Configurable photonic simulator for quantum field dynamics

Abstract

Quantum field simulators provide unique opportunities for investigating the dynamics of quantum fields through tabletop experiments. A primary drawback of standard encoding schemes is their rigidity: altering the theory, its coupling geometry, metric structure, or simulation time typically requires redesigning the experimental setup, which imposes strong constraints on the types of dynamics and theories that can be simulated. Here, we introduce the Optical Time Algorithm (OTA) as a unifying framework, enabling the efficient simulation of large classes of free quantum field dynamics using a single optical circuit design that separates the time from the Hamiltonian's structure. By modifying the parameters of the optical elements, our method allows us to engineer timescales, coupling graphs, spacetime metrics, and boundary conditions, thereby facilitating the implementation of relativistic and non-relativistic, real- and complex-valued, short- and long-range quantum field theories on both flat and curved spacetimes. We exploit the OTA's configurability to investigate the spreading of quantum correlations in space and time for theories with continuously varying coupling ranges. Relevant features predicted by quantum field theory can be observed on systems of $10$ to $20$ modes, which paves the ground for experimental implementations.

Figures (10)

• Figure 1: Pictorial representation of the map between scalar quantum fields (a) and their photonic simulation (b). By tuning the optical parameters, one can realize arbitrary coupling geometries and timescales, thereby enabling the probing of information dynamics in spacetime in general settings.
• Figure 2: Pictorial representation of the OTA for $N=5$. The first and last layers can be further decomposed via standard techniques, such as Reck's Reck1994 and Clements' decompositions Clements2016. We provide an explicit example for the former in App. \ref{['subsubsec:BeamSplitterArray']}.
• Figure 3: For vacuum inputs, the OTA reduces to a complex squeezer layer followed by an interferometer. The complex-valued squeezing parameters $\zeta = \xi e^{i \Phi}$ are time-dependent, while the interferometer remains time-independent. This matches the structure of the Gaussian boson sampler setup.
• Figure 4: (a) Time evolution of the Rényi-2 entanglement entropy \ref{['eq:Renyi2Entropy']} (scaled by $10^2$) of the subregion $[0,\ell]$ following a quench from the optical vacuum \ref{['eq:PreQuenchHamiltonian']} to the relativistic theory \ref{['eq:Relativistic Real Hamiltonian Operator in Real Space']} with $\ell=L/5, m=1, \epsilon=2$ for varying $L=N \epsilon$ against the field-theoretic prediction \ref{['eq:Renyi2EntropyQFT']} (black dashed curve). The light-cone-like motion of quasi-particles generates entanglement (see inset), leading to a linear increase in entanglement entropy until the time $\tau/N = \ell/(2Nv_{\text{max}}) \approx 0.48$ (gray box) that the fastest quasi-particle pairs require to end up in distinct regions. (b) Same analysis for the Rényi-2 mutual information \ref{['eq:Renyi2MI']} between two spatial regions separated by a distance $d=\ell$. Correlations can only emerge after $t \ge \tau$, i.e., when the fastest quasi-particles have bridged the distance $d$. We refer to Fig. \ref{['fig:EntanglementGenerationOBC']} in App. \ref{['subsubsec:OpenBoundaries']} for a treatise of open boundaries.
• Figure 5: Spatiotemporal evolution of the Rényi-2 mutual information \ref{['eq:Renyi2MI']} of two modes ($l=\epsilon$) separated by a distance $d=(M+1) \epsilon$ after quenching the optical vacuum \ref{['eq:PreQuenchHamiltonian']} to the fractional Laplacian theory \ref{['eq:FractionalLaplacianHamiltonian']} with $L=2,m=1,\epsilon=0.1$ fixed. We vary the coupling range [see inset in (a) for the coupling strength $f_M = \lvert f_{j j+M} (\alpha)\rvert$ of the two modes at various distances $M$] from $\alpha=2$ (relativistic) in (b) to $\alpha \to 0$ (all-to-all) in (f). We cut off the mutual information below $0.01$ to probe the first correlation front (points) by fitting the characteristic curves summarized in Tab. \ref{['tab:LightCones']} and shown in (a). The light cones start to bend when $\alpha < 2$, as long-range couplings enable distant modes to become correlated at a faster rate. Correlations between neighboring modes pass the threshold at later times for longer range couplings, see gray regions in (b)--(f), since the nearest-neighbor coupling strength $f_0$ decreases monotonically with $\alpha$, see inset in (a), converging to $f_M \to 1/2$ when $\alpha \to 0$.