Time-dependent Markovian master equation for adiabatic systems and its application to Cooper-pair pumping

Abstract

For adiabatically and periodically manipulated dissipative quantum systems we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper pair pumping and demonstrate the inadequacy of the secular (rotating wave) approximation when calculating the pumped charge. The secular approximation producing a master equation of the Lindblad type approximates well the quantum state (density matrix) of the system, while to determine the pumped charge a non-Lindblad master equation beyond the rotating wave approximation is necessary.

Floquet topological phases coupled to environments and the induced photocurrent

“…[24,36] to outline a theoretical framework to describe the time dependent reduced density matrix by a generalized Lindblad-type equation, up to second order in the coupling with an environment. As a first step, one switches to the interaction picture with respect to the noninteracting Hamiltonians, H(t) =H S (t) +H E , where the time evolution of the interacting system's density matrix ρ(t) is governed simply by the Hamiltonian H SE (t), which we factorize as…”

“…4, visualizing the momentum dependence of the average value of ρ z in the Ohmic case. Note that ρ z can become smaller than −1/2, which is a common feature in other nonsecular approaches as well [24]. The secular approximation clearly breaks down at certain momenta, and is outperformed by the DFA there [45].…”

“…(24) Similar to the current, only the first term survives in the secular approximation, and in further analogy to the quantized current, we define M c = dp 2π 1 2 = 2π ln 2 , which is independent of ω and logarithmically divergent in the cutoff parameter . The crossovers in ρ z as a function of ω are also revealed in transverse magnetization, which can be highlighted by subtracting the low frequency transverse magnetization M c as a reference value (Fig.…”

“…[24], we apply a generalized Lindblad type formulation (the Bloch-Redfield equations) to describe how the environment affects the dynamics of the edge states. In particular, we keep nonsecular terms, which are not captured in the Lindblad equation, but are found to affect the dynamics considerably.…”

Cooper-pair current in the presence of flux noise

“…The experimental access to the parameters J L,R and V g allows for full control of the quantum system and makes it an excellent prototype for different applications. Several steps have been taken in the study of the connection between Cooper-pair pumping and geometric phases, in both its Abelian 9,22 and its non-Abelian version, [23][24][25] the robustness of the ground-state pumping, [17][18][19][36][37][38] and the geometric Landau-Zener-Stückelberg interferometry. 39 Analogous systems have been studied for the relations between pumping and topological phases.…”