On the diagonalization of quadratic Hamiltonians

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can establish the diagonalization of a quadratic Hamiltonian by changing the frame of reference by a unitary transformation. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.

Measurement of the Temperature Using the Tomographic Representation of Thermal States for Quadratic Hamiltonians

The Wigner and tomographic representations of thermal Gibbs states for one- and two-mode quantum systems described by a quadratic Hamiltonian are obtained. This is done by using the covariance matrix of the mentioned states. The area of the Wigner function and the width of the tomogram of quantum systems are proposed to define a temperature scale for this type of states. This proposal is then confirmed for the general one-dimensional case and for a system of two coupled harmonic oscillators. The use of these properties as measures for the temperature of quantum systems is mentioned.

Long time behaviour and turnpike solutions in mildly non-monotone mean field games

We consider mean field game systems in time-horizon (0,T), where the individual cost depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (the aggregation rate of the cost function) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either globally Lipschitz Hamiltonians or quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we give a complete description of the ergodic and long time properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon (0,T), (ii) the convergence of the system from (0,T) towards (0,\infty), (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. We extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.

Invariant Quantum States of Quadratic Hamiltonians

The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.