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Soviet Journal of Quantum Electronics

Quasienergy states of multilevel systems

A. O. Melikyan

Published Feb 28, 1977• 3 citations
Publisher

Abstract

The Hill method is used in an analysis of the Schrodinger equation for a quatum system subjected to a time-periodic nonmonochromatic field. An algebraic equation is obtained and its degree in respect of exp(iET), where E is the quasienergy and T is the period of the external field, is equal to the number of levels. The coefficients of this equation are expanded in powers of the intensity parameter into a series with a factorial convergence even when one of the field frequencies is in exact resonance with one of the transitions. The Fourier components of the density matrix are expressed in terms of quasienergy considered as a function of the parameters of the problem and this gives approximate expressions outside the perturbation theory framework. Various results are obtained for a two-level system and, in particular, a formula is derived for the description of the natural lifetime of a quasienergy state.

DOI
10.1070/qe1977v007n02abeh008974

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The Hill method is used in an analysis of the Schrodinger equation for a quatum system subjected to a time-periodic nonmonochromatic field. An algebraic equation is obtained and its degree in respect of exp(iET), where E is the quasienergy and T is the period of the external field, is equal to the number of levels. The coefficients of this equation are expanded in powers of the intensity parameter into a series with a factorial convergence even when one of the field frequencies is in exact resonance with one of the transitions. The Fourier components of the density matrix are expressed in terms of quasienergy considered as a function of the parameters of the problem and this gives approximate expressions outside the perturbation theory framework. Various results are obtained for a two-level system and, in particular, a formula is derived for the description of the natural lifetime of a quasienergy state.
A method to build various integrable models for description of coherent excitation of multilevel media by laser pulses is suggested. Distribution functions over the energy levels of quantum systems depending on the time and frequency detuning are obtained. The distributions follow from Schrödinger equation exact solutions and give the complete dynamical description of laser-excited quantum multilevel systems. Interpretation based on the Fourier spectra of the probability amplitudes of a quantum system is presented. The spectra are expressed in terms of orthonormal polynomials and their weight functions. Matrix elements of the dipole transitions between levels are equal to coefficients of the recurrence formula for the orthonormal polynomial system. Some examples are presented. The Kravchuk oscillator family as an integrable model is constructed to describe the coherent excitation dynamics of multilevel resonance media. It is based on the use of the Kravchuk orthogonal polynomials. The Kravchuk oscillator excitation dynamics is described by the binomial distribution of energy level populations and the distribution parameter depends on excitation conditions. Two known basic models in quantum physics -- the harmonic oscillator and two-level system are the special representatives of the Kravchuk oscillator family.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization M+(t)=Mx(t)+iMy(t){M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)M+​(t)=Mx​(t)+iMy​(t) of a system of NNN noninteracting spins under the influence of a static field in the zzz direction and an arbitrary number of rf fields in the xxx direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.
Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an NNN-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.
A multilevel system driven strongly by several near-resonant and arbitrary modulated optical fields is considered. The orthonormal basis of the time-dependent quasienergy states is introduced to represent in a matrix form the density operator and the operators describing atom-field interaction and relaxation. The above formalism simplifies the density-matrix equations of motion and provides a uniform treatment of resonance phenomena such as probe wave resonances, parametric resonances, and quantum interference. Special consideration is given to the steady-state and transient-time evolution of two- and three-level systems.
Analytical expressions for the semiclassical dressed states and corresponding quasienergies are obtained for a two-level quantum system driven by a non-resonant and/or strong laser field in a coherent state. These expressions are of first order in a proper perturbative expansion, and already contain all the relevant physical information on the dynamical and spectroscopic properties displayed by these systems under such particular conditions. The influence of the laser field parameters on transition frequencies, selection rules and line intensities can be easily understood in terms of the quasienergy-level diagram and the allowed transitions between the different semiclassical dressed states.
We present exact analytic solutions for non-linear quantum dynamics of \n twolevel \nsystem (TLS) subject to \n periodic-in-time external field. >n constructing the \nexactly solvable models, we use \n approach where the form of external perturbation \nis chosen to preserve \nn integrability constraint, which yields \n single non-linear \ndifferential equation for the ac-field. solution to this equation is expressed in terms \nof Jacobi elliptic functions with three independent parameters that allows n to \nchoose the frequency, average value, and amplitude of the time-dependent field at \nwill. This form of the ac-drive is especially relevant to the problem of dynamics of \nTLS charge defects that cause dielectric losses ?n superconducting qubits.