Lecture 3 BEC at finite temperature

1. Lecture 3BEC at finite temperature Thermal and quantum fluctuations in condensate fraction. Phase coherence and incoherence in the many particle wave function. Basic assumption and a priori justification Consequences Connection between BEC and two fluid behaviour Connection between condensate and superfluid fraction Why BEC implies sharp excitations. Why sf flows without viscosity while nf does not. How BEC is connected to anomalous thermal expansion as sf is cooled. Hoe BEC is connecged to anomalous reduction in pair correlations as sf is cooled.

2. Boltzman factor exp(-Ej / T)/Zj Basic assumption; (√fis amplitude of order parameter) Δf ~1/√N Fj= f± ~ 1/ √N Thermal Fluctuations At temperature T

3. g(E) η(E) ΔE, Δf ~1/ √N-1/2 All occupied states give same condensate fraction Can take one “typical” occupied state as representative of density matrix As T changes band moves to different energy “Typical” state gives different f All occupied states gives same f to ~1/√N Drop subscript j to simplify notation

4. f(s) = F±~1/√N ΔF~1/√N f(s)~ f ± 1/√N F = f ±~1/√N Quantum Fluctuations

5. Weight f to ~1/√N for any state and any s Delocalised function of r (non-zero within volume > f V) J. Mayers Phys. Rev. Lett. 84 314 (2000),Phys. Rev.B 64 224521, (2001) Phase correlations in r over distances ~L otherwise width~ħ/L BEC n(p)

6. Phase incoherent Phase coherent rC No condensate Condensate ~1/rC

7. Temperature dependence At T = 0 , Ψ0(r,s)must be delocalised over volume ~ f0V and phase coherent. For T > TB occupied states Ψj(r,s) must be either localised or phase incoherent. What is the nature of the wave functions of occupied states for 0 < T < TB?

8. BASIC ASSUMPTION • Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s) • Ψ0(r,s) is phase coherent ground state • ΨR(r,s) is phase incoherent in r • b(s)  0 as T  TB for typical occupied state • ΨR(r,s)  0 as T  0 • Gives correct behaviour in limits T  TB, T  0 • True for IBG wave functions. • Bijl-Feynman wave functions have this property • 4. Implications agree with wide range of experiments

9. Bijl-Feynman wave functions J. Mayers, Phys. Rev.B74 014516, (2006) • nk= number of phonon-roton excitations with wave vector k. • M = total number of excitations • sum of NM terms. b(s) is sum of all terms not containing r = r1 Phase coherent in r. ΘR(r,s) is sum of terms containing r Phase incoherent in r rC ~1/Δk ~ 5 Å in He4 at 2.17K Fraction of terms in b(s) is (1-M/N) as N   M  N Θ(r,s) is phase incoherent (T  TB) M  0 Θ(r,s) is phase coherent (T  0)

10. If Δf ~1/N1/2 Consequences Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s)

11. X Macroscopic System Microscopic basis of two fluid behaviour

12. Parseval’s theorem wR = 1- wC. Momentum distribution and liquid flow split into two independent components of weights wC(T), wR(T).

13. Thermodynamic properties split into two independent components of weights wc(T), wR(T) wc(T) = ρS(T) wR(T) = ρN(T) Bijl-Feynman wR determined by number of “excitations” • True to within term ~N-1/2 • Only if fluctuations in f, ρS and ρN are negligible. • Not in limits T 0 T  TB

14. o o T. R. Sosnick,W.M.Snow and P.E. Sokol Phys. Europhys Lett 9 707 (1989). X X H. R. Glyde, R.T. Azuah and W.G. Stirling Phys. Rev. B 62 14337 (2000).

15. PN =PB Superfluid has extra “Quantum pressure”

16. SR-1 S-1 q α < 1 → S less ordered than SR

17. V.F. Sears and E.C. Svensson, Phys. Rev. Lett. 43 2009 (1979). α(T) α0 SR(q)  S0(q) →Ψ0(r,s) and ΨR(r,s)  0 for different s

18. Why is superfluid more disordered? Assume for s where ΨR(r,s)  0 negligible free volume Ground state more disordered Quantitative agreement with measurement at atomic size and N/V in liquid 4He J. Mayers Phys. Rev. Lett. 84 314 (2000) For s where Ψ0(r,s)  0 ~7% free volume

19. s such that Ψ0(r,s) is connected (Macro loops) Quantised vortices, macroscopic quantum effects Phase coherent component Ψ0(r,s)

20. Phase incoherent component ΨR(r,s) s such that ΨR(r,s) is not connected Localised phase incoherent regions. Localised quantum behaviour over length scales rC ~ 5 Å No MQE or quantised vortices

21. Momentum transfer = ħq Energy transfer = ħω Phase incoherent Regions of size ~rC |Aif(q)|2has minimum width Δq ~ 1/rC Excitations Normal fluid - momentum of excitations is uncertain to ~ ħ/rC Superfluid - momentum can be defined to within ~ ħ/L

22. 0 < T < TB ε ( deg K) h/rC q (Å-1) Anderson and Stirling J. Phys Cond Matt (1994)

23. Landau Theory Basic assumption is that excitations with well defined energy and momentum exist. Normal fluid vC= 0 ω q Only true in presence of BEC Landau criterion vC = (ω/q)min

24. Summary BASIC ASSUMPTION Ψ(r,s) = b(s)Ψ0(r,s) + ΨR(r,s) Phase coherent ground state Phase incoherent • Has necessary properties in limits T0, T  TB • IBG, Bijl-Feynman wave functions have this form • Simple explanations of • Why BEC is necessary for non-viscous flow • Why Landau theory needs BEC.

25. Summary Existing microscopic theory does not provide even qualitative explanations of the main features of neutron scattering data This is the only experimental evidence of the microscopic nature of Bose condensed helium. Theory given here explains quantitatively all these features Why the condensate fraction is accurately proportional to the superfluid fraction Why spatial correlations decrease as superfluid helium is cooled Why superfluid helium is the only liquid which contains sharp excitations Why superfluid helium expands when it is cooled