Bounding the Bogoliubov coefficients

TL;DR

This paper develops a rigorous method to establish bounds on Bogoliubov coefficients, which quantify quantum particle production and wave transmission, by relating them to invariants of a matrix derived from the wave equation.

Contribution

It introduces a formal exact solution for second-order linear ODEs using time-ordered exponentials and relates Bogoliubov coefficients to matrix invariants for bounding purposes.

Findings

Derived bounds on Bogoliubov coefficients.

Connected matrix invariants to quantum particle production.

Provided a systematic approach for bounding wave transmission and reflection.

Abstract

While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [Phys. Rev. A 59 (1999) 427-438], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2x2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.