Koopman wavefunctions and classical-quantum correlation dynamics

Abstract

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman-von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical-quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect-the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical-quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

Keywords: Koopman–von Neumann theory; classical–quantumdynamics; quantum density matrix.

Figure 1.

Hybrid evolution of a degenerate two-level quantum system quadratically coupled to a one-dimensional classical harmonic oscillator. The system Hamiltonian is given in (5.1). The depicted dynamics has the exact solution (5.3) with ω = m = 1 (a.u), α = (0.95, 0, 0) (a.u.), and the factorized initial condition (5.5) with β = 10⁵ (a.u.). The classical Liouville density (3.7) for this system is depicted at different times t = 0,  2.4,  5.7,  8.8 (a.u.) in the top panels (a), (b), (c) and (d), respectively. Red corresponds to positive values of the classical density $\mathrm{T}\mathrm{r}\hat{\mathcal{D}}$, whereas white marks vanishingly small values. (e) depicts the trajectory traced by the Bloch vector $\mathit{n}=\mathrm{T}\mathrm{r}(\hat{\mathit{\sigma }}\hat{\rho })$ for the quantum density matrix (3.6) during the evolution. The progression of time is represented by a colour gradation from dark blue to bright yellow along the curve. Since the trajectory lies on the yz plane, only the yz projection is plotted. The dashed black line denotes the surface of the Bloch sphere. (f ) displays the purity $\mathrm{T}\mathrm{r}({\hat{\rho }}^2)=|\mathit{n}{|}^2$ of the quantum density matrix (3.6) as a function of time. In (e) and (f ), the captioned black dots mark time at which figures (a)–(d) are plotted. The colour encoding of time is the same in both (e) and (f ). (Online version in colour.)

Figure 2.

Hybrid evolution (5.6) governed by the AG equation (1.1) with the Hamiltonian given in (5.1) and the initial condition ${\hat{\mathcal{D}}}_0$ in (5.5). The parameters used are the same as in figure 1. (a) depicts the trajectory traced by the Bloch vector for the quantum density matrix $\hat{\rho }=\int \hat{\mathcal{D}}\hspace{0.17em}\mathrm{d}p\mathrm{d}q$ during the evolution. Similarly to figure 1, the progression of time is represented by a color gradation from dark blue to bright yellow. Again, the trajectory lies on the yz plane. However, we emphasize the very different time scale from the evolution displayed in figure 1. (b) displays the purity $\mathrm{T}\mathrm{r}({\hat{\rho }}^2)$ as a function of time. The color encoding of time is the same in both (a) and (b). The classical Liouville density $\mathrm{T}\mathrm{r}\hat{\mathcal{D}}$ is identical to figure 1a–d. (Online version in colour.)