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Canadian Journal of Physics

Two-level quantum mechanical system with decay: density matrix description and shifted frequency approximation

D. C. Morgan, J. E. Grindlay

Published Dec 1, 1985• 0 citations
Publisher

Abstract

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.

DOI
10.1139/p85-254

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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.
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.
Previously derived equations for the expectation values of the dynamical variables of a general two-level dipole system (TLS) coupled to a relaxation mechanism---of which the Bloch equations for a spin-\textonehalf{} magneticdipole system are a special case---are reformulated in the density-matrix formalism in order to point out the explicit difference in the relaxation terms between the present description, which includes electric-dipole systems, and that of conventional treatments that are based on the assumption that any TLS is equivalent to a spin-\textonehalf{} magnetic dipole. The original equations are then solved for a weak monochromatic driving field, and the frequency of maximum power absorption is shown to depend on the type of TLS under consideration. The case of a stronger monochromatic driving field and the resulting saturation effects are studied for a special TLS, a spatially linear electric dipole which may have permanent dipole moment. This case exhibits a strong-field resonance shift that depends on the magnitude of the permanent dipole moment. A driving field consisting of a strong component at resonance and a weak component near resonance is considered in connection with the same TLS. The polarization is shown to contain, in lowest order, a component at the difference frequency with an amplitude determined by the magnitude of the permanent dipole moment.
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 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.
Pulsed radio-frequency transient phenomena of resonant two-level spins are often described by the Bloch equations where the phenomenological dipole (T2{T}_{2}T2​) and population (T1{T}_{1}T1​) decay times appear. A similar description has been carried over recently into the microwave and infrared regions as well. At visible-ultraviolet wavelengths, however, a more general solution is required because the transition levels do not constitute an isolated system and also spontaneous emission can no longer be disregarded. As a consequence, the lower (1) and upper (2) transition levels are characterized by vastly different decay rates (${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$) and the lifetime T1{T}_{1}T1​ is no longer a defined quantity. In view of laser-induced coherent transient phenomena now being observed, we present general solutions of the density-matrix equations of motion for an optically resonant two-level system in contact with a multilevel reservoir where ${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$. Particular and perturbative solutions involving intensity-dependent damping and frequency shifts are included with applications to echo and nutation phenomena.