Abstract
The study of coupled networks with parametric amplification of vacuum fluctuations has garnered increasing interest due to its intricate physics and potential applications. In these systems, parametric interactions lead to beam-splitter coupling and two-mode squeezing, creating quadrature-dependent dynamics. These systems can be modelled as bosonic networks, arrays or lattices, exhibiting exotic effects such as unidirectional amplification and non-Hermitian chiral transport that influence multimode squeezing. However, exploring and controlling these network dynamics experimentally in all-optical systems remains challenging. Recent advances in integrated nonlinear microresonators, known as Kerr microcombs, have enabled the generation and control of broadband high-repetition pulses on microchips. Kerr microcombs exhibit intriguing nonlinear dynamics where coherent photons occupy discrete spectral lines, leading to multimode squeezed vacuum states. Here we explore the lattice dynamics of vacuum fluctuations driven by dissipative Kerr microcombs. We design a photonic chip on which a spontaneously emergent pair of pulses creates extended multimode states of parametrically amplified vacuum fluctuations. These states exhibit oscillatory dynamics, with implications for squeezing and secondary comb formation. By employing integrated micro-heaters, we tune the vacuum fluctuations to eliminate the oscillations, establishing a fundamental connection between non-Hermitian lattice symmetries and Kerr combs, and paving the way for exotic quadrature-dependent optical networks with broad implications for quantum and classical photonic technologies.
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Acknowledgements
We gratefully acknowledge discussions with E. Ng, R. Yanagimoto and K. Yang. This work is funded by the Defense Advanced Research Projects Agency under the QuICC programme and by the NSF QuSeC-TAQS Award ID 2326792, as well as the Vannevar Bush Faculty Fellowship from the US Department of Defense and AFOSR award number FA9550-23-1-0248. E.L. acknowledges the Yad Hanadiv Rothschild fellowship, and the Zuckerman institute Zuckerman fellowship. Part of this work was performed at the Stanford Nanofabrication Facility and the Stanford Nano Shared Facilities. S.F. acknowledges support from a MURI project for the US Air Force Office of Scientific Research (grant number FA9550-22-1-0339). We thank NGK Insulators, Ltd. for the 4H-SiCOI substrates used to fabricate devices in this work.
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E.L., M.A.G. and D.M.L. conceived the original idea, fabricated the device and performed the theoretical analysis. E.L. and M.A.G. performed the experiment with contributions from D.M.L. S.F. and J.V. supervised the project. All authors discussed the results and contributed to the final manuscript.
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Extended data
Extended Data Fig. 1 Micro-ring resonator: fabrication and dispersion.
a. Optical image of the Kerr ring resonator alongside the smaller on-chip filter ring and micro-heaters. b. Normalized transmission over wavelength detuning of a ultra high-Q resonance in our device with intrinsic quality factor of 9 million c. Integrated dispersion of the TE mode of the Kerr ring resonator (blue) and fit to D2 = 1.2MHz, D1 = 152.7 GHz (red). Dispersion defects are avoided mode crossings with other mode families. d. Cold cavity intrinsic quality factors (Qi - red) and coupling quality factors (Qc - blue) of the resonances in the relevant mode-family.
Extended Data Fig. 2 The experimental optical spectrum of the Kerr microcomb.
a. Transmission as a function of pump detuning. The different colored lowercase Roman numerals and color shades correspond to the different stages of the microcomb. b. The spectrum of the different combs specified by lowercase Roman numerals in (a). i and ii correspond to the primary 2-FSR comb which is the main focus of this work, while iii is a 1-FSR comb, and iv-v are chaotic states. Finally vi is a 2-FSR stable soliton crystal state. c Single-photon optical spectrum analyzer (SPOSA) measurement of photon populations in above threshold (red) and below threshold (green) modes of the 2-FSR state.
Extended Data Fig. 3 Intracavity field in real space for different regimes in the comb formation - LLE with perturbed dispersion - non oscillating threshold.
a. Transmission as a function of continuous pump detuning (toward longer wavelengths). This plot is the same as in fig 1(b) but the shaded areas mark different regimes plotted in (b). b. Intracavity field in real space for the transition between the (i): primary comb (2-FSR) and the secondary comb (1-FSR) corresponding to the shaded red region in (a). (ii): the transition from the secondary comb (1-FSR) to a second oscillating 1-FSR comb which oscillates (shaded green region). Experimental evidance of this transition can be found in section 3 of Supplementary Material.
Extended Data Fig. 4 Examination of the simulated threshold between a 2-FSR comb to a 1-FSR comb at 0.115 GHz detuning.
a. Spectrum of the 2-FSR primary Kerr comb (red) and light in odd-numbered modes (green). b. Intracavity field in real space (rolling Turing) c. g(2) correlations of the below threshold light. d. Eigen-spectrum of the comb showing that a non-oscillating supermode crosses threshold, and that the spectrum also has oscillating supermodes with less parametric gain. e. The squeezing RF spectrum, showing the single peak which correspond with the non-oscillating supermode that crosses threshold in d. f. The spectrum of the most squeezed state in the basis of cavity modes.
Extended Data Fig. 5 Principle components of the full experimental setup for quadrature variance measurements.
a. Schematic diagram of the experimental setup. The abbreviations are: EDFA- erbium doped amplifier, EOM- electro-optic modulator, WS- waveshaper, PD- photodiode, pre-amp- low-power EDFA.
Extended Data Fig. 6 2-band model of a quadrature lattice induced by a 2-FSR Kerr comb.
a. Illustration of the above threshold comb (red) and below threshold comb (green) in the model, and of the different terms - pair generation and Bragg scattering that couple the different modes. b. The 1D lattice unit cell, in the generalized quadrature basis c. Different band topologies (different number of exceptional points) of the Brilloin zone of the lattice in b, top plot corresponds to \({A}_{\pm 2}/{A}_{0}=0.25,\Delta \tilde{\omega }=2\), center plot corresponds to \({A}_{2}/{A}_{0}=0.25,{A}_{-2}/{A}_{0}=0.2,\Delta \tilde{\omega }=2\) and bottom plot to \({A}_{\pm 2}/{A}_{0}=0.25,\Delta \tilde{\omega }=0\). d. The number of exceptional points for different 2-FSR Kerr combs on different slices of the 3D parameter space (i-iii). The parameter space is spanned by the two ratios of the sidebands and the overall detuning (bottom right).
Extended Data Fig. 7 Dispersion symmetry and the properties of supermode squeezed states.
a.-c. The real (blue dots) and imaginary (red dots) parts of the eigenvalues of the amplified vacuum supermodes driven by a 2-FSR comb with different values of \({D}_{1}^{{\prime} }\): 0, 3 and 8 MHz respectively. For larger \({D}_{1}^{{\prime} }\) the purely real eigenvalues gradually become complex. Highlighted quadrature supermodes are those close to threshold. d.-g. The frequency(ω)-dependent squeezing spectrum of the supermodes close to threshold (highlighted). Each line represents the supermode spectrum for a linearly swept intra cavity power, and the supermodes can be seen to narrow as they approach threshold. The supermode in f. does not narrow asymptotically because e. reaches threshold first. h.,i. The supermode composition as a function of cavity modes of the multimode squeezed states, for the supermodes in e. and f. respectively (colours correspond to the frequencies).
Extended Data Fig. 8 Existence of non-detuned state in the presence of non-zero D2 and \({D}_{1}^{{\prime} }\).
a. Top: eigenvalue solution of the quadrature Hamiltonian \({\mathcal{M}}\) for a 2-FSR Kerr comb with side band drop ratio of \({\mathcal{R}}=2.25\), and D2/2π = 1.2MHz. Bottom: Zoom-in on the quadrature supermodes with higher-gain showing that all of them are undetuned (imaginary value is 0.) b. Same as a. only with \({D}_{1}^{{\prime} }/2\pi =2.8\,MHz\). c.,d. Numerically counting the un-detuned states as a function of underlying GVD (D2) and detuning of the comb for the symmetric case of \({D}_{1}^{{\prime} }=0\) and the non-symmetric case \({D}_{1}^{{\prime} }/2\pi =2.8\,MHz\).
Supplementary information
Supplementary Information
Supplementary Figs. 1–4 and Sections 1– 5.
Supplementary Data 1
Order of data points from left to right in Fig. 2f.
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Lustig, E., Guidry, M.A., Lukin, D.M. et al. Quadrature-dependent lattice dynamics of dissipative microcombs. Nat. Photon. 19, 1247–1254 (2025). https://doi.org/10.1038/s41566-025-01777-z
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DOI: https://doi.org/10.1038/s41566-025-01777-z
