Nonlinear Interferometers: Quantum-Enhanced Metrology

• Nonlinear interferometers are optical systems that replace linear beamsplitters with nonlinear parametric elements (e.g., OPAs) to generate quantum correlations and achieve enhanced phase sensitivity.
• They employ SU(1,1) transformations, enabling Heisenberg-limited performance and overcoming the shot-noise limit in various metrology, spectroscopy, and sensing applications.
• Experimental architectures, including fiber, integrated, and multistage designs, demonstrate practical quantum-enhanced sensing with improved noise suppression and scalability.

Nonlinear interferometers are optical interferometric systems in which linear mixing elements (e.g., beamsplitters) are replaced or supplemented with nonlinear parametric elements such as optical parametric amplifiers (OPAs) or Kerr media. These devices exhibit quantum correlations and can surpass the standard quantum limit (SQL) of phase sensitivity, enabling quantum-enhanced metrology, spectroscopy, and sensing. Nonlinear interferometers realize distinctive SU(1,1) transformations, as opposed to the SU(2) operations of linear interferometers, leading to phase supersensitivity and noise characteristics unavailable in traditional setups.

1. Fundamental Principles and Architectures

Nonlinear interferometers (NLIs) prototypically replace, partially or entirely, the beamsplitters of a Mach–Zehnder interferometer (MZI) with nonlinear devices. An SU(1,1) interferometer is the canonical example: two-mode parametric amplifiers (PAs), such as OPAs or four-wave mixers, perform quantum-correlated photon pair (twin-beam) generation and recombination (Pooser et al., 2019, Lukens et al., 2018, Luo et al., 2021).

The core physical transformation of an SU(1,1) interferometer is governed by the two-mode squeezing operator: $$S(r) = \exp[r(\hat a \hat b - \hat a^\dagger \hat b^\dagger)]$$ with squeezing parameter $r$. The action on input modes $(\hat a_{\text{in}}, \hat b_{\text{in}})$ yields output modes: $$\hat a_{\text{out}} = \cosh r\, \hat a_{\text{in}} + \sinh r\, \hat b_{\text{in}}^\dagger$$

$$\hat b_{\text{out}} = \cosh r\, \hat b_{\text{in}} + \sinh r\, \hat a_{\text{in}}^\dagger$$

Unlike SU(2) MZIs, where the signal is subject to shot-noise-limited sensitivity scaling as $1/\sqrt{N}$ (photon number), in the SU(1,1) architecture, phase sensitivity can reach the Heisenberg limit ($1/N$ scaling) or better under idealized conditions (Pooser et al., 2019, Luis et al., 2015, Kranias et al., 2023).

In addition to canonical SU(1,1) designs, alternative nonlinear architectures include Sagnac–Michelson geometries with a single PA traversed twice (Lukens et al., 2016), crystal superlattice interferometers (Paterova et al., 2019), and NLIs leveraging Kerr-type nonlinearity for intensity-dependent phase shifts (Luis et al., 2015, Xu et al., 2012).

2. Phase Sensitivity and Quantum Enhancement

SU(1,1) interferometers achieve quantum noise reduction beyond the SQL via quantum correlations generated in the nonlinear gain stages. The amplified quantum correlations enable reduced quadrature noise on output observables that encode the phase, as quantified by the variance

$\sigma^2_{\text{NLI}} = \frac{1}{2G - 1}$

with single-stage gain $G = \cosh^2 r$ (Pooser et al., 2019). This yields a minimum resolvable phase

$\Delta\varphi_{\text{min}} = \frac{1}{\sqrt{2G-1}}$

corresponding to a quantum enhancement factor over the SQL.

Noise suppression at the dark fringe is a hallmark of NLIs: the output photon-number variance at the destructive interferences approaches the shot-noise (Poissonian) limit, while at the bright fringe it becomes super-Poissonian due to amplified quantum fluctuations (Lukens et al., 2018, Lukens et al., 2016).

In the presence of loss, metrological advantage persists only if the internal transmissivity $T$ exceeds a threshold $T_C$, with quantum Fisher information (QFI) analysis showing Heisenberg-like scaling $I_Q\sim N_\phi^2$ (mean photon number through the phase) when $T \sim 1$, but no advantage below $T_C \to 1/2$ for large $N_\phi$ (Kranias et al., 2023). Coherent-state seeding and optimal squeezing parameters can partially restore advantage in high-loss regimes.

3. Nonlinear Interferometer Taxonomy and Coherence

Interferometric architectures may be classified by the number of nonlinear stages in the manipulation layer:

Nonlinearity	Example Architecture	Coherence Signature
0	Standard MZI (SU(2))	First-order only
1	HOM/HOM-like (semilinear)	Second-order only (HOM dip)
2	SU(1,1)	First and second order

Fully nonlinear (SU(1,1)) setups exhibit first-order coherence in the singles and high-visibility interference in coincidences, with fringe period determined by the photon wavelength and envelope width twice that of the HOM dip (Luo et al., 2021).

Multimode effects alter phase sensitivity: in spatially or spectrally broadband NLIs, phase supersensitivity persists only if all Schmidt modes are compensated (i.e., matched gain and proper phase relationships), with sensitivity degradation scaling with effective Schmidt number $K$ (Ferreri et al., 2020, Scharwald et al., 2023).

4. Experimental Implementations and Applications

Truncated and Full SU(1,1) Schemes

Practical quantum sensors, such as quantum-enhanced atomic force microscopy (AFM), use “truncated” NLIs—single squeezing element followed by dual homodyne detection—to combine phase supersensitivity with minimal photon backaction (Pooser et al., 2019). Quantum noise suppression of 3 dB below the SQL (1.7 fm/$\sqrt{\text{Hz}}$ in cantilever displacement) has been demonstrated, with independent control over probe and local oscillator power to manage backaction-detection noise tradeoff.

Fiber, Integrated, and Multistage Architectures

All-fiber SU(1,1) NLIs using four-wave mixing in highly nonlinear fiber achieve 97% visibility over 554 GHz, with dark-fringe noise cancellation tested against the shot-noise limit (Lukens et al., 2018). Integrated KTP-waveguide-based SU(1,1) NLIs support on-chip operation with up to THz bandwidth, near-perfect destructive interference, and phase supersensitivity (Ferreri et al., 2020).

Multistage NLIs (e.g., cascaded χ^2 or fiber-based arrays) generate interference patterns analogous to N-slit gratings, with phase sensitivity and fringe width improving as $1/N$ (number of nonlinear elements) (Paterova et al., 2019, Ma et al., 2020). Binomial weighting of nonlinear section lengths enables near-perfect spectral filtering and the engineering of factorable two-photon states, critical for quantum information processing.

Nonlinear Michelson and Sagnac–Michelson Interferometers

Nonlinear Michelson interferometers leveraging Kerr media enable phase sensitivity scaling as $\Delta x \propto 1/(\chi N^{3/2})$, outperforming linear shot-noise and even Heisenberg scaling, with pulse duration $\tau$ providing an additional optimization lever (Luis et al., 2015). The Sagnac–Michelson SU(1,1) architecture attains high visibility, passive stability, and sub-SQL phase sensitivity in a compact package lacking the path-stabilization challenges of two-arm designs (Lukens et al., 2016).

Quantum Imaging, Metrology, and Sensing

Applications of NLIs address domains ranging from quantum-enhanced readout in scanning probe microscopy (Pooser et al., 2019) and low-dose optical coherence tomography (OCT) (Rojas-Santana et al., 2021), to visible-wavelength characterization of IR metasurfaces via undetected-photon interference (Paterova et al., 2020, Yang et al., 2021). Nonlinear Sagnac geometries have been adopted for ultrafast all-optical switching in photonic fiber loops and potentially for on-chip logic or gyroscopic sensors (Huang et al., 2012).

5. Precision Bounds, Multimode Estimation, and Robustness

The ultimate phase sensitivity of a nonlinear interferometer is set by quantum estimation theory; for multi-parameter displacement estimation, the Holevo Cramér–Rao bound (HCRB) provides the tight precision limit. In SU(1,1) interferometers with pure Gaussian inputs, the HCRB is achieved by dual homodyne (heterodyne) detection and given by $C^{\rm H}=8e^{-2g}$, where $g$ is the squeezing parameter. The symmetric logarithmic derivative (SLD)-CRB is not always saturable at low gain due to measurement incompatibility, but converges to the HCRB at higher squeezing (Zhou et al., 5 Feb 2025).

Robustness to loss depends on squeezing, geometry, and seeding. With strong second-stage squeezing, external loss can be rendered asymptotically negligible ("loss tolerance"), ensuring the persistence of quantum advantage in realistic sensor designs. Seeded NLIs can be optimally adjusted to compensate for internal loss, with a tradeoff between coherent seeding and squeezing (Kranias et al., 2023).

6. Advanced Effects and Hybrid Architectures

Nonlinear interferometry extends to Mach–Zehnder–Fano hybrid devices, combining loop and nonlinear Fano resonances for low-power, high-contrast, all-optical switching via Kerr nonlinearity and resonance hybridization (Xu et al., 2012). Cyclic nonlinear interferometry based on closed-loop Hamiltonian evolution in many-body quantum systems (e.g., spinor condensates) enables deep entanglement and quantum Fisher information maximization without explicit time-reversal, unlocking non-Gaussian metrological gain well beyond the SQL (Liu et al., 2021).

• "Truncated nonlinear interferometry for quantum enhanced atomic force microscopy" (Pooser et al., 2019)
• "Quantum optical coherence: From linear to nonlinear interferometers" (Luo et al., 2021)
• "Interference fringes in a nonlinear Michelson interferometer based on spontaneous parametric down-conversion" (Yang et al., 2021)
• "Holevo Cramér-Rao bound for multi-parameter estimation in nonlinear interferometers" (Zhou et al., 5 Feb 2025)
• "Nonlinear Interference in Crystal Superlattices" (Paterova et al., 2019)
• "Nonlinear interferometry with infrared metasurfaces" (Paterova et al., 2020)
• "A naturally stable Sagnac-Michelson nonlinear interferometer" (Lukens et al., 2016)
• "Engineering the spectral profile of photon pairs by using multi-stage nonlinear interferometers" (Ma et al., 2020)
• "Nonlinear Michelson interferometer for improved quantum metrology" (Luis et al., 2015)
• "Spectral-domain optical coherence tomography based on nonlinear interferometers" (Rojas-Santana et al., 2021)
• "Spectrally multimode integrated SU(1,1) interferometer" (Ferreri et al., 2020)
• "Nonlinearities in Long-Range Compact Michelson Interferometers" (Smetana et al., 2023)
• "Nonlinear interferometry beyond classical limit facilitated by cyclic dynamics" (Liu et al., 2021)
• "Quantum Theory of All-Optical Switching in Nonlinear Sagnac Interferometers" (Huang et al., 2012)
• "Phase sensitivity of spatially broadband high-gain SU(1,1) interferometers" (Scharwald et al., 2023)
• "Metrological Advantages in Seeded and Lossy Nonlinear Interferometers" (Kranias et al., 2023)
• "Nonlinear Mach-Zehnder-Fano interferometer" (Xu et al., 2012)
• "A broadband fiber-optic nonlinear interferometer" (Lukens et al., 2018)