Quantum–Classical Hybrid Systems and Ehrenfest’s Theorem

Abstract

The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum–classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum–classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest’s theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum–classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum–classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum–classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum–classical case. We expect that these theoretical results will be useful both to analyze quantum–classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.

Markovian dynamics for a quantum/classical system and quantum trajectories

“…The T(H)-valued process σ t , defined in (9), satisfies the weak-sense stochastic differential equation (10). Moreover, it is the unique solution of (10) with the initial condition (8). We have also…”

“…The interest in quantum/classical hybrid systems is old, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. One of the main motivations of the study of hybrid systems is that the output of a monitored system is classical; then, implicitly or explicitly, the dynamics of quantum/classical systems is involved in the theory of quantum measurements in continuous time and quantum filtering [1][2][3][4][15][16][17][18][19][20][21][22][23][24][25][26][27][28].…”

Nanowires: Exponential speedup in quantum computing

“…For a while now, there has been a search for a clear explanation of how quantum-classical hybrid systems behave. This search aims to achieve several objectives, such as understanding how mesoscopic systems work, finding a way to combine quantum theory and gravity, and explaining how quantum measurements occur [ [168] , [169] , [170] , [171] , [172] , [173] ]. The utilization of atomic resonance has been employed to precisely control and adjust the emission frequency of quantum dots, resulting in the creation of a highly stable source of single photons.…”