Simulation of Hopfield-like Hamiltonians using time-multiplexed photonic networks

Abstract

We propose a time-multiplexed photonic network architecture based on coupled ring resonators, capable of accurately emulating specific Hamiltonian dynamics. We show that, in the Suzuki-Trotter limit, the resulting stroboscopic evolution reproduces the characteristic dynamics of the bosonized Hopfield model. Furthermore, by incorporating a nonlinear element within the main resonator loop, we outline a scalable route toward optical simulation of both mean-field and quantum nonlinear dynamics associated with the Tavis-Cummings model. Our results establish time-multiplexed resonator networks as a versatile photonic framework for simulating interacting light-matter Hamiltonians and collective many-body phenomena.

Figures (5)

• Figure 1: Time-multiplexed photonic resonator network. (a) Sketch of the proposed architecture. An injection fiber loads a coherent state into the auxiliary cavity. At the evanescent coupling, a proportion $T^2$ of the light intensity is transmitted to the main cavity and the rest ($R^2$) continues to travel inside the auxiliary cavity. The length of the main cavity is N times larger than the one of the auxiliary cavity, defining N temporal sites inside the main cavity, achieving time-multiplexing of the main cavity transverse mode. (b) Representation of the temporal sites in the synthetic dimension. Each site $\hat{a}_i$ exchanges energy with the auxiliary cavity at a given time $t_i$, via a beam-splitter like operation. Each and every sites can store many photons, whose phase can be individually tuned using the intra-cavity phase modulators. One complete loop inside the main cavity is achieved in a time $T_\mathrm{RT} = N \times \Delta T$. In the Trotterized limit, every operation can be considered to happen simultaneously during the coarse-grained integration time $T_\mathrm{RT}$.
• Figure 2: Dynamics and fidelity of the TMN simulator. In all simulation N=3 and the dynamics are initialized with $|\alpha = \sqrt{0.4} \rangle$ in the auxiliary cavity. (a) & (b) Time evolution of the mean population for the TMN (squares) and continuous (dotted line) model. Red (orange) color labels the auxiliary cavity site and shades of grey the main cavity sites. Reflectivity is $R = 0.954$ for (a) and $R = 0.988$ for (b). (c) Time evolution of the Fidelity as a function of the number of Rabi periods elapsed. The two curve corresponds to long-time dynamics of Reflectivity (a) & (b). The red dashed line indicate the time cut-off used in (d). (d) Evolution of the global minimum of the fidelity, after 60 Rabi oscillations, as a function of Reflectivity (or BS angle). Red horizontal line shows the maximal $\mathcal{F} = 1$ value of the fidelity.
• Figure 3: Numerical simulation of Hopfield-like model emulation. Here we take the reference of phases as $\varphi = 0$. For all simulation $\theta_\mathrm{disc} \sim 0.03 \pi~(R=0.99)$, $\theta_\mathrm{disc}/\gamma_\mathrm{RT} = 10$, $F_\mathrm{disc}/\gamma_\mathrm{disc} = 0.1$.(a) Steady-state population of the auxiliary cavity site for varying main cavity/pump and main/auxilliary cavities detuning. The two bright branches corresponds to lower (LP) and upper (UP) polaritons. Dashed line corresponds to Eq. \ref{['eq:polaritons_dispersion']}. (b) Same as (a) for the first main cavity sites. (c) Steady state auxiliary cavity population for several values of main cavity pulses diagonal disorder. Main cavity pulses energies are uniformly distributed in $[\varphi - W, \varphi + W]$. (d) Evolution of the Rabi-splitting of the TMN ($\Omega_\mathrm{TMN}$) with the number of pulses fitting inside the main cavity. The red dashed line represented the theoretical curve $\omega_\sim/\theta_\mathrm{disc} = \sqrt{N}$ curve.
• Figure 4: Cooperativity diagram of a device with a fixed auxiliary cavity length of $L_\mathrm{aux} = 1\,$mm and for various $\theta_\mathrm{disc}$ angles. Here $n=1.5$ . All the part to the left of the cooperativity = 1 line corresponds to the strong coupling domain. The silver part correspond to the optical fiber domain and the light blue part to the Si integrated photonics domain. The left edge of the Si domain indicates the limit of state-of-the-art cavities.
• Figure 5: Non linearity and qubit limit for the TMN simulator. Dotted Lines represents the ST Hamiltonian Eq. \ref{['eq:H_ST']}, continuous faded lines the ST Hamiltonian in the qubit limit and square symbols the TMN. Red (orange) colour labels the auxiliary cavity population and shades of grey the main cavity one. Simulation are done at $\theta_\mathrm{disc} \sim 0.0.3\pi~$($R = 0.99$) and $\theta_\mathrm{disc}/\gamma_{RT} = 5$. When not specified, $F_\mathrm{disc}/\gamma_{disc} = 0.1$. (a) Spectroscopy of the lower polariton of the system for N = 1. Colored arrows corresponds to corresponding curves on (b). Detuning are expressed in units of $\gamma_\mathrm{RT}$. (b) Auxiliary cavity population at steady-state for increasing pump amplitude in the optical limiter (orange) and bistable (purple) cases. Black dotted line represents the same detuning as the optical limiter case, with no non-linearity. Saturation behaviors at high pumping amplitude is dominated by truncation effects, as the Fock spaces are truncated to $|n_\mathrm{max}\rangle = |12\rangle$. (c) Dynamic of the quantum simulator and the ST Hamiltonian until steady-state for $U_\mathrm{Kerr} = \theta_\mathrm{disc}$. Insert are contour plots of the Wigner function for the main cavity site taken at maximum filling (around roundtrip 10). (d) Same as (c) for $U_\mathrm{Kerr} = 10~\theta_\mathrm{disc}$ie in the fermionized (or qubit) limit. (c) & (d) are done with N = 2.