Time-reversal symmetric resolution of unity without background integrals in open quantum systems

TL;DR

This paper introduces a new complete set of states for open quantum systems that respects time-reversal symmetry and allows explicit analysis of decay and growth processes without background integrals, enhancing understanding of quantum dynamics.

Contribution

It presents a novel complete set of states that includes resonant and anti-resonant states, enabling transparent analysis of quantum evolution and symmetry breaking.

Findings

Complete set contains all point spectrum states without background integrals.

Explicit calculation of survival and escape probabilities demonstrates the method's effectiveness.

Time evolution contributions are clearly attributed to specific spectral components.

Abstract

We present a new complete set of states for a class of open quantum systems, to be used in expansion of the Green's function and the time-evolution operator. A remarkable feature of the complete set is that it observes time-reversal symmetry in the sense that it contains decaying states (resonant states) and growing states (anti-resonant states) parallelly. We can thereby pinpoint the occurrence of the breaking of time-reversal symmetry at the choice of whether we solve Schroedinger equation as an initial-condition problem or a terminal-condition problem. Another feature of the complete set is that in the subspace of the central scattering area of the system, it consists of contributions of all states with point spectra but does not contain any background integrals. In computing the time evolution, we can clearly see contribution of which point spectrum produces which time dependence. In the whole infinite state space, the complete set does contain an integral but it is over unperturbed eigenstates of the environmental area of the system and hence can be calculated analytically. We demonstrate the usefulness of the complete set by computing explicitly the survival probability and the escaping probability as well as the dynamics of wave packets. The origin of each term of matrix elements is clear in our formulation, particularly the exponential decays due to the resonance poles.