Ehrenfest dynamics is purity non-preserving: A necessary ingredient for decoherence

Abstract

We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.

Hybrid classical-quantum formulations ask for hybrid notions

“…In all these approaches one starts with initially separated purely classical and quantum sectors and then makes them interact in order to analyze the outcome. Without pretending to be exhaustive, we can classify these approaches in the following categories: (1) approaches that try to maintain the use of quantum states (or density matrices) to describe the quantum sector and trajectories for the classical sector [2,3], (2) those that first formulate the classical sector as a quantum theory [4][5][6] and then work with a formally completely quantum system [7][8][9][10][11], (3) conversely, those that first formulate the quantum sector as a classical theory [12] and then work with a formally completely classical system [13][14][15][16], and (4) approaches that take the quantum and the classical sectors to a common language and then extend it to a single framework in the presence of interactions, for instance, using Hamilton-Jacobi statistical theory for the classical sector and Madelung representation for the quantum sector [17][18][19] or modeling classical and quantum dynamics starting from Ehrenfest equations [20]. This classification is not sharp and in some cases is subject to interpretation, but it may be useful as a way to organize the possible procedures and conceptual viewpoints in the enterprise of constructing a hybrid theory.…”

Nonextensive thermodynamic functions in the Schrödinger-Gibbs ensemble

“…On the other hand, the use of density distributions is a natural option when studying hybrid quantum-classical models such as Ehrenfest models in nonadiabatic molecular dynamics (see [4] and [5]). We leave for future work the study of the consequences of the analysis in this paper for the equilibrium and non-equilibrium statistics of hybrid quantum-classical systems (see [43] for a recent approach to nonequilibrium and irreversibility), where the nonlinear effects on the dynamics produced by the classical subsystem may alter significantly the results we have presented here.…”

“…It is well known (see [4,5] and references therein) that finite dimensional quantum systems defined on C n admit a If one approaches the canonical ensemble from the microcanonical one the origin is clear.…”

“…One may now prove that the average value of an arbitrary quantum observableÔ, can be calculated as (see [5])…”

“…The approach also offers many advantages in the description of hybrid quantum-classical systems, as it can be seen in [4,5]. Nonetheless, it is important to keep in mind that these distributions are ambiguous from the physical point of view.…”