## Abstract

The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light. While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions, these nonlinear effects typically result in long-range interactions along the frequency axis. Thus, it has been difficult to use the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions. Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically modulated ring resonator incorporating ${\chi}^{(3)}$ nonlinearity, provided that the group velocity dispersion of the waveguide forming the ring is specifically designed. As a demonstration we numerically implement a Bose–Hubbard model and explore photon blockade effect in the synthetic frequency space. Our work opens new possibilities for studying fundamental many-body physics in the synthetic space in photonics, with potential applications in optical quantum communication and quantum computation.

© 2020 Chinese Laser Press

## 1. INTRODUCTION

The concept of the synthetic dimension enables one to explore higher-dimensional physics in lower-dimensional systems [1–14]. In photonics [15,16], the synthetic dimension can be achieved by exploiting various degrees of freedom of light such as its frequency [17,18], spatial mode [19], or orbital angular momentum [20], and by specifically controlling the coupling between these degrees of freedom [21–23]. While initially explored mostly theoretically [24–30], very recently there have been a number of important experimental developments, including the first experimental demonstration of topological insulators with synthetic dimensions [19], the observation of the effective gauge potential in two independent synthetic dimensions [31], the measurement of band structure along the frequency axis of light [32], and the control of the light spectrum in synthetic space [33,34].

Most of the existing theoretical and experimental works on the synthetic dimension in photonics concern linear processes without photon-photon interactions. Certainly, it would be of interest to consider nonlinear systems where there is photon-photon interaction. Moreover, since one of the major objectives by going into synthetic dimension is to create a platform to study specific interacting Hamiltonians that are of physical significance, it would be important to develop a strategy to synthesize these interacting Hamiltonians. A very large class of interesting interacting Hamiltonians have local interactions [35–39]. For example, in the Bose–Hubbard Hamiltonian

In this paper, we consider a synthetic frequency dimension of light, formed in a ring resonator incorporating a modulator. To create photon-photon interaction we consider ${\chi}^{(3)}$ processes in the waveguide forming the ring. We show that a local photon-photon interaction in the frequency dimension can be achieved with a careful design of the group velocity dispersion of the waveguide [Fig. 1(a)]. As a demonstration we show that this system can be used to demonstrate a Bose–Hubbard model and achieve photon-blockade effects along the synthetic axis. This work here significantly broadens the range of physics phenomena that can be studied in photonic synthetic space and may lead to new opportunities in quantum simulations and in the manipulation of light.

## 2. MODEL

We start with a brief review of the ring resonator under the dynamic modulation as shown in Fig. 1(b) [24,44], which naturally leads to a synthetic dimension of light along the frequency axis. We consider a ring composed by a single-mode waveguide with an effective refractive index ${n}_{\mathrm{eff}}$ and a length $l$. For simplicity, assuming zero group velocity dispersion in the waveguide, the ring resonator supports resonant modes at the resonant frequencies ${\omega}_{m}={\omega}_{0}+m\mathrm{\Omega}$, where $\mathrm{\Omega}\equiv 2\pi c/{n}_{\mathrm{eff}}l\ll {\omega}_{0}$ is the free spectral range (FSR) for the ring. We place a phase modulator inside the ring and choose the modulation frequency to be equal to the FSR. The transmission of light passing through the modulator can be described by a time-dependent transmission coefficient [45]

where $\alpha $ is the modulation amplitude. We assume that the field experiences small changes for each round-trip. Then, after one round-trip, the change of the field amplitude ${E}_{m}$ for the $m$th resonant mode can be written as [24]Here $J=\alpha \mathrm{\Omega}/2\pi $ is the modulation strength and ${a}_{m}$ (${a}_{m}^{\u2020}$) is the annihilation (creation) operator for the $m$th resonant mode. The Hamiltonian in Eq. (4) describes a one-dimensional tight-binding model along the synthetic frequency axis [see Fig. 1(b)]. With this approach, a wide variety of noninteracting Hamiltonians with different connectivities and topological properties can be created [15]. However, there have been far less works on creating important interacting Hamiltonians.

As an illustration of the general difficulty as well as our approach of creating locally interacting Hamiltonians in the frequency synthetic dimension, we aim to synthesize the effective Bose–Hubbard Hamiltonian ${H}_{1}={H}_{\mathrm{BH}}$ in Eq. (1) in the synthetic frequency dimension. Although nonlinear optical phenomena in many photonic materials have been extensively studied, creating the Hamiltonian of Eq. (1) in the frequency synthetic space is in fact nontrivial [42]. The introduction of the nonlinearity typically leads to long-range interactions over all synthetic lattice sites. For example, the dynamically modulated ring as shown in Fig. 1(b), with a third-order nonlinear susceptibility ${\chi}^{(3)}$, is described by an interacting Hamiltonian [43]

In order to synthesize the Bose–Hubbard model where the interaction is local along the frequency dimension, we propose a ring consisting of two sections of waveguides. The two sections have the same ${\chi}^{(3)}$ nonlinear susceptibility. However, they have opposite GVD as illustrated in Fig. 1(a). In the linear regime of light propagation, for each complete round-trip as light goes around the ring, the dispersion effect in the two waveguide sections cancels. Hence, this ring also features equally spaced resonances along the frequency axis, same as a ring with zero GVD, and a one-dimensional synthetic frequency dimension can be created in this ring by applying the dynamic modulation. On the other hand, in each waveguide section, due to the GVD, the only phase-matched processes are the SPM and XPM processes. Moreover, as we will show below, in any Hilbert space with a fixed photon number, the Hamiltonian including both SPM and XPM processes in fact reduces to a Hamiltonian only including the SPM process. We therefore show that our design supports an effective Bose–Hubbard Hamiltonian [Eq. (1)] along the frequency axis.

In the following, we explain our design in detail: we start with reviewing the light propagating in a dispersive waveguide, followed by considering the FWM process in such a waveguide, and then go to the discussion of the ring composed of dispersive waveguides with nonlinearity. We first consider light propagating inside a single-mode waveguide. The evolution of the field amplitude $A(z,\omega )$ with the frequency $\omega $ at the position $z$ in the waveguide can be described as [47]

where $\beta (\omega )$ denotes the propagation wavevector. $\beta (\omega )$ can be expanded around the reference frequency ${\omega}_{0}$, i.e.,Next, we explore the light propagating through the waveguide section having a positive GVD (${\beta}_{2}>0$) and ${\chi}^{(3)}$ as shown in Fig. 1(a). For simplicity, we consider two incident waves with the frequencies ${\omega}_{1}$ and ${\omega}_{2}$ in the vicinity of the reference frequency ${\omega}_{0}$. We look for the solution for a general FWM process where output fields are generated at frequencies ${\omega}_{3}$ and ${\omega}_{4}$, satisfying the energy conservation relation ${\omega}_{3}+{\omega}_{4}={\omega}_{1}+{\omega}_{2}$ [see Fig. 3(a)]. We also assume the narrowband limit, i.e., $|{\omega}_{0}-{\omega}_{m}|\ll {\omega}_{0}$, $m=1,2,3,4$. The resulting coupled-mode equations under the slowly varying amplitude approximation are [42]

Having treated a light propagation effect in a single waveguide section, we now discuss the ring in Fig. 1(a). The waveguide A with the length $L=l/2$ has a positive GVD (${\beta}_{2,\mathrm{A}}>0$). The waveguide B with the length $L$ has a negative GVD (${\beta}_{2,\mathrm{B}}<0$), and we assume that ${\beta}_{2,\mathrm{B}}=-{\beta}_{2,\mathrm{A}}$. The quadratic term of the phase delay in Eq. (8) due to the dispersion effect is therefore compensated once the light propagates through the entire ring. Hence, this ring supports equally spaced resonant modes at the resonant frequencies ${\omega}_{m}$ with the FSR $\mathrm{\Omega}$. As light goes through each round-trip, its amplitude experiences a small change due to both the nonlinearity and the dynamic modulation. After a round-trip, the change of the field amplitude ${\tilde{A}}_{m}$ for the $m$th mode at frequency ${\omega}_{m}$ can be written from Eqs. (3), (9), and (10):

In a Hilbert space where the total number of photons $N$ is conserved in the system, Eq. (13) can be further simplified by noting that

## 3. RESULTS AND DISCUSSIONS

To demonstrate some of the physics effects along the synthetic frequency dimension in the Hamiltonian Eq. (13), we perform numerical simulations with photon number $N=2$. External waveguides are coupled with the ring, where the photons are input into the ring from the through-port and the detection can be made at the drop-port waveguide [see Fig. 1(c)]. Hence, our system becomes open, and the photon transport property is described by the input-output formalism in the Heisenberg picture [53,54]:

As the input state, we consider a strongly correlated photon pair [55]

To measure the quantum statistics of the photons inside the ring, we define the two-photon correlation function ${G}_{m,n}^{(2)}(t,{t}^{\prime})=\u27e8\varphi |{d}_{\mathrm{out},m}^{\u2020}(t){d}_{\mathrm{out},n}^{\u2020}({t}^{\prime}){d}_{\mathrm{out},n}({t}^{\prime}){d}_{\mathrm{out},m}(t)|\varphi \u27e9$ [56]. We then compute the two-photon correlation probability in which two output photons coincide in time

The simulation procedure follows the standard formalism [55] but in the synthetic frequency dimension. We consider the synthetic lattice in Fig. 1(b) involving 31 resonant modes ($m=-15,\dots ,15$) and set $\gamma =0.2J$, ${t}_{D}=20{J}^{-1}$, and $\mathrm{\Delta}t=6.4{J}^{-1}$. We also choose $\mathrm{\Delta}\omega =g$ to compensate for the overall frequency shift in the effective Bose–Hubbard model. A photon pair $|\varphi (-4,4)\u27e9$ is injected into the waveguide. We consider the cases with $g=0$, $g=2J$, and $g=10J$ in the simulations. Normalized distributions of the two-photon correlation probability ${P}_{m,n}$ for each choice of $g$ are plotted in Figs. 4(a)–4(c). There is no photon interaction in the $g=0$ case. Notice the existence of nonzero probability ${P}_{m,m}$, where the output photons have the same frequency. As the interaction increases, the output photon statistics changes. For a large $g=10J$, the correlation probability ${P}_{m,m}=0$ for all $m$. High nonlinearity thus introduces a large blockade effect at each resonant mode along the synthetic frequency dimension.

We next use the photon pair $|\varphi (-15,-15)\u27e9$ as the input and assume that there is a sharp boundary at ${\omega}_{-16}$ in the synthetic frequency dimension. Such an artificial sharp boundary can be created by adding another ring to strongly couple only to the $-16$th resonant mode but no other modes [30]. In this input state the two photons have the same carrier frequencies. Figures 4(d)–4(f) show the simulated normalized distributions of the two-photon correlation probability ${P}_{m,n}$ for different $g$. At $g=0$, there is a strong probability for the two photons to have the same frequencies. In contrast, with a large nonlinearity $g=10J$, away from the input frequency at ${\omega}_{-15}$, there is very little probability that the output photons exhibit the same frequencies. Again, we see a strong photon-blockade effect in the synthetic frequency dimension.

Simulations in Fig. 4 show the photon-photon interaction along the synthetic frequency dimension, which is an analog to the on-site nonlinear interaction in the spatial dimension [57–59]. Moreover, we note that simulation results in Fig. 4 are the same as results performed by simulating Eqs. (16)–(18) with ${H}_{3}$ replaced by ${H}_{1}$ in Eq. (1), indicating that the Hamiltonian ${H}_{3}$ in Eq. (13) indeed provides an implementation of the Bose–Hubbard model in the synthetic frequency dimension.

Our proposed system requires waveguides with strong nonlinearity and large GVD, which is experimentally challenging but may be achievable with the current state-of-the-art technology. For a ring with the length $\sim 1\text{\hspace{0.17em}}\mathrm{mm}$, $\mathrm{\Omega}/2\pi $ is $\sim 10\text{\hspace{0.17em}}\mathrm{GHz}$. The modulation strength $J/2\pi $ can be tuned at the order of 0.1–1 GHz to make the system operate in the weak modulation regime. As an example, photonic crystal fiber filled with a high-density atomic gas has the potential to create a nonlinearity with $g/2\pi $ up to $\sim 1\text{\hspace{0.17em}}\mathrm{GHz}$ in the few-photon regime [60]. Moreover, the strong nonlinearity in optical resonators is potentially feasible with other nanophotonic systems [36,61–66], for example, nanophotonic waveguides incorporated with rubidium vapors [67–69]. Strong GVD is desired to bring the phase-mismatching condition $|\mathrm{\Delta}k{L}_{A}|=n\pi $, which can be simplified to $|{\beta}_{2}{L}_{A}{\mathrm{\Omega}}^{2}|\sim 2\pi $. The necessary high dispersion ${\beta}_{2}\sim {10}^{-19}\text{\hspace{0.17em}}{\mathrm{s}}^{2}/\mathrm{m}$ is possible with dispersion engineering in the photonic crystal fiber [70] or by exploiting mode crossing in coupled waveguides [71]. The theoretical proposal here may also be implementable in the microwave frequency range using superconducting quantum circuits [72].

## 4. CONCLUSION

In summary, we show that a ring resonator undergoing dynamic modulation can be used to achieve an interacting Hamiltonian where the interaction is local along the synthetic frequency dimension, provided that the ring incorporates ${\chi}^{(3)}$ nonlinearity, and with a specific design of the GVD of the ring. Our work complements other emerging efforts in creating locally interacting Hamiltonians in synthetic space, by showing that such locally interacting Hamiltonians can be achieved in a system that has been widely used in photonics. The possibility of creating locally interacting Hamiltonians in the synthetic frequency dimension is fundamentally important for future studies of photon-photon interactions and many-body physics with synthetic dimensions in photonics, with promising potentials for a variety of quantum optical applications such as quantum information science and quantum communication technology [35,73].

We noticed Ref. [41] when we were in the final stage of preparing this paper.

## Funding

National Natural Science Foundation of China (11974245); National Key Research and Development Program of China (2017YFA0303701, 2018YFA0306301); Natural Science Foundation of Shanghai (19ZR1475700); Air Force Office of Scientific Research (FA9550-18-1-0379); Vannevar Bush Faculty Fellowship from the U. S. Department of Defense (N00014-17-1-3030); National Science Foundation (CBET-1641069).

## Disclosures

The authors declare no conflicts of interest.

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