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Physical Review

Nonlinear Resonance Effects in Multilevel Quantum Systems Weakly Coupled to a Thermal Reservoir

Clayton W. Bates

Published Dec 10, 1969• 6 citations
Publisher

Abstract

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 $N$-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.

DOI
10.1103/physrev.188.529

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Dissipation in Quantum Mechanics. Two-Level System. III

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Previous theory describing the behavior of a general two-level system (TLS) coupled to a loss mechanism and driven by an oscillating field near resonance is generalized to include an arbitrary driving field. Integro-differential equations of motion for the expectation values of the Pauli spin matrices, which describe completely the behavior of a TLS, are derived. The relaxation constants involved in these equations prove to be frequency-dependent, and reactive effects produced by the loss mechanism appear. It is shown that for an approximately monochromatic driving field near resonance, the general equations of motion can be approximated by differential equations. In the case of a magnetic TLS, these differential equations reduce to the Bloch equations if reactive effects are neglected. An exact solution of the general equations of motion for the special, simple case of a magnetic TLS being driven by a transverse rotating field of arbitrary frequency is obtained, and deviation from a Lorentzian resonance shape, caused by the frequency dependence of the relaxation and reactive constants, is found to exist.

Physical Review•Mar 15, 1965•9 citations
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.
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.
The resonant response of a multilevel atomic system to strong coherent radiation is analyzed using an extension of the methods of time-independent scattering theory, adapted to the density-matrix formalism in order to include thermal bath effects. General steady-state expressions are derived for self-attenuation and resonance scattering rates in strong saturating fields, as functions of the coupling strengths, relaxation parameters, and frequency detunings. Formal expressions are given for radiative-damping and collision-broadening relaxation parameters, relating the latter to scattering amplitudes. Modifications of collision-broadening parameters by the presence of very strong fields are discussed. The Bloch equations are derived by transformation to the time domain, under specified conditions.
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.
The effects of incoherent feeding and decay on a coherently driven two-level system are investigated using density-matrix techniques. A closed-form solution that includes the effects of relaxation and spontaneous emission between the two levels is obtained for cases in which the sample size is both small and large compared to the wavelength of the coherent field. The formalism developed herein is somewhat more general than the original approach of Bloch, Bloembergen, Purcell, Pound, and Redfield, insofar as it is applicable to cases where the trace of density matrix is not constant in time. As such, several new phenomena are predicted. Specifically, the steady-state solutions reveal that under appropriate conditions a significant off-resonance coherent component can be maintained indefinitely by the combination of driving field and incoherent feeding. We term this effect "kinetic coherence" and develop its formalism for both optical and radio-frequency fields.
We develop an exact analytical framework for studying the nonequilibrium dynamics of interacting quantum harmonic oscillators coupled to a modelled thermal reservoir and driven by external classical fields. The total Hamiltonian is diagonalized via successive Bogoliubov transformations, enabling a nonperturbative treatment of dissipation and driving. For two coupled oscillators with an external source applied to the first, we derive explicit energy expressions and analyze their dependence on coupling and bath parameters. The phase-space structure is characterized through Husimi QQQ-functions for initial separable coherent and number states. We obtain the reduced density matrix elements in the number-state basis and demonstrate that, in the absence of driving and at zero temperature, the reduced density matrix of the main system satisfies the Lindblad master equation. A generating function for the Husimi function is introduced to facilitate computation of higher-order correlations. The analysis is then generalized to nnn interacting oscillators, with analytical energy expressions provided and the case n=3n=3n=3 examined in detail. Our results establish a versatile and exact framework for driven-dissipative quantum systems, with potential applications in quantum thermodynamics, open quantum systems, and many-body physics.
Previous theory describing the behavior of a general two-level system (TLS) coupled to a loss mechanism and driven by an oscillating field near resonance is generalized to include an arbitrary driving field. Integro-differential equations of motion for the expectation values of the Pauli spin matrices, which describe completely the behavior of a TLS, are derived. The relaxation constants involved in these equations prove to be frequency-dependent, and reactive effects produced by the loss mechanism appear. It is shown that for an approximately monochromatic driving field near resonance, the general equations of motion can be approximated by differential equations. In the case of a magnetic TLS, these differential equations reduce to the Bloch equations if reactive effects are neglected. An exact solution of the general equations of motion for the special, simple case of a magnetic TLS being driven by a transverse rotating field of arbitrary frequency is obtained, and deviation from a Lorentzian resonance shape, caused by the frequency dependence of the relaxation and reactive constants, is found to exist.