The problem of a two-level atom interacting with an external, time-dependent field is solved approximately using Light's perturbation theory. The solutions obtained are compared with accurate numerical solutions for three specific time profiles. The agreement is found to be excellent except for a certain range of detunings, and even in this range the agreement is good. The validity of an adiabatic approximation is also assessed.

Journal of Physics B Atomic and Molecular PhysicsFeb 14, 198714 citations

Effect of relaxation on adiabatic following

Munir Hasan Nayfeh

Physical Review ASep 1, 19767 citations

Physical Review A

Time-dependent collisional approximations for two-level systems

Michael E. Crenshaw, Cyrus D. Cantrell

Published Jun 1, 1989• 0 citations

Publisher

Abstract

DOI: 10.1103/physreva.39.5705

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Time-dependent collisional approximations for two-level systems

Michael E. Crenshaw, Cyrus D. Cantrell

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. 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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. 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However, Rabi oscillations, which can be an important component of the nonlinear response, ' cannot be obtained by finite-order approxiWe 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. 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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. 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Quantum Electronics and Laser Science ConferenceApr 24, 19890 citations

Adiabatic excitation of multilevel systems

G. L. Peterson, Cyrus D. Cantrell

We derive and discuss the behavior of several multilevel models relevant to the problem of molecular multiphoton excitation under the action of a quasimonochromatic laser field, the amplitude of which rises slowly and falls rapidly on a certain internal time scale. Our method is suitable for the numerical solution of problems involving the adiabatic excitation of large multilevel systems. We find adiabatic inversion to be a general characteristic of the models considered. The fact that adiabatic inversion is accompanied by marked qualitative and quantitative changes in the expectation value of the dipole operator makes our results useful for understanding the propagation of laser pulses. An explicit expression that is especially convenient for numerical evaluation is presented for determining the limiting internal time scale, which is determined by the energy levels and transition moments of the system and the laser frequency. We illustrate the criterion for adiabatic inversion by applying it to a particular multilevel model. From a density-matrix treatment we conclude that collisional damping destroys the coherence required for adiabatic inversion. On the basis of our results we believe that the adiabatic approximation may be appropriate for some experiments of current interest.We derive and discuss the behavior of several multilevel models relevant to the problem of molecular multiphoton excitation under the action of a quasimonochromatic laser field, the amplitude of which rises slowly and falls rapidly on a certain internal time scale. Our method is suitable for the numerical solution of problems involving the adiabatic excitation of large multilevel systems. We find adiabatic inversion to be a general characteristic of the models considered. The fact that adiabatic inversion is accompanied by marked qualitative and quantitative changes in the expectation value of the dipole operator makes our results useful for understanding the propagation of laser pulses. An explicit expression that is especially convenient for numerical evaluation is presented for determining the limiting internal time scale, which is determined by the energy levels and transition moments of the system and the laser frequency. We illustrate the criterion for adiabatic inversion by applying it to a particular multilevel model. From a density-matrix treatment we conclude that collisional damping destroys the coherence required for adiabatic inversion. On the basis of our results we believe that the adiabatic approximation may be appropriate for some experiments of current interest.

Physical Review AFeb 1, 198530 citations

Optical resonance of a two-level atomic system

Munir Hasan Nayfeh, Ali Hasan Nayfeh

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Nonlinear polarizations and excitations and their time dependence in discrete multilevel systems

H. B. Puell, C. R. Vidal

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Physical Review ADec 1, 197632 citations

Time-dependent pulses in quantum optics: the use of Light's perturbation theory

Peter Rodgers, S. Swain

Journal of Physics B Atomic and Molecular PhysicsFeb 14, 198714 citations

Effect of relaxation on adiabatic following

Munir Hasan Nayfeh

A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.

Physical Review ASep 1, 19767 citations

{T}_{2}

and

{T}_{1}

. It is shown that the transient one-photon excitation known as adiabatic following, is not only important for a pulse duration $\ensuremath{\tau}<{T}_{2}$, but may also be important for ${T}_{2}\ensuremath{\lesssim}\ensuremath{\tau}<{T}_{1}$. In addition, expressions for the transient two-photon excitation in a multilevel system are derived. Finally, the two-level limit is evaluated to all orders of the perturbation including damping effects, which gives results valid also for large incident light intensities. For the "off-resonance" case an expression is obtained which describes not only the transient adiabatic following process, but also the usual one-photon absorption including saturation effects giving rise to power broadening.