Bose-Einstein Condensation and Superfluidity

1. 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.

2. Existing microscopic theory does not explain the only existing experimental evidence about the microscopic nature of superfluid helium 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

3. Superfluid fraction J. 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 xH. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000). What is connection between condensate fraction and superfluid fraction? Accepted consensus is that size of condensate fraction is unrelated to size of superfluid fraction

4. Superfluid helium becomes more ordered as it is heated Why?

5. Line width of excitations in superfluid helium is zero as T → 0. Why?

6. 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.B64 224521, (2001) 74 014516, (2006)

7. ħ/L Bose-Einstein Condensation T>TB 0<T<TB T~0 D. S. Durfee and W. Ketterle Optics Express 2, 299-313 (1998).

8. T.R. Sosnick, W.M Snow P.E. Sokol Phys Rev B 41 11185 (1989) 3.5K 0.35K Kinetic energy of helium atoms. J. Mayers, F. Albergamo, D. Timms Physica B 276 (2000) 811 BEC in Liquid He4 f =0.07 ±0.01

9. 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

11. ħ/L Quantum mechanical expression for n(p) in ground state What are implications of BEC for properties of Ψ?

12. = overall probability of configuration s = r2, …rN of N-1 particles Define ψS(r) is many particle wave function normalised over r momentum distribution for given s Condensate fraction for given s |Ψ(r,s)|2 = P(r,s)= probability of configuration r,s of N particles |ψS(r)|2isconditional probability that particle is at r, given s

13. Probability of momentum ħp given s Implications of BEC for ψS(r) ψS(r) non-zero function of r over length scales ~ L ψS(r) is not phase incoherent in r – trivially true in ground state

14. 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

15. ψS(r) = 0 if |r-rn| < a ψS(r) = cS otherwise cS =1/√ΩS 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 ΩS is total volume within which ψSis non-zero

16. “free volume” Calculation of Condensate fraction in Feynman model 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) Calculate “free” volume fraction for each randomly generated s with P(s) non-zero Bin values generated.

17. J. Mayers PRL 84 314, (2000) PRB64 224521,(2001) 24 atoms Δf 192 atoms Has same value for all possible s to within terms ~1/√N Periodic boundary conditions. Line is Gaussian with same mean and standard deviation as simulation. f ~ 8% O. Penrose and L. Onsager Phys Rev 104 576 (1956)

18. Gaussian distribution with mean z and variance ~z/√N N=1022 What does “possible” mean? Probability of deviation of 10-9 is ~exp(-10-18/10-22)=exp(-10000)!!

19. 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).

20. In generalψS(r) is non –zero within volume >fV. PRB64 224521 (2001) For any given fψS(r) non-zero within vol >fV Feynman model -ψS(r) is non –zero within volume fV. 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

21. For any possible sψS(r) must connected over macroscopic length scales 1 Loops in ψS(r) over macroscopic length scales 2 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 Loops in ψS(r) over macroscopic (cm) length scales

22. In ground state Rotation of the container creates a macroscopic velocity field v(r) Galilean transformation if BEC is preserved Quantisation of circulation but Superfluidity Macrocopic ring of He4 at T=0 At low rotation velocities v(r)=0

23. BEC Supersolidity ψS(r) in solid Can still be connected over macro length scales if enough vacancies are present But how can a solid flow?

24. In ground state In frame rotating with ring Leggett’s argument (PRL 25 1543 1970) Ω Ω = angular velocity of ring rotation R = radius of ring dR<<R dR R Maintained when container is slowly rotated

25. x is distance around the ring. Simplified model for ψS F=|ψS|2v(x) ρ1=|ψ1|2 ρ2=|ψ2|2 Mass density conserved In ring frame if

26. ρ1=|ψ1|2 ρ2=|ψ2|2 ρ1=ρ2= ρ→F=ρRΩ No mass rotates with ring 100% 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 connectivity suggests f~10% in hcp lattice.

27. liquid O single crystal high purity He4 X polycrystal high purity He4 □ 10ppm He3 polycrystal solid 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 BEC in crystalline solid

28. for all s Superfluidity and Supersolidity Summary BEC in the ground state implies that; ψS(r) is a delocalised function of r. – non zero over a volume ~V Mass flow is quantised over macroscopic length scales