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Physical Review A

Gauge invariance of phenomenological models of the interaction of quantum dissipative systems with electromagnetic fields

Mikhail D. Tokman

Published May 15, 2009• 18 citations
Publisher

Abstract

We discuss specific features of the electrodynamic characteristics of quantum systems within the framework of models that include a phenomenological description of the relaxation processes. As is shown by W. E. Lamb, Jr., R. R. Schlicher, and M. O. Scully [Phys. Rev. A 36, 2763 (1987)], the use of phenomenological relaxation operators, which adequately describe the attenuation of eigenvibrations of a quantum system, may lead to incorrect solutions in the presence of external electromagnetic fields determined by the vector potential for different resonance processes. This incorrectness can be eliminated by giving a gauge-invariant form to the relaxation operator. Lamb, Jr., et al. proposed the corresponding gauge-invariant modification for the Weisskopf-Wigner relaxation operator, which is introduced directly into the Schr\"odinger equation within the framework of the two-level approximation. In the present paper, this problem is studied for the von Neumann equation supplemented by a relaxation operator. First, we show that the solution of the equation for the density matrix with the relaxation operator correctly obtained ``from the first principles'' has properties that ensure gauge invariance for the observables. Second, we propose a common recipe for transformation of the phenomenological relaxation operator into the correct (gauge-invariant) form in the density-matrix equations for a multilevel system. Also, we discuss the methods of elimination of other inaccuracies (not related to the gauge-invariance problem) which arise if the electrodynamic response of a dissipative quantum system is calculated within the framework of simplified relaxation models (first of all, the model corresponding to constant relaxation rates of coherences in quantum transitions). Examples illustrating the correctness of the results obtained within the framework of the proposed methods in contrast to inaccuracy of the results of the standard calculation techniques are given.

DOI
10.1103/physreva.79.053415

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We discuss specific features of the electrodynamic characteristics of quantum systems within the framework of models that include a phenomenological description of the relaxation processes. As is shown by W. E. Lamb, Jr., R. R. Schlicher, and M. O. Scully [Phys. Rev. A 36, 2763 (1987)], the use of phenomenological relaxation operators, which adequately describe the attenuation of eigenvibrations of a quantum system, may lead to incorrect solutions in the presence of external electromagnetic fields determined by the vector potential for different resonance processes. This incorrectness can be eliminated by giving a gauge-invariant form to the relaxation operator. Lamb, Jr., et al. proposed the corresponding gauge-invariant modification for the Weisskopf-Wigner relaxation operator, which is introduced directly into the Schr"odinger equation within the framework of the two-level approximation. In the present paper, this problem is studied for the von Neumann equation supplemented by a relaxation operator. First, we show that the solution of the equation for the density matrix with the relaxation operator correctly obtained ``from the first principles'' has properties that ensure gauge invariance for the observables. Second, we propose a common recipe for transformation of the phenomenological relaxation operator into the correct (gauge-invariant) form in the density-matrix equations for a multilevel system. Also, we discuss the methods of elimination of other inaccuracies (not related to the gauge-invariance problem) which arise if the electrodynamic response of a dissipative quantum system is calculated within the framework of simplified relaxation models (first of all, the model corresponding to constant relaxation rates of coherences in quantum transitions). Examples illustrating the correctness of the results obtained within the framework of the proposed methods in contrast to inaccuracy of the results of the standard calculation techniques are given.
We define three general conditions that the relaxation operator in the von Neumann equation for the density matrix of an open quantum system should satisfy. The first one provides the correct relation between the macroscopic current and polarization in a quantum medium, the second is defined by thermodynamic arguments and the third one is imposed by gauge invariance of the solution. The corrected phenomenological relaxation operator is proposed for any quantum system with a discrete number of basis functions, as well as for a quantum system described by the density matrix in coordinate representation. The dissipative three-dimensional oscillator in an arbitrary oriented magnetic field is considered in detail. We have shown that the minimally modified relaxation operator in comparison with the standard relaxation operator, corresponding to tau-approximation, eventually leads to simpler and noncontradictory equations for the dynamics of averaged values of physical quantities.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in the presence of an electromagnetic field in phase space, which is spanned by position and kinetic momentum. This approach is independent of the chosen potentials, as it depends only on the electric and magnetic field variables. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field based solely on differential operators that simplify existing formulations. A gauge-dependent equation is derived first using Moyal's equation. A transformation of the Wigner function, introduced by Stratonovich, is then used to make the equation gauge-invariant, which gives us a strong formulation of the problem. This equation can be transformed into its weak form, which proves that both formulations are equivalent. An analysis of the different properties of the gauge-dependent and gauge-invariant formulations is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak formulations.
A gauge-invariant Wigner quantum mechanical theory is obtained by applying the Weyl-Stratonovich transform to the von Neumann equation for the density matrix. The transform reduces to the Weyl transform in the electrostatic limit, when the vector potential and thus the magnetic field are zero. Both cases involve a center-of-mass transform followed by a Fourier integral on the relative coordinate introducing the momentum variable. The latter is continuous if the limits of the integral are infinite or, equivalently, the coherence length is infinite. However, the quantum theory involves Fourier transforms of the electromagnetic field components, which imposes conditions on their behavior at infinity. Conversely, quantum systems are bounded and often very small, as is, for instance, the case in modern nanoelectronics. This implies a finite coherence length, which avoids the need to regularize nonconverging Fourier integrals. Accordingly, the momentum space becomes discrete, giving rise to momentum quantization and to a semidiscrete gauge-invariant Wigner equation. To gain insights into the peculiarities of this theory one needs to analyze the equation for specific electromagnetic conditions. We derive the evolution equation for the linear electromagnetic case and show that it significantly simplifies for a limit dictated by the long coherence length behavior, which involves momentum derivatives. In the discrete momentum picture these derivatives are presented by finite difference quantities which, together with further approximations, allow to develop a computationally feasible model that offers physical insights into the involved quantum processes. In particular, a Fredholm integral equation of the second kind is obtained, where the ``power'' of the kernel components, measuring their rate of modification of the quantum evolution, can be evaluated.
The conventional perturbation theory in quantum mechanics for the interaction of electromagnetic radiation with matter, which is based on the time-dependent vector and scalar potentials, is shown to be dependent on the gauge. The problem is reformulated, using the approach of Yang, in a gauge-invariant way, in which the transitions between states are induced by the quantum-mechanical power operator. In order to use an unperturbed Hamiltonian which does not involve the electromagnetic field, the gauge is chosen to give the electric dipole interaction. This choice can be made if the magnetic field is negligible and the electric field is slowly varying over atomic dimensions. For a comparison of the gauge-invariant and conventional formulation, a two-level atom in a radiation field is considered. For the probability that an atom originally in the lower state will make a transition to the upper state when the field is not in resonance, the gauge-invariant formulation gives the correct result, whereas the gauge-dependent formulation gives an incorrect result.