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