Time reflection and refraction in synthetic frequency dimension

The duality of space and time in Maxwell's equations has prompted interest in time boundaries and the accompanying temporal analog of spatial reflection and refraction. However, achieving observable time-boundary effects at optical frequencies in real materials is challenging. In this Letter, we demonstrate that time reflection and refraction can be observed in a two-band model centered around a nonzero reference energy. Our model can be physically implemented in the synthetic frequency dimension as a system of two coupled dynamically modulated ring resonators. We find that modulation at microwave frequencies is sufficient to observe time-boundary effects for optical waves in synthetic frequency dimension. Our Letter shows that implementing multiband models in synthetic dimensions opens a new avenue for further exploration of time boundaries.

Figure 1

(a) Schematic of the two-leg ladder lattice model. Blue and red spheres represent $a$ and $b$ lattice sites, respectively. $C$ and $-C$ are the inter-rung coupling constants for the $a$ and $b$ sublattices, respectively. $\kappa$ is the interleg coupling constant. (b) Band structure of Hamiltonian given by Eq. (1) where blue indicates before the time-boundary ($t<0$) with coupling constant $C/\kappa =1$, whereas red indicates after the time-boundary ($t>0$) with $C/\kappa =-4$. Dots denote the states on the band structures with central wave-vector $k_0=\pi /3$. (c) Temporal evolution of the system initialized in the state described by Eq. (4). The system consists of $N=400$ total lattice sites, composed of $M=200$ unit cells of $a$ and $b$ sites. The even and odd lattice sites are $a$ and $b$ sites, respectively. The coupling constant $C$ changes instantaneously from $C/\kappa =1$ to $C/\kappa =-4$ at the time-boundary $t=0$. Simulation employs the matrix exponential method.

Figure 1

(a) Schematic of the two-leg ladder lattice model. Blue and red spheres represent $a$ and $b$ lattice sites, respectively. $C$ and $-C$ are the inter-rung coupling constants for the $a$ and $b$ sublattices, respectively. $\kappa$ is the interleg coupling constant. (b) Band structure of Hamiltonian given by Eq. (1) where blue indicates before the time-boundary ($t<0$) with coupling constant $C/\kappa =1$, whereas red indicates after the time-boundary ($t>0$) with $C/\kappa =-4$. Dots denote the states on the band structures with central wave-vector $k_0=\pi /3$. (c) Temporal evolution of the system initialized in the state described by Eq. (4). The system consists of $N=400$ total lattice sites, composed of $M=200$ unit cells of $a$ and $b$ sites. The even and odd lattice sites are $a$ and $b$ sites, respectively. The coupling constant $C$ changes instantaneously from $C/\kappa =1$ to $C/\kappa =-4$ at the time-boundary $t=0$. Simulation employs the matrix exponential method.

Figure 2

Numerical demonstration of complete time reflection. Shown here are the temporal evolutions of the state described by (a) Eq. (4) and (b) Eq. (8). In both panels, the Hamiltonian parameters are $C=5$ and $\kappa =2$ before the time boundary. At the boundary $t=0$, the Hamiltonian instantaneously changes according to Eq. (7).

Figure 3

Numerical demonstration of smoothly varying time boundary. (a)–(d) Temporal evolution of the wave function initialized in the state described by Eq. (4), using the same parameters as Fig. 1. In the Hamiltonian, $C$ is varied smoothly from $C=\kappa$ to $C=-4\kappa$, where $C\left(t\right)$ is given by Eq. (9). The band structures for $C=\kappa$ and $C=-4\kappa$ are shown in Fig. 1. Simulations were performed using the fourth-order Runge-Kutta method. (e)–(h) Change in coupling constant $C$ as a function of time, depicted below its corresponding dynamical simulation.

Figure 4

Synthetic frequency dimension implementation using a dynamically modulated coupled two-ring resonator system. $V_a\left(t\right)$ and $V_b\left(t\right)$ are the applied voltages, EOM is the electro-optical modulator, and $\kappa$ represents the directional coupler between the rings. Arrows indicate the counterpropagating modes needed to implement the two-band model.