Quantum-Coherent Nonlinear Interferometry

• Quantum-coherent nonlinear interferometry is a technique that uses nonlinear dynamics to create and manipulate interference between alternative pair-generation events.
• It employs methods such as SPDC, SU(1,1) parametric amplification, and Kerr-phase schemes to achieve enhanced phase sensitivity and robust metrological performance.
• This approach spans optical, atomic, and integrated platforms, offering practical insights for infrared metrology, spectral-state engineering, and quantum-enhanced sensing.

Quantum-coherent nonlinear interferometry denotes interferometric schemes in which the phase-sensitive amplitudes are created, transformed, or read out by nonlinear quantum dynamics rather than exclusively by passive linear mixing. In the literature, this includes induced-coherence interferometers based on spontaneous parametric down-conversion (SPDC), SU(1,1) interferometers built from parametric amplifiers, Kerr-phase Michelson schemes driven by coherent states, interaction-based many-body interferometers in spinor condensates, and related architectures in which coherence resides in pair-generation alternatives, nonlinear phase accumulation, or cyclic interacting dynamics (Yang et al., 2021, Flórez et al., 2022, Liu et al., 2021). The subject is unified by a common principle: the measured fringe is controlled by coherent superposition of alternatives generated by a nonlinear Hamiltonian, so visibility, phase response, and metrological utility depend on indistinguishability in spectral, spatial, temporal, and internal degrees of freedom.

1. Conceptual scope and classification

A useful classification organizes interferometers by the number of nonlinear elements in the preparer–sample–analyzer chain: linear interferometers contain only linear mixers, seminonlinear interferometers contain one nonlinear and one linear mixing stage, and nonlinear interferometers contain two nonlinear stages (Luo et al., 2021). In the fully nonlinear case, the interferometer is no longer merely a device that manipulates a pre-existing photon; it becomes part of the generation process itself. This distinction is central to quantum-coherent nonlinear interferometry because interference can then arise between different generation events, not only between different propagation paths.

Within this classification, several major families appear. Induced-coherence and low-gain SPDC devices interfere amplitudes for pair creation in separate nonlinear interactions (Yang et al., 2021). SU(1,1) interferometers replace beam splitters with active parametric amplifiers or squeezers and use phase-sensitive gain as the interferometric resource (Flórez et al., 2022). Interaction-based atomic interferometers use nonlinear many-body evolution itself as both probe preparation and readout, including closed-loop protocols in which the second stage returns the system close to its initial state without explicit Hamiltonian sign reversal (Liu et al., 2021). Other realizations replace a linear beam splitter by an anharmonic quantum scatterer, as in Jaynes–Cummings Hong–Ou–Mandel interferometry (Oehri et al., 2014), or embed non-Hermitian transport for one mode inside a nonlinear interferometer, as in passive PT-symmetric integrated couplers (Kumar et al., 2022).

The scope is broader than “nonclassical input plus nonlinear medium.” A Kerr Michelson interferometer analyzed with coherent-state probes remains quantum-coherent because the relevant generator is nonlinear in photon number, $G_j=\hat N_j+\frac{\chi}{2}\hat N_j^2$, even though the probe is classical coherent light (Luis et al., 2015). Conversely, some recent “quantum-like” nonlinear interferometers deliberately reproduce phase super-resolution and wavelength-translated sensing with coherent states and classical nonlinear optics, while remaining bounded by the shot-noise limit; these occupy a boundary region rather than the core of genuinely quantum-enhanced nonlinear interferometry (Dalidet et al., 2024).

2. Coherence mechanisms and mathematical structure

In SPDC-based induced-coherence devices, the canonical low-gain mechanism is the coherent addition of two pair-creation amplitudes. For one interaction,

$$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$

and after propagation between two interactions,

$$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$

The detected signal count is then

$\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$

so single-photon interference appears because the pair-creation amplitudes are coherent and indistinguishable (Yang et al., 2021). In the multimode case, the same mechanism survives only if signal and idler remain matched in the relevant spectral and spatial modes.

For active SU(1,1) interferometers, the nonlinear element is the two-mode squeezing operator

$$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$

with Bogoliubov transformation

$$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$

Here interference is governed by the relative phase between two nonlinear interactions rather than by beam-splitter recombination (Flórez et al., 2022).

In Kerr-based Michelson interferometry, the phase generator becomes intensity dependent. The mean signal near the compensated operating point is

$$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$

and in the small-signal regime $\chi Nkx\ll 1$,

$$\langle M\rangle \simeq \eta N\,kx\left(1+\chi\frac N2\right).$$

The enhanced slope comes from nonlinear phase accumulation rather than from nonclassical probe statistics (Luis et al., 2015).

In many-body interferometers, coherence can be organized around recurrence rather than inversion. For the spinor-condensate closed-loop protocol, the essential condition is not $U_2=U_1^\dagger$ globally, but

$$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$0

so a small inserted phase disrupts this closure and becomes measurable in a simple population observable (Liu et al., 2021). This suggests that quantum-coherent nonlinear interferometry is best understood at the level of state-centered coherent maps, not only at the level of explicit time reversal.

3. Canonical architectures and platforms

The field now spans optical, atomic, condensed-matter, and integrated-photonic platforms. The architectures differ, but all use nonlinear dynamics to create the interfering amplitudes.

Platform	Nonlinear element	Interferometric mechanism
Michelson SPDC QNI (Yang et al., 2021)	One second-order crystal traversed twice	Signal counts depend on undetected idler phase
Seeded or unseeded SU(1,1) (Flórez et al., 2022)	Two OPAs / squeezers	Pair generation in stage $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$1 versus $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$2
Truncated NLI for AFM (Pooser et al., 2019)	First FWM stage only	Nonlinear resource generation plus dual homodyne readout
Spin-1 $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$3Rb closed loop (Liu et al., 2021)	Spin-mixing dynamics	Forward nonlinear evolution closes a loop in Hilbert space
Jaynes–Cummings HOM (Oehri et al., 2014)	Anharmonic cavity–qubit scatterer	Exchange interference modified by two-photon interaction
PT-symmetric integrated QNI (Kumar et al., 2022)	Two SPDC sections plus passive PT idler coupler	Signal fringes encode idler loss and transfer

The Michelson SPDC interferometer is the clearest induced-coherence realization in a one-output geometry. A single nonlinear crystal is traversed twice; visible signal photons are detected, while infrared idler photons remain undetected but control the phase through their optical path (Yang et al., 2021). The paper explicitly stresses that this is not conventional output-port energy redistribution: interference modulates the total pair-generation probability.

SU(1,1) variants span low-gain spontaneous devices, seeded interferometers, and truncated forms. In the seeded low-gain theory, seeding the undetected mode boosts the useful interfering term without adding a large direct background at the detected port, which is why coherent-state and number-state seeding become effectively equivalent in low gain for visibility, contrast, phase sensitivity, and signal-to-noise ratio (Flórez et al., 2022). In the truncated AFM realization, the second nonlinear stage is removed and replaced by dual homodyne detection, while the first stage still supplies the two-mode squeezed resource (Pooser et al., 2019).

Beyond photonics, the three-mode spinor-condensate realization shows that nonlinear interferometry can be implemented in a deep many-body regime with about $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$4 of atoms transferred out of the pump mode at the sensing time, well beyond the undepleted-pump approximation (Liu et al., 2021). In disordered metals, two coherent terahertz pulses generate a current echo at $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$5, interpreted as time-domain interferometry of weak localization through interference of a self-intersecting electron path and its time-reversed partner (Li et al., 2024). A plausible implication is that “nonlinear interferometer” has become a platform-independent concept tied to coherent nonlinear pathway control rather than to any single optical geometry.

4. Metrology, spectroscopy, and experimentally demonstrated performance

The Michelson SPDC quantum nonlinear interferometer provides a detailed benchmark for visible readout of infrared information. With a $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$6 nm CW pump, a $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$7-mm PPKTP crystal, approximately $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$8 nm signal, and $$| \psi_1 \rangle = |0\rangle + \varepsilon |1_s,1_i\rangle,$$9 nm idler, fitting the visibility envelope gave a maximum visibility of $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$0 and a coherence length $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$1 mm, in good agreement with the theoretical $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$2 mm (Yang et al., 2021). The same experiment measured an equivalent wavelength $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$3 nm from equal-inclination fringes, inferred an infrared refractive index $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$4 for a $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$5 mm BBO crystal, estimated a wedge angle $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$6, and reported $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$7 relative RMS intensity fluctuation over $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$8 one-second ICCD frames. These results established nonlinear Michelson interferometry as a practical platform for infrared metrology with visible detection.

In the spinor-condensate closed-loop interferometer, a total of $$|\psi_{\text{out}}\rangle \sim |0\rangle + \varepsilon e^{i k_p l_p}\left(1+e^{-i\phi}\right)|1_s,1_i\rangle, \qquad \phi = k_p l_p-k_s l_s-k_i l_i.$$9 atoms yielded a metrological gain of $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$0 dB over the classical limit, with readout performed through the fractional population $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$1 rather than through single-atom-resolved detection (Liu et al., 2021). The experiment is notable because the probe state at the sensing time was already highly non-Gaussian, yet the output near the working point remained readable through first and second moments.

In truncated nonlinear interferometry for atomic-force microscopy, the first practical sensing application of an SU(1,1)-type architecture achieved quantum noise reduction of up to $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$2 dB below the standard quantum limit, corresponding to a quantum-enhanced beam-displacement measurement of $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$3 (Pooser et al., 2019). The phase-sum squeezing measured in the actual AFM readout was $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$4–$\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$5 dB below the shot-noise limit, and the experiment demonstrated how a weak squeezed probe can minimize photon backaction while high-power local oscillators preserve readout sensitivity.

Seeded SU(1,1) theory addresses a different experimental bottleneck: robustness to internal loss and detector inefficiency. For $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$6, $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$7, $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$8, and $\langle n_s\rangle = 2|\varepsilon|^2(1+\cos\phi),$9, the visibility rises from about $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$0 in the unseeded case to $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$1 with only a few tens of seed photons, reaching $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$2 at $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$3 (Flórez et al., 2022). The same analysis reports that the minimum phase variance drops below the unseeded ideal quantum-limited reference at around $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$4, while about $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$5 recovers the ideal unseeded sensitivity in the lossy, inefficient device.

The multiphoton-absorption SU(1,1) proposal extends nonlinear interferometry from phase estimation to estimation of nonlinear material parameters. For weak $$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$6-photon absorption, it predicts improvement of the asymptotic variance scaling from

$$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$7

for coherent-light transmission to

$$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$8

for the imbalanced SU(1,1) scheme, at fixed photon number incident on the sample (Panahiyan et al., 2022). The paper attributes this to an optimal amplitude-squeezed probe produced by the first OPA and coherently anti-squeezed by the second.

5. Spectral-state engineering, generalized interference, and formal developments

A major branch of the subject treats the interferometer as a coherent state-engineering device. In multi-stage SU(1,1) spectral engineering, the nonlinear interferometer acts as an active filter: undesired spectral components are suppressed by destructive interference of generation amplitudes rather than by post-generation loss. In the pulsed-pump fiber system analyzed in detail, both modal purity and efficiency are improved simultaneously, and a multi-stage nonlinear interferometer is proposed for near-ideal single-mode operation and near-unity efficiency (Cui et al., 2018). This is a distinct shift in viewpoint: the interferometer is used not merely to estimate phase but to sculpt the joint spectral function itself.

A related but more recent development is mixed-order nonlinear interference between second- and third-order processes. In the integrated microring proposal, SPDC and SFWM coherently contribute to the same final two-photon state,

$$\hat S(\xi)=e^{\xi \hat c_1^\dagger \hat c_2^\dagger-\xi^\ast \hat c_1\hat c_2},$$9

so the total pair rate oscillates with the relative pump phase and the biphoton spectral amplitude can be reshaped directly (Stefano et al., 26 Jan 2026). For $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$0 ps pump pulses, SPDC-pump pulse energy $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$1 pJ, SFWM-pump peak pulse energy $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$2 pJ, and repetition rate $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$3 MHz, the individual pair-generation rates were computed as $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$4 kHz for both processes, with an estimated visibility $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$5. The conceptual novelty is that the interfering alternatives are different nonlinear orders, not two copies of one process.

The formal structure of multipath nonlinear interferometry has also been clarified through nonlinear–linear duality. For monochromatic nondegenerate PDC networks, partial time reversal maps the nonlinear setup to a linear network in which each PDC is replaced by a hypothetical wavelength-shifting beam splitter with $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$6 and $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$7, while cascaded PDCs become cavity-like systems composed by the Redheffer star product (Zheng et al., 19 Aug 2025). In the low-gain limit this reduces to an advanced-wave picture; beyond low gain, additional normalization factors encode looping-photon contributions inside the dual cavities. This suggests a unifying analytical language in which complicated nonlinear postselection amplitudes can sometimes be treated as linear scattering amplitudes with altered boundary conditions.

Entangled-photon nonlinear spectroscopy provides a complementary formalism in Liouville space. There the observable is a time-resolved coincidence count after a Michelson-type preparation stage and a HOM detection stage, and the signal is written as a coherent sum over four detection pathways $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$8 and five nonlinear matter pathways $$\begin{bmatrix} \hat c_1' \ \hat c_2'^\dagger \end{bmatrix} = \begin{bmatrix} u & v \ v^\ast & u \end{bmatrix} \begin{bmatrix} \hat c_1 \ \hat c_2^\dagger \end{bmatrix}, \qquad u=\cosh G,\quad v=e^{i\varphi}\sinh G.$$9 (Asban et al., 2022). In the short-entanglement-time regime, only a few processes survive, and the preparation delay $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$0, HOM delay $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$1, and coincidence lag $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$2 become pathway-selective coordinates. The result is a version of nonlinear interferometry in which coherence of the source, coherence of the detection alternatives, and coherence of matter pathways are manipulated simultaneously.

6. Boundaries, misconceptions, and open problems

Several recurrent misconceptions are addressed explicitly in the literature. First, not all nonlinear interferometers are SU(1,1) devices. Induced-coherence Michelson systems rely on coherent superposition of pair-creation alternatives in the low-gain regime, whereas SU(1,1) interferometers rely on active parametric gain and hyperbolic mode mixing (Yang et al., 2021, Flórez et al., 2022). Second, not all “quantum-like” nonlinear interferometers are quantum-enhanced in the metrological sense: the frequency-engineered classical-light scheme reproduces super-resolution and dispersion-cancellation-like behavior, but the paper states that its precision remains bounded by the shot-noise limit (Dalidet et al., 2024). Third, explicit time reversal is not a prerequisite: cyclic closed-loop dynamics can replace $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$3 when the system naturally returns close to its initial state (Liu et al., 2021).

The dominant technical limitations are also architecture specific. For SPDC Michelson interferometers, high visibility requires excellent mode matching on both passes, pump coherence much longer than the relevant path lengths, low optical loss, and path-length balance within the SPDC coherence length; angular-spectrum-dependent fringes can appear because the SPDC bandwidth is broad and angle dependent (Yang et al., 2021). In seeded SU(1,1), external loss can be largely neutralized by high second-stage gain, but internal loss remains fundamental because it degrades the squeezed state before readout (Flórez et al., 2022). In Kerr Michelson metrology, the analysis is based on error propagation rather than quantum Fisher information, assumes $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$4, and, for ordinary weak Kerr media, the operating-point condition can imply impractically large arm lengths unless compensation is introduced (Luis et al., 2015). In the spinor-condensate closed-loop experiment, the main limitations are atom loss, RF noise, and the quasi-periodic rather than exactly periodic nature of the many-body dynamics (Liu et al., 2021).

Some controversies concern what exactly should count as the interferometric object. In the weak-localization echo proposal, the delayed current signal is specifically attributed to interference between a self-crossing electron path and its time-reversed partner; the paper distinguishes this from generic nonlinear transport and from strong-localization echoes that do not require time-reversal interference (Li et al., 2024). In passive PT-symmetric integrated nonlinear interferometry, the sharp signal-fringe jump at critical idler loss is explicitly distinguished from PT-symmetry breaking itself; it occurs when the nonlinear-interference transfer amplitude $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$5 crosses zero, not when eigenvalues coalesce at $$\langle M\rangle \simeq \eta N e^{-N\chi^2k^2x^2/8} e^{-\sigma^2/2} \sin\!\left[kx\left(1+\chi\frac N2\right)\right],$$6 (Kumar et al., 2022). In active-medium electron–phonon interferometry, the central open problem is how the predicted two-stage buildup of signal–idler and then idler–idler entanglement survives realistic decoherence, propagation loss, and thermal occupation, none of which are included in the closed coherent model (Ogiri et al., 30 Oct 2025).

The resulting picture is not a single device class but a research program. Quantum-coherent nonlinear interferometry now includes low-gain induced-coherence metrology, active SU(1,1) sensing, many-body closed-loop readout, non-Hermitian and mixed-order pathway control, and formal dualities that connect nonlinear quantum networks to effective linear ones. What remains common is the operational criterion: the observable fringe must be governed by coherent superposition of alternatives generated or transformed by nonlinear quantum dynamics.