Quantum stochastic equation for test particle interacting with dilute Bose gas

TL;DR

This paper derives a quantum stochastic equation for a test particle interacting with a dilute Bose gas using the stochastic limit method, emphasizing correlator-based derivation and state-independent algebraic structures.

Contribution

It introduces a novel correlator-based derivation of the quantum white noise equation for the system, applicable to arbitrary low density states of the Bose gas.

Findings

Derived quantum white noise equation driven by quantum Poisson process

Established state-independent algebra of master fields and Ito table

Proved convergence of correlators to master field correlators in the low density limit

Abstract

We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form-factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity $S-1$, where $S$ is the one-particle $S$ matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a Fock-antiFock (or Gel'fand-Naimark-Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state.