Master equation and conversion of environmental heat into coherent electromagnetic energy

Abstract

We derive a non-Markovian master equation for the long-time dynamics of a system of Fermions interacting with a coherent electromagnetic field, in an environment of other Fermions, Bosons, and free electromagnetic field. This equation is applied to a superradiant p–i–n semiconductor heterostructure with quantum dots in a Fabry–Perot cavity, we recently proposed for converting environmental heat into coherent electromagnetic energy. While a current is injected in the device, a superradiant field is generated by quantum transitions in quantum dots, through the very thin i-layers. Dissipation is described by correlated transitions of the system and environment particles, transitions of the system particles induced by the thermal fluctuations of the self-consistent field of the environment particles, and non-local in time effects of these fluctuations. We show that, for a finite spectrum of states and a sufficiently weak dissipative coupling, this equation preserves the positivity of the density matrix during the whole evolution of the system. The preservation of the positivity is also guaranteed in the rotating-wave approximation. For a rather short fluctuation time on the scale of the system dynamics, these fluctuations tend to wash out the non-Markovian integral in a long-time evolution, this integral remaining significant only during a rather short memory time. We derive explicit expressions of the superradiant power for two possible configurations of the superradiant device: (1) a longitudinal device, with the superradiant mode propagating in the direction of the injected current, i.e. perpendicularly to the semiconductor structure, and (2) a transversal device, with the superradiant mode propagating perpendicularly to the injected current, i.e. in the plane of the semiconductor structure. The active electrons, tunneling through the i-zone between the two quantum dot arrays, are coupled to a coherent superradiant mode, and to a dissipative environment including four components, namely: (1) the quasi-free electrons of the conduction n-region, (2) the quasi-free holes of the conduction p-region, (3) the vibrations of the crystal lattice, and (4) the free electromagnetic field. To diminish the coupling of the active electrons to the quasi-free conduction electrons and holes, the quantum dot arrays are separated from the two n and p conduction regions by potential barriers, which bound the two-well potential corresponding to these arrays. We obtain analytical expressions of the dissipation coefficients, which include simple dependences on the parameters of the semiconductor device, and are transparent to physical interpretations. We describe the dynamics of the system by non-Markovian optical equations with additional terms for the current injection, the radiation of the field, and the dissipative processes. We study the dependence of the dissipative coefficients on the physical parameters of the system, and the operation performances as functions of these parameters. We show that the decay rate of the superradiant electrons due to the coupling to the conduction electrons and holes is lower than the decay rate due to the coupling to the crystal vibrations, while the decay due to the coupling to the free electromagnetic field is quite negligible. According to the non-Markovian term arising in the optical equations, the system dynamics is significantly influenced by the thermal fluctuations of the self-consistent field of the quasi-free electrons and holes in the conduction regions n and p, respectively. We study the dependence of the superradiant power on the injected current, and the effects of the non-Markovian fluctuations. In comparison with a longitudinal device, a transversal device has a lower increase of the superradiant power with the injected current, but also a lower threshold current and a lesser sensitivity to thermal fluctuations.