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Physical Review A

Two-level system in a strong laser field: A comparative analysis between the Landau-Lifshitz solution and an exact numerical one

Raanan Bavli, Yuri I. Dakhnovskii

Published Aug 1, 1993• 9 citations
Publisher

Abstract

We numerically solve the problem of a two-level system in a near-resonance time-dependent electric field of arbitrary amplitude. The obtained radiation spectrum of the dipole moment is compared both to that of a Landau-Lifshitz (LL) textbook solution [Quantum Mechanics (Pergamon, New York, 1987), p. 145] and to an off-resonance analytical solution [Dakhnovskii and Bavli (unpublished)] obtained recently for both optical and tunneling initial conditions. We show that the LL solution is valid at low field intensities while the off-resonance analytic solution qualitatively describes the intermediate- and high-intensity regimes.

DOI
10.1103/physreva.48.r886

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We numerically solve the problem of a two-level system in a near-resonance time-dependent electric field of arbitrary amplitude. The obtained radiation spectrum of the dipole moment is compared both to that of a Landau-Lifshitz (LL) textbook solution [Quantum Mechanics (Pergamon, New York, 1987), p. 145] and to an off-resonance analytical solution [Dakhnovskii and Bavli (unpublished)] obtained recently for both optical and tunneling initial conditions. We show that the LL solution is valid at low field intensities while the off-resonance analytic solution qualitatively describes the intermediate- and high-intensity regimes.
Linearized version of the Riccati-type differential equation for the ratio of population amplitudes in a laserdriven two-level system is used to calculate analytically the induced electric dipole moment.The formula found for the dipole moment is valid for weak excitations by smooth-shape pulses of arbitrary off-resonant frequencies, i.e., those producing neither exact one-photon nor odd multiphoton resonances.For a given pulse shape, the formula allows to express the Fourier components of the induced dipole moment as functions of both laser frequency and intensity and to find the field-dependent refractive index of the system versus laser frequency.
The theory of coherent radiation emission in two-level quantum systems with permanent dipole moments at the multiphoton resonant excitation by moderately strong laser fields is presented. A simple analytic formula for the expectation value of a dipole moment operator is obtained. The spectrum corresponding to harmonic generation and low-frequency radiation on the generalized Rabi frequency is investigated both analytically and numerically. The effect of dynamic Stark shift on a dipole spectrum under direct multiphoton resonant excitation is considered.
In the limits of weak and strong excitation, we solve analytically the Riccati-type differential equation for a two-level system in the field of two different lasers and then use these solutions to find easy to handle formulae for the induced electric-dipole moment of the system.We apply these formulae to describe the effects of strong excitation in both the higher harmonics generation and multiphoton frequency mixing, namely the diminishing of the generation of harmonics of a given laser when an additional laser is turned on and the diminishing of the radiation at mixed frequencies with increasing strength of lasers under the condition of multiphoton excitation.
In a previous paper, we have already considered the system composed by a two level atom interacting with a coherent external electromagnetic field. No application whatsoever has been made of the rotating wave approximation. Being specially interested in the problem of higher harmonic generations for the case of very intense laser fields, we have developed in this letter a much more efficient way to obtain these solutions as well as to carry out some calculations in a range in which the parameters take extreme values. Also the formalism allows us now to provide analytic expressions in the WKB regime for the electric dipole moment and the population inversion. The spectrum can be decomposed in periodic and non-periodic contributions. Only the latter depends upon the Floquet exponent and can be responsible of the main complexities of the observed Rabi revivals and the hyper-Raman shift.
The possibility of the phase control for the excitation of a quantum system in the discrete spectrum under the action of a short high-intensity laser pulse with a controlled phase is theoretically analyzed. The analysis is performed using the example of the simplest two-level system (TLS) employing a numerical solution to equations for the optical Bloch vector. It is demonstrated that the excitation probability and the mean dipole moment of the quantum system, which determine the normal luminescence and the superradiance in the TLS ensemble, respectively, substantially depend on the phase of the laser pulse for relatively short and powerful laser pulses.
The rate of absorption and emission of radiation is calculated for two laser fields interacting with a homogeneously broadened, three-level molecular system. The time-dependent Schrödinger equation is used in the density matrix formalism and the laser fields are treated classically. The laser fields, at frequencies ωpand ωs, are off resonance by δpand δs, respectively, such that|\delta_{p}|, |\delta_{s}| \ll \omega_{p}, \omega_{s}. Analytic expressions are derived for the imaginary part of the electric susceptibility,\chi"(\omega_{p})and\chi"(\omega_{s}). The present work differs from previous calculations in that the laser fields may have arbitrary values of field strength and each field may be on or off resonance. A discussion is presented of the properties of the solutions for\chi"(\omega_{p})and\chi"(\omega_{s}), for both strong and weak laser fields, based on computer evaluation of the general analytic expressions. It is shown that the solutions may be divided into single-photon and two-photon, or Raman, contributions. Conditions for optimizingdI(\omega_{s})/dzand the photon conversion efficiency are derived. These results may be applied to analyzing the pulsed, optically pumped submillimeter laser and to studying two-photon absorption in a gas.