Photon transport in a dissipative chain of nonlinear cavities

By means of numerical simulations and the input-output formalism, we study photon transport through a chain of coupled nonlinear optical cavities subject to uniform dissipation. Photons are injected from one end of the chain by means of a coherent source. The propagation through the array of cavities is sensitive to the interplay between the photon hopping strength and the local nonlinearity in each cavity. We characterize photon transport by studying the populations and the photon correlations as a function of the cavity position. When complemented with input-output theory, these quantities provide direct information about photon transmission through the system. The position of single-photon and multiphoton resonances directly reflects the structure of the many-body energy levels. This shows how a study of transport along a coupled cavity array can provide rich information about the strongly correlated (many-body) states of light even in presence of dissipation. The numerical algorithm we use, based on the time-evolving block decimation scheme adapted to mixed states, allows us to simulate large arrays (up to 60 cavities). The scaling of photon transmission with the number of cavities does depend on the structure of the many-body photon states inside the array.

Figure 1

A sketch of the one-dimensional cavity array. Neighboring cavities are coupled by photon hopping. Nonlinearities in the cavities may produce an effective repulsion between the photons leading to an anharmonic spectrum. We consider a Kerr-type nonlinearity. Photons in the cavities have a finite lifetime therefore the cavities are pumped with an external coherent drive. Here, we suppose that only the leftmost cavity is pumped, in order to study photon transport through the system.

Figure 2

The population in the $M\text{th}$ cavity $n_M$ in the NESS as a function of the detuning $({\omega }_p-{\omega }_0)/\gamma$ for different values of the driving strength. The dashed vertical lines are the spectral positions of the peaks in the fermionized limit ($|k_5\rangle , |k_5,k_6\rangle$, and $|k_6\rangle$ from left to right). The parameters are $U/J=\infty , J/\gamma =20, {\omega }_0/\gamma =1$, and $M=10$.

Figure 3

The population in the $M\text{th}$ cavity $n_M$ in the NESS as a function of the pump amplitude $F/\gamma$, when the one-photon resonance condition is satisfied for the states $|k_1\rangle$ and $|k_2\rangle$. Symbols denote the numerical data, while solid lines are the outcomes of the OBM (or equivalently of the TLM). Dashed lines indicate a behavior $n_T\propto {(F/\gamma )}^2$, and are plotted to guide the eye. The parameters are the same as in Fig. 2.

Figure 4

Photon occupations of each site in the NESS, when the one-photon resonance condition is satisfied for different states (as indicated in each panel) and for different system sizes (respectively, $M=6,10,20,40,60$). Symbols denote the numerical data, solid lines are the outcomes of the OBM, and dashed lines are the results of the TLM (shown for $M=6,10$). The parameters are the same as in Fig. 2, but with $F/\gamma =2$ for $M=6,10$ and $F/\gamma =1$ in the other panels.

Figure 5

Typical level scheme on resonance with a two-body state $|k_p,k_q\rangle$. The solid arrows stand for coherent Hamiltonian couplings, while wavy arrows indicate the decay induced by the Lindbladian.

Figure 6

Top panel: the population in the $M\text{th}$ cavity as a function of the pump amplitude, when the two-photon resonance condition (12) is satisfied. Symbols denote the numerical data, while solid lines are the TBM predictions. The dashed line indicates a behavior $n_T\propto {(F/\gamma )}^2$, and is plotted to guide the eye. The parameters are the same as in Fig. 2. Bottom panels: photon occupations of each site in the NESS, when the two-photon resonance condition (12) is satisfied. Symbols are the numerical data, while solid lines are the TBM predictions. Left panel: $F/\gamma =1$ and the target state is $|k_2,k_3\rangle$. Right panel: $F/\gamma =0.1,0.3,0.5,0.7,1,1.4$ (from bottom to top, respectively) and the target state is $|k_1,k_2\rangle$. In both panels, the remaining parameters are set as in Fig. 2.

Figure 7

Normalized two-body correlation function when the two-photon resonance condition (12) is satisfied. Symbols denote the numerical data, while solid lines are the results of the TBM. The data are taken for two values of $F/\gamma$, as indicated in the legend. Top panels refer to the target state $|k_1,k_2\rangle$, while the lower ones to $|k_2,k_3\rangle$. The remaining parameters are set as in Fig. 2. On the left plots the $g^{\left(2\right)}$ function is plotted starting from the center of the array, while on the right plot we considered the $M\text{th}$ cavity as reference.

Figure 8

Total population (top panel) and population in the $M\text{th}$ cavity (bottom panel) as a function of laser detuning. The various symbols stand for different values of $U/J$, as indicated in the legend. The dashed vertical lines are $|k_5\rangle , |k_5,k_6\rangle$, and $|k_6\rangle$, respectively. In both panels $F/\gamma =2, J/\gamma =20, {\omega }_0/\gamma =1$, and $M=10$.

Figure 9

The normalized two-body correlation function on resonance with the $U/J$ dependent two-photon peak (see Fig. 8). Here, the cavity on the opposite side with respect to the driving laser is taken as reference. In the inset, the autocorrelation of the $M\text{th}$ cavity is shown as a function of $U/J$. The parameters are set as in Fig. 8.

Figure 10

The absolute value of the matrix element $\langle k_n\left|{\widehat{b}}_m\right|k_p,k_q\rangle$ for the target two-photon state $|k_5,k_6\rangle$ as a function of $k_n$ and $k_m$ for $M=10$.

Figure 11

Graphical representation of the density matrix in a tensor-network language. Each block is a multi-index tensor, open links represent the free indexes, while connected links stand for the contracted indexes.

Figure 12

The spectrum of the Schmidt coefficients ${\lambda }_{\alpha }^{[M/2]}$ for a symmetric bipartition. We consider various system sizes and different bond-link dimensions: $\chi =50$ (dotted lines), 80 (dashed lines), 100 (continuous lines). The laser is resonant with the single-particle state $|k_{M/2}\rangle$ and the parameters are $U/J=\infty ,J/\gamma =20,{\omega }_0/\gamma =1$, and $F/\gamma =1$.

Figure 13

Local density (top panel) and normalized two-body function (lower panel) for different values of the bond-link dimension $\chi$. As indicated in the legend, the solid lines are the predictions of the TBM (see Appendix pp1). Here, the laser is resonant with the two-photon state $|k_2,k_3\rangle$ of an array of $M=10$ cavities. The parameters are $U/J=\infty ,J/\gamma =20,{\omega }_0/\gamma =1$, and $F/\gamma =1$. The data for $\chi =100$ are shown in the left lower panel of Fig. 6 (local density) and in the right lower panel of Fig. 7 (two-body function).

Figure 14

Evolution toward the NESS value of the total population for different system sizes and target states, as indicated in the legend. The initial state is random and the parameters are set as in Fig. 12.