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

Physical Review Letters

Condensation of N Bosons and the Laser Phase Transition Analogy

Marlan O. Scully

Published May 17, 1999• 83 citations
Publisher

Abstract

A simple analytic expression for the ground state of a dilute gas of $N$ ideal bosons in a 3D harmonic potential at temperature $T$ is derived from the steady state solutions of nonequilibrium equations of motion. The $N$ particle constraint plays the important role of introducing the essential nonlinearity yielding a Ginzburg-Landau free energy. The present analysis has much in common with the quantum theory of the laser, and with the laser phase transition analogy.

DOI
10.1103/physrevlett.82.3927

Related papers

Condensation ofNbosons. II. Nonequilibrium analysis of an ideal Bose gas and the laser phase-transition analogy

Vitaly V. Kocharovsky, Marlan O. Scully, Shi-Yao Zhu +1 more

A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.

Physical Review A•Jan 20, 2000•64 citations

Bose-Einstein Condensation and Superfluidity in 3D and 2D bosons

Andrea Tononi

The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.

•Sep 1, 2018•1 citation

Condensation of N bosons: Microscopic approach to fluctuations in an interacting Bose gas

Anatoly A. Svidzinsky, Marlan O. Scully

We present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.

Physical Review A•Dec 28, 2010•9 citations

Quantum Gases

J. R. Dorfman, Henk van Beijeren, T. R. Kirkpatrick

At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.

Cambridge University Press eBooks•Jun 18, 2021•0 citations

The condensation of a trapped dilute Bose gas with three-body interactions

Phan Thành Nam, Julien Ricaud, Arnaud Triay

We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$. In the limit $N\to \infty$, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schrödinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the $N$-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schrödinger minimizer is unique up to a complex phase.

arXiv:2110.08195•Probability and Mathematical Physics•Mar 29, 2023•15 citations

BEC IN NONEXTENSIVE STATISTICAL MECHANICS: SOME ADDITIONAL RESULTS

LUCA SALASNICH

In a recent paper1 we discussed the Bose–Einstein condensation (BEC) in the framework of Tsallis's nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction.

arXiv:cond-mat/0101350•International Journal of Modern Physics B•Apr 10, 2001•7 citations
A simple analytic expression for the ground state of a dilute gas of NNN ideal bosons in a 3D harmonic potential at temperature TTT is derived from the steady state solutions of nonequilibrium equations of motion. The NNN particle constraint plays the important role of introducing the essential nonlinearity yielding a Ginzburg-Landau free energy. The present analysis has much in common with the quantum theory of the laser, and with the laser phase transition analogy.
A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.
The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.
We present a microscopic derivation of the master equation for the condensate density matrix for an interacting Bogoliubov-Bose gas of N atoms. We choose the interaction Hamiltonian in a special way that substantially simplifies the master equation, yielding no coupling between diagonal and off-diagonal terms. The present formulation allows us to solve the problem analytically in a steady state and obtain the expression for the distribution function and equilibrium condensate fluctuations. For the first two central moments, our results are equivalent to those obtained in the canonical-ensemble quasiparticle formalism [V. V. Kocharovsky, Vl. V. Kocharovsky, and M. O. Scully, Phys. Rev. Lett. 84, 2306 (2000); Phys. Rev. A 61, 053606 (2000)], in the low-temperature range where these papers are valid, but also give an accurate description at high temperatures. The present analysis for an interacting Bose gas is as accurate as the master equation approach of Kocharovsky et al. [Phys. Rev. A 61, 023609 (2000)] is for an ideal gas.
At low temperatures the De Broglie wavelength of the gas particles becomes on the order of their average separation, and the effects of their indistinguishability become important. In the absence of a phase transition in the gas, the quantum mechanical Wigner distribution function for a dilute gas of fermions or bosons satisfies the Uehling-Uhlenbeck equation. This equation satisfies an H- theorem with equilibrium solutions being ideal boson or ideal fermion distributions. Navier-Stokes equations can be derived by standard methods. A low temperature gas of weakly interacting bosons undergoes a Bose-Einstein condensation with a macroscopically occupied ground state. A different approach is required to describe the non-equilibrium processes in such a situation. A kinetic equation can be derived for the Bogoliubov excitations in the gas at very low temperatures. The associated hydrodynamic equations are the Landau-Khaltnikov, two fluid equations, and explicit expressions are obtained for the six associated transport coefficients.
We consider a trapped dilute gas of NNN bosons in R3\mathbb{R}^3R3 interacting via a three-body interaction potential of the form N V(N1/2(x−y,x−z))N\, V(N^{1/2}(x-y,x-z))NV(N1/2(x−y,x−z)). In the limit N→∞N\to \inftyN→∞, we prove that every approximate ground state of the system is a convex superposition of minimizers of a 3D energy-critical nonlinear Schrödinger functional where the nonlinear coupling constant is proportional to the scattering energy of the interaction potential. In particular, the NNN-body ground state exhibits complete Bose--Einstein condensation if the nonlinear Schrödinger minimizer is unique up to a complex phase.