Floquet-driven light transport in programmable photonic processors via discretized evolution of synthetic magnetic fields

Abstract

Photons, unlike electrons, do not couple directly to magnetic fields, yet synthetic gauge fields can impart magnetic-like responses and enable directional transport. Discretized Floquet evolution provides a controlled route, where the time-ordered sequencing of non-commuting Hamiltonians imprints complex hopping phases and breaks time-reversal symmetry. However, stabilizing such driven dynamics and observing unambiguous signatures of these effects on a reconfigurable platform has remained challenging. Here we demonstrate synthetic gauge fields for light on a programmable photonic processor by implementing discretized Floquet drives that combine static and dynamic phases. The resulting dynamics exhibit chiral circulation that reverses under drive inversion, flux-controlled interference with high visibility, and robust directional flow stabilized by optimizing the driving period. We further characterize the system using a first-harmonic phase as an order parameter, whose per-period winding number quantifies angular drift and reverses sign with the drive order. These results establish discretized Floquet evolution as a versatile framework for driven photonics, providing a programmable route to engineer gauge fields, stabilize driven phases, and probe winding-number signatures of chiral transport.

Figures (5)

• Figure 1: Discretized Floquet evolution on a programmable photonic processor. A state $|\psi(0)\rangle$ injected on the left bus enters a programmable photonic processor consisting of a reconfigurable MZI mesh (black box) that implements the time-evolution operator $U(t)$ on $N$ spatial waveguide modes, yielding $|\psi(t)\rangle=U(t)|\psi(0)\rangle$ at the right bus. a, CW drive: one period $T$ is decomposed into three sequential substeps $U_1(T/3)$ (red), $U_2(T/3)$ (green), and $U_3(T/3)$ (blue). These time-ordered operations constitute the Floquet operator $U_F^{\text{CW}}(T)=U_3U_2U_1$. Dashed guide lines indicate that this schedule is programmed onto the chip; horizontal dots indicate repetition over successive periods. At intermediate times (e.g., $t=T/3$), only the substep executed up to that moment has acted. b, CCW drive obtained by reversing the order of the substeps, giving $U_F^{\text{CCW}}(T)=U_1U_2U_3$. c, Scalable $N$-site lattice representation. Lattice sites (white circles) map to physical waveguides, while solid colored bonds represent the tunable couplings realized by the MZI mesh active in the respective substeps (colors match panels a–b); dashed bond extensions indicate tiling to larger lattices. Curved arrows highlight the induced chiral circulation in each triangular plaquette, illustrating how the programmed drive order generates a synthetic magnetic flux.
• Figure 2: Experimental setup. Coherent light at 1550 nm from a pulsed diode laser passes through a polarization controller set for TE polarization and is directed to the common port of a $1\times12$ MEMS optical switch. The switch can be programmed to route the light to any of its twelve multiplexed ports. These ports couple via edge couplers to the corresponding input waveguides of a 12-mode programmable photonic chip, thereby selecting the input. The chip consists of a reconfigurable MZI mesh driven by multichannel current drivers. Light at the chip output is coupled via edge couplers into a second switch operated in reverse; its common port is connected to an optical power meter. Control and acquisition are handled by a PC, which commands the switches, sets the heater currents to realize the unitaries, and logs readings from the power meter. A thermoelectric cooler beneath the chip, regulated by a PID temperature controller, maintains a fixed room-temperature setpoint.
• Figure 3: Chiral photon transport in a three-site plaquette. Each row shows four snapshots of the system's state, taken at the switching boundaries $t=0,\;T/3,\;2T/3$ and $T$. For every snapshot, the colored surface represents the experimental normalized population, rendered as a single smooth surface by summing three Gaussian peaks centered at the lattice sites. The black wireframe overlays the simulated population. In the triangular schematics, a colored bond marks the active substep, with $U_1(T/3)$ red, $U_2(T/3)$ green, and $U_3(T/3)$ blue. The arrow direction indicates the transport directionality, and gray bonds are off. a, For clockwise (CW) modulation, the population transfers sequentially along the path $1 \rightarrow 2 \rightarrow 3 \rightarrow 1$. b, For counter-clockwise (CCW) modulation, the sequence reverses to $1 \rightarrow 3 \rightarrow 2 \rightarrow 1$. The close surface-wireframe overlap indicates excellent agreement. Reversing the drive clearly reverses the circulation.
• Figure 4: Flux-tuned interference in a four-site lattice.a, Schematic of two triangles A and B with a shared edge (green), driven by the CW sequence $U_1$ (red), $U_2$ (green), and $U_3$ (blue). Triangles A and B have loop fluxes $\Phi_A$ and $\Phi_B$, respectively. Interference at the shared vertices (sites 2 and 4) is governed by their flux difference $\Delta \Phi \equiv \Phi_A - \Phi_B$. The initial state is $(|1\rangle + |3\rangle)/\sqrt{2}$, launching light into both loops. b, Simulated mid-period ($t=T/2$) populations $P_2$ (black) and $P_4$ (red, dashed) versus $\Delta \Phi$; the traces are complementary and cross at $\Delta\Phi=0$. c-e, Simulated dynamics over one period $t=T$ for $\Delta \Phi=0$ (CW/CW), $+\pi$ (CW/CCW), and $-\pi$ (CCW/CW). Horizontal planes mark $t=0,\,T/3,\,2T/3,\,T$ and the colored labels indicate the active substep $U_1,\,U_2,\,U_3$ between planes. Insets use $\otimes/\odot$ to denote flux into/out of the triangle plane (drawn above population columns). f-h, Measured dynamics for the same settings, showing balanced splitting at $\Delta \Phi=0$, constructive interference at site 2 for $+\pi$, and at site 4 for $-\pi$, in agreement with simulations. In the latter two cases, destructive interference suppresses the green link, so the effective dynamics proceed primarily through the red and blue couplings. Color scale indicates normalized population.
• Figure 5: Floquet-period optimization and directional transport in a seven-site hexagon. Top-right: schematic of the lattice with color-coded bonds (red/green/blue for $U_1/U_2/U_3$). The system is initialized in state $|1\rangle$. a, Minimal quasi-energy splitting $\Delta \varepsilon_\text{min}$ of the three-site Floquet operator $U_F(T)$ versus drive period $T$ (dimensionless; $J=\hbar=1$). Peaks at $\tau$, $3\tau$, $5\tau$ mark the Rabi $\pi$ condition and its odd multiples with the maximum splitting reaching $\Delta \varepsilon_\text{min} = 0.125$. The dips indicate near-degeneracy of quasi-energies. b, In the seven-site case, operating points are identified at $T_1=6.7574$, $T_2=13.6647$ (used in sim/exp, $\Delta \varepsilon_\text{min} = 0.119$), and $T_3=25.3791$. c,d, CW modulation: simulated and experimental population dynamics over three driving periods ($t=3T$). Oscillatory exchanges are observed within each substep with an overall CW circulation. Violet boxes mark the multi-peak packet and cyan boxes the single-peak packet, with an overlaid arrow indicating the trajectory; these two contributions form a superposition and both advance in the CW sense. e,f, For CCW modulation, the circulation reverses: the same guides show the multi-peak and single-peak packets advancing CCW. g, First harmonic Fourier phase $\phi^{(1)}/2\pi$ from the measured and simulated dynamics. Solid (sim) and dashed (exp) are overlaid. Gray guide lines with slopes $\nu_1^{\text{CCW}}=0.276$ and $\nu_1^{\text{CW}}=-0.222$ indicate the extracted per-period winding numbers. Traces show opposite overall trends for CW (red) versus CCW (blue), consistent with broken time-reversal symmetry. Movies of evolutions are provided in the Supplementary Materials.