) is many particle wave function normalised over r
2
).exp()( rrprS di ssp dPV
n )(1
)( momentum distributionfor given s
2
)(1
)0( rrSSS dV
nf Condensate fraction for given s
ψS(r) non-zero function of r over length scales ~ L
)(pSn
Implications of BEC for ψS(r)
ψS(r) is not phase incoherent in r – trivially true in ground state
2
).exp()(1
)( rrprp SS diV
n
Probability of momentum ħp given s
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
VV
cd
Vf SSS
SS rr
22
)(1
ΩS is total volume within which ψS is non-zero
SSS rr 221)( cd cS =1/√ΩS
Calculation of Condensate fraction in Feynman model
ss SdfPf )(
Take a=hard core diameter of He atom
N / V = number density of He II as T → 0
Bin values generated.
Calculate “free” volume fraction for each randomly generated s with P(s) non-zero
“free volume”
Generate random configurations s(P(s) = constant for non-overlapping spheres, zero otherwise)
f ~ 8%O. Penrose and L. Onsager Phys Rev 104 576 (1956)
J. Mayers PRL 84 314, (2000) PRB64 224521,(2001)
24 atoms
192atoms
Periodic boundary conditions.Line is Gaussian with same mean and standard deviation as simulation. rrS d)(
Has same value for allpossible s to within terms~1/√N
2)(1 rrSS d
Vf
f
f
N/1
Δf
What does “possible” mean?
Gaussian distribution with mean z and variance ~z/√N N=1022
Probability of deviation of 10-9 is
~exp(-10-18/10-22)=exp(-10000)!!
Pressure dependence of f in Feynman model
Experimental pointstaken from T. R. Sosnick,W.M.Snow and P.E. Sokol
Europhys Lett 9 707 (1989).
In general ψS(r) is non –zero within volume >fV. PRB 64 224521(2001)
1)(2 V drrS
2
)(1V
dV
f rrS
Assume ψS(r) is non zero within volume Ω
ψS = constant within Ω → maximum value of f = Ω/V
For any given f ψS(r) non-zero within vol >fV
Any variation in phase or amplitude within Ω gives smaller condensate fraction.
eg ideal Bose gas → f=1 for ψS(r) =constant
Feynman model - ψS(r) is non –zero within volume fV.
ψS(r) must be non-zero within volume >fV.
ψS(r) must be phase coherent in r in the ground stateIn any Bose condensed system
For any possible s ψS(r) must connected over macroscopic length scales
2
1
Loops in ψS(r) over macroscopiclength scales
Superfluidity
Macrocopic ring of He4 at T=0
0.)(
r
r
rS d
In ground state
Rotation of the container creates a macroscopic velocity field v(r)
r
rrr SS .
)()()(
mv
i Galileantransformation
At low rotation velocities v(r)=0
nd 2.
)(
rr
rS if BEC is preserved
but
nhdmv rr).( Quantisation of circulation
ψS(r) in solid
Can still be connected over macro length
scales if enough vacancies are present
But how can a solid flow?
BEC
Supersolidity
Ω = angular velocity of ring rotation
R = radius of ring
dR<<R
RdR
Ω
Leggett’s argument (PRL 25 1543 1970)
Maintained whencontainer is slowly rotated
rr
rd.
)(
In frame rotating with ring
2)2( Rm
0.)(
r
r
rd
In ground state
2211 vv
Mass density conservedIn ring frame if
m
hdxxv
R2
0)(
21
21
Rm
hF
Rdx
dx
d2
0
S
x is distance around the ring.
dx
d
m
hxv S
)(
F=|ψS|2v(x)ρ1=|ψ1|2
ρ2=|ψ2|2
Simplified model for ψS
21
21
Rm
hF
ρ1= ρ2= ρ → F=ρRΩ
No mass rotates with ring100% supersolid.
ρ2 → 0 → F=0
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 ψ is non-zero
ψ1→ 0 → 50% supersolid fraction in model
ρ1=|ψ1|2
ρ2=|ψ2|2
connectivity suggests f~10% in hcp lattice.
O single crystal high purity He4 X polycrystal high purity He4□ 10ppm He3 polycrystal
solid
liquid
J. Mayers, F. Albergamo, D. Timms Physica B 276 (2000) 811
M. A. Adams, R. Down ,O. Kirichek,J Mayers Phys. Rev. Lett. 98 085301 Feb 2007
Supersolidity not due to BECin crystalline solid
Summary
ψS(r) is a delocalised function of r. – non zero over a volume ~V
NV
fd
V
11)(
1rrS for all s
Mass flow is quantised over macroscopic length scales
BEC in the ground state implies that;
Superfluidity and Supersolidity