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Journal of Modern Optics

Optical Lindblad operator in non-equilibrium thermo field dynamics

Masashi Ban

Published Feb 14, 2009• 10 citations
Publisher

Abstract

The Lindblad operator of the Markovian quantum master equation for an oscillator system is diagonalized by means of linear transformations of annihilation and creation operators in non-equilibrium thermo field dynamics. When the linear attenuation and amplification processes are considered, the left and right eigenstates of the Lindblad operator are constructed in terms of the transformed operators. In the linear attenuation process, the results are compared with those obtained by the damping base of the Lindblad superoperator. The base describing the time-evolution of the linear amplification process is derived and compared with the damping base. The method for diagonalizing the Lindblad operator is useful for calculating average values of physical quantities in the quantum Markovian process.

DOI
10.1080/09500340802696571

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The Lindblad operator of the Markovian quantum master equation for an oscillator system is diagonalized by means of linear transformations of annihilation and creation operators in non-equilibrium thermo field dynamics. When the linear attenuation and amplification processes are considered, the left and right eigenstates of the Lindblad operator are constructed in terms of the transformed operators. In the linear attenuation process, the results are compared with those obtained by the damping base of the Lindblad superoperator. The base describing the time-evolution of the linear amplification process is derived and compared with the damping base. The method for diagonalizing the Lindblad operator is useful for calculating average values of physical quantities in the quantum Markovian process.
We analyze the Markovian dynamics of a quantum system involving the interaction of two quantized fields at finite temperature decay. Utilizing superoperator techniques and applying two non-unitary transformations, we reformulate the Lindblad master equation into a von Neumann-like equation with an effective non-Hermitian Hamiltonian. Furthermore, an additional non-unitary transformation is employed to diagonalize this Hamiltonian, enabling us to derive an exact solution to the Lindblad master equation. This method provides a framework to calculate the evolution of any initial state in a fully quantum regime. As a specific example, we present the photon coincidence rates for two indistinguishable photons initially interacting within a cavity.
Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalization procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented which reflect, to varying degrees, the appearance of negative decoherence rates in the Lindblad-like form of the master equation. We therefore propose using the negative decoherence rates themselves, as they appear in the canonical form of the master equation, to completely characterize non-Markovianity. The advantages of this are especially apparent when more than one decoherence channel is present. We show that a measure proposed by Rivas et al. [Phys. Rev. Lett. 105, 050403 (2010)] is a surprisingly simple function of the canonical decoherence rates, and give an example of a master equation that is non-Markovian for all times t>0t>0t>0, but to which nearly all proposed measures are blind. We also give necessary and sufficient conditions for trace distance and volume measures to witness non-Markovianity, in terms of the Bloch damping matrix.
In a unitary approach generally used in quantum optics, we consider Lindblad's Markovian master equation and the non- Markovian master equation of Ford, Lewis and O'Connell. We show that the second-order master equation in the hierarchy obtained from a Krylov-Bogoliubov expansion corresponds to the Born approximation. By time averaging, and neglecting the rapidly varying terms, Lindblad's master equation is obtained. With these two equations, we calculate the decay spectrum. We find that for rather low dissipated energies, only the non-Markovian master equation provides correct results. Based on the independent oscillator model of the dissipative coupling, explicit expressions of the dissipative coefficients are obtained.
Out of thermal equilibrium state, the vacuum is unstable and evolves in time.Consequently, the annihilation operators associated with the unstable vacuum depend on time.This dissipative time-evolution of quantum systems can be systematically treated, within the canonical operator formalism referred to as non-equilibrium thermo-field dynamics.Given is an alternative route to derive the time-dependent annihilation operators within the formalism.As an example, time-dependent annihilation operators for the systems of bosonic and fermionic semi-free fields are derived.
We discuss the mathematical structure of quantum states on which the master equation generates a semigroup of irreversible time evolution. As an illustration of an open system we choose to treat the harmonic oscillator, in which case we can perform the construction of a complete set of eigenelements describing both the attenuator and the amplifier. We show that the corresponding eigenelements belong to adjoint spaces, and their orthogonality and completeness is shown explicitly. We also show that our results are in agreement with earlier work by Briegel and Englert. We illustrate the use of our results by deriving previously known time-dependent results for some simple cases.