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arXiv:1311.0018 · Physical Review A

Master equation for dissipative interacting qubits in a common environment

Jader P. Santos, F. L. Semião

Published Feb 28, 2014• 30 citations
arXivHTMLPDF

Abstract

In this paper, we derive a microscopic master equation for a pair of $XY$-coupled two-level systems interacting with the same memoryless reservoir. In particular, we apply this master equation to the case of a pair of two-level atoms in free space where we can clearly contrast the predictions made with the microscopic master equation obtained here and the phenomenological approaches where the atom-atom coupling is included just a posteriori, i.e., not taking into account in the derivation of the open system equation of motion. We show, for instance, that the phenomenological approach fails completely in the assessment of the role played by the symmetric and antisymmetric decay channels. As a consequence, the predictions related to collective effects such as superradiance, for instance, are misleading in the phenomenological approach. We also obtain the fluorescence spectrum using the microscopic model developed here.

DOI
10.1103/physreva.89.022128

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 9, 2002•1 citation
In this paper, we derive a microscopic master equation for a pair of XYXYXY-coupled two-level systems interacting with the same memoryless reservoir. In particular, we apply this master equation to the case of a pair of two-level atoms in free space where we can clearly contrast the predictions made with the microscopic master equation obtained here and the phenomenological approaches where the atom-atom coupling is included just a posteriori, i.e., not taking into account in the derivation of the open system equation of motion. We show, for instance, that the phenomenological approach fails completely in the assessment of the role played by the symmetric and antisymmetric decay channels. As a consequence, the predictions related to collective effects such as superradiance, for instance, are misleading in the phenomenological approach. We also obtain the fluorescence spectrum using the microscopic model developed here.
We investigate a system of two interacting qubits having one of them isolated and the other coupled to a thermal reservoir. We consider two different models of system-reservoir interaction: i) a model, in which the master equation is derived taking into account the interaction between the two subsystems (qubits), ii) a naive model, in which the master equation consists of a dissipative term added to the unitary evolution term. We study the dynamics of quantities such as bipartite entanglement, quantum discord and the linear entropy of the isolated qubit for both strong and weak coupling regimes (qubit-qubit interaction) as well as for different temperatures of the reservoir. We find significant disagreements between the results obtained from the two models even in the weak coupling regime. For instance, we show that according to the phenomenological model, the isolated qubit would approach a maximally mixed state more slowly for higher temperatures (unphysical result), while the microscopic model predicts the opposite behaviour (correct result).
We investigate the dynamics of a composite quantum system, comprised of coupled subsystems, of which only one is significantly interacting with the environment. The validity of the conventional ad hoc approach---assuming that relaxation terms can be extracted directly from the master equation of the subsystem interacting with the reservoir---was examined. We derived the equation of motion for the composite system's reduced density matrix---applying only the factorization approximation, but not the conventional sequence of Markoff, coarse grain, and secular approximations. From our analysis, we concluded that the conventional ad hoc approach is applicable to zero-temperature reservoir, but fails for finite temperatures. It is further shown that at finite temperatures, the standard procedure does not even yield a master equation for the composite system, and its dynamics has to be studied by the equations of motion which are developed here. For demonstration we considered a system of a three-level atom, the two excited states are coupled to each other, and only one of them communicates with the ground state via a radiation reservoir.
Abstract We find analytical solutions to the evolution of interacting two-level atoms when the master equation is symmetric under the permutation of atomic labels. The master equation includes atomic independent dissipation. The method to obtain the solutions is: first, we use the system symmetries to describe the evolution in an operator space whose dimension grows polynomially with the number of atoms. Second, we expand the solutions in a basis composed of eigenvectors of the dissipative part of the master equation that models the independent dissipation of the atoms. This atomic damping basis is an atomic analog to the damping basis used for bosonic fields Briegel and Englert (1993 Phys. Rev. A 47 3311–29). The solutions show that the system decays as a sum of sub- and super-radiant exponential terms.
We study in this thesis composite quantum systems under the influence of external environments.In the first part of this work, we investigate a two qubit interacting system having one of them isolated and the other coupled to a thermal bath.We analyze the effect of the temperature of the bath on the dynamics of the two qubit system.In order to do that, we consider two different models of system-reservoir interaction: i) a "microscopic" model, in which the master equation is derived taking into account the interaction between the two subsystems (qubits), ii) a "phenomenological" approach, in which the master equation consists of a dissipative term added to the unitary evolution term.We show that in the strong coupling regime between the subsystems (qubits), the expected thermal equilibrium steady state for the two-qubit system naturally arises in the framework of the microscopic model, while in the phenomenological approach it is obtained a steady state density operator which is not correct.Furthermore, the differences are even more profound in the weak coupling regime, when the models give rise to opposite behaviors with regard to the linear entropy of qubit 1.At the context of quantum systems coupled to environments with few degrees of freedom, we solve analytically the matter-radiation interaction model known as two atom Tavis-Cummings Model.With the solutions at hand, achieved in general circumstances until the present not found in literature, in which the constituent atoms may be coupled with different strengths to the field and also have different frequency detunings, we study the effects that arise from the interaction of a small environment (atom in a statistical mixture state) with the other atom coupled to an oscillator (cavity mode).We show that nonclassical features associated to the main system may be significantly degraded by the action of the small environment, if atom 2 is resonantly coupled to the field.We also demonstrate that the nonclassical behaviour of the system may be restored if we detune the field from the transition frequency of atom 2, the environment.
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. By employing the exact solution for the model, analytical expressions are determined for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows an explicit comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, the structure of widely used phenomenological master equations with a memory kernel may be incompatible with a nonperturbative treatment of the underlying microscopic model. Several physical implications of the results on the microscopic analysis and the phenomenological modeling of non-Markovian quantum dynamics of open systems are discussed.
We derive a quantum master equation for a system of femions coupled to the blackbody radiation field and to many other excitations of the surrounding particles. In comparison with other master equations for many-level systems, the new master equation depends on microscopic coefficients (as functions of physical constants, matrix elements, densitie of the environment states, and occupation probabilities of these states), and includes two additional terms: (1) a term depending on the selfconsistent field of the collective motion of the system or of other surrounding particles, and (2) a non-Markovian term.