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High-dimensional frequency crystals and quantum walks in electro-optic microcombs

Synthetic optical structures have enabled studies of various physical processes and discoveries of new phenomena in a controllable manner. Past realizations, using for example coupled optical waveguides, were however limited in size and dimensionality. Here we theoretically propose and experimentally demonstrate optical frequency crystals of arbitrary dimensions, formed by hundreds of coupled spectral modes within an on-chip electro-optic frequency comb. We show a direct link between the measured optical transmission spectrum and the density-of-state of frequency crystals, validated by measurements of one-, two-, three- and four-dimensional crystals with no restrictions to further expand the dimensionality. We measure signatures of quantum random walks and Bloch oscillations in the frequency domain of light in one- and two-dimensional synthetic lattices. Our work employs a quantum description of phase modulation11 to describe resonant electro-optic frequency combs, and shows that they enable the study of exotic crystal properties, topologies and quantum dynamics, while providing a versatile platform to manipulate quantum information in the optical frequency domain.

Here we investigate the quantum properties of electro-optic frequency combs 10 and show that they can be modeled by means of a generalized tight-binding model. In agreement with the theoretical predictions, we measure the generation of synthetic optical frequency crystals in high-dimensional space. In our approach, discrete lattice points are formed by spectral modes of an optical microring resonator realized in thin-film lithium niobite (LN) integrated photonic platform 18 , while the coupling between lattice points is controlled by electro-optic phase modulation 11 , enabled by the second-order nonlinearity of LN. Light coupled into the frequency crystals experiences quantum coherent scattering and interference at different lattice points, in direct analogy to electron behaviour in solid-state crystals. We show that it is possible to directly measure the density-ofstates (DOS) of the frequency crystals and furthermore measure signatures of quantum coherent scattering processes such as Bloch oscillations and multi-dimensional quantum random walks. Our results show that electro-optic frequency combs can be used for frequency-domain quantum information processing [19][20][21][22][23] .
The tight-binding model is one of the most fundamental models in solid state physics 24 , which assumes that particles (such as electrons or here photons) are localized at specific positions of the crystal lattice, and that they can hop between neighboring lattice points while preserving phase coherence. Optical tight-binding systems have in the past been realized using spatial modes in coupled optical waveguides [2][3][4]25 , or temporal modes in coupled resonators with different roundtrip times 8,26 . In contrast, here we experimentally realize synthetic crystal 15,[27][28][29] utilizing the discrete frequency modes of a LN microring resonator 10 , see Fig. 1 and Methods. By applying an electronic radio-frequency (RF) signal with a frequency equal to the separation between adjacent frequency modes (known as free spectral range, FSR), optical coupling can be initiated, where the coupling strength can be adjusted by the strength of the electric driving signal. Such an electro-optic resonator driven by a single-tone RF signal, commonly referred to as an electro-optical frequency comb 10,30 , can also be described in a quantum model as an one-dimensional tight-binding lattice 28 with a hopping rate related to the applied RF power. We here show that such a tight-binding frequency crystal representation is not limited to one-dimensional realizations. For example, using two RF signals (both only very slightly detuned from the resonator FSR), a photon placed in one optical resonance (Fig. 1b) can be frequency-shifted to neighboring resonance by the driving RF signal. Such a system can be described by a two-dimensional tight-binding model, see Methods.
Remarkably, we demonstrate that the same principle can be extended to three, four and many more dimensions using different RF driving signal. Here, each additional RF frequency-tone can span an additional spectral dimension, see Fig. 1b and Methods. Importantly, individual frequency modes within the microring resonators can be unambiguously mapped to individual lattice points within the crystal's synthetic frequency space, see Fig. 1. This high level of control and one-toone mapping of spectral modes in real frequency space to lattice points in synthetic frequency space enables the experimental investigation of crystal structures in high-dimensions.
Using the quantum mechanical tight-binding model to describe the electro-optic frequency comb, we show that the frequency crystal's density-of-states (DOS) can be directly measured by probing the optical transmission spectrum of the resonator, see Methods. In particular, using input-output theory of optical resonators, we established a direct relationship between the optical transmission ( ) and density of states ( ) of the frequency crystals, see Methods, which is given by: where Δ is wavelength detuning from the resonance center, κ e is the external coupling rate, d is the number of RF tone, and is the total number of cavity resonance mode coupled by one RF tone, see Methods. The expression can be understood by considering that a larger DOS leads to a the larger the number of optical modes that are coupled through electro-optic modulations, and due to the interference condition in the resonator, this leads to a larger the effective optical "absorption" within the resonator, and thus smaller transmission. These results show that measuring the transmission spectrum of cavity resonance, using a tunable continuous waver laser at telecom wavelengths, represents a direct measurement of the DOS of the frequency crystal.
Leveraging this analogy, we experimentally probed the DOS of one-, two-, three-, and fourdimensional frequency crystals, using up to four RF drives to drive an electro-optic ring resonator with 10 GHz free spectral range, where the RF tones were detuned by 1 MHz with respect to each other. We found excellent agreement with the analytical solution for the DOS of the crystals, see A particularly important feature described by a tight-binding model is the occurrence of quantum random walks, which arise from the phase-coherent step-wise propagation of particles in a lattice.
Therefore, exciting our frequency crystal with a photon that has a narrow spectral linewidth (i.e. driving a single lattice point of the crystal in synthetic space) is expected to give rise to the quantum random walk dynamics in frequency domain, resulting in spectral spreading for each roundtrip in the resonator (Fig. 3a). However, a photon that is spectrally narrow enough to only excite a single resonance intrinsically has to have a temporal duration much longer than the round-trip time of the resonator. For this reason, a description using individual round-trips cannot adequately describe the dynamics. We find that instead of experiencing discrete steps, multiple steps of the quantum walk coherently interfere over the coherence time of the photon, forming a steady-state output, see Methods and Fig. 3c. This result of our tight-binding model, predicting an exponential envelope in the output spectrum, agrees with classical model of electro-optic frequency combs 10 , and therefore provides a means to model electro-optical combs at the single photon level. Considering two-and higher-dimensional frequency crystals, quantum random walks are expected to appear in the synthetic frequency space (schematic in Fig. 1b), which maps into measurable signatures in real frequency space (schematic in Fig. 1c). In a two-dimensional frequency crystal (formed by two RF modulation frequencies), a photon positioned in a single lattice point can "hop" to one of the four nearest-neighbour lattice points per roundtrip. Fig. 4a shows examples of possible paths that a photon can take if placed at a single lattice point (e.g. at the center of Fig. 4a) within a 2D synthetic lattice. In classical random walks, the different possible paths are independent, leading to a Gaussian probability distribution. In contrast, in quantum random walks, all possible paths to individual lattice points are phase coherent and interfere, leading to a non-Gaussian probability distribution. Making use of the tight-binding model, we numerically simulated the probability distribution of a single photon propagating in a frequency crystal, finding clear signatures of quantum random walks, see Fig. 4b. Furthermore, as discussed in Fig. 3, when frequency crystal is excited with spectrally narrow photons, multiple random-walk steps coherently interfere, leading to a steady-state photon distribution in synthetic frequency space, see Device fabrication was performed at the Harvard University Center for Nanoscale Systems, a member of the National Nanotechnology Coordinated Infrastructure Network, which is supported by the NSF under ECCS award no. 1541959. Fig. 1: Optical frequency crystals generation in electro-optic frequency combs. a) Schematic of the electro-optically modulated resonator used to generate the frequency crystals. The device consists of a waveguide coupled race-track resonator with electrodes placed around it. The highdimensional frequency crystals are formed by modulating the device with multiple, slightly detuned, RF signals (here two RF signals are shown for illustration). This gives rise to multiple excitations of each race-track resonance, each representing a crystal lattice point. b) The tightbinding crystals can be represented in synthetic frequency space as fixed lattice points (yellow circle), where coupling between neighboring lattice points is mediated by electro-optic modulation due to applied RF field. By modulating simultaneously with different frequencies, highdimensional lattices can be generated in synthetic frequency space. c) As a result, each optical resonance (mode of a resonator) represents one lattice point, one crystallographic direction, or one crystal plane of the synthetic frequency crystal when one, two or three RF tones are applied, respectively. Within each resonance, the spacing between optical excitations is determined by the frequency difference between the RF driving signals.

Fig. 2: Density-of-states of high-dimensional frequency crystals.
By scanning an excitation laser through the optical resonances, it becomes possible to directly measure the density-of-states of the frequency crystals. Shown is the measured normalized optical transmission (blue trace) for one-to four-dimensional crystals, superimposed with the analytical model (orange trace) based on the density-of-states of the frequency crystal. Very good agreement is observed, thus confirming that the density-of-states can be measured directly.

Fig. 3: Quantum random walks and Bloch oscillations in one-dimensional frequency crystals. a)
In the absence of RF fields, CW laser excitation of the ring resonator leads to the excitation of one spectral mode (mode number 0), which is equivalent to the excitation of a single lattice point of a synthetic crystal. With RF fields applied, photon can hop to neighboring optical modes, giving rise to a quantum random walk and spectral broadening. Numerical simulations show that spectral modes with increasing mode number can be excited as light completes more roundtrips (RT) inside the resonator. Here, the frequency of the RF drive was perfectly matched to the free spectral range (FSR) of the resonator. b) If the RF driving signal is detuned form the resonator FSR, an effective linear force is imposed, which leads to Bloch oscillations in the frequency domain. c) If the frequency crystal is excited with photons that are spectrally narrow enough to excite only a single lattice point, their temporally duration has to be larger than multiple round trips. In such narrowband excitation, all roundtrips of the random walks coherently interfere over the coherence time of the photon, forming a steady state output with characteristic exponentially decaying spectrum. d) In the presence of Bloch oscillations, a sharp cut-off in the optical output spectrum is measured, which arises from the oscillations in the random walks. The insets in c) and d) show numerical simulations for different RTs to illustrate the effect arising from the coherent addition of multiple quantum random walk roundtrips. The distribution in synthetic space can be mapped to measurable real frequency space, see Fig. 1. The emission spectrum in real frequency space was measured with an optical spectrum analyzer d), while the spectral content within individual resonances was measured using homodyne detection e-h). As the spectral content within individual resonances are traces through the steadystate in synthetic space, broad spectra were expected and measured close to the excitation e-f), while narrow spectra were measured further away from the excitation, g-h), showing excellent agreement with simulations.

Methods:
Experimental setup. The electro-optic resonators were fabricated on x-cut single crystalline 600 nm thick lithium niobate (LN) layer on top of 2 µm thermally gown SiO2 layer, on top of silicon substrate (wafers by NANOLN). The patterns were defined using standard electro-beam lithography, and were transferred into the LN layer using argon plasma etching. Gold electrodes were placed next to the optical waveguides to enable electro-optic modulation. The chips were diced and the facet polished to enable end-fire optical coupling implemented with two lensed optical fibers, and the electrical signal was fed to the microelectrodes using a contact probe. The microwave driving signals were generated by four radio-frequency (RF) synthesizers (two HP 83711B and two Hittite HMC-T2240), which were phase-locked via a common 10 MHz clock and can operate independently at different frequencies. The different RF signals were combined using RF power splitters and were amplifier by an electrical power amplifier (Pasternack) and passed through a circulator with a 50 Ohm terminated third port, before being connected with the contact probe. In all presented measurements, we used a tunable laser (Santec TSL-510) to pump the resonator. For the measurements shown in Fig, 2 the laser was scanned through the resonances and detected the output with a 125 MHz photodiode (New Focus 1811). For the measurements shown in Fig. 3, the excitation laser was into the center of the resonance and the output spectrum was measured with a standard grating based optical spectrum analyzer with 20 pm resolution.
For the heterodyne detection measurements shown in Fig. 4, the excitation laser was tuned to the center of the resonance as for the measurements in Fig. 3, and the light was sent to a 3dB fiber direction coupler and overlapped with light emitted from a second tunable laser (Santec TSL-510), used as a local oscillator, before being detected with a 12 GHz photodiode (New Focus 1544A). The photodiode signal was analyzed with a radio frequency spectrum analyzer to record the beat-note of the local oscillator with the individual lines emitted from the electro-optic resonator.
Generalized tight-binding model of electro-optic frequency combs driven by one and multiple radio-frequency signals.
For one-dimensional frequency crystal, one electro-optic modulation Ω cos is applied to the ring resonator. This modulation can induce the hopping between different resonant modes of the cavity, which can be described by the following Hamiltonian: Here ω j is the frequency of the j-th resonant modes and for now we assume that there is no group velocity dispersion so that all resonant modes have equal spacing with free spectral range (FSR) ω F = +1 − . We choose ω 0 (i.e. j=0) as the mode that is optically excited with a pump laser, and we consider N t = 2 + 1 resonant modes. In two-dimensional case, there is more than one mode within a resonant peak of the resonator because light at any given frequency can be modulated to the neighbor resonance with two different choices, related to the different RF frequency. In order to make the mode that represents individual lattice points more clearly, we define the frequency crystal modes that corresponding to each lattice point in a two-dimensional crystal as c 1 , 2 = 1 √ 1 + 2 ( 1 1 + 2 2 ) Which can also be written as In real frequency spectrum if two modes (e.g. c 1,1 and c 2,0 ) are in the same resonant peak, then the summation of their own subscript are same (that means 1 + 1 = 2 + 0) because ∝ ∑ .
It can be seen that c satisfies c 1 , 2