All-optical dissipative discrete time crystals

The paper addresses whether periodically driven, dissipative photonic systems can realize discrete time crystals (DTCs) that spontaneously break discrete time-translation symmetry (TTS) and exhibit temporal long-range order suitable for coherent technologies. Building on the broader concept of spontaneous symmetry breaking (SSB) from condensed matter to high-energy physics, time crystals extend symmetry breaking to the temporal domain. While early theoretical efforts focused on continuous TTS in closed systems, research has shown that periodically driven (Floquet) systems can spontaneously develop responses with periods that are integer multiples of the drive. Understanding DTCs in dissipative open systems is particularly important for applications such as quantum memory and computation because environmental coupling can stabilize periodic steady states. The study reports an all-optical, room-temperature realization of DTCs using a Kerr-nonlinear microresonator driven by two continuous-wave lasers, demonstrating discrete-time SSB via robust subharmonic generation and Kerr soliton formation.

The work situates itself in the evolution from Wilczek’s proposal of time crystals and subsequent no-go theorems toward Floquet DTCs in driven systems, where interactions and disorder can enable long-lived time-crystalline order. It highlights ongoing efforts to understand ergodicity-breaking mechanisms and notes that dissipative DTCs in open quantum systems have only recently gained momentum. Prior photonic proposals include continuous-time SSB and boundary time crystals in coupled Kerr cavities, subharmonic entrainment phenomena distinct from DTCs, and extensive studies of Kerr microcombs and soliton crystals under monochromatic pumping. However, monochromatic pumping with higher-order dispersion or avoided mode crossings does not establish discrete TTS and thus cannot host DTCs. The present work fills a gap by demonstrating discrete-time SSB in a single Kerr resonator with dichromatic pumping and self-injection locking, enabling big DTCs and defect-carrying DTCs.

The platform is a high-Q Kerr-nonlinear optical microresonator driven by two independent CW lasers. Conceptually, two pumps at frequencies fp1 and fp2 separated by M free spectral ranges (FSRs) create a rotating, periodic modulated background (lattice) with period T = 1/(fp2 − fp1) = TR/M (TR is the cavity round-trip time). Dissipative Kerr solitons form and become trapped on the lattice; depending on soliton number and arrangement, the output pulse train period becomes mT (1 ≤ m ≤ M), realizing DTCs when m > 1. Numerical modeling: Dynamics are described by a two-pump variant of the Lugiato–Lefever equation (LLE), a damped, driven NLSE including detuning, damping, and dispersion. In a rotating frame, the equation includes a two-tone drive term with spatial modulation e−iMθ and temporal beating at (ω2 − ω1 + D2 M^2/2). The drive enforces discrete TTS via invariance under t → t + 1/(fp2 − fp1) or θ → θ + 2π/M. The model is non-dimensionalized (time normalized to twice the cavity photon lifetime; detunings and dispersion normalized to the HWHM linewidth; field and pump normalized to sideband threshold). Simulations used split-step Fourier propagation and adaptive-step Runge–Kutta on coupled-wave equations, with agreement between methods. Hard excitation with high-energy initial fields was employed; for soft excitation, vacuum noise terms were added. Parameters matched the experiment (e.g., D2 ≈ 2π × 6.8 kHz). Experimental setup: A prism-coupled MgF2 whispering-gallery-mode resonator (radius ≈ 1.06 mm, FSR ≈ 32.8 GHz) with loaded linewidth ~2κ = 2π × 200 kHz was pumped by two DFB lasers near 1545 nm. BK7 prism couplers provided >60% coupling efficiency. Up to ~3 mW coupled into the resonator; group velocity dispersion β2 ≈ 4.9 ps^2/km (D2 ≈ 2π × 6.8 kHz). The output was monitored with an optical spectrum analyzer for sech-shaped combs and with a fast photodiode and RF spectrum analyzer for microwave beat signals at the repetition rate. Self-injection locking: Each laser was locked to its cavity mode via resonant Rayleigh backscattering; phase delay tuning set optimal detunings. Operating point selection involved reducing pump currents from high power until stable subharmonic generation and soliton microcombs emerged. Typically, one pump power was set above the parametric threshold and the other slightly below; neither alone generated a soliton comb. Frequency-domain signatures (narrow RF beat at ~32.8 GHz, subharmonic generation between pumps) confirmed coherent pulse trains and DTC behavior.

• Demonstration of all-optical dissipative discrete time crystals (DTCs) in a Kerr microresonator driven by two CW lasers. Subharmonic generation between the pumps accompanies period multiplication of the soliton pulse train, evidencing discrete TTS breaking.
• Numerical phase diagrams (comb energy vs. detuning) for M = 5 and M = 8 show step-like multistability regions corresponding to 1…M solitons per round trip; DTC type depends on soliton arrangement relative to the lattice.
• Temporal long-range order: Simulated DTC states (e.g., for M = 5 and M = 8) remain stable over hundreds of cavity photon lifetimes (thousands of drive periods). Frequency spectra show multiple subharmonics between pumps with sech-shaped soliton envelopes.
• Experimental observations: Using a 32.8 GHz-FSR MgF2 resonator, narrow RF signals at 32.8 GHz confirm coherent pulse trains. When pumps lock to non-adjacent modes (e.g., M = 3), subharmonic generation and soliton trapping further narrow the RF peak due to frequency division and stronger coupling between harmonics.
• Spectral and temporal evidence of m-tupling DTCs: Period-quadrupling (M = 4) and period-tripling (M = 3) DTCs were observed; experimental spectra match numerical modeling, and reconstructed time-domain pulse trains per round trip show m-fold period multiplication.
• Big DTCs and defects: The platform supports big DTCs (m ≫ 1 in principle) and defective DTCs (vacancies, dislocations, interstitials) with distinct Fourier signatures, demonstrated numerically.
• Robustness: DTC formation occurs over ranges of pump powers and detunings without fine-tuning; states recur across experimental runs once parameters are appropriately chosen.

The system fulfills key DTC criteria for a periodically driven dissipative many-body system: (1) Discrete TTS is explicitly present in the equations via the two-tone drive; (2) Stable steady states emerge spontaneously with periods that are integer multiples of the drive period (m-tupling), persisting robustly over parameter ranges; (3) The photonic system operates in the thermodynamic limit (many photons) and is accurately captured by a mean-field LLE model. Subharmonic generation in frequency accompanies period multiplication in time, providing practical frequency-domain diagnostics of DTC order. Compared to monochromatically pumped microcombs, dichromatic pumping establishes a clear discrete TTS and traps solitons in a rotating lattice, enabling direct observation of discrete-time SSB. The platform’s versatility suggests extensions to other resonator geometries and mode-locked lasers, and provides a route to studying DTC phase transitions, interactions, and defect dynamics using accessible photonic techniques.

This work demonstrates a room-temperature, chip-compatible photonic platform in which two self-injection-locked lasers drive a Kerr cavity to realize dissipative DTCs via robust subharmonic generation and soliton trapping in a rotating lattice. The experiments and matching theory confirm m-tupling DTCs (including tripling and quadrupling), temporal long-range order, multistability, and the feasibility of big and defect-carrying DTCs. The approach bridges time-crystalline physics and Kerr microcomb technology, offering potential applications in frequency division, precision metrology, quantum information, and timekeeping. Future directions include deterministic state preparation via synchronous pumping and fine laser tuning, exploration of phase transitions and interactions between DTC phases, extension to integrated platforms with tailored dispersion, and comprehensive time-domain probes of rigidity and coherence.

• The experimental demonstrations focus on pump separations of a few FSRs; broader exploration of larger M and the detailed role of the pump beatnote is left for future work.
• Evidence of rigidity and long-range order is primarily inferred from frequency-domain measurements and stable operation over long times; single-shot temporal diagnostics are suggested for deeper verification.
• Theoretical modeling assumes negligible higher-order dispersion and avoided mode crossings near the pumps; while efforts were made to engineer the cavity accordingly, residual effects are not exhaustively characterized.
• Phase transitions and mutual interactions between distinct DTC phases are proposed as accessible in this platform but were not experimentally demonstrated here.