Conditional wave theory of environmental interactions with a quantum particle.

We present an alternative formulation of quantum decoherence theory using conditional wave theory (CWT), which was originally developed in molecular physics (also known as exact factorisation methods). We formulate a CWT of a classic model of collisional decoherence of a free particle with environmental particles treated in a long-wavelength limit. In general, the CWT equation of motion for the particle is non-linear, where the non-linearity enters via the CWT gauge fields. For Gaussian wave packets the analytic solutions of the CWT equations are in exact agreement with those from the density matrix formalism. We show that CWT gauge terms that determine the dynamics of the particle's marginal wave function are related to a Taylor series expansion of the particle's reduced density matrix. Approximate solutions to these equations lead to a linear-time approximation that reproduces the ensemble width in the limits of both short and long times, in addition to reproducing the long-term behaviour of the coherence length. With this approximation, the non-linear equation of motion for the particle's marginal wave function can be written in the form of the logarithmic Schr\"odinger equation. The CWT formalism may lead to computationally efficient calculations of quantum decoherence, since it involves working with wave-function level terms instead of evolving a density matrix via a master equation.