Disorder effects in tunable waveguide arrays with parity-symmetric tunneling

We investigate the effects of disorder on single-particle time evolution and two-particle correlations in an array of evanescently coupled waveguides with position-dependent tunneling rates. In the clean limit, the energy spectrum of such an array is widely tunable. In the presence of a Hermitian on-site or tunneling disorder, we find that the localization of a wave packet is highly sensitive to this energy spectrum. In particular, for an input confined to a single waveguide, we show that the fraction of light localized to the original waveguide depends on the tunneling profile. We compare the two-particle intensity correlations in the presence of Hermitian tunneling disorder and non-Hermitian, parity- and time-reversal ($\mathcal{PT}$) symmetric, on-site potential disorder. We show the two-particle correlation function in both cases is qualitatively similar since both disorders preserve the particle-hole symmetric nature of the energy spectrum.

Figure 1

(left) The steady-state intensity localized to the initial waveguide as a function of $\alpha$ for an array with $N=100$ waveguides, $N_r={10}^6$ disorder realizations, and input state $|\psi \left(0\right)\rangle =|m_0\rangle$. The input state is given by $m_0=50$ (blue circles) and $m_0=15$ (red squares), and the disorder strength is higher than the bandwidth $\hslash \sigma /{\Delta }_{\alpha }^{\left(0\right)}=3$. We see that the localized fraction is weakly dependent upon $\alpha$. (right) The corresponding results for an array with $N=37$ waveguides, $N_r={10}^5$ disorder realizations, and a weaker disorder $\hslash \sigma /{\Delta }_{\alpha }^{\left(0\right)}=1$. We see that the localized fraction as a function of $\alpha$ depends acutely on different initial input states, $m_0=11$ (blue squares) and $m_0=5$ (red circles), when the input locations are relatively close to the boundary.

Figure 2

The $\alpha$-dependent disorder averaged intensity $I(j,t)$ for an input state $|\psi \left(0\right)\rangle =|m_0=15\rangle$ in an array with $N=100$ waveguides, a weak disorder $\hslash \sigma /{\Delta }_{\alpha }^{\left(0\right)}=0.05$, and $N_r={10}^6$ disorder realizations. The horizontal axis in each panel indicates time in units of ${\tau }_{\alpha }$. (a) Exponential localization with a single peak at $m_0$ for an array with uniform tunneling $\alpha =0$. Corresponding results for (b) $\alpha =1$ and (c) $\alpha =2$. In each case, ballistic expansion and reconstruction are followed by the emergence of a steady-state intensity profile $I\left(j\right)$ that has two peaks, one at the input waveguide $m_0$ and the other at its mirror-symmetric counterpart, $N+1-m_0$. The relative weights at the two peaks can be tuned by varying the weak disorder and the input position $m_0$.

Figure 3

(left) Intensity $I(j,t)$ as a function of $\theta$ for an array with $N=60$ waveguides, disorder $\hslash \sigma /{\Delta }_{\alpha }^{\left(0\right)}=0.05$, and $N_r={10}^5$ disorder realizations. The initial input state is $|\psi \left(0\right)\rangle =\left(\right|20\rangle +e^{i\theta }|40\rangle )/\sqrt{2}$. The top, middle, and bottom panels correspond to $\theta =0$, $\theta =\pi /2$, and $\theta =\pi$, respectively. The interference structure at short times $t/{\tau }_{\alpha }\lesssim 100$ is replaced by a steady-state double-peak intensity profile at times $t/{\tau }_{\alpha }⩾300$. The peak intensity is twice the average intensity $I_a=1/N=0.0167$ per waveguide. (right) The corresponding steady-state intensity $I\left(j\right)$ at $t/{\tau }_{\alpha }=600$ as a function of phase $\theta$. The intensity near the waveguide array center shows enhancement for $\theta =0$ (blue solid line) and suppression for $\theta =\pi$ (black dotted line) when compared with the corresponding intensity for $\theta =\pi /2$ (red dashed line). As expected, this intensity difference vanishes for a strong disorder.

Figure 4

(left) The disorder-averaged, steady-state, classical correlation matrix ${\Gamma }_{jk}$ for a uniform array with $N=20$ waveguides and a weak disorder $\hslash \sigma /{\Delta }_{\alpha }=0.02$; the results are averaged over $N_r={10}^4$ disorder realizations. (a) The matrix for on-site, $\mathcal{PT}$-symmetric, non-Hermitian disorder; (b) the matrix for tunneling-rate, Hermitian disorder. (right) The correlation functions $g\left({\Delta }_r\right)$ extracted from the steady-state, classical correlation matrix for (c) on-site, $\mathcal{PT}$-symmetric disorder and (d) off-diagonal tunneling disorder. The similarity between the two results shows that the particle-hole symmetry of the disordered energy spectrum is instrumental in the correlation function properties.