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The Journal of Chemical Physics

Exact Solutions for Many-Level Multiple-Resonance Problems

Karl F. Freed

Published Aug 15, 1965• 23 citations
Publisher

Abstract

The matrix elements of the time-development operator are obtained exactly for the influence of many resonant or near-resonant fields on a system with a finite number of levels for arbitrary initial conditions in a wide class of problems. Perturbative techniques are presented which enable the exact treatment of all perturbations to a desired order. As examples, a three-level double-resonance and two eight-level triple-resonance systems are given. The applications of these exact solutions to the treatment of perturbations, including relaxation phenomena, are presented using both standard-perturbation-theory and density-matrix techniques.

DOI
10.1063/1.1696891

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The matrix elements of the time-development operator are obtained exactly for the influence of many resonant or near-resonant fields on a system with a finite number of levels for arbitrary initial conditions in a wide class of problems. Perturbative techniques are presented which enable the exact treatment of all perturbations to a desired order. As examples, a three-level double-resonance and two eight-level triple-resonance systems are given. The applications of these exact solutions to the treatment of perturbations, including relaxation phenomena, are presented using both standard-perturbation-theory and density-matrix techniques.
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.
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.
The author presents a quantum mechanical method capable of providing exact non-perturbative solutions in problems involving the interaction of atoms and molecules with the electromagnetic field. To illustrate the method two examples of double-resonance phenomena are considered. In the first example the author studies the interaction of a near-resonance rf field with the Zeeman sublevels of atomic hydrogen. In the second the interaction of a three-level molecule with two applied electromagnetic fields is studied. Both the molecular levels and the electromagnetic field are quantized. The method eliminates the deficiencies of a semiclassical treatment, yet does not require the sophisticated computations of the resolvant formalism of Cohen-Tannoudji.
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.
Solutions of the time-dependent Schrödinger equation, in the semi-classical electric dipole approximation, are developed for the interaction of a two-level system, possessing both transition and diagonal dipole matrix elements, with applied static and sinusoidal electric fields. Exact solutions can only be obtained subject to conditions involving the orientations and strengths of the applied fields. These, and perturbative extensions, are used to discuss some of the features of multi-photon processes that depend on the existence of non-zero diagonal dipole matrix elements. Some of the results apply, in the limit of vanishing permanent moments, to the atomic two-level problem.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.