Quantum entanglement and the self-trapping transition in polaronic systems

We revisit from a quantum-information perspective a classic problem of polaron theory in one dimension. In the context of the Holstein model we show that a simple analysis of quantum entanglement between excitonic and phononic degrees of freedom allows one to effectively characterize both the small and large polaron regimes as well as the crossover in between. The small (large) polaron regime corresponds to a high (low) degree of bipartite quantum entanglement between the exciton and the phonon cloud that clothes the exciton. Moreover, the self-trapping transition is clearly displayed by a sharp drop of exciton-phonon entanglement.

Figure 1

The $J\text{‐}g$ phase diagram for the Toyozawa Ansatz. A thin sword-shaped regime identifies with the polaronic self-trapping line, across which the effective mass of the polaron decreases drastically as $g$ is decreased. The top (solid) line of the sword-shaped regime indicates the onset of bifurcation at $K=0$ when the polaronic structure goes through a sudden change as one travels vertically downward in the phase diagram, and the bottom (dashed) line indicates where the state of the highest $K$ (in the vicinity of $K=0$) for which a discontinuous change in the polaronic structure is observed acquires two solutions as one travels vertically upward.

Figure 2

Heteroentanglement between the exciton and the phonons as measured by the linear entropy $1-{\mathrm{Tr}}_e[{({\rho }_e^{K=0})}^2]$ is displayed in the upper panel for the entire $J\text{‐}g$ phase diagram. The entanglement is calculated for the lowest energy state with zero crystal momentum $K=0$. The solid line in the lower panel is the edge of the cliff which separates the large and small entanglement regions.