Quantum Entanglement from Classical Trajectories

A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin $\frac{1}{2}$. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.

Figure 1

(a) Sketch of a path of spin vectors (red arrows on the sphere) and their centroid (black arrow inside the sphere). The spin orientations corresponding to a pure state of the underlying two-level system are shown as blue circles at $s_z=\pm \frac{1}{2}$. (b) Weight of $\overline{\mathit{s}}$ [Eq. (8)], with positive contributions in blue and negative in red. The negative regions grow in importance for increasing number of spins, $N$. (c)–(d) Distribution of the $x$, $y$, and $z$ components of the spin centroid. The distributions become peaked around the quantum-mechanical eigenvalues $\pm \frac{1}{2}$ with increasing $N$, and the relative peak heights approach the corresponding expectation values of the density matrix, here plotted for $\widehat{\rho }=\frac{2}{3}|1⟩⟨1|+\frac{1}{3}|2⟩⟨2|$.

Figure 2

Probability distribution of the nuclear momentum after an avoided crossing (model I). The inset shows the diabatic potentials (solid lines) and coupling (dashed line). A wave packet enters from the left on the lower surface with a narrow distribution of kinetic energies at roughly 1.5 (left panels) or 5 (right panels) times the asymptotic energy difference. Ehrenfest (MFT) gives a single peak around the average momentum, while linearized spin mapping ($N=1$) envelopes the exact wave-packet distribution. For higher $N$, the spin path-integral method correctly reproduces the wave-packet branching.

Figure 3

Impurity in the extended coupling model (model III). The inset shows the adiabatic surfaces (solid) and nonadiabatic coupling (dashed). A wave packet enters from the left and is partly reflected. Only the spin path-integral method with $N=4$ is able to correctly describe the second crossing of the interaction region (results for $N=8$, not shown, overlay with $N=4$).

Figure 4

Transmission and reflection probabilities onto the adiabatic states in the extended coupling model (model III). While surface hopping (FSSH) suffers from erroneous oscillations, the spin path-integral method appears to be converging smoothly with increasing $N$ towards the exact scattering probabilities from the log-derivative method [47].