Long-range entanglement and quantum correlations in a multi-frequency comb system

TL;DR

This work addresses the challenge of generating and controlling quantum correlations across multiple, spectrally diverse frequency combs by leveraging cascaded three-wave mixing mediated by a common idler comb in a multimode cavity. It develops a theoretical framework showing long-range multipartite entanglement and strong two-mode squeezing that spans from UV to mid-IR, and demonstrates how the covariance matrix of the multimode state can be engineered through dissipation, pump, and dispersion design. The study combines steady-state linearized quantum noise analysis with ultrafast pulse simulations (GNLSEs and QSA) to show robust time-frequency squeezing and entanglement buildup, including across modes not directly coupled by the nonlinear interactions. The findings suggest practical routes for broadband quantum resources, enabling applications such as ghost spectroscopy and spectral multiplexing in continuous-variable quantum information, with potential path to chip-scale implementations and integration with topological photonics concepts. All mathematical notation in the paper is used with proper delimiters, reflecting rigorous treatment of multimode quantum correlations across cascading nonlinear processes.

Abstract

Frequency combs are multimode photonic systems that underlie countless precision sensing and metrology applications. Since their invention over two decades ago, numerous efforts have pushed frequency combs to broader bandwidths and more stable operation. More recently, quantum squeezing and entanglement have been explored in single frequency comb systems for quantum advantages in sensing and signal multiplexing. However, the production of quantum light across multiple frequency combs remains unexplored. In this work, we theoretically explore a mechanism that generates a series of nonlinearly coupled frequency combs through cascaded three-wave upconversion and downconversion processes mediated by a single idler comb. We show how this system generates inter- and intracomb two-mode squeezing and entanglement spanning a very large spectral range, from ultraviolet to mid-IR frequencies. Finally, we show how this system can be engineered to produce on-demand multimode quantum light through covariance matrix optimization. Our findings could enable tunable broadband ghost spectroscopy protocols, squeezing-enhanced pump-probe measurements, and broadband entanglement between spectrally-multiplexed quanta of information.

Figures (5)

• Figure 1: Cascaded comb generation in a multimode cavity. (a) A multimode system is pumped with a comb centered at frequency $\omega_0$ and seeded with either a single frequency or comb centered at frequency $\omega_1$. DFG between the pump and seed generates an idler comb at frequency $\omega_T$, which is then recycled to generated "subcombs" through cascaded sum and difference frequency generation (SFG/DFG) processes. Taken together, the subcombs comprise a "primary comb" with frequency spacing $\omega_T$. This system can be realized in a multimode cavity that is synchronously pumped/seeded by a high repetition rate laser, which gives rise to the subcomb frequency spacing $\omega_m\ll\omega_T$. Cascaded nonlinear processes produce the primary comb with spacing $\omega_T$ centered around the pump subcomb at frequency $\omega_0$. (b) Cascaded comb generation in a multimode cavity with $I=7$ subcombs, each with $J=11$ modes. (c) Heatmap of dispersion-induced phase matching factor between individual modes $\eta_{iji'j'}=|\mathrm{sinc}(\Delta k_{iji'j'}L)|$ ($\Delta k_{iji'j'}=k_{ij}-k_{i'j'}-(\omega_{ij}-\omega_{i'j'})/c$ is the wavevector mismatch, $L$ is the crystal length, and lithium niobate dispersion is assumed zelmon1997infrared). (d) Depletion of the pump pulse due to the cascaded nonlinear interactions. All comb modes have quality factor $Q_0=5\times 10^6$, the idler comb has quality factor $Q_T=10^4$, and $\beta_0=3\times 10^{-4}$ J$^{-1/2}$. The pump wavelength is $\lambda_{0}=465$ nm, the idler wavelength is $\lambda_T=4.07$$\mu$m ($\omega_T=2\pi\cdot 73.7$ THz), and the intracomb spacing is $\omega_m=2\pi\cdot 1$ GHz. The subcombs lie at wavelengths 708, 603, 525, 465, 417, 378, and 346 nm. In (b), "T" denotes the idler comb.
• Figure 2: Quantum correlations and entanglement in a multi-frequency comb system generated by cascaded second-order nonlinear processes. (a) Schematic illustration of cascaded comb generation in a nonlinear (NL) multimode cavity pumped by an electro-optic modulated (EOM) quasi-continuous wave source. (b) Energy and output intensity noise (relative to shot noise limit) for individual comb modes. (c) Two-mode intensity difference noise and entanglement of the first 1,000 bipartitions as measured by the positive partial transform (PPT) criterion simon2000peres. $\tilde{\nu}_\mathrm{min}$ denotes the minimum symplectic eigenvalue. The bipartition between the mid-IR idler comb and the other frequency modes is denoted by a star. Here, $I=5$ subcombs are simulated, each with $J=3$ modes; all comb modes have quality factor $Q_0=5\times 10^6$, the idler comb has quality factor $Q_T=10^5$, and $\beta_0=3\times 10^{-4}$ J$^{-1/2}$. The pump wavelength is $\lambda_{0}=465$ nm, $\lambda_T=4.07$$\mu$m. The pump and seed powers are $|s_{0j}|^2 = 10$ kW, $|s_{1j}|^2=1$ mW. The modulation frequency is $\omega_m=2\pi\cdot 100$ MHz. The subcombs lie at wavelengths 603, 525, 465, 417, and 378 nm.
• Figure 3: Covariance optimization through dissipation, pump, and dispersion engineering. (a) Illustrative optimization of dissipation via $Q$ factor (dissipation) spectrum $Q(\omega)$, pump spectrum $s(\omega)$, and dispersive refractive index $n(\omega)$ to achieve desired covariance matrix. (b) Two optimizations of the covariance matrix using a three parameter (endpoints of $Q$ factor spectrum and pump power of seed comb) constrained optimization. In Optimization 1, we maximize twin beam squeezing of the highest and lowest frequency modes, achieving 15 dB squeezing relative to the initial conditions. This optimization also achieves full comb-comb squeezing between the lowest and highest frequency optical combs, as shown by the green box. In Optimization 2, we maximize full comb-comb squeezing between the highest frequency optical comb and the mid-IR comb (green box), also achieving two-mode squeezing between a frequency mode in subcomb 2 and every other mode in the system ("full spectrum squeezing"). System parameters are identical to those in Fig. \ref{['fig:2']}. In (b), "T" denotes the idler comb.
• Figure 4: Quantum correlations in a femtosecond pumped system with cascaded three-wave mixing processes. (a) Schematic illustration of a femtosecond mode-locked laser pumping a $\chi^{(2)}$ nonlinear crystal (NL) phase matched for cascaded DFG and SFG. (b) Mean field spectra for the weak seed and strong seed limits. The idler pulse is not shown here. Spectra were calculated by integrating the system of nonlinear coupled generalized Schrodinger partial differential equations describing the equation of motion for each pulse. (c) Corresponding twin beam intensity difference noise extracted using quantum sensitivity analysis, where "T" denotes the idler pulse. Each square in the demarcated grid spans a range $\pm 32$ THz (resolution $0.52$ THz) around the center frequency of the mode. Pump wavelength: $\lambda_0=465$ nm, idler wavelength: $\lambda_T=4.07$$\mu$m. Simulated nonlinear strengths are second-order nonlinearity $\beta^L\sim 6\times 10^{-17}$ W$^{-1/2}$m$^{-1}$s and third-order (Kerr) nonlinearity $\Gamma^L\sim 10^{-22}$ W$^{-1}$m$^{-1}$s. Dispersion was modeled through the refractive index profile of lithium niobate zelmon1997infrared. Pump pulse parameters: average power 1 W, repetition rate 200 kHz, pulse duration 210 fs. Seed pulse parameters are identical, except for "weak seed" cases in which the average power is 10 mW. The propagation length in the nonlinear crystal is $L=10$ cm.
• Figure 5: Time-frequency quantum correlations in ultrafast cascaded three-wave mixing. As the nonlinear interaction length increases, inter- and intrapulse quantum correlations begin to develop across the entire multi-pulse system in both the time and frequency domains. Simulation parameters are identical to those used in Fig. \ref{['fig:3']} for the "weak seed" case. Each dashed square spans a range $\pm 0.75$ ps (time bins, resolution 0.012 ps) or $\pm 32$ THz (frequency bins, resolution 0.52 THz) around the center of the pulse.