The Classical-Quantum Limit

Abstract

The standard notion of a classical limit, represented schematically by <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mi>ℏ</a:mi><a:mo stretchy="false">→</a:mo><a:mn>0</a:mn></a:math>, 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><e:mi>τ</e:mi></e:math>, we demonstrate that a double scaling limit in which <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><h:mi>ℏ</h:mi><h:mo stretchy="false">→</h:mo><h:mn>0</h:mn></h:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><l:mi>τ</l:mi><l:mo stretchy="false">→</l:mo><l:mn>0</l:mn></l:math> such that the ratio <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><p:msub><p:mi>E</p:mi><p:mi>f</p:mi></p:msub><p:mo>=</p:mo><p:mi>ℏ</p:mi><p:mo>/</p:mo><p:mi>τ</p:mi></p:math> 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. Published by the American Physical Society 2024