In this study we investigate the lattice dynamics of optical vacuum fluctuations driven by dissipative Kerr microcombs. We designed a microchip on which a pair of emerging pulses parametrically amplifies and induces transport of the vacuum fluctuations, which subsequently exhibit lattice dynamics. We observe that the vacuum fluctuations generally oscillate, and that these oscillations are associated with a mismatch between the microcomb repetition rate and the underlying dispersion. By using integrated thermo-optic tunability, we are able to completely eliminate the oscillations and reach a stable, non-oscillatory (or stationary) regime of the amplified vacuum fluctuations. To explain this phenomenon, we draw parallels with classical parity-time (PT) symmetric lattices 37,38,39,40,41. Furthermore, we demonstrate how the oscillatory nature of the amplified vacuum fluctuations affects threshold processes, leading to the onset (or lack) of radiofrequency (RF) beat notes and subsequent chaos in our microresonator. This advancement opens the door for modelling and using bosonic network dynamics in an all-optical integrated platform with implications for multimode squeezing and entanglement between many frequency modes.
To study the lattice dynamics of vacuum fluctuations driven by dissipative Kerr combs, we injected single-frequency (continuous wave, CW) light into a high-quality factor (Q) microresonator made of thin-film silicon carbide on an insulator (more information on the device is provided in 'High-Q 4H-SiC integrated microrings with integrated filters' in the Methods) and spontaneously generate dissipative Kerr microcombs (Fig. 1a). When the input CW laser beam is red-tuned towards a resonance, a sequence of nonlinear transitions occurs, leading to the formation of different microcombs. Figure 1b shows the Lugiato-Lefever equation (LLE) simulation of the power inside the resonator and corresponds to our device (see 'Additional data on the LLE simulation' in the Methods for more details of the simulation). The discontinuous steps in the transmission are nonlinear state transitions that correspond to different comb spectra (see 'Additional details of the 2-FSR microcombs' in the Methods for experimental comb spectra). For now, we focus on the first generated comb: the primary comb that appears after the first nonlinear threshold. Figure 1c illustrates that in our device this microcomb corresponds to the spontaneous formation of two pulses via the symmetry-breaking phenomenon of Turing rolls. Accordingly, the spectrum of our two pulses forms a frequency comb with discrete frequency lines separated by twice the free spectral range (FSR), which is also the pulse repetition rate (blue lines in Fig. 1c). The FSR in our system is 150 GHz.
To experimentally achieve the spontaneous formation of combs that include multiple frequency lines within a relatively narrow bandwidth (1 THz), we required strong anomalous dispersion. The dispersion of the material forming the waveguide is normal, and the strong anomalous dispersion required could not be achieved by geometric dispersion via tuning the cross-section of the waveguide. Instead, we chose a multimode cross-section and selected devices that feature an avoided mode crossing of two mode families: transverse electric and transverse magnetic. The mode crossing strongly detunes the resonance close to the pump frequency (see 'High-Q 4H-SiC integrated microrings with integrated filters' in the Methods), which enables an optical parametric oscillation (OPO) threshold between cavity modes that are separated in frequency by only 300 GHz, which is twice the FSR. This leads to the emergence of the 2-FSR primary Kerr comb, as illustrated in Fig. 1c.
The choice of a 2-FSR primary Kerr comb with a relatively narrow bandwidth confines the multimode dynamics and simplifies the geometry of the driven vacuum fluctuations. The driven vacuum fluctuations that are the focus of our study are in the modes that are not populated with coherent light. To identify these modes we assigned numbers μ to the resonances in our cavity. We define the pumped resonance as μ = 0 and thus the 2-FSR microcomb exists on all even-numbered modes (even modes hereafter), which are predominantly populated by coherent light. In contrast, light in the odd-numbered modes (odd modes hereafter) is dominated by vacuum fluctuations, which are driven (pair generated) by the microcomb (indicated by red areas in Fig. 1c). This occurs as long as the cavity loss in the odd modes is greater than the parametric gain generated by the microcomb. The light remains in a driven vacuum state until the parametric gain increases above the cavity loss, leading to a secondary threshold that generates more coherent comb lines. Next, we discuss the connectivity of the driven vacuum fluctuations and their modelling as a network of sites (cavity modes) and, more specifically, a lattice. Figure 1c,d presents the network model for our below-threshold light in the odd modes, which has a lattice geometry. Transport, which is the result of Bragg scattering, primarily connects odd cavity modes to their neighbouring odd cavity modes, rather than to the even modes. This aspect makes our network a one-dimensional lattice, rather than a more general network of cavity modes populated by amplified vacuum states of light.
The lattice geometry is also related to the pair generation in our system. In Fig. 1c (lower panel), we plot the frequency axis in a 'folded' form. In the folded form, modes -μ and μ are brought close together. The pair generation process is driven primarily by the pump mode μ = 0 together with the sidebands in cavity modes μ = ±2 (Supplementary Section 1). As a result of energy conservation, pair generation occurs primarily between modes that are in proximity to one another within the folded geometry. Thus, if a photon is generated (or annihilated) in mode μ, its pair will probably generate (or annihilate) at modes -μ, (-μ + 2) or (-μ - 2). From similar considerations, Bragg scattering will be governed by transport of photons from mode μ to a close mode at μ = (μ + 2) or (μ - 2). The simplified lattice model in the folded geometry is presented in Fig. 1d. Taking all interactions into account, we can define a periodic unit cell consisting of four bosonic field operators: , (creation operators) and â, â (annihilation operators). The unit cell is the defining feature of the lattice model.
In the following we solve the dynamics numerically, without relying on the simplified lattice model presented in Fig. 1c,d. The dynamics of the below-threshold light at the odd modes, centred around the pump at μ = 0, is governed by the following linearized Hamiltonian:
where A are above-threshold amplitudes (the Kerr comb) in the even cavity modes μ, are the creation (annihilation) operators of quantum light in the odd cavity mode μ, which are described in the rotating frame of reference of the Kerr comb where ΔΩ is the frequency spacing (repetition rate) of the Kerr microcomb. Δω is the frequency detuning of the cavity mode μ from the Kerr comb rotating frame of reference, g is the nonlinear coefficient, the Kronecker δ reflects photon azimuthal wave number conservation in the system. The indices μ, ν, j, k sum over the even cavity modes, the circumflex (hat) marks operators and h.c is the Hermitian conjugate.
The Hamiltonian in equation (1) can be numerically solved for a given microcomb amplitudes A by obtaining the covariance matrix associated with a discrete set of supermodes, which in our case extends across multiple odd resonances. Each supermode is characterized by a complex eigenvalue, where the real part is the parametric gain (that is, the squeezing and anti-squeezing of the light) and the imaginary part is the oscillations of the supermode. The complex eigenvalues come from the non-Hermitian dynamical matrix that describes the quadrature-dependent evolution in time. is calculated by writing the Hamiltonian in equation (1) in the quadrature basis , where , N is the number of modes, T is transpose, and defining as the matrix that satisfies the Heisenberg equation: , where t is time. The quadrature-dependent dynamics described by carry some analogies to classical non-Hermitian lattices, as we will discuss later in this work.
In principle, Bragg scattering can generate extended lattices and supermodes over arbitrarily large spectral distances. In our system the mechanism that limits the spectral distance, and hence the supermode spectral bandwidth, is the anomalous dispersion (D/2π = 1.2 MHz, where D is the second-order dispersion) of our cavity (see 'Finite model analysis' in the Methods for details of the finite model numerical analysis). In this context, it is worth noting that the cavity modes in our system function as a lattice in the synthetic frequency dimensions, where transport is induced by Bragg scattering. The supermodes are essentially Floquet lattice modes at twice the repetition rate frequency confined by D, which acts as an effective confining potential.
We began the experimental investigation of the below-threshold light by measuring the RF spectrum of a single extended supermode on the odd cavity modes. Our measurements were performed close to the OPO threshold, which means that at least one supermode has high gain that almost compensates the cavity loss. Close to the OPO threshold, typically one or two supermodes are dominant and amplified above all others. To enable the detection of vacuum fluctuations in the presence of the relatively intense Kerr microcomb in the cavity, we used both on-chip and off-chip filtering (see 'Additional data on the LLE simulation' in the Methods). Such filtering is made possible by to the large FSR (150 GHz) between resonances in our microresonator.
We used a custom-built single-photon optical spectrum analyser (SPOSA), which provides >100 dB of dynamic range with single-photon sensitivity, to simultaneously capture the photon populations above and below the threshold (see 'Additional details on the 2-FSR microcombs' in the Methods). Mapping out the two-photon correlation matrix of the below-threshold state can reveal the intermodal connectivity in the Hamiltonian and provide indications of the formation of the supermodes. We performed pairwise second-order photon correlation measurements, , where τ is the time delay, on each pair of below-threshold modes (Fig. 2a). The resulting matrix (both simulation and experiment) is shown in Fig. 2b,c, respectively. The most notable feature is the presence of a square near the centre of the correlation map. This square is related to the presence of a dominant single extended supermode, as will be discussed in more detail below. The square indicates all-to-all correlation in a finite set of coupled resonances. In our case, the group velocity dispersion limits the lattice dynamics to an area of 12 resonances (odd modes between -9 and 13), which is the size of the square.
The square shape observed in the cross-correlation data in Fig. 2c suggests the presence of extended all-to-all correlations across 12 cavity modes, resulting from photon transport and pair generation (Supplementary Section 1). As mentioned, close to the OPO threshold there should be a single dominant supermode. This allows us to identify the RF spectrum of the single extended supermode that reached threshold (which corresponds to the eigenvalue of that supermode). To resolve the RF spectrum, we exploited a notable feature of the multimode vacuum fluctuations generated by microcombs: they oscillate close to the threshold. This can be seen by comparing the time-dependent correlations of squeezed vacuum produced by a CW laser near threshold (Fig. 2d) with vacuum fluctuations amplified by multi-frequency light (Fig. 2e). The key distinction between Fig. 2d and Fig. 2e is the presence of an underlying RF beat note in the multimode scenario. Mathematically, these oscillations represent the imaginary part of the eigenvalue of the dominant supermode. The appearance of the oscillations in the time-dependent correlation of Kerr combs close to threshold was theoretically predicted in ref. , and we confirm them experimentally in this work.
Before examining the origin of the oscillations of the supermode near threshold, we demonstrate that the oscillations are shared by all autocorrelations and cross-correlations among the central 12 resonances, consistent with a single dominant supermode with a complex eigenvalue. Figure 2f presents the oscillations Δ (red data points) and the bandwidth (range in blue) of the autocorrelations and cross-correlations in the odd cavity modes from cavity modes -9 to 13. All of the autocorrelations and cross-correlations exhibit frequency beating in the range of Δ = 140-160 MHz with a bandwidth of approximately 10 MHz. This bandwidth is an order of magnitude narrower than the resonance linewidth and more than an order of magnitude broader than the laser linewidth, which shows that all modes are below the OPO threshold and are consequently squeezed by the optical drive.
By ordering the measured frequencies in Fig. 2f according to acquisition time (Supplementary Data 1), spanning several hours, it is evident that the majority of the non-uniformity between the different pairs of modes comes from gradual shifts in time in the hot-cavity dispersion. As mentioned, each supermode's complex eigenvalue consists of a real part, which represents the parametric gain (and correspondingly the bandwidth, as indicated by the blue bars in Fig. 2f) and an imaginary part, which represents the oscillations (depicted by the red dots in Fig. 2f). Therefore, the consistency of the oscillations and the bandwidth across various correlations demonstrates that light in these cavity modes corresponds to the same eigenvalue. If there were two supermodes, their distinct frequencies would create a fast beat note close to threshold. This effectively resolves the RF spectrum of a single extended supermode of an emerging dissipative microcomb near threshold.
Next we studied the evolution of the supermode as a function of proximity to threshold. By tuning closer to resonance, we amplified the oscillating supermode described in Fig. 2 such that the parametric gain of one supermode exceeded the loss, resulting in a secondary threshold. By performing multi-frequency homodyne measurements as illustrated in Fig. 3a (see also 'Additional details of the experimental set-up' in the Methods for the experimental configuration and Supplementary Section 4), we were able to measure multimode anti-squeezing of the vacuum fluctuations associated with the transition from Turing rolls to multiple RF beat notes (Fig. 3b). The secondary threshold leads to the appearance of multiple classical RF beat notes (in addition to the two peaks shown in the left upper panel Fig. 3b). While the onset of beat notes was studied in ref. , observing the vacuum fluctuations before their appearance sheds additional light on this physical phenomenon. The onset of multiple classical RF beat notes after a secondary threshold may be attributed to multiple incommensurate combs broadening and merging together after threshold. By tracking this phenomenon in the below-threshold regime, we observe that, in our case, the onset of oscillations arises from a single oscillating supermode, which results from the incommensurate nature of vacuum fluctuations below threshold.
The theoretical plot in Fig. 3b illustrates that the oscillating supermode becomes a coherent RF beat note after crossing threshold, which then generates additional beat notes, eventually leading to lines filling the entire spectrum. Measurements in Fig. 3c confirm this by tracking the noise variance of each resonance through the process of threshold crossing. By adjusting the local oscillator to correspond with different cavity modes, we observed a uniform RF beat note identical to that of the below-threshold light in Fig. 2. At the threshold, the entire supermode increases in intensity and narrows in bandwidth to generate a uniform coherent beat note. In the same manner, the g(τ) correlations broaden (Supplementary Section 5). Even before reaching the OPO threshold, the explanation given here is simplified, as the supermodes oscillate at frequencies that can vary as the microcomb evolves with the change in the proximity to threshold. While these variations can in principle be large, in our case, they are observed to be small. At the point of the secondary OPO threshold, the experimental results (upper plots in Fig. 3c) show that the below-threshold oscillations turn into a classical beat note signifying the emergence of additional combs that beat with the original 2-FSR comb. These additional combs, in turn, generate even more beat notes together with the original 2-FSR comb. The process of generating new beat notes is the onset of chaos, and we will show how to prevent it by thermally tuning the below-threshold light.
Before moving to our control of the oscillations, we wanted to observe the quadrature variance for different quadratures in the double-peak structure that we measured with our noise spectroscopy detection in Fig. 3c. Given that the supermodes are below threshold and other noise sources are relatively small, tuning the measured phase allowed us to measure the variance of one of the quadratures at the noise level of the coherent vacuum state. To show this, we performed homodyne measurements using a local oscillator composed of two different frequencies corresponding to resonances: μ = -1, 1. This configuration more accurately measures a single supermode's quadrature variances than a single-frequency local oscillator (while not requiring active phase locking between the local oscillator and the microcomb). Figure 3d presents the quadrature-dependent variance of the oscillating supermode as we temporally swept the phase of the local oscillator (LO). Modes -1 and 1 each contain four peaks, which are part of the approximately 24 peaks in the entire supermode. To observe the below-threshold quadrature-dependent variance, we adjusted the power ratio of our two-tone LO for modes -1 and 1 to P = βP, where β is a ratio that maximizes the overlap with a projection on the amplified quadrature supermode, and P is a local oscillator amplitude corresponding to mode n. Without using at least two tones in our homodyne detection, and adjusting their ratio, the oscillation variance would not dip to shot noise levels for any phase. The shot noise level was measured both by blocking the light emitted from the resonator and by measuring the noise level outside the resonance bandwidth. Figure 3e shows the reconstructed Wigner quasi-probability distribution associated with the pairs of peaks marked by red and blue arrows in Fig. 3c. Each pair requires its own optimized LO: LO1 and LO2 (β = 0.1, β = 0.16); q and p in Fig. 3e are the quadratures that correspond to the optimized LO1 and LO2 for each subplot. These pairs are not separated by the native spacing of the microcomb in the even modes. In performing these measurements, we showed how a below-threshold supermode transitions to classical light and impacts secondary comb formation.
After showing that the oscillations of the supermode affect the below-threshold quadrature variance, and comb evolution, it is important to understand the origins of the oscillations and control them. In our system, Bragg scattering primarily transports photons between resonances separated by a frequency spacing of 2-FSR, generating the depicted lattice dynamics in our below-threshold light (Fig. 1c,d). The below-threshold oscillations that we observe can be attributed to the absence of anti-parity-time (APT) symmetry in the lattice model (see 'Band model for the lattice dynamics induced by a Kerr comb' in the Methods). In scenarios involving squeezing originating from a single pump, or in single-mode squeezing, APT symmetry is inherent. For this reason oscillations are not commonly observed in squeezing experiments near threshold (Fig. 2d). However, in the multi-pump, multimode scenario that we explore here, the presence of two different frequency spacings (the repetition rate and the resonance spacing (dispersion)) generically are not equal to one another and therefore violates this symmetry, leading to the observed oscillations.
To understand the impact of APT symmetry on the lattice dynamics, we simplified the number of parameters by expressing the Hamiltonian for the odd cavity modes in a dimensionless form. The minimal model requires the consideration of only two parameters: the first parameter, α, which defines how the simplified comb amplitudes decrease at a constant rate according to and reaches α = 0.3548 in our experiment. The second parameter, , represents the dimensionless frequency detuning (obtained by dividing by , where g = 6.2 rad sec in our experiment) of the cavity modes from a rotating frame of reference aligned with the Kerr comb. To the first order of α, we obtained the following form for the lattice Hamiltonian:
where c ≡ a, d ≡ a for μ > 0, and d ≡ c, c ≡ d. We note that the higher-order terms introduce Bragg coupling and pair generation terms to the n nearest neighbours, thereby preserving the lattice structure. In this folded representation, if the detuning is symmetric around μ = 0, the Hamiltonian in equation (2) forms a degenerate pseudo-APT-symmetric lattice. This results in an eigenspectrum with eigenvalues that are either purely real or purely imaginary (see 'Band model for the lattice dynamics induced by a Kerr comb' in the Methods for the symmetry analysis). As discussed earlier, the real part of the eigenvalues corresponds to parametric amplification, while the imaginary part corresponds to oscillations. Consequently, when APT symmetry exists, extended supermodes that have a gain close to the threshold may not oscillate at all (purely real eigenvalues). However, in our microcombs, APT symmetry does not exist in general, mainly due to a mismatch between the repetition rate of the comb and the group velocity . Building on this realization, by adjusting the repetition rate of our microcomb, we can tune the oscillations of the supermode to absolute zero. We confirm the existence of extended non-oscillating supermodes generated by Kerr combs numerically (see 'Finite model analysis' in the Methods) and experimentally, as we will now show.
The repetition rate can be tuned by adjusting the micro-heater voltage V. This adjustment detunes the transverse electric and transverse magnetic modes differently, thereby altering the location of the mode crossing. Since the phase matching that initiates the 2-FSR microcomb depends on the mode-crossing location, this directly modifies the conditions necessary for the first OPO threshold, subsequently affecting the microcomb's repetition rate continuously.
To illustrate the transition between oscillating and non-oscillating regimes of the below-threshold light, we continuously tuned the resonator. By varying the voltage applied to the micro-heaters, we observed that repetition rate and the supermode oscillations changed, reaching a non-oscillatory regime as illustrated in Fig. 4a,b. Figure 4c presents the RF spectrum of the oscillating supermode at threshold exhibiting a reduction in the oscillation frequency to below 20 MHz, in comparison with the previous range of 100-150 MHz shown in Fig. 3c. Further increasing the voltage led to the complete elimination of the oscillation rate to zero within a finite temperature range, all without the need for fine-tuning (Fig. 4d). Increasing the voltage beyond this point caused oscillations to resume at approximately 20 MHz, as depicted in Fig. 4e, showcasing the different oscillation regimes achieved through tuning.
The three oscillation regimes -- fast, slow and non-oscillating -- were predicted through numerical investigations of the lattice dynamics spectrum of supermodes (see 'Finite model analysis' in the Methods). The cessation of oscillations over a finite temperature range is unexpected, suggesting that when the APT symmetry is not fully satisfied, but is close to being satisfied, supermodes with real eigenvalues (non-oscillating) are still present. This behaviour agrees with classical PT-symmetric systems, which exhibit completely real eigenvalues even in non-ideal experimental conditions. Our numerical investigations, detailed in 'Finite model analysis' in the Methods indicate that this robustness is supported by our theoretical model and reveal that non-oscillatory states only begin to oscillate after a substantial deviation from APT-symmetric conditions. This finding demonstrates that non-oscillatory multimode squeezed states produced by dissipative Kerr combs can be reliably maintained without stringent control over system parameters.
The behaviour of non-oscillatory supermodes can be further understood experimentally by observing the transition from a dominant oscillating supermode to a non-oscillating supermode. The transition is not a continuous change in the frequency of the supermode, instead, it occurs when the identity of the supermode with the highest gain changes, therefore, the transition point occurs when the two supermodes to have similar parametric gains. Figure 4f shows exactly that: the below-threshold anti-squeezing of two co-existing supermodes at the transition point. At this transition point, small changes in the temperature or input power result in either the oscillating supermode going above threshold or the non-oscillating going above threshold. If the non-oscillating supermode reaches threshold, additional beat notes would not form (as in Fig. 3b), instead a single Kerr comb with a repetition rate of 1-FSR would emerge (see Supplementary Section 3 for additional experimental measurements of this phenomenon).