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Synthesising Keldysh and Lindblad: Correlated Gain and Loss in Higher Order Perturbation Theory
Clemens Muller and Thomas M. Stace
ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, Saint Lucia, Queensland 4072, Australia
Motivated by correlated decay processes driving gain, loss and lasing in driven semiconductor quantum-dots [1–3], we develop a theoretical technique using Keldysh diagrammatic perturbation theory to derive a Lindblad master equation that goes beyond the usual second order perturbation theory . We demonstrate the method on the driven dissipative Rabi model, including terms up to fourth order in the interaction between the qubit and both the resonator and environment. This results in a large class of Lindblad dissipators and associated rates which go beyond the terms that have previously been proposed to describe similar systems. All of the additional terms contribute to the system behaviour at the same order of perturbation theory. We then apply these results to analyse the phonon-assisted steady-state gain of a microwave field driving a double quantum-dot in a resonator. We show that resonator gain and loss are substantially affected by dephasing- assisted dissipative processes in the quantum-dot system. These additional processes, which go beyond recently proposed polaronic theories, are in good quantitative agreement with experimental observations .
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 Y Y Liu, J Stehlik, Christopher Eichler, M J Gullans, Jacob M Taylor, and J R Petta, “Semiconductor double quantum dot micromaser,” Science 347, 285–287 (2015).
 M J Gullans, Y Y Liu, J Stehlik, J R Petta, and Jacob M Taylor, “Phonon-Assisted Gain in a Semiconductor Double Quantum Dot Maser ,” Phys. Rev. Lett. 114, 196802 (2015).
 Clemens Muller and Thomas M Stace, “Deriving Lindblad master equations with Keldysh diagrams: Correlated gain and loss in higher order perturbation theory” Physical Review A 95 (1), 013847