Quantum matched filtering: breaking time-energy separability by 12 orders of magnitude

TL;DR

This work addresses the limitation of classical time–energy sensing by exploiting time–energy entanglement in broadband biphotons and implementing a global quantum detector via coherent SFG. By seeding the SFG process within a Sagnac SU(1,1) interferometer, the authors achieve near-unity SFG efficiency and measure both the time-difference and frequency-sum of entangled photon pairs across an octave-spanning bandwidth, obtaining a product $\Delta(t_1-t_2)\Delta(\nu_1+\nu_2) \approx 2\times10^{-13}$ that dramatically violates the classical bound by over 12 orders of magnitude. The result, demonstrated with $\Delta\nu\approx113$ THz and a high SNR, establishes a practical quantum matched filter that outperforms traditional CV entanglement witnesses and holds promise for quantum illumination (radar) and other CV sensing/communication applications. The approach combines a fully quantum model of multi-mode three-wave mixing with coherent SFG detection, enabling efficient global readout of entanglement and opening avenues for robust quantum sensing in noisy, lossy environments.

Abstract

Detection of signals buried in noise is the major challenge for sensing. Classically, the optimal detector is a matched filter, whose sensitivity meets the classical limit of correlation between the filter target and the measured signal within the noise. For classical signals, the correlation is limited by the separability criterion in frequency-time. Quantum states, however are not necessarily separable, and the correlation between entangled particles can surpass the classical limits. Specifically, time-energy entangled photons can be simultaneously correlated in time difference and frequency sum with no minimum limit, potentially leading to a drastic enhancement of sensitivity for diversified sensing applications. Yet, to enjoy this quantum enhancement, a unique, global detector is needed that can recover the complete information of entanglement in a single shot, i.e. measure the combined correlated variables of time-difference and frequency-sum without measuring the individual frequencies or times. Such a global measurement could, in principle, be realized using the reverse disentangling interaction, such as sum-frequency generation (SFG), but nonlinear interactions at the single-photon level have long been prohibitively inefficient, significantly restricting practical implementations. Here we overcome this barrier: We measure simultaneously and efficiently both the frequency-sum (SFG spectrum) and the time-difference (relative group delay/dispersion) by stimulating the SFG recombination with a strong pump. We generate biphotons with extreme time-energy entanglement (octave-spanning spectrum of 113THz) and measure a relative uncertainty of time-difference and frequency-sum that violates the classical separability bound by >12 orders of magnitude. Our experiment and supporting theory pave the way for quantum sensing applications, such as quantum illumination (radar).

Figures (6)

• Figure 1: (a) Detection methods of time-energy entanglement: Coincidence detection with HOM interference, Direct SFG and SU(1,1) Interference are shown in the top three rows, in comparison to the new Coherent SFG (fourth row) that we present now. In all schemes, a pump laser (black line) interacts with a nonlinear medium ($\chi^{(2)}$) to generate entangled photon pairs—signal (blue) and idler (red), followed by manipulation and measurement. The manipulation can includes removal of the pump or phase modulation (or both). (b) Pros and cons of each method: Four critical aspects are addressed -- access to the time correlation $t_1-t_2$, access to the energy correlation $\omega_1+\omega_2$, detection efficiency, and whether the detectable flux of entangled photons is limited by the speed of the photo-detectors.
• Figure 2: The experimental configuration for Coherent SFG, emphasizing the key elements. Laser sources (green) include a CW pump laser at 885 nm (single frequency) and a seed laser at 1560 nm combined together through a dichroic mirrors (DM1) towards the PPKTP crystal. and the Sagnac Interferometer around it (orange), which enables pumping both forward and backward directions through the crystal. The Sagnac also provides a stable local oscillator to coherently measure the SFG contribution: The Sagnac beam splitter (BS) divides the pump into two paths, each interacting with the nonlinear medium in opposite directions, enabling the generation of entangled photons (forward) and efficient SFG (backwards). The SFG detector is placed at the dark port of the Sagnac interferometer, which prevents detector saturation. Phase Control (blue): A prism-based spectral-shaper and piezoelectric mirror (PZT) ensure precise phase adjustments, compensating for dispersion and maintaining phase coherence across the spectrum. Once the phase is optimized, the entangled photon pairs are reflected back into the PPKTP crystal for a second pass, enabling further nonlinear interaction. Diagnostics (purple): The down-converted spectrum is monitored using a self-assembled spectrometer (prism + CCD camera) and an InGaAs detector at the seed frequency is used to generate feedback for the phase-stabilization of SPDC spectrum relative tothe backwards pump. SFG Detection (yellow): The SFG photons that exit the Sagnac interferometer, measured with a lock-in detection.
• Figure 3: Results. Simultaneous Observation of Energy-Sum and Time-Difference Correlations. The central panel (time correlation) presents the measured (red points) and calculated (black curve) SFG lock-in signal as a function of the GDD applied to the SPDC spectrum in the spectral shaper. The top panel (energy-correlation) shows the RF spectrum of the lock-in SFG signal, highlighting the sharp peak at the dithering frequency of 173 kHz with SNR of 37 dB, which reflects the successful SFG of entangled photon pairs. The blue graph corresponds to the maximum lock-in SFG value near zero GDD (point A in the center panel) whereas the violet graph corresponds to a high GDD value (point B), where the lock-in SFG signal is nearly zero. The bottom panel displays the corresponding SPDC spectra after the second pass through the crystal, showing the spectral SU(1,1) interferogram for the same two GDD values (A- GDD compensated, B- large GDD).
• Figure 4: Residue phase sensitivity. Experimental measurements (black points) and theoretical simulations (red lines) of the SFG signal intensity as a function of the GDD applied to the PDC spectrum by the spectral shaper. Each graph corresponds to a different phase of the pump residue: 0, 0.0015$\pi$, 0.037$\pi$, and 0.4$\pi$. The measured signal captures the time-difference correlation across the entire entangled photon spectrum, where dispersion serves as an analog to time-difference. The strong dependency on dispersion suggests a significant contribution from the spontaneous part of the entangled photons, as the stimulated contribution, which is narrowband, remains unaffected by such variations. This sensitivity is maximized when the phase of the pump residue is near zero, where the contrast of the observed interference patterns is highest.
• Figure 5: The complex contributions to the Coherent SFG field. The diagram illustrates the contributions of different phase components in the SFG process, highlighting the roles of the Sagnac residue, spontaneous SFG, and stimulated SFG fields. (a) A diagram showing the decomposition of the total SFG into its spontaneous (red) and stimulated (green) parts, with their phases $\phi_M$ and $\phi_{sd}$, respectively. The black vector represents the pump, which stimulates the second nonlinear interaction. (b) An illustration emphasizing the consequences of misaligned Sagnac, leading to small residue field (blue) with phase $\varphi_r$. (c) The measured SFG signal (pink) resulting from the interference between these fields, influenced by the phase relationships. Dithering (curved arrow) modulates the phase of the spontaneous SFG and the stimulated SFG relative to the pump, directly influencing the detected signal.