Relaxation Dynamics of Quantum Composite Systems : Validity of Markovian Master Equation Approach

Ting Yu, Lajos Diósi, Nicolas Gisin +1 more

Stefan K. Possanner, Benjamin A. Stickler

We consider the dynamics of composite quantum systems in the particular case that the state operator relaxes towards the Born approximation. For this we augment the von Neumann equation by a relaxation operator imposing a finite relaxation time ${\ensuremath{\tau}}_{r}$. Under the premise that the relaxation is the dominant process we obtain a hierarchy of non-Markovian master equations. The latter arises from an expansion of the total state operator in powers of the relaxation time ${\ensuremath{\tau}}_{r}$. In the Born-Markov limit ${\ensuremath{\tau}}_{r}\ensuremath{\rightarrow}0$ the Lindblad master equation is recovered. Higher-order contributions enable a systematic treatment of correlations and non-Markovian dynamics in a recursive manner.

Fumika Suzuki

Gen Kimura, Shigeru Ajisaka, Kyouhei Watanabe

In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation. In this paper, we show universal constraints on the relaxation times valid for any d-level GKLS master equations, which is a generalization of the well-known constraints for 2-level systems. Specifically, we show that any relaxation rate, the inverse-relaxation time, is not greater than half of the sum of all relaxation rates. Since the relaxation times are measurable in experiments, our constraints provide a direct experimental test for the validity of the GKLS master equations, and hence for the conditions of the complete positivity and Markovianity.

Yuan‐Chung Cheng, R. Silbey

In this paper, we examine the validity of the Markovian approximation and the slippage scheme used to incorporate short time transient memory effects in the Markovian master equations (Redfield equations). We argue that for a bath described by a spectral function, J(omega), that is dense and smoothly spread out over the range omega(d), a time scale of tau(b) approximately 1/omega(d) exists; for times of t > tau(b), the Markovian approximation is applicable. In addition, if J(omega) decays to zero reasonably fast in both the omega --> 0 and omega --> infinity limits, then the bath relaxation time, tau(b), is determined by the width of the spectral function and is weakly dependent on the temperature of the bath. On the basis of this criterion of tau(b), a scheme to incorporate transient memory effects in the Markovian master equation is suggested. Instead of using slipped initial conditions, we propose a concatenation scheme that uses the second-order perturbation theory for short time dynamics and the Markovian master equation at long times. Application of this concatenation scheme to the spin-boson model shows that it reproduces the reduced dynamics obtained from the non-Markovian master equation for all parameters studied, while the simple slippage scheme breaks down at high temperatures.

Manuel Gessner