Efficient determination of the Markovian time-evolution towards a steady-state of a complex open quantum system

Abstract

Master equations are commonly used to describe time evolution of open systems. We introduce a general computationally efficient method for calculating a Markovian solution of the Nakajima-Zwanzig generalized master equation. We do so for a time-dependent transport of interacting electrons through a complex nano scale system in a photon cavity. The central system, described by 120 many-body states in a Fock space, is weakly coupled to the external leads. The efficiency of the approach allows us to place the bias window defined by the external leads high into the many-body spectrum of the cavity photon-dressed states of the central system revealing a cascade of intermediate transitions as the system relaxes to a steady state. The very diverse relaxation times present in the open system, reflecting radiative or non-radiative transitions, require information about the time evolution through many orders of magnitude. In our approach, the generalized master equation is mapped from a many-body Fock space of states to a Liouville space of transitions. We show that this results in a linear equation which is solved exactly through an eigenvalue analysis, which supplies information on the steady state and the time evolution of the system.

Cavity‐Photon‐Induced High‐Order Transitions between Ground States of Quantum Dots

“…[18] Subsequently, we apply vectorization [35] and Kronecker tensor products together with a Markovian approximation to transform the GME from the Fock space of many-body states to the Liouville space of transitions. [36,37] We include 128 Fock states in our transport calculations and thus end up with 16 384 transitions in the Liouville space. The increased space size is counteracted by efficient parallelization and GPU processing.…”

“…The increased space size is counteracted by efficient parallelization and GPU processing. [36] The weak photon dissipation is derived with a Markovian and rotating wave approximation, where the creation (annihilation) operator for the cavity photons needs to be rid of fast rotating annihilation(creation) terms when transformed to the fully interacting basis |̆) in order for vacuum processes to be correctly described in the model. [15,[38][39][40] Due to vectorization, [35] the Markovian master equation in Liouville space assumes the form of a simple linear first-order differential equation [36,41] t vec( S ) = −i vec( S )…”

“…vec( S (t)) = { [ exp ( −i diag t )] } vec( S (0)) (19) with the left and right eigenvector matrices of the non-Hermitian Liouville operator satisfying = I and = I. [36] The imaginary part of the eigenspectrum of the Liouville operator reveals the 16384 relaxation coefficients at work in the Markovian time evolution of the system.…”

Electroluminescence Caused by the Transport of Interacting Electrons through Parallel Quantum Dots in a Photon Cavity

“…The coupling to the leads depends on the geometry of the wavefunctions of the single‐electron states in the “contact area” of the leads and the central system defined to extend approximately into each subsystem. In terms of creation and annihilation operators of single‐electron states in the leads () and the central system () the coupling Hamiltonian is where the index q stands for the combined continuous lead momentum quantum number and a discrete subband index , i labels the single‐electron states in the central system, and is the state‐dependent coupling tensor with . The temperature of the electron reservoirs in the leads is K.…”

“…Here, the effective density operator is with H the Hamiltonian of the total system, its density operator, and Tr res the trace operator with respect to the variables of the reservoir. In the Liouville space the solution of the Master equation is and the two‐time average or the correlation function becomes Here, is an approximation of the Liouville operator of the total system, is the complex diagonal matrix corresponding to it in the Liouville space of transitions, and is the matrix of its left eigenvectors, and the matrix of its right eigenvectors . Tr S is the trace operation with respect to the state space of the central system.…”