Overcoming detection loss and noise in squeezing-based optical sensing

The study addresses how to preserve and exploit the quantum advantage provided by squeezed states in optical sensing when detection is inefficient and/or detection noise is high. While squeezed states are practical and robust relative to non-Gaussian states, their metrological advantage is severely degraded by loss and detection noise, often requiring very high detection efficiencies (greater than ~50%) to beat the shot-noise limit. The authors propose and test a solution: applying strong, phase-sensitive, noiseless parametric amplification to the interferometer output (pre-detection) to protect the phase-sensitive quadrature from loss and noise. They aim to demonstrate sub-shot-noise phase sensing under substantial detection loss and significant detection noise, thereby broadening the applicability of squeezing-enhanced metrology across platforms where efficient detection is challenging.

The paper situates its contribution within extensive prior work on quantum-enhanced metrology and sensing using squeezed light, which has improved sensitivities in gravitational-wave detectors and various sensing modalities (spectroscopy, imaging, polarimetry, magnetometry). Prior proposals include Caves’ seminal idea of injecting squeezed vacuum into interferometers and the use of SU(1,1) (nonlinear) interferometers that can be loss-tolerant but have limitations such as a narrower phase-sensitivity range and greater implementation complexity, especially for multimode operation. Noiseless pre-amplification has been used in microwave quantum-state detection, bandwidth enhancement in homodyne detection, tomography of non-Gaussian optical states, and atom interferometry, but not yet implemented in optical sensing with squeezing. The authors build on theoretical analyses of detection-inefficiency limits and strategies (including squeeze-factor unbalancing and approaches for sub-shot-noise imaging/absorption) to realize a practical, loss- and noise-tolerant scheme via an added phase-sensitive amplifier after a conventional SU(2) interferometer.

Concept and phase-space picture: The output of a squeezing-assisted SU(2) interferometer is phase-sensitively amplified by a degenerate optical parametric amplifier (DOPA). The DOPA amplifies the quadrature carrying phase information, rendering it anti-squeezed and thus less vulnerable to subsequent loss. In the idealized picture, sufficiently large anti-squeezing can nearly eliminate the impact of detection loss and mitigate detection noise without degrading the input signal-to-noise ratio.

Experimental setup: A polarization-based Mach–Zehnder interferometer is implemented using a single half-wave plate (HWP). The two interferometer arms correspond to right- and left-circular polarizations; the inputs and outputs are linear polarizations. A rotation of the HWP by angle θ introduces a phase shift φ = 4θ between the arms. The HWP is a dual-wavelength waveplate (HWP at 800 nm; full-wave at 400 nm).

Sources and interferometer inputs: A horizontally polarized coherent state at 800 nm (Spectra Physics Spitfire Ace, 1.5 ps pulses, 5 kHz rep rate) is attenuated to N_c ≈ 1500 photons/pulse at the interferometer input. Vertically polarized squeezed vacuum (SV) is generated in BBO1 (2 mm, type-I, degenerate, collinear) pumped by the 400 nm second harmonic of the same laser. BBO1 is unseeded to generate SV, with measured squeeze factor G1 = 1.7 ± 0.3 (inferring ~15 dB initial squeezing). The coherent beam is injected via BS2 and is not phase matched in BBO1.

Output phase-sensitive amplification: The interferometer output (vertically polarized) is amplified by BBO2 (2 mm BBO, type-I, degenerate) acting as a DOPA (phase-sensitive amplifier). The amplification/de-amplification phase is set by the crystal spacing and a piezo actuator; only the vertical polarization is amplified. The DOPA transformations are X_out = e^G X_in, P_out = e^{-G} P_in, and for a coherent state N_out = N_in e^{2G} + sinh^2 G. In this experiment, squeeze factors up to G ≈ 3.6 are used, corresponding to ~1340-fold intensity gain and ~31 dB quadrature variance anti-squeezing. Two operating regimes are emphasized: measured G2 ≈ 3.1 ± 0.3 (≈490× photon-number amplification) and G2 ≈ 3.6 ± 0.3; model fits use G2 = 2.7 and G2 = 3.2.

Detection and filtering: The pump is removed with a dichroic mirror (DM); a Glan polarizer (GP) selects the vertical polarization. Spatial filtering uses a lens (f = 1.5 m) and 200 μm pinhole to select an angular bandwidth ≈130 μrad; spectral filtering uses a 3 nm bandpass filter (BF) around 800 nm. Photodetector PD1 measures photons per pulse with dark noise ≈500 photons/pulse. Internal loss between BBO1 and BBO2 is ~3% (uncompensable). Post-amplification detection efficiency η is varied (e.g., 50%, 29%, 15% in one set; 50%, 16%, 6% in another) using a HWP and GP before the collection optics. Spatial/spectral filtering defines the measurement bandwidth and is not treated as loss; transmission within the passband is included in η. Overall η = 0.50 ± 0.03 in the nominal configuration.

Characterization and stabilization: The amplification and squeezing are calibrated by measuring vacuum-amplified photon numbers versus pump power P and fitting N ∝ sinh^2 G with G ∝ B √P. G1 is reduced relative to G2 by detuning BBO1 from perfect phasematching. Mode matching for the coherent beam in BBO2 is optimized by waist sizing (~80 ± 10 μm, near the first Schmidt mode). Pump intensity fluctuations (initially ~2% RMS) are monitored via BS1 and suppressed via high-gain down-conversion amplification and post-selection to ~0.3% RMS. The relative phase between coherent and SV inputs is locked (details in Supplementary Information).

Phase sensitivity measurement: The phase-dependent mean photon number N(φ) at the output is recorded, and phase sensitivity is evaluated as Δφ = ΔN / |d⟨N⟩/dφ|, with ΔN including detector dark noise. The derivative is obtained from fits to N(φ). The shot-noise limit (SNL) is defined as Δφ_SNL = 1/√(N_c + N_SV), with N_c determined at the HWP from measured counts corrected by total efficiency (N_c ≈ 1500 ± 100 photons/pulse) and N_SV ≪ N_c estimated from G1.

Experimental conditions and parameters explored: Two amplification settings: (a) G2 ≈ 3.1 ± 0.3 with η = 50%, 29%, 15%; (b) G2 ≈ 3.6 ± 0.3 with η = 50%, 16%, 6%. Model fits incorporate coherent-beam excess noise via g^(2) − 1 ≈ 0.003–0.004 (measured relative excess noise ≈0.0020 ± 0.0005). Sub-shot-noise phase ranges and sensitivity improvements are extracted and compared to theory including internal loss μ ≈ 3% and detection efficiency η.

• Demonstrated that strong phase-sensitive pre-detection amplification restores sub-shot-noise phase sensitivity under significant detection loss and noise in a squeezing-assisted SU(2) interferometer.
• With 50% detection efficiency and detection noise exceeding the squeezed-light level by >50× (detector dark noise ≈500 photons/pulse; SV ≈10 photons/pulse), the phase sensitivity surpasses the SNL by 6 ± 1 dB (for higher gain setting, G2 ≈ 3.6).
• Sub-shot-noise phase sensitivity persists up to 87% total loss (η ≈ 13%) when the output photon number is amplified by ~600× (fit G2 ≈ 3.2); experimentally, SNL is overcome at η = 16% (G2 ≈ 3.6) and marginally at η = 29% (G2 ≈ 3.1).
• Output amplifier gains: measured G2 up to 3.6 ± 0.3 (≈1340× intensity amplification; ≈31 dB anti-squeezing). Another regime: G2 ≈ 3.1 ± 0.3 (≈490× amplification).
• The sub-shot-noise phase range achieved is up to ~0.4π, broader than typical SU(1,1) implementations, though reduced from the ideal 0.5π due to coherent-beam excess noise (g^(2) ≈ 1.003–1.004) and detector dark noise near φ = 0.
• Theoretical analysis shows that with internal loss μ ≈ 3% and initial ~15 dB squeezing (Q0 ≈ 17), increasing G2 recovers the perfect-detection quantum advantage: normalized advantage Q/Q0 approaches unity as G2 grows. Practical gains G2 ≥ 5 could overcome ≳98% loss.
• The method provides robustness to detection noise: despite substantial pulsed-detection noise (~500 photons/pulse), pre-amplification enables sub-SNL sensitivity where unamplified detection would fail.
• Input squeezing characterized as G1 = 1.7 ± 0.3 (~15 dB initial), with uncompensable internal loss between crystals of ~3%.

The findings directly address the central challenge of loss and detection noise degrading the quantum advantage from squeezing. By relocating the phase information into an anti-squeezed quadrature via phase-sensitive amplification at the interferometer output, the relevant quadrature becomes less sensitive to downstream loss. Experiments confirm substantial recovery of quantum advantage: sub-SNL performance by up to 6 dB at 50% detection efficiency, and persistence of advantage down to η ≈ 13–16% with moderate gains. This validates the long-standing proposal that noiseless amplification can render squeezed-light metrology tolerant to detection inefficiencies. The results have broad relevance: many precision-measurement platforms (e.g., gravitational-wave detectors) suffer output-path loss that limits the benefit of injected squeezing; pre-amplification could recover much of the lost advantage, translating into significant improvements in detection rates. The approach is compatible with both direct and homodyne detection (the latter already resilient to dark noise). Furthermore, the strategy can be extended to multimode fields for sub-shot-noise imaging and to absorption measurements where high detector efficiency is otherwise critical. The demonstrated robustness to high detection noise in pulsed regimes underscores applicability to spectral regions (mid-IR, THz) and microscopy settings where detectors are noisy or collect only a fraction of the optical modes.

The work demonstrates a practical and effective method to overcome detection loss and noise in squeezing-enhanced optical sensing by inserting a phase-sensitive parametric amplifier at the output of a conventional interferometer. The authors achieve up to 6 dB improvement beyond the SNL at 50% detection efficiency and maintain sub-SNL sensitivity down to ~13–16% efficiency with moderate output gains, while operating in the presence of substantial detection noise. This establishes a viable route to restore the quantum advantage of squeezing under realistic, lossy, and noisy detection conditions. Future directions include: implementing higher-gain pre-amplification (e.g., G2 ≥ 5) to tolerate even larger losses (approaching ~98%); adapting the scheme to continuous-wave operation and integrating it into gravitational-wave detectors; extending to multimode pre-amplification for wide-field sub-shot-noise imaging; applying to precise absorption/loss metrology; and deploying in spectral regions (mid-IR, THz) and microscopy where detector inefficiency/limited collection angles are major constraints.

• Internal loss between the input and output amplifiers (≈3%) is uncompensable and limits the achievable quantum advantage.
• The coherent input was not strictly shot-noise-limited (measured excess noise, g^(2) − 1 ≈ 0.002–0.004), reducing the sub-SNL phase range and overall advantage compared to the ideal case.
• Significant detector dark noise (~500 photons/pulse) degrades sensitivity near φ = 0 and prevents reaching the perfect-detection quantum advantage observed in theory; while mitigated by amplification, it still constrains performance.
• Sub-shot-noise phase range achieved (~0.37–0.4π) is below the ideal 0.5π due to practical imperfections (excess noise, dark noise).
• For extremely low detection efficiencies, the required amplification may be unrealistically high in practice.
• Proof-of-principle setup simulates an SU(2) interferometer with a HWP and uses spatial/spectral filtering; full multimode operation and broader bandwidth implementation are not demonstrated here.