The Classical-Quantum Limit

The standard notion of a classical limit, represented schematically by $\hslash \to 0$, provides a method for approximating a quantum system by a classical one. In this work, we explain why the standard classical limit fails when applied to subsystems, and show how one may resolve this by explicitly modeling the decoherence of a subsystem by its environment. Denoting the decoherence time by $\tau$, we demonstrate that a double scaling limit in which $\hslash \to 0$ and $\tau \to 0$ such that the ratio $E_f=\hslash /\tau$ remains fixed leads to an irreversible open-system evolution with well-defined classical and quantum subsystems. The main technical result is showing that, for arbitrary Hamiltonians, the generators of partial versions of the Wigner, Husimi, and Glauber-Sudarshan quasiprobability distributions may all be mapped in the above-mentioned double scaling limit to the same completely positive classical-quantum generator. This provides a regime in which one can study effective and consistent classical-quantum dynamics.

Popular Summary

Although the laws of nature appear to be fundamentally quantum mechanical, in many contexts a classical model provides a good approximate description. Approximating quantum models with classical ones, often referred to as taking the classical limit, is an important theoretical tool to allow a complex quantum dynamics to be replaced with a simpler classical one. However, what if we wanted only part of a quantum system to be approximated classically? In this work, we extend the notion of a classical limit to subsystems, allowing part of a quantum system to be treated as effectively classical, while retaining a quantum description of the remaining degrees of freedom.

To provide a consistent limit of this kind, which we refer to as a classical-quantum limit, we build on the existing notion that classicality can be induced by interactions with an external environment. Introducing two scales, one corresponding to the rate of information loss to the environment and the other quantifying the scale at which quantum features are important, we identify a particular scaling relationship between the two that allows us to construct a consistent classical-quantum limit. In such a limit, the evolution laws are irreversible and stochastic, with the parts that remain quantum able to consistently influence or “backreact” on the effectively classical parts.

Given the ubiquity of effective classical models in physics, it is hoped that the current work provides a theoretical basis for future explorations of consistent classical-quantum methods for use in fields such as quantum information science and quantum chemistry. Our work also provides a rigorous framework for understanding semiclassical regimes, which may be of interest in the study of limits of quantum gravity.