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Canadian Journal of Physics

The quantum-mechanical two-level system with dissipation and feedback: application to nonlinear dielectric response

J. E. Grindlay

Published Jun 1, 1986• 1 citation
Publisher

Abstract

A power-series expansion for long-time solutions is obtained for the density-matrix equations of motion describing a quantum-mechanical two-level system with dissipation and feedback. The resonances in the first- and third-order dipole contributions and the second-order upper level occupation probability are shown to be significantly affected by the feedback.The results are applied to the case of a uniformly polarized, finite, crystalline dielectric made up of two-level systems. Polarizabilities relating the dipole moment to each of the applied field, the internal field, and the macroscopic field are introduced and relations between them derived. The dipole-moment – macroscopic-field polarizabilities are shown in particular to be shape independent. Susceptibilities are defined and related to the single-system polarizabilities through generalized Clausius–Mossotti relations.

DOI
10.1139/p86-127

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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.

Physical Review•Dec 10, 1969•6 citations
A power-series expansion for long-time solutions is obtained for the density-matrix equations of motion describing a quantum-mechanical two-level system with dissipation and feedback. The resonances in the first- and third-order dipole contributions and the second-order upper level occupation probability are shown to be significantly affected by the feedback.The results are applied to the case of a uniformly polarized, finite, crystalline dielectric made up of two-level systems. Polarizabilities relating the dipole moment to each of the applied field, the internal field, and the macroscopic field are introduced and relations between them derived. The dipole-moment – macroscopic-field polarizabilities are shown in particular to be shape independent. Susceptibilities are defined and related to the single-system polarizabilities through generalized Clausius–Mossotti relations.
A two-level dissipative quantum-mechanical system subjected to a small applied multimode electric field plus a feedback term proportional to the induced dipole moment is discussed. The long-term behaviour of the induced dipole moment and of the upper level occupation probability is represented by a series expansion in terms of the applied electric-field amplitudes. First-order and third-order internal, external, and intrinsic polarizabilites are introduced and calculated for the model. Certain ratios of third-order to first-order polarizabilities are shown to have a common value for the internal, external, and intrinsic cases. Clausius–Mossotti relations for the linear and third-order susceptibilities are derived.
The shifted frequency approximation is used to solve the equations of motion for the density matrix describing a quantum mechanical two-level system, with decay, in an applied monochromatic electric field. Expressions are obtained for the transient and the long-term linear electric dipole moment and occupation probability, and the long-term third-order dipole moment. The results are valid across the frequency spectrum, including resonance, provided the phenomenological decay times for the system are not too large or too close in value.
The theory of a previous article, dealing with two types of two-level systems coupled to a loss mechanism (LM), is extended. The first extension consists of the consideration of the most general type of two-level system (TLS), in which the dipole moment is expanded in terms of the three Pauli spin matrices and unit matrix, the expansion coefficients being vectors (dipole vectors). The second extension consists of the addition to the thermal-reservoir type of LM of a large number of systems identical to the TLS under consideration. The TLS is described in terms of the time development of the Pauli matrices and differential equations are obtained for their expectation value in the presence of arbitrary driving fields. The Bloch equations for a magnetic dipole of spin \textonehalf{} are exhibited as a special case of these equations, corresponding to a particular combination of the dipole vectors. All other combinations describe electric dipole systems. Equations for two simple special cases of such systems are presented, one treated in the previous article and the other having permanent dipole moment. The frequency of oscillation of a freely decaying TLS is derived and shown to be shifted by an amount that depends on the relationship between the dipole vectors. It is pointed out that the commonly held belief that any TLS can be represented as a magnetic dipole of spin \textonehalf{} is only approximately correct in the presence of dissipation. The conditions under which the differential equations for the expectation values of the dynamical variables of the TLS can be converted into differential equations for macroscopic variables are discussed.
The study of a two-level system (TLS) coupled to a loss mechanism (LM), the fluctuations of which are taken into account, is motivated by relating the problem to spin-lattice relaxation, radiation damping, and Brownian motion. A TLS of the electric dipole type (coupled to the LM through one variable only) is discussed first. The problem is formulated in terms of the Pauli spin matrices, and the Langevin equations for Brownian motion of a TLS are derived. Their solution---made possible by taking expectation values in LM space---contains the Weisskopf-Wigner exponential decay formula of radiation theory, as well as a second-order shift in frequency of the TLS produced by the (nondispersive) LM. A driving force is added to the problem and expressions are obtained for a driven lossy TLS of the electric dipole type. The same analysis is applied to a TLS of the magnetic dipole type and differential equations for the spin matrices---for which the expectation values in LM space has been taken---are obtained. Taking expectation values in TLS space converts these equations into the Bloch equations. Unlike the electric dipole case, no frequency shift is present.
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.