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Physica Scripta

Solution of the post-Markovian master equation for a damping harmonic oscillator

X. L. Huang, Jing Nie, Jiao Chen, X. X. Yi

Published Jul 25, 2008• 3 citations
Publisher

Abstract

We solve the post-Markovian master equation for a damping harmonic oscillator by virtue of the damping basis approach. With our solution, we study some features for the harmonic oscillator in non-Markovian channel. The results show that the non-Markovian effect governed by the post-Markovian master equation keeps the semi-positivity and slows the energy decay to environment. Moreover, different from the Markovian process, at zero temperature, a coherent state under time evolution does not remain a coherent state.

DOI
10.1088/0031-8949/78/02/025001

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We solve the post-Markovian master equation for a damping harmonic oscillator by virtue of the damping basis approach. With our solution, we study some features for the harmonic oscillator in non-Markovian channel. The results show that the non-Markovian effect governed by the post-Markovian master equation keeps the semi-positivity and slows the energy decay to environment. Moreover, different from the Markovian process, at zero temperature, a coherent state under time evolution does not remain a coherent state.
We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D 45, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the system is driven by $\ensuremath{\pi}$ pulses. For coherence protection to be effective, our numerical results indicate that kicking period should be shorter than memory time of the bath. The effects of using soft pulses in an ohmic bath are also discussed.
The different conditions that could be imposed on a Markovian master equation for a nonlinear oscillator weakly coupled to a thermal reservoir are formulated. They concern preservation of trace and positivity of a density matrix, return to a proper equilibrium state, and the detailed balance condition. It is shown that only one of the known master equations satisfies all of these conditions. Then the validity of the Markovian approximation is reanalyzed using certain non-Markovian weak-coupling approximations, and the existence of different stages of evolution associated with different time scales of the Hamiltonian dynamics is predicted. The consequences of these facts for the description of a damped nonlinear oscillator are discussed.
The interrelationship between the non-Markovian stochastic Schr\"odinger equations and the corresponding non-Markovian master equations is investigated in the finite-temperature regimes. We show that the general finite-temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite-temperature reservoir in the rotating-wave approximation. The exact convolutionless master equation for the damped harmonic oscillator is obtained by averaging the quantum trajectories, relying upon no assumption of coupling strength or time scale. The master equation derived in this way automatically preserves the positivity, Hermiticity, and unity.
An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
Some aspects of quantum damped harmonic oscillator (DHO) obeying a Markovian master equation are considered in the absence of thermal noise. The continuity equation is derived and Bohmian trajectories are constructed. As a solution of the master equation, we take a single coherent state and compute analytically the relative entropy of coherence, [Formula: see text], in the energy, position and momentum bases. Although [Formula: see text] is constant in both the position and the momentum bases, it is a decreasing function of time in the energy basis becoming zero at long times, revealing its role as the preferred basis. Then, quantum coherence is computed for a superposition of two coherent states, a cat state, and also a superposition of two cat states in the energy basis as a function of separation, in the complex plane, between the two superposed states. It is seen that the quantum coherence increases with this separation. Furthermore, quantum coherence of superposition is compared to that of decomposed states in the superposition. Finally, by considering a system of two noninteracting DHOs, the effect of quantum statistics is studied on the coherence of reduced single-particle states, the joint detection probability and the mean square separation of particles. Our computations show that the single-particle coherence for antisymmetric states is always less than that of symmetric ones. Furthermore, boson anti-bunching and fermion bunching is seen in this open system. This behavior of bosons is the matter-wave analogue of photon anti-bunching seen in a modified Hanbury Brown–Twiss (HBT) interferometer.