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

Quantum statistics of a laser cooled ideal gas

J. I. Cirac, Maciej Lewenstein, P. Zoller

Published May 9, 1994• 47 citations
Publisher

Abstract

We study the dynamics of a system of bosonic or fermionic atoms in a microscopic trap undergoing laser cooling. We show that the stationary state can be described by a Bose-Einstein or Fermi-Dirac distribution, respectively. Fluorescence from the system reflects quantum statistical properties.
DOI
10.1103/physrevlett.72.2977

Related papers

Quantum dynamics of a laser-cooled ideal gas

J. I. Cirac, Maciej Lewenstein, P. Zoller

We study a system of bosonic or fermionic atoms in a microscopic trap undergoing laser cooling. We derive a master equation governing the evolution of such a system, and show that the stationary state can be described by Bose-Einstein or Fermi-Dirac distributions. The quantum-statistical character of the atoms exhibits itself in the dynamical behavior of the system and in the statistical properties of fluorescence photons emitted in the stationary state.

Physical Review A•Oct 1, 1994•33 citations

Quantum ideal gases: Fermi-Dirac and Bose-Einstein statistics

Mark E. Tuckerman

Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed

•Aug 1, 2023•0 citations

On quantum statistics for ensembles with a finite number of particles

Evgenii Dmitrievich Trifonov

The well-known Bose–Einstein and Fermi–Dirac quantum distributions can be considered as stationary solutions of kinetic equations for the mean occupation numbers in an ideal gas of an arbitrary finite number of identical particles.

Physics-Uspekhi•Jul 31, 2011•5 citations

Quantum field theory of atoms interacting with photons: Foundations

Maciej Lewenstein, Li You, J. Cooper +1 more

We discuss theoretical foundations of the emerging area of quantum-optical studies of many-body systems: a quantum field theory of atoms interacting with photons. This theory is designed to describe quantum-statistical systems of cooled and trapped atoms undergoing spontaneous and stimulated emission processes. Formulated in the framework of the second quantization, this theory accounts fully for quantum-statistical properties of atoms as bosons or fermions. In particular, it applies to a description of the formation of the Bose-Einstein condensate in bosonic systems, as well as the spectroscopy and other quantum optical properties of the Bose-Einstein condensate. We derive the general second quantized Hamiltonian of this theory and discuss in detail various approximations.

Physical Review A•Sep 1, 1994•106 citations

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

Fermions

C. J. Pethick, H. Smith

The laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.

Cambridge University Press eBooks•Sep 11, 2008•335 citations
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We discuss theoretical foundations of the emerging area of quantum-optical studies of many-body systems: a quantum field theory of atoms interacting with photons. This theory is designed to describe quantum-statistical systems of cooled and trapped atoms undergoing spontaneous and stimulated emission processes. Formulated in the framework of the second quantization, this theory accounts fully for quantum-statistical properties of atoms as bosons or fermions. In particular, it applies to a description of the formation of the Bose-Einstein condensate in bosonic systems, as well as the spectroscopy and other quantum optical properties of the Bose-Einstein condensate. We derive the general second quantized Hamiltonian of this theory and discuss in detail various approximations.
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 laser cooling mechanisms described in Chapter 4 operate irrespective of the statistics of the atom, and they can therefore be used to cool Fermi species. The statistics of a neutral atom is determined by the number of neutrons in the nucleus, which must be odd for a fermionic atom. Since alkali atoms have odd atomic number Z , their fermionic isotopes have even mass number A . Such isotopes are relatively less abundant than those with odd A since they have both an unpaired neutron and an unpaired proton, which increases their energy by virtue of the odd–even effect. In early experiments, 40 K and 6 Li atoms were cooled to about one-quarter of the Fermi temperature. More recently, fermionic alkali atoms have been cooled to temperatures well below one-tenth of their Fermi temperature, and in addition a degenerate gas of the fermionic species 173 Yb (with I = 5/2) has been prepared. In the classical limit, at low densities and/or high temperatures, clouds of fermions and bosons behave alike. The factor governing the importance of quantum degeneracy is the phase-space density ϖ introduced in Eq. (2.24), and in the classical limit ϖ ≪ 1. When ϖ becomes comparable with unity, gases become degenerate: bosons condense in the lowest single-particle state, while fermions tend towards a state with a filled Fermi sea. As one would expect on dimensional grounds, the degeneracy temperature for fermions – the Fermi temperature T F – is given by the same expression as the Bose–Einstein transition temperature for bosons, apart from a numerical factor of order unity.