Symplectic analogs of polar decomposition and their applications to bosonic Gaussian channels

We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms

Dynamics of moments of arbitrary order for stochastic Poisson squeezings

The explicit dynamics of the moments for the GKSL equation is obtained. In our case the GKSL equation corresponds to Poisson stochastic processes which lead to unitary jumps. We consider squeeze operators as the unitary jumps

Non-perturbative effects in corrections to quantum master equation arising in Bogolubov-van Hove limit

We study the perturbative corrections to the weak coupling limit type Gorini-Kossakowski-Sudarshan-Lindblad equation for the reduced density matrix of an open system. For the spin-boson model in the rotating wave approximation at zero temperature we show that the perturbative part of the density matrix satisfies the time-independent Gorini-Kossakowski-Sudarshan-Lindblad equation for arbitrary order of the perturbation theory if all the moments of the reservoir correlation function are finite. But the initial condition for perturbative part of the density matrix does not only differ from that for the whole density matrix, but also fails to be a density matrix under certain resonance conditions

Effective Heisenberg equations for quadratic Hamiltonians

We discuss effective quantum dynamics obtained by averaging projector with respect to free dynamics. For unitary dynamics generated by quadratic fermionic Hamiltonians we obtain effective Heisenberg dynamics. By perturbative expansions we obtain the correspondent effective time-local Heisenberg equations. We also discuss a similar problem for bosonic case

Time-convolutionless master equations for composite open quantum systems

In this work we consider the master equations for composite open quantum systems. We provide purely algebraic formulae for terms of perturbation series defining such equations. We also give conditions under which the Bogolubov-van Hove limit exists and discuss some corrections to this limit. We present an example to illustrate our results. In particular, this example shows, that inhomogeneous terms in time-convolutionless master equations can vanish after reservoir correlation time, but lead to renormalization of initial conditions at such a timescale