New perspectives on classical field simulations of ultracold Bose gases

Transcription

1 New perspectives on classical field simulations of ultracold Bose gases Piotr Deuar Joanna Pietraszewicz Tomasz Świsłocki Igor Nowicki Karolina Borek Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Collaboration: Nick Proukakis University of Newcastle, United Kingdom Peter Drummond King Ng Swinburne University of Technology, Melbourne, Australia N

2 Outline Introduction: a little story The quantitative limits of the classical field regime A nonclassical field with quantum fluctuations 2/27

3 Some things can only be done with c-fields e.g. Defects in a single realisation from thermal equilibrium Karpiuk, PD, Bienias, Witkowska, Pawłowski, Gajda, Rzążewski, Brewczyk, PRL 109, (2012) Full range of quantum gas temperatures Single realization of evaporative cooling density fluctuations Witkowska, PD, Gajda, Rzążewski, PRL 106, (2011) t x T 3/27

4 Classical fields approximation in a nutshell Full quantum field Ensemble of complex-fields Assume highly occupied modes Replace mode amplitude operators with complex number amplitudes Quantum field theory, without discretized particles Evolution: nonlinear Schrodinger equation Developed by many authors: A. Sinatra, M. Brewczyk, M. Gajda, M. Davis, K. Rzazewski, K. Burnett, E. Witkowska, (no order implied) Useful Reviews: M. Brewczyk et al, J. Phys B 40, R1 (2007); P. Blakie et al. Adv. Phys. 57, 363 (2008) 4/27

5 Some disappointing caveats Sensitivity to cutoff No quantum fluctuations Witkowska, Gajda, Rzazewski, PRA 79, (2009) 5/27

6 The cutoff.. The cutoff kc is a very important parameter. Recommendations differ, though: Study Cutoff energy suggestions Ideal gases Canonical ensemble Consideration of number of excited atoms Uniform: 0.30 kbt in 1D Witkowska, Gajda, Rzazewski, PRA 79, (2009) Other values in 2D, 3D SGPE calculations of interacting gas Match particle number in truncated Wigner description to ideal gas Cockburn, Negretti, Proukakis, Henkel, PRA 83, (2011) Brewczyk etal Brewczyk, Gajda, Rzążewski, J. Phys. B 40, R1 (2007) Consideration of damping rates Trapped: 1.0 kbt in 1D Match energy in high E modes to kbt (equipartition) ~ 1 particle in high E modes Sinatra, Lobo, Castin, J. Phys. B 35, 3599 (2002) ~< kbt ~ 10 particles in mode below cutoff Widely used rule of thumb kbt + chemical potential 6/27

7 PART 1: The classical-field region AIMS: Quantitatively determine the conditions under which the physics is that of classical waves. When can classical field predictions be trusted quantitatively? Untangle the reasons behind the different conflicting prescriptions for the cutoff. Give a prescription for the cutoff Want good predictions of many observables simultaneously. 7/27

8 Generic case: uniform section of gas Local Density approximation (LDA) Grand Canonical ensemble (rest of gas acts as a reservoir) Thermodynamic limit for these sections of the gas (i.e. ignore finite-size effects) Plan: (1) understand the ideal gas case (2) benchmark 1D with exact solution Yang, Yang, J. Math. Phys. 10, 1115 (1969) Dimensionless temperature Interaction strength Dimensionless cutoff 8/27

9 Cutoff optimum for different observables matched cutoff fc (ideal gas case) 1D 3D 2D = T / Td Pietraszewicz, PD, PRA 92, (2015) = T / Td Ekin Kinetic energy per particle varn / N Coarse-grained fluctuations lpg phase grain volume (~ coherence length lφ) Half-width of g(1)(x) ρ0 condensate fraction Tc = T / Td Most extreme behaviour 9/27

10 Global accuracy Pietraszewicz, PD, PRA 92, (2015) Single observable error T = Td T = Td T = 0.8 Td Ekin var N / N Ekin cutoff cutoff var N / N cutoff Kinetic energy and coarse-grained fluctuations capture most extreme behaviour In the interacting gas need also Total energy for dynamics to be sensible use these for an indicator Error in any observable will be < RMS 10/27

11 Accuracy RMS( Ekin, varn / N ) 1D T = 8 Td accuracy T = 1.5 Td T = 0.08 Td = T / Td optimum cutoff T = Td cutoff = T / Td Recommendation: Accuracy better than 10% for T < Td Use fc = 0.65 ( Energy cutoff = 1.3 kbt ) Pietraszewicz, PD, PRA 92, (2015) 11/27

12 Accuracy RMS( Ekin, varn / N ) 2D accuracy T = 8 Td T = 1.4 Td T = 0.08 Td 0.33! = T / Td optimum cutoff and less cutoff = T / Td Recommendation: Don't use classical fields, at the least not near the ideal gas regime Pietraszewicz, PD, PRA 92, (2015) 12/27

13 3D Accuracy RMS( Ekin, varn / N ) T = 2.6 Tc 1 c accuracy Tc Tc = T / Td optimum cutoff T= T= T= 0.8 5T c T= T 5.0 Tc cutoff = T / Td Recommendation: Accuracy better than 10% for T < 0.49 Tc Use fc = 0.78 ( Energy cutoff = 1.9 kbt ) Pietraszewicz, PD, PRA 92, (2015) 13/27

14 * Interacting gas: accuracy min[ RMS ] Pietraszewicz, PD, arxiv:1607.xxxxx Generated by Metropolis sampling RMS=0.1 quantum quasicondensate μ<<t Interaction strength RMS=0.1 e soliton gas qu as th ic erm on de al ns at temperature = T / Td As per Witkowska, Gajda, Rzążewski, Opt. Commun. 283, 671 (2010) Generated by SGPE Sampled from Bogoliubov spectrum As per Mora, Castin, PRA 67, (2003) 14/27

15 * Interacting gas: best cutoff Pietraszewicz, PD, arxiv:1607.xxxxx value of fc at min[rms] temperature = T / Td Interaction strength 100 Interaction strength 15/27

16 PART 2: Quantum fluctuations The issue: * Semiclassical field theories do not include quantum fluctuations. antibunching shot-noise zero point phase fluctuations Usually are not sensitive to N, only gn Stopgap measure: truncated Wigner * GPE evolution, but ½ virtual particle per mode to emulate quantum fluctuations. * Good for short times :) * Bad for long times/stationary state :( Thermalizes to classical field ensemble, regardless of initial state. Virtual particles convert to heat. Temperature of this ensemble set by cfield energy + vacuum energy + cutoff 16/27

17 Stochastic GPE alternative classical field model 17/27

18 SGPE Naturally generates a Grand canonical ensemble at long times coupling strength to high energy tails Still no quantum fluctuations. However... it turns out that the several derivations always begin with the full quantum Duine, Stoof, PRA 65, (2003) Wigner representation. Gardiner, Davis, J. Phys. B. 36, 4731 (2003) Then why no quantum fluctuations? because they were always removed by hand by everyone to make the equation easier to handle. SO, what happens when we leave them be? 18/27

19 Wigner SGPE with quantum fluctuations Skip 15 pages, and voila! Small correction * GPE Coupling to High energy tails Energy dependent diffusion In x-space in k-space Minimal diffusion at low energy White noises 19/27

20 * Observables Need to be done properly in Wigner representation: Allows e.g. for antibunching 20/27

21 Sensitivity to absolute particle number Classical fields: GPE / SGPE is unchanged under the transformation: BUT quantum fluctuations rise with γll = g/n ~ λ2 Small absolute particle numbers can be obtained n(x) x 21/27

22 * Antibunching appears PD, Pietraszewicz, Proukakis, Swisłocki, arxiv:1608.xxxxx g(2)(0,x) Classical field (SGPE) 1.01 bunching x antibunching Increasing γll = g / n 0.98 WSGPE γll = g/n ~ λ2 22/27

23 Phase fluctuations with quantum contribution PD, Pietraszewicz, Proukakis, Swisłocki, arxiv:1608.xxxxx As interaction rises, keeping constant chemical potential * μ and T/Td,. Small γll = g / n Larger γll = g / n ~ g(1)(0,x) Largest γll = g / n ~ 0.01 Phase coherence length The expected drop in phase correlation length Thermal fluctuations: Quantum+thermal fluctuations: Coherence length lc Mora, Castin, PRA 67, (2003) quantum thermal classical fields γll = g / n 23/27

24 Strong soliton regime τ / γll 1.0, μ = 22.4 SGPE γll 0, N, e.g. N /27

25 Strong soliton regime τ / γll 1.0, μ = 22.4 WSGPE γll N /27

26 Strong soliton regime τ / γll 1.0, μ = 22.4 WSGPE γll 0.01 N /27

27 Wrap-up We have determined the range of conditions for which a consistent and accurate classical field description exists. It is most necessary to take into account at least: density, density fluctuations, and kinetic energy. A nonclassical field theory has been derived which allows for treatment of quantum and thermal fluctuations together. Can see quantum depletion, antibunching,... This Wigner SGPE has similar numerical properties to the standard SGPE and is not much more numerically intensive. Thank you! 27/27