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Canadian Journal of Physics

The quantum-mechanical two-level system with dissipation and feedback: multimode applied electric-field case

J. E. Grindlay

Published Apr 1, 1987• 0 citations
Publisher

Abstract

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.

DOI
10.1139/p87-052

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International Journal of Modern Physics E•Feb 1, 2000•8 citations
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.
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.
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.
We consider a system of Z fermions coupled to a dissipative environment through a two-body potential. We represent the system in a basis of single-particle, two-particle, … Z-particle excitated states. Using a procedure for averaging the rapid oscillations of the reduced density matrix in the interaction picture, the master equation of the system takes the form of a series expansion of powers of the dissipative potential matrix elements. The term of the second-order describes single-particle transitions, while the higher-order terms correspond to correlated transitions of the system particles. For the second- and the third-order terms, we derive microscopic expressions of the dissipative coefficients. For dissipative systems, when the state collectivity is broken into pieces through quantum diffusion, we use the quantum master equation of the second-order approximation. This equation satisfies basic physical conditions: particle conservation, Fermi–Dirac or Bose–Einstein distributions as asymptotic solutions of the populations, and entropy increase. On this basis, the decay of a Fermi system interacting with a many-mode electromagnetic field is described in terms of microscopic quantities: the matrix elements of the dissipative potential, the densities of the environment states, and the occupation probabilities of these states. A near-dipode–dipode interaction of the system with other neighbouring systems is taken into account. In addition to the coupling of the polarization with the population, included in the usual equations for two-level systems as a non-linear detuning, in equations for N-level systems two new couplings of the polarizations appear: a coupling due to the proximity potential, and a coupling due to the local field corrections, as a renormalization of the Rabi frequencies.