Nonlinearity in the Medium for Electromagnetic Wave Propagation Induced by Quantum Mechanical Interaction of the Field With Finite-State Atoms

Abstract

We consider the interaction between a classical time-varying electric field and the atomic electrical dipole moment operator of a finite state atom and calculate approximately the mixed state evolution of the atom with such an interaction Hamiltonian up to quadratic orders in the external classical electric field using Schrodinger’s equation for mixed state evolution. From this approximate expression for the state evolution, we evaluate the average atomic electric dipole moment as a linear-quadratic function of the electric field. Later on, we also take into account Brownian motion bath noise that couples to the atomic dynamics, so that the Schrodinger equation dynamics gets modified to the Lindblad dynamics of an open quantum system, i.e., the master equation. From the average atomic dipole moment, we obtain an expression for the average polarization field of atoms constituting the medium through which the classical electromagnetic wave propagates. This average polarization is a linear-quadratic causal functional of the electric field. The resulting Maxwell equations for wave propagation, taking into account this polarization current density, thus acquire a quadratic nonlinearity which can be used to explain higher harmonic frequencies at the output of an optical fiber when the input is excited by the electric field coming from a monochromatic laser. We are then also able to calculate the mean square fluctuations in the polarization field using the evolving atomic state, and hence the electric field propagating through the medium of atoms acquires quantum fluctuations which can be used to explain time-varying random shifts in the spectral lines. As mentioned above, we include in this article an extension of this problem to the case when there is white Gaussian noise corrupting the state evolution dynamics of the atom and also the case when there is a pairwise interaction between the spin dipoles of atoms at two different locations. In the former case, we derive an approximate formula for the polarization field as well as its temporal statistical correlations, and we show how this computation can be extended to the latter case too. When there is a pairwise interaction between the atomic spins, we use a version of the approximate quantum Boltzmann equation to calculate the first and second order marginals of the atomic states, which is then used to compute the average polarization as well as its spatio-temporal correlations. Temporal correlations using second marginals of the atomic states cannot be computed using classical probabilistic methods owing to Heisenberg uncertainty, so we propose a sequential measurement-based strategy involving computation of the joint probabilities of the atomic spin components at two different times by first measuring one component of the spin, allowing for state collapse, and then following it up with nonlinear time evolution from the collapsed two-particle state using the quantum Boltzmann equation, followed further by measurement of another component of the spin at the second time.