Classical field techniques for finite temperature Bose gases

Transcription

1 Classical field techniques for finite temperature Bose gases N 0 /N = 0.93 N 0 /N = 0.45 N 0 /N = 0.02 Matthew Davis ARC Centre of Excellence for Quantum-Atom Optics, University of Queensland, Brisbane, Australia.. p.1

2 Australian Centre for Quantum-Atom Optics One of eight Australian Centre s of Excellence funded in Australian National University (Canberra): Rb BEC and atom laser, He BEC, atom-light entanglement, quantum imaging, theory. Swinburne University of Technology (Melbourne): Rb BEC on atom chip, quantum degenerate Fermi gas. University of Queensland (Brisbane): Main theory node: quantum dynamics and correlations.. p.2

3 UQ theory group Eric Calvcanti, Joel Corney, Karén Kheruntsyan, Hui Hu, Murray Olsen, Margaret Reid. MJD, Xia-Ji Liu, Peter Drummond, Ashton Bradley. Absent: Chris Foster, Andy Ferris, Scott Hoffman, Piotr Deuar.. p.3

4 Overview Finite temperature Bose gases. Introduction to classical fields. Simulation of classical fields. Application: Shift in T c for interacting Bose gases. Quantum dynamics with classical fields.. p.4

5 The challenge for theorists Can we come up with a practical non-equilibrium formalism for finite temperature Bose gases? Desirable features: Can deal with inhomogeneous potentials. Can treat interactions non-perturbatively. Calculations can be performed on a reasonable time scale (say under one week).. p.5

6 Potential applications Topics of interest include: Condensate formation. Vortex lattice formation and dynamics. Low dimensional systems (fluctuations important). Correlations. Heating effects. Atom lasers.... p.6

7 Classical fields for matter waves. p.7

8 The original classical field for Bose gases Assumes all particles are in the same quantum state. N 0 1: so quantum effects can be ignored. Gross-Pitaevskii equation i ψ(x) t = H sp ψ(x) + U 0 ψ(x) 2 ψ(x), with U 0 = 4π 2 a/m. We are all aware of how successful this has been.. p.8

9 Finite temperature classical field approximation An example: the classical theory of electromagnetic radiation resulted in the Rayleigh-Jeans law. Based on the equipartition theorem : Each oscillator mode has energy k B T in equilibrium. Lord Rayleigh Sir James Jeans. p.9

10 The UV catastrophe But we all know it doesn t work... So Planck says: Classical fields are no good (?) Energy T = 2500 K Planck law RJ law λ [µm] 0.8 Max Planck Mode occupation RJ 0.2 Planck λ [µm]. p.10

11 However... For the infra-red modes the RJ law is a good approximation. Quantum and classical results are similar for modes with energy E k k B T N k = Essential features: 1 e E k/k BT 1 k BT E k many particles per mode (N k > 5)? high energy cutoff. Energy Mode occupation x T = 2500 K RJ law 1 Planck law λ [µm] RJ law Planck λ [µm]. p.11

12 Example from electroweak theory. p.12

13 Classical fields for matter waves Massive bosons are conserved must introduce µ. Validity requirements: E k µ k B T N k k BT E k µ. However µ is large and negative away from BEC. So only a limited temperature range for which there are classical modes.. p.13

14 Classical region for Bose gases 87 Rb, N = , harmonic trap with ν = 100 Hz. T c 10 6 BEC mode Mode occupation GPE: Ultracool : T 0 T > 0 T T c 10th 35th 120th 600th 2000th Kinetic: T >> T c 5 atoms Temperature [nk]. p.14

15 Outline of classical field formalism Define a projection operator for classical region C: P{F (x)} = φ k (x) d 3 x φ k(x )F (x ), Q = 1 P. k C Projections of Bose field operator ˆΨ(x): ˆψ(x) = P{ ˆΨ(x)}, ˆη(x) = Q{ ˆΨ(x)}. Classical field approximation: ψ(x) ˆψ(x) i ψ t = H sp ψ + U 0 P { ψ 2 ψ } + U 0 P { 2 ψ 2 ˆη + ψ 2 ˆη } + U 0 P { ψ ˆηˆη + 2ψ ˆη ˆη + ˆη ˆηˆη }.. p.15

16 Classical field for matter waves The Projected Gross-Pitaevskii equation (PGPE): i dψ(x) dτ = H sp ψ(x) + C nl P { ψ(x) 2 ψ(x) }, C nl = 8πaN L. All modes assumed to be highly occupied. Projection stops higher energy modes becoming occupied: P{F (x)} = φ k (x) d 3 x φ k(x )F (x ) prevents UV catastrophe. k C Advantages: 1. Relatively easy (i.e possible!) to simulate in 3D. 2. Method is non-perturbative. However: Experimental comparisons require atoms above cutoff.. p.16

17 Classical field simulations. p.17

18 Behaviour of PGPE simulations Begin simulations with randomised initial conditions: ψ(x, t = 0) = k C c k φ k (x). PGPE conserves normalisation and energy (microcanonical): N = k C c k 2, E = k C ɛ k c k 2 + U 0 2 ijkl C We find time evolution gives thermal equilbrium. c i c jc k c l ij kl PGPE system appears ergodic: time average ensemble average A = lim N 1 N j=1 A j = lim θ θ 0 A(t)dt.. p.18

19 Homogeneous gas M. J. Davis et al. PRL 87, (2001); PRA 66, (2002). Plane wave basis, 3D, k < 15 2π/L Condensate population versus time 10 3 Averaged quasiparticle distributions E = 6000 E = 7500 E = 9000 E = N 0 (τ) / N N k / N tot τ k (2π/L) Agrees with second order theories of BEC. Morgan J. Phys. B 33, 3847 (2000), also Fedichev & Shylapnikov, PRA 58, 3146 (1998); Giorgini, PRA 61, (2000). See also eg Gòral et al. PRA 66, (2002), Sinatra et al. PRL 87, (2001), Stoof and Bijlsma, J. Low. Temp. Phys 124, 431 (2001).. p.19

20 Snapshots of the trapped gas Blakie and Davis, condmat/ What does a classical matter wave look like? ψ(x, y, 0) 2, log scale. N 0 /N = 0.93 N 0 /N = 0.45 N 0 /N = 0.02 Low energy = high energy. p.20

21 Experimental snapshots in trap Dark ground imaging, MIT p.21

22 Toy simulation of evaporative cooling TOP trap geometry, E cut = 31 ω x (1739 modes). Begin at T > T c. Atoms removed beyond z > 9x 0. Column density Density slice. p.22

23 Time-averaged column densities Φ(k) (a) f c = 0.00 Momentum space (b) f c = Ψ(x) 2 k z (d) k x (c) f c = 0.87 z x (e) (f) Real space. p.23

24 Theorists criterion for BEC: Penrose-Onsager Single-particle density matrix has a macroscopic eigenvalue. Given ψ(x, t) = k c k(t)φ(x) we can calculate ρ ij = c i c j lim T 1 T T 0 c i (t)c j (t)dt Typically have 2000 states below cutoff. This can easily be diagonalized on a workstation. [We have a non-perturbative, microcanonical measure of T, µ] Condensate fraction Temperature Cnl = 0 Cnl = 5e2 Cnl = 1e3 Cnl = 2e3 Cnl = 5e3. p.24

25 Experimentalists measure of BEC Fit a bimodal distribution to column density. Compare the two measures from an evaporative cooling calculation. dk x Φ(k x,k y,k z ) 2 (a) k y k z dk y Φ(k x,k y,0) (b) k y [1/x 0 ]. p.25

26 Fluctuations For the Bose field operator, we define correlation functions as g (2) (x, x ) = ˆΨ (x) ˆΨ (x ) ˆΨ(x) ˆΨ(x ) ˆΨ (x) ˆΨ(x) ˆΨ (x ) ˆΨ(x ) and similarly for g (3) (x, x, x ). For the classical field method we calculate g (2) (x, x) = ψ(x) 4 time ave. [ ψ(x) 2 time ave. ] 2, Standard results: g (2) (x) = 1 for condensate, = 2! for thermal. g (3) (x) = 1 for condensate, = 3! = 6 for thermal.. p.26

27 Results Calculated for TOP trap simulations along radial axis n 0 = n 0 = n 0 = 0.01 g 2 (r) g 3 (r) r r r. p.27

28 Shift of T c with interaction strength. p.28

29 The ideal gas BEC N 0 / N tot = 1 ( T / T c ) 3/2 Condensate fraction Temperature / T c. p.29

30 Shift in T c for the homogeneous gas: a long history Hartree-Fock prediction: shift in µ c, no shift in T c. First order shift is due to critical fluctuations. First predictions from Lee and Yang: 1957 : δt c a 1958 : δt c a Several other calculations, giving a wide variety of results. Many attempts use perturbation theory. However: condensation is governed by long wavelength physics: inherently non-perturbative.. p.30

31 The debate has recently been settled Baym et al. used effective field theory to show δt c T 0 c = c a n 1/3 In 2001 Monte Carlo calculations gave: c = 1.32 ± 0.02: Arnold and Moore et al., PRL 87, (2001). c = 1.29 ± 0.05: Kashurnikov et al., PRL 87, (2001). Many other results in broad agreement 1/N expansions (Baym et al., Arnold and Tomasik) Summation of various diagrams (Baym et al.) Variational perturbation theory (Kastening, Kleinert,... ) Renormalization group approaches (Ledowski et al.) See Jens O. Andersen, Rev. Mod. Phys 76, 599 (2004) for a recent review.. p.31

32 The classical field approach to shift in T c The Projected Gross-Pitaevskii equation: i ψ(x) τ = H sp ψ(x) + C nl P { ψ(x) 2 ψ(x) }, C nl = 8πaN L. Procedure: Choose momentum cutoff. Generate randomised initial ψ(x). Evolve to equilibrium. Measure T, µ. (Davis and Morgan, 2003.) Condensate fraction Cnl = 0 Cnl = 5e2 Cnl = 2e3 Cnl = 5e3 Cnl = 10e3 Cnl = 15e3 Cnl = 20e4 0 We find c = 1.3 ± Temperature. p.32

33 The trapped Bose gas is qualitatively different Several competing phenomena: Mean field effects Giorgini et al., Phys. Rev. A 54, R4633, (1996). δt c T 0 c ( ) a 1.33 N 1/6 a ho Finite size effects Grossmann and Holthaus, Phy. Lett. A 208, 188 (1995). δt c T 0 c Critical fluctuations (+ve) 0.24 N 1/3 [ ωx + ω y + ω z (ω x ω y ω z ) 1/3. ].. p.33

34 An early experiment on thermodynamics by JILA T c /T 0 c = 0.06 ± 0.05 Ensher et al., Phys. Rev. Lett. 77, 4984 (1996). Error bars as big as mean field shift. Similar in other experiments.. p.34

35 Other shift in T c calculations for the trapped gas Second order calculation using lattice result. P. Arnold and B. Tomasik, Phys. Rev. A 64, (2001). Mean field shift for power law potentials. O. Zobay, J. Phys. B 37, 2593 (2004). Renormalization group approach for power law potentials. O. Zobay, G. Metikas and G. Alber, Phys. Rev. A 69, (2004). Variational perturbation theory for power law potentials. O. Zobay, G. Metikas and H. Kleinert, Phys. Rev. A 71, (2005). All of these are in the thermodynamic limit. Also no comparison with experiment.. p.35

36 Measurement by Gerbier et al., PRL 92, (2004) Procedure: vary final rf frequency of evaporative cooling ramp. 4 T (nk) N0 (10 ) (a) (b) Tc 6 N (10 ) (c) Nc Trap depth ν rf -ν 0 (khz) 120. p.36

37 Measurement by Gerbier et al., PRL 92, (2004) 700 Critical temperature (nk) Experimental data Expt. one sigma fit Ideal gas Finite size ideal gas Critical atom number (10 6 ) Finite size with mean field shift. p.37

38 Are critical fluctuations important for the trapped gas? PGPE for Bose gas in TOP trap. We compare results to mean-field HFB-Popov calculations for the same basis set. Condensate fraction Cnl = 0 Cnl = 5e2 Cnl = 2e3 Cnl = 10e3 HFB Popov 0 Answer: perhaps? Temperature Must take account of the modes ABOVE the cutoff.. p.38

39 Calculating T c for an experimental system Orsay trap: 87 Rb, ν = 413 Hz, ν z = 8.69 Hz. Must first identify appropriate parameters. Fix N total Determine Tc 0, µ 0 c for ideal gas. Select cutoff condition: e.g. N cut 5 Mean occupation 10 3 Classical region Ntot = 2e6 Tc = 640 nk modes 10 1 Ecut = 325 hν x Ncut = This determines N below, E cut Energy (hν x ) N total (10 6 ) Tc 0 (nk) N cut E cut ( ω x ) Modes N below p.39

40 PGPE: Run simulations, measure thermodynamics N 0 / Nbelow Critical region µ (hν x ) T (nk) Energy (hν ) x Energy (hν ) x Energy (hν ) x 10 3 Ecut = 325 hν x Remember: Only below cutoff atoms Mean ccupation Different number of atoms above cutoff for each T Energy (hν x ). p.40

41 Use Hartree-Fock theory for above cutoff atoms Critical point determined by comparison with ideal gas N 0 (T 0 c ). Simulation gives n below (x), T, µ. N 0 (10 3 ) HFB Popov PGPE T c Ideal gas N 0 (T c 0 ) N tot = d 3 x n tot (x), n tot (x) = n below (x) + n above (x) T (nk) Solve self consistently for n above (x): n above (x) = 1 h 3 E HF >E cut d 3 p { ( ) exp EHF (p,x) µ k B T 1} 1, E HF (p, x) = p 2 /2m + V trap (x) + 2gn tot (x). N tot (10 6 ) N c Ideal gas p.41 T (nk)

42 Comparison with experiment 700 Critical temperature (nk) Critical atom number (10 6 ) Experimental data Expt one sigma fit Finite size ideal gas Analytic estimate GPE + semiclassical HFB + semiclassical Classical field Experimental error bars must improve to distinguish theories. p.42

43 Quantum simulations with classical fields. p.43

44 Phase space techniques Represent ˆρ with a quasi-probability distribution. Glauber P -distribution: ˆρ = P (α) α α d 2 α. Wigner distribution: W (α) = 2 π P (α ) exp( 2 α α 2 )d 2 α. Can convert master equation for ˆρ into phase space equation of motion for W [ψ(x)] (multi-mode) i W t = d 3 x [ δ δψ ( Hsp + U 0 ( ψ 2 δ(x)) ) ψ U 0 4 δ 3 ] δ 2 ψδψ W + c.c. ψ Can neglect third-order terms for short times or N M. Then: Fokker-Planck equation equivalent Langevin SDE. i ψ t = H spψ + U 0 ( ψ 2 δ C )ψ.. p.44

45 Not quite so simple... Expectation values are given by {(â ) m â n } sym QM (α ) m α n stoch. e.g. ˆn = â â = 1 2 â â + ââ 1 2. So for a vacuum state we must have: (α ) m α n stoch = 1 2. This means that modes with no real particles still contain 1/2 a particle of quantum noise. Still many things to be careful of.... p.45

46 Application: Condensate collisions Noise stimulates spontaneous events. Norrie, Ballagh and Gardiner, PRL 94, (2005).. p.46

47 Current applications Bosenova: still some quantitative discrepancies in GPE solutions. Correlations in down-converted molecules. Instabilities and heating in an optical lattice. Rapid heating observed by Florence, Otago groups. Population 6 x Total populations (1D) k < 0 k > 0 k x (m 1 ) x Log10 mode occupations (1D) Time (sec) Time (sec) 2. p.47

48 Stochastic GPE approach Gardiner and Davis, J. Phys. B 36, 4731 (2003). Split field operator as earlier: ˆΨ = ˆψ + ˆη. Treat high-lying modes of ˆη as a bath: µ, T. Derive Wigner equations for classical region. GPE with growth terms and thermal driving noise. Similar to approach of Stoof, derived via Keldysh formalism. Applications: Condensate growth. Formation of vortex lattices. See forthcoming work from Bradley and Gardiner.. p.48

49 Summary Finite temperature Bose gases. Introduction to classical fields. Simulations of classical fields via PGPE. Shift in T c for interacting Bose gases. Quantum simulations with classical fields. Thanks to: Blair Blakie, Ashton Bradley, Chris Foster, Andy Ferris, Crispin Gardiner, Sam Morgan, Keith Burnett.. p.49