Master equation for open quantum systems: Zwanzig-Nakajima projection technique and the intrinsic bath dynamics

Abstract

The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath ̺ B (t) and the system ̺ S (t) are written down. A formal solution of the equation for ̺ B (t), having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in ̺ S (t), is related to the intrinsic bath dynamics and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we applied the generalized ZN projection scheme to a simple dephasing model. It is shown that consideration of the lowest order in interaction is insufficient to describe time evolution of the system coherence adequately. We explain this fact by analyzing the exact and approximate forms of ̺ B (t) and give some hints how to take the dynamic correlations (which originate from the spin-bath coupling) into account.

Decoherence in open quantum systems: influence of the intrinsic bath dynamics

“…has been drawn to a very recent paper [36] where the problems of intrinsic bath dynamics were studied in the framework of the so-called correlation picture, which connects a correlated bipartite state to its uncorrelated counterpart. Some of the results and conclusions are very close to ours, obtained in the current paper and previously in [26], namely: i) generalization of the projecting operators; ii) the chain of equations for the reduced density matrices of the system and the bath. However, there are also some distinctive points: i) the models utilized in [36] in order to verify the robustness of their approach differ from ours; ii) we have separated the initial correlations and the BD contributions, yielding different forms of the corresponding time-convolutionless master equations.…”

“…After some manipulations with the r.h.s. of equation (3.2), which are described in detail in the Appendix A of our recent paper [26], we obtain the final master equation in the interaction picture:…”

“…2It should be noted that in the original ZN scheme [14] it is supposed that [ 𝐻 𝐵 , 𝜚 𝐵 (0) ] = 0, i.e., the bath is initially in thermal equilibrium. If this initial state differs from thermal equilibrium, it is natural to write the decomposition of 𝜚 (𝑡 ) in the form 𝜚 (𝑡 ) = 𝜚 𝑆 (𝑡 ) 𝜚 In the weak coupling limit (see Appendix B in [26] for details), the ZN master equation looks as follows:…”

“…Having calculated all commutators in equation ( 4.3) and having performed thermal averaging (see Appendixes C and D in [26] for details), we can write down the final kinetic equation for the generalized coherence:…”

“…In our recent paper [26] we solved the generic non-Markovian equation (4.4) numerically. We considered the Ohmic and super-Ohmic (with 𝑠 = 3/2) qubit-bath coupling and studied the qubit dynamics at both low and high temperatures.…”