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arXiv:1504.06154 · Physical Review A

Classical field records of a quantum system: Their internal consistency and accuracy

Joanna Pietraszewicz, Piotr Deuar

Published Dec 10, 2015• 16 citations
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Abstract

We determine the regime where the widespread classical field description for quantum Bose gases is quantitatively accurate in one dimension (1D), 2D, and 3D by a careful study of the ideal gas limit. Numerical benchmarking in 1D shows that the ideal gas results carry over unchanged into the weakly interacting gas. The optimum high-energy cutoff is in general shown to depend strongly on the observable in question (e.g., energy, density fluctuations, phase coherence length, condensate fraction). This explains the wide range of past results. A consistent classical field representation with less than 10% deviation in all typical observables can be given for systems at temperatures below $0.0064$ degeneracy temperature in 1D, and $0.49$ critical temperature in 3D. Surprisingly, this is not possible for the two-dimensional ideal gas even at zero temperature because mean density, density fluctuations, and energy cannot be simultaneously matched to the quantum results.

DOI
10.1103/physreva.92.063620

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We determine the regime where the widespread classical field description for quantum Bose gases is quantitatively accurate in one dimension (1D), 2D, and 3D by a careful study of the ideal gas limit. Numerical benchmarking in 1D shows that the ideal gas results carry over unchanged into the weakly interacting gas. The optimum high-energy cutoff is in general shown to depend strongly on the observable in question (e.g., energy, density fluctuations, phase coherence length, condensate fraction). This explains the wide range of past results. A consistent classical field representation with less than 10% deviation in all typical observables can be given for systems at temperatures below 0.00640.00640.0064 degeneracy temperature in 1D, and 0.490.490.49 critical temperature in 3D. Surprisingly, this is not possible for the two-dimensional ideal gas even at zero temperature because mean density, density fluctuations, and energy cannot be simultaneously matched to the quantum results.
To finalize information about the accuracy of the classical field approach for the one-dimensional (1D) Bose gas, the lowest-temperature quasicondensate is studied by comparing the extended Bogoliubov model of Mora and Castin [Mora and Castin, Phys. Rev. A 67, 053615 (2003)] to its classical field analog. The parameters for which the physics is well described by matter waves are now presented for all 1D regimes and, concurrently, the optimal cutoff that best matches all observables together is also provided. This cutoff rises strongly with density when the chemical potential is higher than the thermal energy to account for kinetic energy. As a consequence, clouds that reach this coldest quantum fluctuating regime are better described using a trap basis than plane waves. This contrasts with higher-temperature clouds for which the basis choice is less important. In passing, estimates for chemical potential, density fluctuations, and kinetic and interaction energy in the low-temperature quasicondensate are obtained up to several leading terms.
We first consider an exactly solvable classical field model to understand the coherence properties and the density fluctuations of a one-dimensional (1D) weakly interacting degenerate Bose gas with repulsive interactions at temperatures larger than the chemical potential. In a second part, using a lattice model for the quantum field, we explain how to carefully generalize the usual Bogoliubov approach to study a degenerate and weakly interacting Bose gas in 1D, 2D or 3D in the regime of weak density fluctuations. In the last part, using the mapping to an ideal Fermi gas in second quantized formalism, we calculate and discuss physically the density fluctuations and the coherence properties of a gas of impenetrable bosons in 1D.
We present a novel norm preserving stochastic evolution equation for a Bose field at finite temperature. Ensemble averages are quantum expectation values in the canonical ensemble. This numerically very stable equation suppresses high-energy fluctuations exponentially, which prevent cutoff problems from occurring. We present 3D simulations for an ideal gas in various trapping potentials and focus on ground-state occupation numbers and spatial correlation functions for a wide range of temperatures above and below the critical temperature. Although rigorously valid for non-interacting bosons only, we argue that weakly interacting Bose gases may also be amenable to this approach, in the usual mean-field approximation.
We study the temperature regimes of the one-dimensional interacting gas to determine when the matter wave (c-field) theory is, in fact, correct and usable. The judgment is made by investigating the level of discrepancy in many observables at once in comparison to the exact Yang-Yang theory. We also determine what cutoff maximizes the accuracy of such an approach. Results are given in terms of a bound on accuracy, as well as an optimal cutoff prescription. For a wide range of temperatures the optimal cutoff is independent of density or interaction strength and so its temperature-dependent form is suitable for many cloud shapes and, possibly, basis choices. However, this best global choice is higher in energy than most prior determinations. The high value is needed to obtain the correct kinetic energy, but does not detrimentally affect other observables.
We present a novel norm preserving stochastic evolution equation for a Bose field. Ensemble averages are quantum expectation values in the canonical ensemble. This numerically very stable equation suppresses high-energy fluctuations exponentially, preventing cutoff problems to occur. We present 3D simulations for an ideal gas in various trapping potentials focussing on ground state occupation numbers and spatial correlation functions for a wide range of temperatures above and below the critical temperature. Although rigorously valid for non-interacting Bosons only, we argue that weakly interacting Bose gases may also be amenable to this approach, in the usual mean-field approximation.
We investigate the long-range phase coherence of homogeneous and trapped Bose gases as a function of the geometry of the trap, the temperature, and the mean-field interactions in the weakly interacting limit. We explicitly take into account the (quasi)condensate depletion due to quantum and thermal fluctuations, i.e., we include the effects of both phase and density fluctuations. In particular, we determine the phase diagram of the gas by calculating the off-diagonal one-particle density matrix and discuss the various crossovers that occur in this phase diagram and the feasibility of their experimental observation in trapped Bose gases.