Strongly correlated photon transport in nonlinear photonic lattices with disorder: Probing signatures of the localization transition

We study the transport of few-photon states in open disordered nonlinear photonic lattices. More specifically, we consider a waveguide quantum electrodynamics (QED) setup where photons are scattered from a chain of nonlinear resonators with on-site Bose-Hubbard interaction in the presence of an incommensurate potential. Applying our recently developed diagrammatic technique that evaluates the scattering matrix (S matrix) via absorption and emission diagrams, we compute the two-photon transmission probability and show that it carries signatures of the underlying localization transition of the system. We compare the calculated probability to the participation ratio of the eigenstates and find close agreement for a range of interaction strengths. The scaling of the two-photon transmission probability suggests that there might be two localization transitions in the high energy eigenstates corresponding to interaction and quasiperiodicity respectively. This observation is absent from the participation ratio. We analyze the robustness of the transmission signatures against local dissipation and briefly discuss possible implementation using current technology.

Figure 1

Two waveguides are coupled to each end of the lattice. We input two photons with momenta $k_1$ and $k_2$ from waveguide $\mathcal{W}1$ and investigate the statistics of the scattered photons at waveguide $\mathcal{W}N$ as a function of disorder for a fixed value of the on-site interaction. The sum of the momenta $k_1+k_2$ is resonant with the eigenstate $|E_{\alpha }^{\left(2\right)}\rangle$. We also consider how losses in the one-dimensional chain affect the statistics of the output light. (a) If $|E_{\alpha }^{\left(2\right)}\rangle$ is delocalized over the system, it will couple strongly to both waveguides $\mathcal{W}1$ and $\mathcal{W}N$, allowing the two input photons to be transmitted at $\mathcal{W}N$. (b) If $|E_{\alpha }^{\left(2\right)}\rangle$ is localized, it will couple weakly to both waveguides. Hence, the two-photon transmission at $\mathcal{W}N$ is strongly suppressed. (c) A diagram showing the choices of $k_1$ and $k_2$. To maximize the probability of the first photon entering the system, $k_1$ is set to be resonant with one of the one-particle eigenstates $|E_{\mu }^{\left(1\right)}\rangle$. The second photon then populates $|E_{\alpha }^{\left(2\right)}\rangle$ from $|E_{\mu }^{\left(1\right)}\rangle$. However, since we are only interested in the properties of $|E_{\alpha }^{\left(2\right)}\rangle$, we later reduce the contribution of $|E_{\mu }^{\left(1\right)}\rangle$ by varying $k_1$ to be resonant with all possible $|E_{\mu }^{\left(1\right)}\rangle$, and averaging the two-photon transmission probability measured at $\mathcal{W}N$. The resonant condition for the input state also ensures that two-photon scattering has the largest contribution in the output.

Figure 2

Diagrammatic method to calculate the scattering elements by summing up all possible optical absorption and emission paths. Scattering diagrams for (a) two-point Green's function and (b) four-point Green's function.

Figure 3

$U=0$: Participation ratio vs transmission spectra. (a) The participation ratio ${log}_{d_2}R\left(\alpha \right)$ and (b) the two-photon transmission probability $T^{\left(2\right)}\left(\alpha \right)$, for $U=0$. Here, $N=15$ and $b=\frac{\sqrt{5}-1}{2}$. Both diagrams show a transition from finite value to zero at $h/J=2$, with the latter being an open system with a realistic waveguide system coupling of $\kappa =0.25J$.

Figure 4

$U=3.5J$: Participation ratio vs transmission spectra. Comparison between (a) the participation ratio ${log}_{d_2}R\left(\alpha \right)$ and (b) the two-photon transmission probability $T^{\left(2\right)}\left(\alpha \right)$, for $U=3.5J$. Here, $N=15$ and $b=\frac{\sqrt{5}-1}{2}$. For (b), $\kappa =0.25J$. Both quantities show similar behavior. The delocalization-localization transition happens at different values of $h/J$ for different eigenstates.

Figure 5

$U=10J$: Participation ratio vs transmission spectra. Comparison between (a) the participation ratio ${log}_{d_2}R\left(\alpha \right)$ and (b) the two-photon transmission probability $T^{\left(2\right)}\left(\alpha \right)$, for $U=10J$. Here, $N=15$ and $b=\frac{\sqrt{5}-1}{2}$. For (b), $\kappa =0.25J$. Both quantities show similar behavior. The delocalization-localization transition happens at different values of $h/J$ for different eigenstates.

Figure 6

Scaling with $N$: Participation ratio vs effective TIP's localization length. (a)–(c) $U=0$ (AA model), (d)–(f) $U=3.5J$, and (g)–(i) $U=10J$ for three different eigenstates: $\alpha =1, \lfloor d_2/2\rfloor$, and $d_2$. Here, $b=\frac{\sqrt{5}-1}{2}$ and $\kappa =0.25J$. Both quantities ${log}_{d_2}R$ and ${\mathrm{\Lambda }}_2$, approach the same transition point as $N$ increases. Magenta dashed lines indicate the AA model's predicted transition at $h/J=2$. For plots of ${\mathrm{\Lambda }}_2$, gray dashed lines labeled AA satisfy the equation $1/{\mathrm{\Lambda }}_2=2ln(h/2J)$ for $h/J>2$. It is the infinite $N$ limit of the effective TIP's localization length in the AA model. For the highest energy eigenstate $\alpha =d_2$, cyan lines indicate the theoretical transition of $h/J=2J/U$ in the strongly interacting limit of $U\gg h,J$.

Figure 7

Effect of local losses: Plots of two-photon transmission probability, $T^{\left(2\right)}\left(\alpha \right)$, for two different local loss rates of (a) $\gamma =0.0001J$ and (b) $\gamma =0.1J$ for the same waveguide system coupling of $\kappa =0.25J$. Here, $N=15, U=3.5J$, and $b=\frac{\sqrt{5}-1}{2}$. The presence of local losses washes out most of the features observed in the lossless case. However, when the loss rate is much smaller than the coupling strength, $\gamma /\kappa \ll 1$, the mobility edge is still clearly visible.

Figure 8

Coherent state input: Plot of the two-photon transmission probability with identical photon momenta input (a small coherent state), $T_{\text{coh}}^{\left(2\right)}\left(\alpha \right)$. Here, $N=15, U=3.5J, b=\frac{\sqrt{5}-1}{2}$, and $\kappa =0.25J$. $T_{\text{coh}}^{\left(2\right)}$ behaves very differently from $T^{\left(2\right)}$ and ${log}_{d_2}R$.