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.
The problem of a two-level atom interacting with a non-monochromatic radiation pulse is studied in the resonant approximation for arbitrary detunings, and without the resonant approximation. By using the time-dependent theory of perturbation the analytical expressions for the transition probability and induced dipole moment of the atom are obtained. It is shown that the upper level population of the atom exhibits oscillations that are absent in the resonant approximation. The temporal shape of the pulse is shown to affect significantly the atomic transition probability. The behaviour of the transition probability and dipole moment spectrum for arbitrary pulse shapes, phases and detunings is investigated. The interference between the spectral components arising in nonlinear processes is shown to play a considerable role in the formation of a dipole moment spectrum.
The behavior of a nonlinear oscillator (NO) coupled to a radiation field is investigated. The NO considered is an angular momentum oscillator, of energy $\ensuremath{\hbar}\ensuremath{\omega}{L}_{3}$, that describes the collective effect of a number of identical two-level (or spin-\textonehalf{}) systems under given idealized conditions, a large total angular momentum quantum number
L0 corresponding to a large number of two-level systems. The field is described by a set of modes. Free decay---with the NO initially excited and the radiation field in the ground state---and forced oscillation---with the NO subject to a prescribed resonant driving field---are studied. The analysis is performed both classically and quantum mechanically, using the classical and Heisenberg equations of motion, respectively, so as to display explicitly the difference in the results. Interest in the comparison between the two formalisms is motivated by the expectation that for large
L0, the NO should behave essentially classically, except near its highest-energy state in the absence of a driving field. The general equations of motion are reduced to equations for the NO variables only. In the classical analysis of free decay, expressions for the energy and oscillating coordinates are derived. The decay time is shown to approach infinity as
L3(0) approaches
L0 (the limiting condition being that of unstable equilibrium) and the radiative frequency shift is shown to be approximately proportional to
L3(t). It is also shown that use of the rotating-wave approximation alters qualitatively the expression for the frequency shift. In the classical analysis of forced oscillation, approximate results are obtained for a weak driving field and a strong driving field, the NO being initially in the ground state. The weakfield results exhibit a monotonic approach of
L3(t) to a constant (negative) value---or steady state---at which the power absorbed equals the power radiated; in the strong-field case,
L3(t) oscillates periodically between the limits $\ifmmode\pm\else\textpm\fi{}{L}_{0}$, the coupling to the radiation field having negligible effect on the frequency of this oscillation. In the quantum-mechanical analysis, the equations of motion become simplified for
L0=21, and this case is treated first. The free-decay results are essentially similar to those of the Weisskopf-Wigner theory, exhibiting an exponential decay of
〈L3(t)〉 and a radiative frequency shift in the oscillating coordinates. Under forced oscillation, with $〈{L}_{3}(0)〉=\ensuremath{-}{L}_{0}, 〈{L}_{3}(t)〉$ approaches a constant value either monotonically, if the driving field is sufficiently weak, or by means of a damped oscillation, if the driving field is strong. For
L0>21, the free decay is treated by a method that involves the derivation of a set of expressions for the
kth derivative of
〈L3(t)〉 as an expectation value of a polynomial in
L3 of order
k+1. This set, together with the eigenvalue equation for
L3, is shown to lead to a solution for all the moments of
L3. The method is used to obtain complete solutions for several low values of
L0, and also to obtain initial derivatives of
〈L3(t)〉 as polynomials in
L0 for
〈L3(0)〉=L0. For large
L0, a comparison of classical and quantum-mechanical equations shows that only the condition $〈{L}_{3}(0)〉\ensuremath{\approx}{L}_{0}$ requires quantum-mechanical treatment, and that for a short time only. The free-decay problem with initial condition
〈L3(0)〉=L0 solved quantum mechanically up to such a time
t1 by means of the initial derivatives previously derived, and then this solution if used to provide initial conditions determining a classical solution for $t\ensuremath{\ge}{t}_{1}$. The statistical aspects introduced by the quantum mechanics are preserved in the classical solution, and their significance is discussed, with several examples, comparison is also made with a nonstatistical approximation. In the case of forced oscillation, the behavior of
〈L3(t)〉---for arbitrary
L0---is examined in detail for a strong field, and turns out to be described by a damped oscillation, the existence of the damping being independent of
L0. The apparent inconsistency of this result with the expectation that a large system subject to strong forces should behave classically is discussed, and
〈L23(t)〉 is examined. It is concluded that in a single experiment, the NO energy oscillates approximately like the classical energy, without damping; quantum mechanics introduces, however, a slight randomness (or unpredictability) in the frequency of this oscillation, because of the coupling with the radiation field. Since
〈L3(t)〉 describes the average over an ensemble of which each member consists of a NO coupled to a radiation field, the random frequency variation among the members accounts for the damping of the average. The usefulness of the combination of classical and quantum-mechanical analyses in achieving and interpreting theoretical results for a class of phenomena involving the collective interaction between a number of atoms and a radiation field is pointed out.
Abstract We analyze the response of a single anharmonic diatomic molecule to a monochromatic time dependent electric field ω(t) and evaluate the temporal evolution of the dipole moment, phase space trajectories, and several transition probabilities as a function of the intensity of the field. We define deformed boson operators A† = f( )a†, A = af( ) in terms of the usual harmonic oscillator creation and annihilation operators a, a†, = a†a and choose the number operator function f( ) such that the energy spectrum of a harmonic oscillator‐type Hamiltonian written in terms of the deformed operators yields an energy spectrum similar to that of a Morse potential. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
Numerical experiments with a nonlinear (λχ4) oscillator which has its harmonic frequency changing randomly with time reveal certain interesting features of its dynamics of quantum evolution. When λ = 0, the level populations are seen to oscillate. But, as the nonlinear coupling is switched on (λ > 0), a threshold is reached at λ = λc when the evolution is seen to be characterized by an abrupt transition dominantly to the highest available state of the unperturbed (initial) oscillator. It is shown that this transition probability is maximized at a particular value of λ. The time threshold for this transition decreases with increasing nonlinear coupling strength. The numerically obtained structures of the underlying quantum-phase spaces of the linear and nonlinear random oscillators are examined. Possible use of these results in a problem of chemical origin is explored. © 1997 John Wiley & Sons, Inc.