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

Physica Scripta

Atomic Brownian Motion in a Light Wave

Jean Dalibard

Published Jan 1, 1986• 2 citations
Publisher

Abstract

We derive a kinetic equation describing the motion of slow atoms in a light wave. This equation is of the Kramers-Fokker-Planck type. The diffusion terms are due to the fluctuations in momentum exchanges between the atom and the field, and the friction term is responsible for radiative cooling. Connections with semi classical approaches to this problem and with usual Brownian motion theory are established.

DOI
10.1088/0031-8949/1986/t12/004

Related papers

Atomic motion in laser light: connection between semiclassical and quantum descriptions

Jean Dalibard, Claude N. Cohen-Tannoudji

The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.

Journal of Physics B Atomic and Molecular Physics•Apr 28, 1985•151 citations

The light pressure force and the friction and diffusion coefficients for atoms in a resonant nonuniformly polarized laser field

A. V. Bezverbnyĭ, Прудников Олег Николаевич, А. В. Тайченачев +2 more

The motion of slow atoms with degenerate energy levels in a resonant, nonuniformly polarized laser field is described by the Fokker-Planck equation for the atomic distribution function in phase space in terms of the semiclassical approach. Field gradient expansions are used for the spatially nonuniform coefficients of the equation. For closed atomic transitions J g =J→J e =J+1 (J g and J e are the total angular momenta of the ground and excited states, respectively), new analytical results are presented for the light pressure force and the friction and diffusion coefficients in momentum space. These results allow the kinetic effects (laser cooling, localization in optical potential wells, etc.) in a field of arbitrary one-, two-, or three-dimensional configuration to be investigated. In several cases, the new contributions to the friction coefficient are interpreted qualitatively.

Journal of Experimental and Theoretical Physics•Mar 1, 2003•20 citations

Brownian Motion of a Quantum Oscillator

G. S. Agarwal

The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

Physical Review A•Aug 1, 1971•208 citations

Brownian motion of a quantum particle

Deepak Kumar

Proposed here is an exactly soluble model for the field-free motion of a quantum particle interacting with a heat bath. The heat-bath interaction includes a fluctuating random force and a frictional force of Langevin type. The momentum and position distributions obtained in the model have a close relation to the corresponding solutions of the Fokker-Planck equation of a free classical particle.

Physical Review A•Mar 1, 1984•21 citations

Quantum-Mechanical Fluctuations of the Resonance-Radiation Force

Richard J. Cook

The influence on atomic motion of quantum-mechanical fluctuations of the resonance-radiation force is investigated theoretically. Fluctuations due to both induced and spontaneous absorption-emission processes give rise to diffusion of atomic momentum, here described by a Fokker-Planck equation. It is shown that quantum-mechanical fluctuations of the radiation force place a lower bound on the temperature achievable by radiation cooling, inhibit cooling in a strong standing wave, and lead to finite, often short, confinement times for atoms in radiation traps.

Physical Review Letters•Apr 14, 1980•40 citations

Laser cooling theory of multistate atoms

Juha Javanainen

In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations
We derive a kinetic equation describing the motion of slow atoms in a light wave. This equation is of the Kramers-Fokker-Planck type. The diffusion terms are due to the fluctuations in momentum exchanges between the atom and the field, and the friction term is responsible for radiative cooling. Connections with semi classical approaches to this problem and with usual Brownian motion theory are established.
The quantum kinetic equation describing slow atomic motion in laser light is derived by an operational method which provides mathematical expressions with a transparent physical structure. The authors prove in a general way that the coefficients appearing in this equation, which is of a Fokker-Planck type, are simply related to the mean value and to the correlation functions of the Heisenberg radiative force of the semiclassical approach, where the atomic position is treated classically. They derive in particular a new theoretical expression for the damping force responsible for radiative cooling and they interpret it in terms of linear response theory. The authors also obtain a new crossed r-p derivative term, which does not appear in semiclassical treatments, but which they find to be very small in most situations. Finally, all the theoretical expressions derived are valid for any J g to J e transitions and are not restricted to two-level atoms.
The motion of slow atoms with degenerate energy levels in a resonant, nonuniformly polarized laser field is described by the Fokker-Planck equation for the atomic distribution function in phase space in terms of the semiclassical approach. Field gradient expansions are used for the spatially nonuniform coefficients of the equation. For closed atomic transitions J g =J→J e =J+1 (J g and J e are the total angular momenta of the ground and excited states, respectively), new analytical results are presented for the light pressure force and the friction and diffusion coefficients in momentum space. These results allow the kinetic effects (laser cooling, localization in optical potential wells, etc.) in a field of arbitrary one-, two-, or three-dimensional configuration to be investigated. In several cases, the new contributions to the friction coefficient are interpreted qualitatively.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
The influence on atomic motion of quantum-mechanical fluctuations of the resonance-radiation force is investigated theoretically. Fluctuations due to both induced and spontaneous absorption-emission processes give rise to diffusion of atomic momentum, here described by a Fokker-Planck equation. It is shown that quantum-mechanical fluctuations of the radiation force place a lower bound on the temperature achievable by radiation cooling, inhibit cooling in a strong standing wave, and lead to finite, often short, confinement times for atoms in radiation traps.
In the limit of asymptotically small photon momentum, a Fokker–Planck equation for the center-of-mass Wigner function suffices to describe laser cooling of an atom. We have devised a procedure to compute the light pressure force and the diffusion tensor for an atom with an arbitrary level structure, traveling with an arbitrary velocity in any given three-dimensional light field. The scheme is based on solving a set of differential equations closely related to the density matrix equations of motion encountered in conventional recoilless laser spectroscopy. Computer programs have been written that automatically construct and solve the required equations from broad specifications of the problem, such as energies and angular momenta of the atomic levels. A Langevin equation stochastic simulation has been implemented to solve the Fokker-Planck equation of laser cooling using the computed force and diffusion. Comparisons of optical-molasses experiments and simulations carried out for a model sodium atom with a J = 2→J = 3 transition show qualitative similarities such as sub-Doppler limit temperatures, but the quantitative agreement of theory and experiment is not satisfactory.