Description of the Simplest Non-Markov Process Using a Differential Equation for the Quantum State Vector

The Jaynes–Cummings model with one atom and a photon is considered. A photon leaks out of the cavity (optical resonator). An atom can be in an excited and ground state. Usually, the dynamics of the probability of finding a photon in a cavity is considered using the basic quantum Lindblad equation, in which the density matrix acts as an unknown function. The Lindblad equation describes a quantum Markov random process. The article attempts to replace the equation from the density matrix with an ersatz of the Lindblad equation, which is a differential equation from the state wave vector. The quantum master equation involves the use of a matrix with a dimension equal to the dimension of the state space, which increases the complexity of the calculations, since it requires a quadratically large memory. For example, for the dimension of the main space equal to a billion, the memory required to solve the basic quantum equation will be about a quintillion, which is a problem even for supercomputers. Whereas a billion-long column fits easily into the memory of a personal computer and can be easily processed on a personal laptop. The ersatz of the quantum master equation, which we are constructing, cannot accurately describe the dynamics of the density matrix and therefore cannot serve as an exact replacement for the quantum master equation. Our ersatz will describe a special process of exchange with the environment.