1. Lecture 2. Why BEC is linked with single particle quantum behaviour over macroscopic length scales Interference between separately prepared condensates of ultra-cold atoms Quantised vortices in 4 He and ultra-cold trapped gases http://cua.mit.edu/ketterle_group/

2. Single particle behaviour over macroscopic length scales is a consequence of the delocalisation of Ψ(r 1, r 2 …r N ) This delocalisation is a necessary consequence of BEC. Delocalisation leads to; A new thermodynamic quantity – the order parameter. Factorisation of Ψ over macroscopic length scales A macroscopic single particle Schrödinger equation. Macroscopic single particle behaviour Outline of Lecture

3. Ground state of 4 He. ψ S (r) is MPWF Ψ(r,s), normalised over r ψ S (r) is non-zero within volume of at least fV (7% of total volume in 4 He) ψS(r)ψS(r) ψ S (r) occupies the spaces between particles at s ψ S (r) = 0 if |r-r n | < a ψ S (r) = constant otherwise f = volume of white regions Feynman Model

4. 192 atoms has same value for all possible s in macroscopic system Probability distribution for f S becomes narrower and more Gaussian for large N ΔfΔf Width of Gaussian is ~1/√√N J. Mayers PRL 84 314, (2000) PRB64 224521,(2001) 24 atoms Feynman model

5. The independence of s of the integral is a general property of the ground state wave function of any Bose condensed system Other physically relevant integrals over ψ S (r) are also independent of s to ~1/√N Due to delocalisation of the wave function in presence of BEC Leads to single particle behaviour over macroscopic length scales

6. 3. Why independence of s ? Volume of spaces between particles similarly becomes independent of s independent of s Similar to physical reason why number of particles in large volume Ω of fluid is independent of s. Rigorous result of liquids theory

7. Basic Assumptions 3. Pair correlations extend only over distances of a few interatomic spacings. definition of a fluid 4. Interactions between particles extend only over a few interatomic spacings true for atoms - i.e. liquid helium and ultra-cold trapped gases. implicit in assumption 3. 2. Fluid of uniform macroscopic density. 1. ψ S (r) is a delocalised function of r- Necessary consequence of BEC

8. , How doesvary with s?Divide V into N cells of volume V/N Average of 1 atom/cell Define Integral over single cell

9. Uniform density - all cells give the same average contribution Cell fluctuation with s i j Arrangement of atoms near cell i is not correlated with that near widely separated cell j. Short range interactions + Form of ψ S (r) within cell i not correlated with that within cell j Δg i (s), Δg j (s) uncorrelated

10. For cells of size V/N, N Ω ~1 Random walk Gaussian distribution for g(s) Sign of Δg i (s) varies randomly with i

11. Argument fails if ψ S (r) is localised function of r Only ~1 cell contributes to integrals No cancellation of fluctuations from large number of cells Consequence of delocalisation of ψ S (r)

12. No correlations in fluctuations of widely separated cells i ≠ j Second Demonstration g(s) = G ± ~G/√N

13. Potential energy All n make same contribution is mean potential energy of each particle =N<v>=N<v> Independent of s

14. Kinetic Energy =Nκ κ is mean kinetic energy/atom Kinetic and potential energy can be accurately calculated in macroscopic system by calculating single particle integral for any possible s Independent of s

15. Non-uniform particle density.r Cell of volume Ω centred at r assume density varies sufficiently slowly that it is constant within cell within single cell Contains on average N Ω >>1 atoms ΔN Ω / N Ω ~1/√N Ω

16. Integrals over cell at can be treated in same way as integrals over total volume V at constant density 1/√N Ω fluctuations in N Ω do not change this Same for all possible s if N Ω is large Same for all possible s if N Ω is large

17. Coarse grained average of potential energy Normalisation factor Coarse grained average of kinetic energy Mean potential energy of particle in Ω(r) Mean kinetic energy of particle in Ω(r)

18. Localised ψ S (r) X Integrals over Ω are not independent of s if ψ S (r) is localised S X S'S'

19. The order parameter Single particle density matrix if Penrose-Onsager Criterion for BEC α(r) is the “order parameter”

20. Coarse grained average of SPDM Means equal to within terms ~ 1/√N Ω

21. Order parameter is coarse grained average of ψ S (r). Valid for averages over macroscopic regions of space New thermodynamic variable created by BEC

22. Microscopic density Macroscopic density Same for all s to ~1/√N Ω Macroscopic density is integral of ψ S (r) over Ω (r) for any possible s

23. Coarse grained average of many particle wave function Integral of each coordinate over cube of volume Ω Same is true for r 2, r 3 etc Single particle behaviour over macroscopic length scales

24. Coarse grained average of N particle Schrödinger equation 1 2 3 4 Ψ is real 4

25. Consider term n=1 Potential energy of interaction between particles. 3

26. Kinetic energy Every particle satisfies 1 3

27. Derivation neglects contribution to kinetic energy due to long range variation in particle density. This contributes extra term

28. Microscopic kinetic and potential energy gives effective single particle potential is mean kinetic energy/particle at uniform macroscopic density is mean potential energy/particle Both depend upon Non-linear single particle Schrödinger equation

29. Limits of validity Accurate to within ~1/√N Ω where N Ω is number of atoms within resolution vol. Ω Describes time evolution of particle density if this is a meaningful concept Valid providing Ψ(r,s) is delocalised over macroscopic length scales. BEC implies that all particles satisfy the same non-linear Schrödinger equation on macroscopic length scales

30. Single particle Schrödinger equation is valid for any Bose condensed system irrespective of size of condensate fraction The order parameter is not the condensate wave function

31. Calculations in a dilute Bose gas give Reduces to Gross-Pitaevski Equation in weakly interacting system

32. Gross-Pitaevski equation My Equation Can only be derived in weakly Derivation valid for any Interacting system strength of interaction Requires presence of BEC Requires delocalisation (implied by BEC) Valid only in weakly Valid for any Interacting system strength of interaction Existing derivations assume Derivation valid for fixed particle number is not fixed or variable N

33. Delocalisation is necessary consequence of BEC Order parameter is integral of ψ S (r) over macroscopic region of space. Delocalisation implies that integrals over r of quantities involving ψ S (r) are independent of s Coarse grained average of single particle wave function factorises Coarse grained average of many particle Schrödinger equation gives non-linear single particle equation Summary BEC implies single particle behaviour over macroscopic length scales True for any size of condensate fraction

34. Division of Κ tot (r) Contribution due to short range structure in ψ S (r) Contribution due to long range variation in average density over V

35. Mean kinetic energy of particle in gnd state at constant density const within Ω(r) Proportional to mean momentum of atoms in gnd state at constant density =0 ±~1/√N Ω const within Ω(r) Mean kinetic energy of particle in system at constant density Kinetic energy due to structure of density on macroscopic scales

36. Denote average over s as S Standard Theory T=0 Quantum average over field operator Here Quantum average of ψ S over possible particle positions s Order Parameter Order Parameter suggests

37. Integrate over r,r / Penrose criterion for BEC If Ω is sufficiently largeMakes negligible contribution must be independent of s Hence this definition is consistent with proven properties of ψ S (r) in ground state |r-r’| ρ 1 (r-r’) f 1 d

38. Finite T Notation Standard Theory Order Parameter Quantum and thermal average over field operator Order Parameter suggests Here Quantum average over s given j Thermal average over states j must be the same for all j and s

39. Finite T ψ jS (r) = √ρ S exp[iφ j (s)] ψ S (r) + ψ SR (r) ψ S (r) is phase coherent ground state ψ SR (r) is phase incoherent in r Phase φ j (s) must be the same for all j and s

40. Physical interpretation When BEC first occurs particular N particle state j is occupied with a random value of φ j (s) Delocalisation of wave function implies thermally induced transition to states with different phase must occur simultaneously over macroscopic volume Therefore very unlikely – like transition to different direction of M in ferromagnet. Hence broken symmetry – states of different phase are degenerate but only one particular phase is accessible Not broken gauge symmetry. Particle number is fixed.

41. Interference between condensates Not necessary to assume that interference fringes are created by observation Total number of particles is fixed, but necessary to assume that condensates exchange particles ΔN 1 = ΔN 2 ~√N Only superfluid component contributes to interference effects New testable prediction