Enhancing delocalization and entanglement in asymmetric discrete-time quantum walks

Abstract

In this paper, we investigate the enhancement of delocalization and coin-position entanglement in asymmetric discrete-time quantum walks (DTQWs). The asymmetry results from asymmetric coin operations, asymmetric initial states, and asymmetric polarization-dependent losses. By varying these asymmetry factors, the inverse participation ratio and entanglement entropy of the walker are numerically calculated for different coin and loss parameters, both for symmetric and asymmetric initial states. We then experimentally implement a 16-step asymmetric DTQW using a time-multiplexing fiber loop structure. By choosing an asymmetric initial state, both coin-position entanglement and delocalization are simultaneously enhanced under specific coin parameters. Moreover, we observe that with finite asymmetric polarization-dependent loss, the photon probability on the left side decreases significantly, while that on the right side increases and becomes more localized. Interestingly, under specific coin parameters, the entanglement and delocalization exhibit improved robustness against polarization-dependent loss. These results demonstrate that the DTQWs constitute an ideal platform for investigating photonic delocalization and hybrid entanglement.

Figures (7)

• Figure 1: (a) Coin-position entanglement entropy $S_{\mathrm{ E}}$ versus coin parameter $\theta$ and initial state parameter $\phi$. (b) IPR of the walker versus coin parameter $\theta$ and initial state parameter $\phi$.
• Figure 2: (a) Coin-position entanglement entropy $S_{E}$ versus coin parameter $\theta$ and loss parameter $\gamma$. (a) Initial state parameter $\phi=0$. (b) Initial state parameter $\phi=\pi /4$. The red, blue, and wine lines correspond to numerical simulations with the loss parameter $\gamma=0$, $\gamma=0.1$, $\gamma=0.2$, respectively.
• Figure 3: (a) IPR of the walker versus coin parameter $\theta$ and loss parameter $\gamma$. (a) Initial state parameter $\phi=0$. (b) Initial state parameter $\phi=\pi /4$. The red, blue, and wine lines correspond to numerical simulations with the loss parameter $\gamma=0$, $\gamma=0.1$, $\gamma=0.2$, respectively.
• Figure 4: Schematic of the experimental setup. The laser beam is attenuated to single-photon levels by a neutral-density (ND) filter and coupled into the loop via a 90/10 beam splitter (BS). HWP: half-wave plate; QWP: quarter-wave plate; PBS: polarization beam splitter; APD: single-photon avalanche photodetector.
• Figure 5: Experimental and numerical probability distributions of polarized photons in the 16-step quantum walk with the initial state $\left| H \right\rangle \otimes \left| 0 \right\rangle$, as a function of position $x$ for a loss parameter $\gamma = 0$, with (a) $\theta = 37^{\circ}$, (b) $\theta = 48^{\circ}$, and (c) $\theta = 59^{\circ}$.