Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1707.01776 · Physical Review A

Classical fields in the one-dimensional Bose gas: Applicability and determination of the optimal cutoff

Joanna Pietraszewicz, Piotr Deuar

Published Aug 17, 2018• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.98.023622

Related papers

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

Joanna Pietraszewicz, Piotr Deuar

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.

arXiv:1504.06154•Physical Review A•Dec 10, 2015•16 citations

Complex wave fields in the interacting one-dimensional Bose gas

Joanna Pietraszewicz, Piotr Deuar

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.

arXiv:1808.10251•Physical Review A•May 11, 2018•7 citations

Properties of the stochastic Gross–Pitaevskii equation: finite temperature Ehrenfest relations and the optimal plane wave representation

Ashton S. Bradley, P. B. Blakie, Crispin W. Gardiner

We present Ehrenfest relations for the high temperature stochastic Gross–Pitaevskii equation description of a trapped Bose gas, including the effect of growth noise and the energy cutoff. A condition for neglecting the cutoff terms in the Ehrenfest relations is found which is more stringent than the usual validity condition of the truncated Wigner or classical field method—that all modes are highly occupied . The condition requires a small overlap of the nonlinear interaction term with the lowest energy single particle state of the noncondensate band, and gives a means to constrain dynamical artefacts arising from the energy cutoff in numerical simulations. We apply the formalism to two simple test problems: (i) simulation of the Kohn mode oscillation for a trapped Bose gas at zero temperature, and (ii) computing the equilibrium properties of a finite temperature Bose gas within the classical field method. The examples indicate ways to control the effects of the cutoff, and that there is an optimal choice of plane wave basis for a given cutoff energy. This basis gives the best reproduction of the single particle spectrum, the condensate fraction and the position and momentum densities.

arXiv:cond-mat/0410653•Journal of Physics B Atomic Molecular and Optical Physics•Nov 14, 2005•43 citations

Weakly Interacting Bose Gas in the One-Dimensional Limit

P. Krüger, Sebastian Hofferberth, I. E. Mazets +2 more

We prepare a chemically and thermally one-dimensional (1D) quantum degenerate Bose gas in a single microtrap. We introduce a new interferometric method to distinguish the quasicondensate fraction of the gas from the thermal cloud at finite temperature. We reach temperatures down to $kT\ensuremath{\approx}0.5\ensuremath{\hbar}{\ensuremath{\omega}}_{\ensuremath{\perp}}$ (transverse oscillator eigenfrequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$) when collisional thermalization slows down as expected in 1D. At the lowest temperatures the transverse-momentum distribution exhibits a residual dependence on the line density ${n}_{1\mathrm{D}}$, characteristic for 1D systems. For very low densities the approach to the transverse single-particle ground state is linear in ${n}_{1\mathrm{D}}$.

arXiv:1012.3048•Physical Review Letters•Dec 28, 2010•75 citations

Simple theoretical tools for low dimension Bose gases

Yvan Castin

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.

arXiv:cond-mat/0407118•Journal de Physique IV (Proceedings)•Oct 1, 2004•117 citations

Condensation and quasicondensation in an elongated three-dimensional Bose gas

Michael C. Garrett, Tod M. Wright, Matthew J. Davis

We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body $T$ matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.

arXiv:1209.3297•Physical Review A•Jun 12, 2013•14 citations
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 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.
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 Ehrenfest relations for the high temperature stochastic Gross–Pitaevskii equation description of a trapped Bose gas, including the effect of growth noise and the energy cutoff. A condition for neglecting the cutoff terms in the Ehrenfest relations is found which is more stringent than the usual validity condition of the truncated Wigner or classical field method—that all modes are highly occupied . The condition requires a small overlap of the nonlinear interaction term with the lowest energy single particle state of the noncondensate band, and gives a means to constrain dynamical artefacts arising from the energy cutoff in numerical simulations. We apply the formalism to two simple test problems: (i) simulation of the Kohn mode oscillation for a trapped Bose gas at zero temperature, and (ii) computing the equilibrium properties of a finite temperature Bose gas within the classical field method. The examples indicate ways to control the effects of the cutoff, and that there is an optimal choice of plane wave basis for a given cutoff energy. This basis gives the best reproduction of the single particle spectrum, the condensate fraction and the position and momentum densities.
We prepare a chemically and thermally one-dimensional (1D) quantum degenerate Bose gas in a single microtrap. We introduce a new interferometric method to distinguish the quasicondensate fraction of the gas from the thermal cloud at finite temperature. We reach temperatures down to $kT\ensuremath{\approx}0.5\ensuremath{\hbar}{\ensuremath{\omega}}_{\ensuremath{\perp}}$ (transverse oscillator eigenfrequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$) when collisional thermalization slows down as expected in 1D. At the lowest temperatures the transverse-momentum distribution exhibits a residual dependence on the line density n1D{n}_{1\mathrm{D}}n1D​, characteristic for 1D systems. For very low densities the approach to the transverse single-particle ground state is linear in n1D{n}_{1\mathrm{D}}n1D​.
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 study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body TTT matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.