Objective trajectories in hybrid classical-quantum dynamics

Abstract

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

Path integrals for classical-quantum dynamics

“…of continuous master equations [7,10], the most general form of which was introduced in [17]. We find these path integrals have a natural decomposition into a pure classical part, representing the stochastic nature of the classical degrees of freedom, a pure quantum part, which includes a Feynman-Vernon term, and a classical-quantum partwhich acts to exponentially suppress the paths which deviate from the averaged equations of motion -as summarized by Table I.…”

“…Under certain conditions, namely when the classical-quantum action (20) is quadratic in momenta, one can arrive at a configuration space path integral, where paths deviating from the ± averaged Euler-Lagrange equations as suppressed (see Equation ( 64)). In order for the dynamics to be completely positive, the decoherence-diffusion trade-off 4D 2 D −1 0 must be satisfied [9,17], where D −1 0 is the generalized inverse of D 0 , which must be positive semi-definite.. As a consequence, there must be both a Feynman-Vernon term D 0 , and deviation from paths away from their expected drift due to the diffusion coefficient D 2 , and both effects cannot be made small. When the trade-off is saturated, the path integral preserves purity of the quantum state, conditioned on the classical degree of freedom [35] (see Section IV C).…”

“…Recently, there has been progress in understanding the dynamics of classical-quantum systems, where one system can be treated as classical and the other quantum mechanically. Examples of consistent classical-quantum (CQ) master equations, originally introduced in [1,2], have since been studied in a variety of different contexts [3][4][5][6], including gravity [5,[7][8][9], and can be shown to be completely positive, trace preserving (CPTP), and preserve the split between classical and quantum degrees of freedom [1,6,10,11]. The CPTP condition is required for the dynamics to respect positivity and normalisation of probabilities [12].…”

“…Other approaches to classical-quantum dynamics include sourcing classical degrees of freedom via feedback and measurement of quantum matter [13][14][15][16], which also lead to completely positive classical-quantum master equations. The most general form of CPTP classical-quantum dynamics has been derived [5,17], and takes an analogous form to that of the GSKL or Lindblad equation [18,19] in open quantum systems and the rate equation in classical dynamics.…”

“…In this work we shall make use of the recent developments in the understanding of classical-quantum master equations to write down a classical-quantum path integral, equivalent to dynamics which is CPTP. Specifically, using the most general form of CPTP classical-quantum dynamics introduced in [5,17], we associate a path integral to any CQ master equation, from which the conditions on complete positivity can easily be read off. The general result is given by Equation (20).…”