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Journal of Experimental and Theoretical Physics Letters

Model for the Description of the Relaxation of Quantum-Mechanical Systems with Closely Spaced Energy Levels

I. V. Vovchenko, Vladislav Yu. Shishkov, A. A. Zyablovsky, E. S. Andrianov

Published Jul 1, 2021• 12 citations

Abstract

Existing Markov models of the relaxation of quantum-mechanical systems are applicable in the approximations where the difference between eigenfrequencies is either much larger than the relaxation rates (global approach) or much lower than these rates (local approach). In this work, a model has been proposed to describe relaxation in systems where the difference between some eigenfrequencies is close to the relaxation rates. It has been shown that the master equation for the density matrix contains coefficients explicitly depending on the time in the interaction representation. Equations for the occupation numbers of the modes of the system have been derived by example of two interacting oscillators. It has been shown that occupation numbers in the proposed approach in the limits of small and large differences between eigenfrequencies compared to the relaxation rates asymptotically match the occupation numbers in the local and global approaches, respectively.

DOI: 10.1134/s0021364021130099

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Existing Markov models of the relaxation of quantum-mechanical systems are applicable in the approximations where the difference between eigenfrequencies is either much larger than the relaxation rates (global approach) or much lower than these rates (local approach). In this work, a model has been proposed to describe relaxation in systems where the difference between some eigenfrequencies is close to the relaxation rates. It has been shown that the master equation for the density matrix contains coefficients explicitly depending on the time in the interaction representation. Equations for the occupation numbers of the modes of the system have been derived by example of two interacting oscillators. It has been shown that occupation numbers in the proposed approach in the limits of small and large differences between eigenfrequencies compared to the relaxation rates asymptotically match the occupation numbers in the local and global approaches, respectively.

We consider a two-level system, , coupled to a general n level system, , via a random matrix. We derive an integral representation for the mean reduced density matrix ρ( t ) of in the limit n → ∞, and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ρ(∞). We also consider an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of ρ( t ) on an appropriate time scale.

For sufficiently low reservoir temperatures, it is known that open quantum systems subject to decoherent interactions with the reservoir relax toward their ground state in the weak coupling limit. Within the framework of quantum master equations, this is formalized by the Born-Markov-secular (BMS) approximation, where one obtains the system Gibbs state with the reservoir temperature as a stationary state. When the solution to some problem is encoded in the (isolated) ground state of a system Hamiltonian, decoherence can therefore be exploited for computation. The computational complexity is then given by the scaling of the relaxation time with the system size

n

. We study the relaxation behavior for local and nonlocal Hamiltonians that are coupled dissipatively with local and nonlocal operators to a bosonic bath in thermal equilibrium. We find that relaxation is generally more efficient when coherences of the density matrix in the system energy eigenbasis are taken into account. In addition, the relaxation speed strongly depends on the matrix elements of the coupling operators between initial state and ground state. We show that Dicke superradiance is a special case of our relaxation models and can thus be understood as a coherence-assisted relaxation speedup.

Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.

Master equation for density matrix of an open many–particle system is derived in the occupation number representation. The Born approximation with respect to system–bath interaction is utilized and the fast relaxation within the system is assumed to be fulfiled. The reduction of a linear master equation to a nonlinear set of kinetic equations for one–particle distribution functions is carried out at the condition of strong particle–particle interaction. As an example, the procedure of derivation of kinetic equations for description of electron transfer through specific molecular nanostructures like molecular wires is demonstrated with taking into consideration the strong Coulomb repulsion between the transferred electrons.