http://cua.mit.edu/ketterle_group/ Quantised vortices in 4He and ultra-cold trapped gases Interference between separately prepared condensates of ultra-cold atoms Lecture 2.Why BEC is linked with single particle quantum behaviour over macroscopic length scales
Macroscopic single particle behaviour Outline of Lecture Single particle behaviour over macroscopic length scales is a consequence of the delocalisation of Ψ(r1,r2…rN) This delocalisation is a necessary consequence of BEC. Delocalisation leads to; A new thermodynamic quantity – the order parameter. Factorisation of Ψover macroscopic length scales A macroscopic single particle Schrödinger equation.
Feynman Model ψS(r) = 0 if |r-rn| < a ψS(r) = constant otherwise f = volume of white regions ψS(r) Ground state of 4He. ψS(r) is MPWF Ψ(r,s), normalised over r ψS(r)occupies the spaces between particles at s ψS(r)is non-zero within volume of at least fV (7% of total volume in 4He)
J. Mayers PRL 84 314, (2000) PRB64 224521,(2001) Feynman model 24 atoms Δf 192 atoms Width of Gaussian is ~1/√√N has same value for all possible s in macroscopic system Probability distribution for fS becomes narrower and more Gaussian for large N
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 ψS(r) are also independent of s to ~1/√N Due to delocalisation of the wave function in presence ofBEC Leads to single particle behaviour over macroscopic length scales
Similar to physical reason why number of particles in large volume Ω of fluid is independent of s. Rigorous result of liquids theory independent of s 3. Why independence of s ? Volume of spaces between particles similarly becomes independent of s
Basic Assumptions 1.ψS(r) is a delocalised function ofr-Necessary consequence of BEC 2. Fluid of uniform macroscopic density. • 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.
How does vary with s? Define Divide V into N cells of volume V/N Average of 1 atom/cell Integral over single cell ,
Uniform density - all cells give the same average contribution i Cell fluctuation with s Arrangement of atoms near cell i is not correlated with that near widely separated cell j. + j Short range interactions Form of ψS(r) within cell i not correlated with that within cell j Δgi(s), Δgj(s) uncorrelated
For cells of size V/N, NΩ~1 Sign of Δgi(s) varies randomly with i Random walk Gaussian distribution for g(s)
Only ~1 cell contributes to integrals No cancellation of fluctuations from large number of cells Consequence of delocalisation of ψS(r) Argument fails if ψS(r) is localised function of r
g(s) = G ± ~G/√N Second Demonstration No correlations in fluctuations of widely separated cells i ≠ j
<v> is mean potential energy of each particle =N<v> Independent of s Potential energy All n make same contribution
Independent of s Kinetic Energy κis mean kinetic energy/atom =Nκ Kinetic and potential energy can be accurately calculated in macroscopic system by calculating single particle integral for any possible s
assume density varies sufficiently slowly that it is constant within cell within single cell Non-uniform particle density Cell of volume Ω centred at r Contains on average NΩ>>1atoms ΔNΩ/ NΩ~1/√NΩ .r
Same for all possible s if NΩis large Same for all possible s if NΩis large Integrals over cell at can be treated in same way as integrals over total volume V at constant density 1/√NΩfluctuations in NΩ do not change this
Normalisation factor Coarse grained average of potential energy Mean potential energy of particle in Ω(r) Coarse grained average of kinetic energy Mean kinetic energy of particle in Ω(r)
S S' Localised ψS(r) X Integrals over Ω are not independent of s if ψS(r) is localised X
Penrose-Onsager Criterion for BEC if Single particle density matrix The order parameter α(r) is the “order parameter”
Means equal to within terms ~ 1/√NΩ Coarse grained average of SPDM
Order parameter is coarse grained average of ψS(r). • Valid for averages over macroscopic regions of space • New thermodynamic variable created by BEC
Same for all s to ~1/√NΩ Microscopic density Macroscopic density Macroscopic density is integral of ψS(r) over Ω(r) for any possible s
Coarse grained average of many particle wave function Integral of each coordinate over cube of volume Ω Same is true for r2, r3 etc Single particle behaviour over macroscopic length scales
Ψ is real 4 Coarse grained average of N particle Schrödinger equation 1 2 3 4
Consider term n=1 Potential energy of interaction between particles. 3
3 Every particle satisfies Kinetic energy 1
This contributes extra term Derivation neglects contribution to kinetic energy due to long range variation in particle density.
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
BEC implies that all particles satisfy the same non-linear Schrödinger equation on macroscopic length scales Limits of validity Accurate to within ~1/√NΩwhere NΩis 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.
The order parameter is not the condensate wave function Single particle Schrödinger equation is valid for any Bose condensed system irrespective of size of condensate fraction
Calculations in a dilute Bose gas give Reduces to Gross-Pitaevski Equation in weakly interacting system
Gross-Pitaevski equation My Equation Requires presence of BEC Requires delocalisation (implied by BEC) Can only be derived in weakly Derivation valid for any Interacting system strength of interaction Existing derivations assume Derivation valid for fixed particle number is not fixed or variable N Valid only in weakly Valid for any Interacting system strength of interaction
BEC implies single particle behaviour over macroscopic length scales Summary Delocalisation is necessary consequence of BEC Delocalisation implies that integrals over r of quantities involving ψS(r) are independent of s Order parameter is integral of ψS(r) over macroscopic region of space. Coarse grained average of single particle wave function factorises Coarse grained average of many particle Schrödinger equation gives non-linear single particle equation True for any size of condensate fraction
Contribution due to short range structure in ψS(r) Contribution due to long range variation in average density over V Division of Κtot(r)
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/√NΩ const within Ω(r) const within Ω(r) Kinetic energy due to structure of density on macroscopic scales Mean kinetic energy of particle in system at constant density
Standard Theory T=0 Here Quantum average of ψS over possible particle positions s Quantum average over field operator suggests Order Parameter Order Parameter Denote average over s as < >S
Integrate over r,r/ ρ1(r-r’) 1 d f If Ω is sufficiently large Makes negligible contribution |r-r’| must be independent of s Penrose criterion for BEC Hence this definition is consistent with proven properties of ψS(r) in ground state
Here Standard Theory Order Parameter suggests Order Parameter Quantum average over s given j Thermal average over states j Quantum and thermal average over field operator must be the same for all j and s Finite T Notation
Finite T • ψjS(r) = √ρSexp[iφj(s)] ψS(r) + ψSR(r) • ψS(r) is phase coherent ground state • ψSR(r) is phase incoherent in r Phase φj(s) must be the same for all j and s
Physical interpretation When BEC first occurs particular N particle state j is occupied with a random value of φj(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.
Only superfluid component contributes to interference effects New testable prediction 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 ΔN1 =ΔN2~√N