Decoherence-free subspaces in the noisy dynamics of discrete-step quantum walks in a photonic lattice

Abstract

We study the noisy dynamics of periodically driven, discrete-step quantum walks in a one-dimensional photonic lattice. We find that in the bulk, temporal noise that is constant within a Floquet period leads to decoherence-free momentum subspaces, whereas fully random noise destroys coherence in a few time-steps. When considering topological edge states, we observe decoherence no matter the type of temporal noise. To explain these results, we derive a non-perturbative master equation to describe the system's dynamics and experimentally confirm our findings in a discrete mesh photonic lattice implemented in a double-fibre ring setup. Surprisingly, our results show that a class of bulk states can be more robust to a certain type of noise than topological edge states.

Figures (7)

• Figure 1: (a) Scheme of the experimental setup with beam splitters BS, variable beam splitter VBS, electrooptic modulator EOM, phase modulator PM, photodiodes PD, amplifier G and frequency shifter FS to create a local oscillator for the measurement of the eigenvectors and eigenvalues. The $\alpha$ and $\beta$ rings have a length of 45.34 m and 44.63 m, respectively. (b) Discrete-step lattice after time-demultiplexing of the pulses in the double ring with $\theta$ and $\varphi$ correspond to couplings from VBS and phase from PM, respectively.
• Figure 2: Measured light intensity in the $\beta$ ring after single-site injection under different types of step noise on a discrete-step lattice with $\theta_1=0$, $\theta_2=0.25\pi$, and $\varphi=0$. (a) Noiseless evolution ($\sigma=0$). (b) Evolution under large random noise ($\sigma=0.4\pi$). (c) Evolution under large stroboscopic noise ($\sigma=0.4\pi$). Each panel displays the coherent mean intensity ($|\mathrm{mean}(\beta)|^2$), averaged over 100 independent noise realizations.
• Figure 3: Measured dispersions under different types of noise for a lattice with $\theta_1=0$, $\theta_2=0.25\pi$, and $\varphi=0$. The intensity is computed by $|\tilde{\alpha}|^2 + |\tilde{\beta}|^2$, where the tilde indicates the Fourier amplitudes of the $\alpha$ and $\beta$ sublattices, and then it is averaged over 100 independent noise realizations. The lower panel shows the Gaussian-fitted full width at half maximum (FWHM) of the upper band for each quasimomentum $k$ . (a) Noiseless evolution ($\sigma=0$). (b) Large random noise ($\sigma=0.4\pi$). (c) Large stroboscopic noise ($\sigma=0.4\pi$).
• Figure 4: Measured (dots) averaged occupation probability $\overline{|\langle j | \psi_M \rangle|^2}$ as a function of step $M$ after initial excitation of the left edge site in a lattice with $N=44$ sites and averaged over 100 independent realizations. The couplers in the lattice are set to $\theta_1=0.5\pi$, $\theta_2=0.0\pi$, and $\varphi=0.2\pi$, such that the dispersion has two flatbands. Dots represents the edge state $| e_L \rangle$ return probability ($j=1$). Red dots display the case without noise, green dots with random noise and blue dots with stroboscopic noise with noise strength $\sigma=0.12\pi$ for both cases. Square blue dots show the population dynamics of bulk sites in $\alpha$ ($j=3$ and $5$ respectively). For comparison, the gray dashed line indicates purely exponential decay with rate $\sigma^2$. The solid black lines represent the numerical occupation probabilities derived from Eq. \ref{['eq:occProb']}.
• Figure 5: (a) Zoom on the first time steps of the measured time trace of the signal intensity at the output of the $\alpha$ fiber loop. (b) Spatio-temporal diagram of the $\alpha$ ring reconstructed from the time trace shown in (a). (c) Corresponding stroboscopic spatio-temporal diagram obtained by sampling only odd time steps from (b).