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Physical Review A

Rabi oscillations in the scattering-state representation

Ahmet Elçi, David A. Depatie

Published Aug 1, 1990• 0 citations
Publisher

Abstract

This paper presents an analysis of the Rabi oscillations of two-level atoms in the scattering-state representation. A collision complex obtained from a pair of identical two-level atoms is coupled to a field mode and it is found that there can be three distinct Rabi frequencies. These reduce to a single Rabi frequency for a photon distribution that is sharply peaked around an average number n\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\gg}1. One then finds that, in addition to the three frequency components in the off-diagonal element of the atomic density matrix, there are two components that are symmetrically displaced around the mode frequency at intervals of twice the Rabi frequency. They represent the collisionally induced cooperative motion of the atoms. When various observable quantities are averaged over the orientation of the complex, one obtains results that apply to many-atom systems in the binary collision approximation. Collisional effects do not enter the above results as lifetimes due to the fact that scattering states have precisely defined energies and correspond to entire particle histories. The collision potential opens up new radiative coupling channels, which collectively represent spectral line broadening. The collision potential and the radiative coupling combine to produce static and quasistatic dipole moments for the complex. At the threshold for atomic population inversion, the density matrix for the complex, as well as for an atom, becomes a constant multiple of the identity matrix, demonstrating that the threshold of stimulated emission is also a critical point for collisional effects, within the approximations of this paper.

DOI
10.1103/physreva.42.1569

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This paper presents an analysis of the Rabi oscillations of two-level atoms in the scattering-state representation. A collision complex obtained from a pair of identical two-level atoms is coupled to a field mode and it is found that there can be three distinct Rabi frequencies. These reduce to a single Rabi frequency for a photon distribution that is sharply peaked around an average number n\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\gg}1. One then finds that, in addition to the three frequency components in the off-diagonal element of the atomic density matrix, there are two components that are symmetrically displaced around the mode frequency at intervals of twice the Rabi frequency. They represent the collisionally induced cooperative motion of the atoms. When various observable quantities are averaged over the orientation of the complex, one obtains results that apply to many-atom systems in the binary collision approximation. Collisional effects do not enter the above results as lifetimes due to the fact that scattering states have precisely defined energies and correspond to entire particle histories. The collision potential opens up new radiative coupling channels, which collectively represent spectral line broadening. The collision potential and the radiative coupling combine to produce static and quasistatic dipole moments for the complex. At the threshold for atomic population inversion, the density matrix for the complex, as well as for an atom, becomes a constant multiple of the identity matrix, demonstrating that the threshold of stimulated emission is also a critical point for collisional effects, within the approximations of this paper.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.
The authors have studied resonance collisions amongst atoms dressed by a strong laser field. The equations of motion have been solved numerically to yield collisional transition rates for various values of the Rabi frequency (V) due to the laser and its detuning ( Delta ) from the atomic transition frequency ( omega 0 ). The atoms are assumed to follow classical rectilinear paths and only undergo binary collisions. The resonance interaction is taken to be the first-order dipole-dipole interaction acting within the ground and resonance levels of a pair of atoms. A qualitative explanation is offered for the results obtained by splitting the Hamiltonian into parts that represent the collision-free evolution of the system at the Rabi frequency, the 'normal' resonance interaction, and the 'dressed interaction' responsible for opening new scattering channels. The latter interaction depends upon the relative strengths of the resonance interaction (R), the generalized Rabi frequency ( Omega ) and the ratio V/ Delta .
General expressions are derived for the spectral line shapes of resonance absorption and scattering of coherent radiation in collision-broadened gases, taking into account effects of coherent excitation of two or more atoms (or molecules), as steady-state solutions of a hierarchy of master equations described in a previous publication (paper I). Coupling between the coherent motions of the atoms, provided by a Bethe-Salpeter---type effective interaction, in the binary-collision approximation, forms the essential mechanism for introducing cooperative coherent effects into the steady-state spectra. Explicit expressions are given for the effects of two-atom coherence in the binary-collision approximation, in which the Bloch-type dressed-atom self-energy superoperator is modified by the presence of collisions in which both atoms retain memory of their coherent propagation before the collision. The self-energies include the effects of resonance exchange symmetrization in self-broadening, and are renormalized by the coincidence of radiative transitions during the collisions. The impact (near-resonance) and the quasistatic (line-wing) limits of the applied-frequency detunings are discussed. In the quasistatic limit, coherent many-atom excitations become irrelevant; however, interactions of both collision partners with the radiation during the collision accounts for such phenomena as collision-induced absorption or radiative collisions. In the impact limit, the inclusion of the Bethe-Salpeter interactions allows for the appearance of two-atom resonances. Magnitude estimates of these effects are discussed. Effects of higher-rank (many-body) coherences are formally discussed with the help of a diagrammatic method, leading into implicit bootstrap equations that can be solved by iterative or other procedures.
We derive two types of perturbative approximation for the response of collisionally broadened two-level atoms to a time-dependent field. The perturbative approximations for the inversion and polarization are valid to any desired order in the derivatives of the field. Comparison with a numerical solution of the full density-matrix equations of motion demonstrates that the new approximations significantly improve upon the adiabatic-following and collisional steady-state approximations for time-dependent pulses in the appropriate damping regime. Application of the approximations to pulse propagation, sideband instability, and population pulsations is discussed. The collisional steady-state and adiabatic-following approximations find extensive use in many areas of nonlinear optics. The deficiencies of these approximations are well known: The collisional steady-state approxirnation is valid only for monochromatic (cw) radiation and the adiabatic-following approximation is valid only in the absence of damping and only if the product of the detuning and a characteristic time is much larger than one. In spite of their manifest faults, the collisional steady-state and adiabatic-following approximations are often exceptionally good approximations, in the appropriate damping regime, for the local response of a vapor of two-level atoms to a slowly varying field. However, timedependent nonlinear processes modify a propagating field zuch that it varies less slowly and the neglected portions of the atomic response become more important. ' Therefore improved approximations for the damped atomic response to a time-dependent field are desirable. In this paper we present two improved perturbative approximations for the inversion and polarization of collisionally broadened two-level atoms induced by a time-dependent field. The adiabatic-following approximation for two-level systems was derived phenomenologically by Grischkowsky, who used it to explain certain experimental results of near-resonance laser-pulse propagation. Motivated by the work of Grischkowsky, Crisp provided an analytic derivation for the adiabatic-following approximation and obtained a quadrature for the error. The essential feature of Crisp's derivation is the expansion of the kernel of the integral representation of the off-diagonal density-matrix element p2, in a Fourier series with integration proceeding term by term. Both the adiabatic-following approximation and the collisional steady-state approximation were obtained in first order. Both Crisp and Grischkowsky used the adiabatic inversion to derive a correction to the adiabatic-following approximation for the polarization. We have shown how Crisp's method can be extended to obtain corrections to the adiabatic inversion and to improve corrections to the polarization in the absence of damping. In this paper we show how the integrated Fourier series can be used to obtain two types of perturbative approximation for the inversion and polarization of collisionally broadened twolevel systems that are valid to any desired order in the derivatives of the field. The two types of perturbative approximation to the inversion are (i) a formal solution of the rate equation for the inversion which is obtained from the density-matrix equations for the populations; and (ii) an approximation obtained using the approximation (p22 — p») —I — 4p2, P, 2, valid for weak damping. The approximations for the polarization are obtained by substituting either (i) or (ii) for the inversion in the integrated-Fourier-series expansion of p2, . The approximation obtained using the first type of approximate inversion converges most rapidly for strong damping and reduces to the collisional steady-state approximation when the magnitude and phase of the field are constant. The approximation obtained using the second type of approxirnate inversion converges most rapidly for weak damping and in first order reduces to the adiabaticfollowing approximation for zero damping. Comparison with a numerical solution of the density-matrix equations demonstrates that retention of higher-order derivatives by retaining more terms in the integrated Fourier series results in a significant improvement in accuracy for a time-dependent field. However, Rabi oscillations, which can be an important component of the nonlinear response, ' cannot be obtained by finite-order approxi
Variations in the spatial intensity distribution of light caused by coherent interaction with two-level atoms are determined by semi-classically calculating a term for interference between incident light and spherical radiation from the atom. The interference term is divided into two components: one oscillates at the Rabi frequency corresponding to absorption and stimulated emission, and the other contributes to dispersion. The spatial distribution of both components involves alternation of constructive and destructive parts in the form of parabolic surfaces where light energy flows. Although the time average over the Rabi oscillation of the former component vanishes at every position, that of the latter remains. The light-related reaction of a dipole force acting on an atom is considered to appear in the non-vanishing latter component.
The evaluation of the binary-collision operator Phi ( omega ), pertaining to collisional redistribution of strong laser radiation, involves the computation of a time-ordered exponential, which only reduces to a scalar exponential in the medium-coupling limit. In this approximation one neglects the alteration of the collisional dynamics due to the presence of the laser field. A two-state atom is considered, for which the authors expand Phi ( omega ) in the ratio of the Rabi frequency Omega to the detuning from resonance. This parameter vanishes in the medium-coupling case, and is small for common gas-phase experiments. It is shown that the correction to medium coupling can again be expressed in a scalar function, which represents the temporal correlation between the interaction potential and the full time evolution during a single collision. The decay constants for collisional transitions between dressed states, which are germane to the width of the spectral lines, appear to be sensitive to the deviation from medium coupling. Quite remarkably, a non-vanishing collisional contribution survives in the limit Omega to 0 of Phi ( omega ).