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

Documentation of work developing theoretical methods for treating atomic excitation in a cavity: Progress report

M. J. Shaw, B. Shore, J R Johnson

Published Oct 10, 1988• 0 citations
OpenAlex

Abstract

This memo documents a portion of our work developing methods to treat selective excitation (and subsequent ionization) of atomic vapor in a cavity. Specifically, it describes a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We present this method and the algorithm that forms the basis of a computer code for solving these equations. We present examples of solutions for a simple model of stationary two-state atoms in the limit of very low density. These examples illustrate both the usefulness and the limitations of spatial quasi-Fourier expansion techniques. In particular, the examples presented here show the importance of retaining a succession of spatial harmonics when the excitation is coherent (as evidenced by Rabi oscillations). 12 refs., 5 figs.

Related papers

Coherent Atomic Excitation in a Cavity. I. Low Density

M.J. Shaw, Bruce W. Shore

We present a procedure for treating coherent selective excitation (and subsequent ionization) of atomic vapour in a radiation-filled cavity. Specifically, we describe a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We discuss the resulting coupled equations and present examples of solutions for a simple model of stationary and moving two-state atoms in the limit of very low density. The atomic excitation exhibits a travelling wave behaviour, but with significant modulation.

Journal of Modern Optics•May 1, 1990•5 citations

Nonlinear dynamics in Fabry–Perot cavities

Luigi Lugiato, Franco Prati, Massimo Brambilla

In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.

Cambridge University Press eBooks•Feb 28, 2015•0 citations

Cavity-modified Maxwell-Bloch equations for the vacuum Rabi splitting

Changxin Wang, Reeta Vyas

We derive a set of cavity-modified Maxwell-Bloch equations for a two-level atom in a cavity driven by a coherent field from the single-atom Fokker-Planck equation in the bad-cavity limit. These equations have the same form as the Maxwell-Bloch equations for a two-level atom in free space interacting with a coherent field. The presence of a cavity is reflected in three cavity-modified parameters: decay rate ${\ensuremath{\gamma}}_{\mathrm{eff}}$, Rabi frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{eff}}$, and detuning ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{eff}}$. We show that the cavity-modified Maxwell-Bloch equations provide an easy way to study cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, and vacuum Rabi splitting.

Physical Review A•Jan 1, 1997•16 citations

Bloch-state treatment of an atom in a standing-wave cavity

W. Ren, H. J. Carmichael

We use a basis of Bloch wave functions to solve the problem of spontaneous emission from an atom inside a standing-wave cavity. In this way we incorporate the effects of the atom’s quantized center-mass-motion in the periodic light-force potential of the one-quantum dressed states during the spontaneous emission process. We calculate emission spectra and the momentum distribution of the atom. Under strong-coupling conditions, “vacuum” Rabi spectra become asymmetric, with an increased relative weight given to the red side of the spectrum because of the transfer of energy to the center-of-mass motion of the atom. Because of the nonperturbative interaction between the atom and the cavity mode, many momentum quanta are transferred to the atom from the cavity mirrors during the irreversible emission of just one photon. Thus, the atom is diffracted by its own radiation reaction field. We analyze these effects using the Bloch-state basis. Use of the secular approximation provides a simple rate equation description of transitions from the Bloch states excited in the initial state to the atom-cavity ground state. We compare this approach with other ways of doing the calculations and discuss the accuracy of the secular approximation. We consider the extension of these calculations to treat the effects of photon statistics on the momentum distribution of atoms traversing a standing-wave cavity.

Annual Meeting Optical Society of America•Jan 1, 1992•0 citations

Cavity quantum electrodynamics

THOMAS W. MOSSBERG

Cavities are employed to effect various changes in the electromagnetic reservoir to which specimen atoms are coupled. Reservoirs of diverse character ranging from single-modelike to strongly frequency dependent can be realized. Generally speaking, atoms coupled to perturbed electromagnetic reservoirs display novel dynamics that can be of practical as well as fundamental importance.

Annual Meeting Optical Society of America•Jan 1, 1991•6 citations

Solution of the time-dependent Liouville-von Neumann equation: dissipative evolution

M Berman, Ronnie Kosloff, Hillel Tal‐Ezer

A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative Liouville superoperator has been developed. The framework has been designed to incorporate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, the new method has been successfully applied to dissipation under the constraint of selection rules. More specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied.

Journal of Physics A Mathematical and General•Mar 7, 1992•112 citations
This memo documents a portion of our work developing methods to treat selective excitation (and subsequent ionization) of atomic vapor in a cavity. Specifically, it describes a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We present this method and the algorithm that forms the basis of a computer code for solving these equations. We present examples of solutions for a simple model of stationary two-state atoms in the limit of very low density. These examples illustrate both the usefulness and the limitations of spatial quasi-Fourier expansion techniques. In particular, the examples presented here show the importance of retaining a succession of spatial harmonics when the excitation is coherent (as evidenced by Rabi oscillations). 12 refs., 5 figs.
We present a procedure for treating coherent selective excitation (and subsequent ionization) of atomic vapour in a radiation-filled cavity. Specifically, we describe a technique for solving the coupled Maxwell-Bloch equations for atoms in a one-dimensional cavity by means of a spatial quasi-Fourier expansion that introduces a sequence of increasing spatial harmonics for atomic populations and coherences. We discuss the resulting coupled equations and present examples of solutions for a simple model of stationary and moving two-state atoms in the limit of very low density. The atomic excitation exhibits a travelling wave behaviour, but with significant modulation.
In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.
We derive a set of cavity-modified Maxwell-Bloch equations for a two-level atom in a cavity driven by a coherent field from the single-atom Fokker-Planck equation in the bad-cavity limit. These equations have the same form as the Maxwell-Bloch equations for a two-level atom in free space interacting with a coherent field. The presence of a cavity is reflected in three cavity-modified parameters: decay rate ${\ensuremath{\gamma}}_{\mathrm{eff}}$, Rabi frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{eff}}$, and detuning ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{eff}}$. We show that the cavity-modified Maxwell-Bloch equations provide an easy way to study cavity-modified spontaneous emission, cavity-induced radiative energy level shifts, and vacuum Rabi splitting.
We use a basis of Bloch wave functions to solve the problem of spontaneous emission from an atom inside a standing-wave cavity. In this way we incorporate the effects of the atom’s quantized center-mass-motion in the periodic light-force potential of the one-quantum dressed states during the spontaneous emission process. We calculate emission spectra and the momentum distribution of the atom. Under strong-coupling conditions, “vacuum” Rabi spectra become asymmetric, with an increased relative weight given to the red side of the spectrum because of the transfer of energy to the center-of-mass motion of the atom. Because of the nonperturbative interaction between the atom and the cavity mode, many momentum quanta are transferred to the atom from the cavity mirrors during the irreversible emission of just one photon. Thus, the atom is diffracted by its own radiation reaction field. We analyze these effects using the Bloch-state basis. Use of the secular approximation provides a simple rate equation description of transitions from the Bloch states excited in the initial state to the atom-cavity ground state. We compare this approach with other ways of doing the calculations and discuss the accuracy of the secular approximation. We consider the extension of these calculations to treat the effects of photon statistics on the momentum distribution of atoms traversing a standing-wave cavity.
A mathematical and numerical framework has been worked out to represent the density operator in phase space and to propagate it in time under dissipative conditions. The representation of the density operator is based on the Fourier pseudospectral method which allows a description both in configuration as well as in momentum space. A new propagation scheme which treats the complex eigenvalue structure of the dissipative Liouville superoperator has been developed. The framework has been designed to incorporate modern computer architecture such as parallelism and vectorization. Comparing the results to closed-form solutions exponentially fast convergence characteristics in phase space as well as in the time propagation is demonstrated. As an example of its usefulness, the new method has been successfully applied to dissipation under the constraint of selection rules. More specifically, a harmonic oscillator which relaxes to equilibrium under the constraint of second-order coupling to the bath was studied.