High-Parametric-Gain SU(1,1) Interferometer

• The SU(1,1) interferometer is a nonlinear system that replaces passive beam splitters with parametric amplifiers, using Bogoliubov transformations for phase-dependent readout.
• Key innovations include noise suppression through homodyne detection and gain unbalancing, enabling enhanced signal amplification and robust handling of internal and external losses.
• High-gain configurations are applied in rotational sensing, spectroscopy, and integrated photonics, achieving near Heisenberg-limited phase sensitivity in challenging detection environments.

Searching arXiv for recent and foundational work on high-parametric-gain SU(1,1) interferometers and related implementations. A high-parametric-gain SU(1,1) interferometer is a nonlinear interferometric architecture in which the passive beam splitters of an SU(2) interferometer are replaced by parametric amplifiers, so that phase-dependent readout is mediated by Bogoliubov transformations rather than photon-number-conserving mixing. In the high-gain regime, the relevant gain parameters satisfy $G=\cosh r$, $g=\sinh r$, with $G^2-g^2=1$, and observables acquire strong gain dependence through factors such as $(G+g)^2$, $G^2+g^2$, $e^{\pm 2g}$, or $e^{\pm 4g}$, depending on the specific configuration and detection scheme (Zhao et al., 2022, Natan et al., 4 Feb 2026). Across rotational sensing, stochastic phase estimation, optical coherence tomography, undetected-photon spectroscopy, and multimode integrated implementations, a recurring result is that high parametric gain can amplify the phase-to-signal transfer function while either preserving shot-noise-limited readout or relocating the dominant noise to a quadrature or stage that does not directly limit estimation (Zhao et al., 2022, Zheng et al., 2020, Machado et al., 2020, Hashimoto et al., 2024).

1. Group-theoretic structure and parametric transformations

The defining transformation of an SU(1,1) interferometer is the action of one or two parametric amplifiers on one mode in the degenerate case or on signal–idler mode pairs in the non-degenerate case. For a two-mode parametric amplifier, the Heisenberg transformation can be written as

$$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$

or, with $\gamma=0$, as $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$, $g=\sinh r$0, where $g=\sinh r$1, $g=\sinh r$2 (Jana et al., 21 May 2026). In the degenerate case, a single mode obeys $g=\sinh r$3 (Zhao et al., 2022).

This active transformation contrasts with SU(2) beam-splitter evolution. In the wide-field formulation, the SU(1,1) generator is written through the two-mode squeezing unitary

$g=\sinh r$4

where photon number is not conserved (Frascella et al., 2019). The same structure appears in double-pass interferometers based on spontaneous parametric down-conversion,

$g=\sinh r$5

with corresponding Bogoliubov maps $g=\sinh r$6, $g=\sinh r$7 (Zotti et al., 3 Sep 2025).

A useful distinction is between the full SU(1,1) interferometer with two parametric amplifiers and the truncated configuration with only the first amplifier followed by direct or homodyne detection. For joint homodyne detection, the truncated interferometer can be formally equivalent in sensitivity to the full interferometer when the gains are equal, for arbitrary input states (Jana et al., 21 May 2026). This suggests that, from the standpoint of quantum Fisher information or homodyne-accessible classical Fisher information, the second amplifier often functions primarily as a readout stage rather than as an essential state-preparation resource. A closely related conclusion is reached in the analysis of bright- and vacuum-seeded SU(1,1) interferometers, where the truncated scheme saturates the QFI bound under optimized homodyne detection (Anderson et al., 2017).

2. High-gain regime and scaling laws

High parametric gain is characterized by $g=\sinh r$8, so that $g=\sinh r$9, $G^2-g^2=1$0, and $G^2-g^2=1$1 (Zhao et al., 2022). In single-mode squeezing formulations, the output quadrature variances scale as $G^2-g^2=1$2, while phase variances or signal terms often scale as $G^2-g^2=1$3, $G^2-g^2=1$4, or their reciprocals, depending on detection strategy (Natan et al., 4 Feb 2026).

For squeezing-enhanced Sagnac sensing with an in-loop OPA, the output dark-port quadratures in the ideal lossless case are

$G^2-g^2=1$5

so that for small $G^2-g^2=1$6, $G^2-g^2=1$7, while $G^2-g^2=1$8 at $G^2-g^2=1$9 (Natan et al., 4 Feb 2026). In the high-gain, high-seed, low-loss regime, both direct detection and parametric homodyne attain

$(G+g)^2$0

and, in the unseeded parametric-homodyne configuration,

$(G+g)^2$1

which the source identifies as Heisenberg-like scaling in the total photon number generated in the interferometer (Natan et al., 4 Feb 2026).

In bright- and vacuum-seeded SU(1,1) interferometers, high gain similarly drives Heisenberg-like behavior in the spontaneous-photon resource. For the bright-seeded case, the QFI scales as

$(G+g)^2$2

so that at large gain $(G+g)^2$3, with $(G+g)^2$4 (Anderson et al., 2017). For the vacuum-seeded case, the optimal intensity-detection phase variance is

$(G+g)^2$5

hence $(G+g)^2$6 at high gain (Anderson et al., 2017).

In stochastic phase estimation, gain must be optimized rather than maximized. The SU(1,1) photocurrent has signal $(G+g)^2$7 and noise variance $(G+g)^2$8, so the SNR is non-monotonic in gain (Zheng et al., 2020). The optimal high-gain asymptotics satisfy

$(G+g)^2$9

thereby reaching the stochastic Heisenberg scaling for an Ornstein–Uhlenbeck phase process (Zheng et al., 2020). A plausible implication is that “high gain” is application-dependent: in static or weakly varying phase sensing it is often advantageous to push gain as high as losses permit, whereas in stochastic estimation there exists a finite optimum set by the interplay of phase diffusion and amplified measurement noise.

3. Interferometer architectures and readout strategies

Several distinct high-gain SU(1,1) topologies appear in the literature.

In rotational sensing, one architecture nests two SU(1,1) interferometers inside a Sagnac loop, one for each propagation direction, and reads out the dark port with homodyne detection (Zhao et al., 2022). The motivation is that a naive SU(1,1)-Sagnac hybrid is rotation blind because SU(1,1) measures a phase sum, whereas counterpropagating Sagnac modes acquire opposite phases, $G^2+g^2$0 (Zhao et al., 2022). The nested geometry avoids this cancellation by arranging both entangled fields contributing to the SU(1,1) phase sum to propagate in the same direction within each loop (Zhao et al., 2022).

A distinct Sagnac implementation places a single degenerate OPA inside the loop, so that both clockwise and counterclockwise beams are squeezed by the same in-loop OPA, and optionally adds a second OPA at the dark port for parametric homodyne (Natan et al., 4 Feb 2026). This design preserves compatibility with standard Sagnac configurations and exploits automatic pump–signal phase stability because the pump co-propagates in the loop (Natan et al., 4 Feb 2026).

In undetected-photon spectroscopy and low-coherence interferometry, the dominant realization is a double-pass, single-crystal SU(1,1) interferometer. The first pass generates signal–idler pairs; the sample acts only on the idler; the second pass phase-sensitively re-amplifies the returning fields; only the signal is detected (Hashimoto et al., 2024, Zotti et al., 3 Sep 2025, Machado et al., 2020). This architecture can operate in a high-gain regime with output powers in the nW–$G^2+g^2$1W range, detected by standard visible photodiodes or power meters, while the idler power at the sample remains much smaller (Hashimoto et al., 2024, Machado et al., 2020).

Integrated SU(1,1) interferometers generalize the two-crystal configuration to two periodically poled waveguides separated by a linear section, or to double-pass waveguide designs. In the two-colour multimode integrated case, two identical periodically poled KTP waveguides of length $G^2+g^2$2 are separated by an unpoled KTP section of length $G^2+g^2$3, and polarization converters compensate first-order dispersion (Ferreri et al., 2022). This suggests that high-gain integrated SU(1,1) interferometry is less constrained by conceptual scaling than by the engineering of spectral and group-delay matching.

Measurement strategy is equally central. The principal observables are summarized below.

Configuration	Measurement	Key feature
Seeded SU(1,1)	Joint or single-port homodyne	Can saturate QCRB under optimized weighting (Anderson et al., 2017, Jana et al., 21 May 2026)
Vacuum-seeded SU(1,1)	Intensity sum	Saturates QCRB in the ideal case (Anderson et al., 2017)
High-gain Sagnac SU(1,1)	Direct detection or parametric homodyne	Parametric homodyne is robust to detector inefficiency (Natan et al., 4 Feb 2026)
SISNI / nested SU(2)-in-SU(1,1)	Output homodyne	Combines high SU(2) power with SU(1,1) loss tolerance (Du et al., 2020)

The comparative literature increasingly favors homodyne-based readout for seeded high-gain operation. A general homodyne framework for arbitrary input states shows that single-output-mode homodyne with equal gains is highly robust to very high internal losses, while a boosted-gain second amplifier is advantageous for mitigating external losses (Jana et al., 21 May 2026). In bright-seeded interferometers, a reweighted homodyne observable

$G^2+g^2$4

with $G^2+g^2$5 saturates the QFI for all gains in the truncated configuration (Anderson et al., 2017).

4. Signal enhancement, phase sensitivity, and the role of noise

A recurring theme is that high gain amplifies the phase-dependent mean more strongly than the measured noise, or at least amplifies both in a way that preserves a favorable ratio after unavoidable losses.

In the quantum-entangled Sagnac interferometer, the homodyne SNR for the dark-port phase quadrature is

$G^2+g^2$6

with $G^2+g^2$7 the classical-loop phase and $G^2+g^2$8 the quantum-loop phase (Zhao et al., 2022). The classical contribution is not gain amplified, whereas the quantum-loop contribution is multiplied by $G^2+g^2$9 (Zhao et al., 2022). In the high-gain, quantum-dominated limit,

$e^{\pm 2g}$0

so for fixed total phase-sensing photon number $e^{\pm 2g}$1, the SU(1,1) scheme gains an extra factor $e^{\pm 2g}$2 over a classical Sagnac interferometer (Zhao et al., 2022).

In the in-loop OPA Sagnac design, the direct-detection sensitivity satisfies

$e^{\pm 2g}$3

and parametric homodyne with balanced gains gives

$e^{\pm 2g}$4

(Natan et al., 4 Feb 2026). The qualitative message is that direct detection demands a large seed

$e^{\pm 2g}$5

to suppress amplified vacuum contamination, whereas parametric homodyne attains the same $e^{\pm 2g}$6 scaling with much smaller seed and even works unseeded (Natan et al., 4 Feb 2026).

In seeded SU(1,1) interferometers with degenerate amplifiers, homodyne phase sensitivity near the optimal point has the generic form

$e^{\pm 2g}$7

with

$e^{\pm 2g}$8

where $e^{\pm 2g}$9 and $e^{\pm 4g}$0 are internal and external transmissions (Manceau et al., 2016). This expression makes explicit that high gain in the second amplifier suppresses the impact of external losses through the factor $e^{\pm 4g}$1, but internal losses set an irreducible floor (Manceau et al., 2016). The same paper shows that the supersensitive phase range narrows approximately as

$e^{\pm 4g}$2

so larger input squeezing improves $e^{\pm 4g}$3 while shrinking the usable operating interval (Manceau et al., 2016). This is one of the clearest statements of the tension between ultimate sensitivity and practical lock range in high-gain SU(1,1) devices.

A broader homodyne theory for arbitrary inputs reaches a similar conclusion from another angle. For coherent-vacuum input $e^{\pm 4g}$4, balanced single-port homodyne yields at the dark fringe

$e^{\pm 4g}$5

while joint homodyne and unbalanced single-port detection yield

$e^{\pm 4g}$6

in the lossless high-gain limit (Jana et al., 21 May 2026). Yet when internal losses are included, the balanced single-port configuration becomes the most robust, which the source identifies as a surprising result (Jana et al., 21 May 2026).

5. Loss, decoherence, excess noise, and gain unbalancing

Loss models in SU(1,1) interferometers are typically implemented by fictitious beam splitters mixing vacuum into the signal and idler modes. Internal loss acts between the two parametric amplifiers; external loss acts after the second amplifier, including detector inefficiency (Jana et al., 21 May 2026, Manceau et al., 2016, Manceau et al., 2017).

The distinction is crucial because a high-gain second amplifier can strongly mitigate external loss but not internal loss. In the general homodyne theory, balanced single-port detection with symmetric internal loss $e^{\pm 4g}$7 gives, for coherent-vacuum input,

$e^{\pm 4g}$8

so the asymptotic scaling softens from $e^{\pm 4g}$9 to $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$0 because the vacuum introduced by internal loss is amplified by the second PA (Jana et al., 21 May 2026). By contrast, external-loss penalties in the same configuration are gain independent, and in an unbalanced interferometer with $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$1, they can be suppressed as $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$2 (Jana et al., 21 May 2026).

This logic underlies gain unbalancing. Theoretical work on squeezing-assisted interferometers shows that for a fixed first-stage gain, increasing the magnitude of the second-stage gain improves both the minimum phase sensitivity and the supersensitive phase interval under detection loss (Manceau et al., 2016). In the idealized formulas, the external-loss contribution enters as

$$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$3

so large $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$4 restores the performance of a perfect detector (Manceau et al., 2016). In direct-detection experiments with an unseeded SU(1,1) interferometer composed of two degenerate OPAs, a strongly unbalanced second amplifier preserved supersensitivity even with detection losses as high as 80%, and the device experimentally beat the shot-noise limit by 2.3 dB (Manceau et al., 2017).

That experiment uses a phase-sensitive amplitude

$$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$5

with mean output photon number

$$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$6

showing explicitly how the second-stage gain amplifies the phase-dependent signal (Manceau et al., 2017). The measured supersensitivity persisted down to $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$7, corresponding to 83% detection loss, when $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$8 (Manceau et al., 2017).

A separate but related source of degradation is excess noise intrinsic to the nonlinear medium. In the SISNI analysis, a minimal FWM noise model with internal gain and thermal auxiliary modes explains why SNR peaks at moderate PA gains and then saturates or worsens (Du et al., 2020). This suggests a general practical caveat: the mathematical “high-gain limit” of ideal SU(1,1) interferometry is often unattainable because physical gain media become increasingly non-ideal as gain rises.

6. Multimode, spectral, spatial, and application-specific realizations

High-gain SU(1,1) interferometry has diversified into several technically distinct regimes.

Spectral and modal engineering

A two-crystal SU(1,1) interferometer with a dispersive medium between the crystals can act as a lossless spectral filter for bright squeezed vacuum. In the experiment on group-velocity-dispersion engineering, the frequency bandwidth was narrowed from $$\begin{pmatrix} a_s^{(o)} \ a_i^{\dagger(o)} \end{pmatrix} = \begin{pmatrix} \cosh r & e^{i\gamma}\sinh r \ e^{-i\gamma}\sinh r & \cosh r \end{pmatrix} \begin{pmatrix} a_s^{(i)} \ a_i^{\dagger(i)} \end{pmatrix},$$9 THz to $\gamma=0$0 THz, and the number of effective frequency modes was reduced from approximately 50 to $\gamma=0$1 (Lemieux et al., 2016). The mechanism is gain selective: after Schmidt decomposition

$\gamma=0$2

the high-gain mode weights become

$\gamma=0$3

so larger gain exponentially favors the dominant Schmidt modes (Lemieux et al., 2016). The same work predicts that for $\gamma=0$4, the effective mode number can approach $\gamma=0$5, nearly single-mode (Lemieux et al., 2016).

Integrated two-colour spectrally multimode SU(1,1) interferometers expose another aspect of high-gain operation: in the absence of loss, dispersion alone can destroy supersensitivity at sufficiently high gain. In that setting, the effective gain parameter is

$\gamma=0$6

and sub-SNL performance survives up to $\gamma=0$7, when up to $\gamma=0$8 photons are generated (Ferreri et al., 2022). First-order dispersion compensation using polarization converters is required to achieve strong destructive interference at $\gamma=0$9 and maintain useful visibility (Ferreri et al., 2022).

Spatially multimode and wide-field operation

The wide-field SU(1,1) interferometer uses high-gain parametric down-conversion in a double-pass geometry with a focusing element to support multimode spatial operation over an angular field of approximately 20 mrad (Frascella et al., 2019). The reported gains are $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$0 and $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$1, and the device demonstrates $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$2 dB quadrature squeezing for a single pixel and $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$3 dB over the full frame (Frascella et al., 2019). The interference phase is flat across the angular field, which is essential for phase-front sensing (Frascella et al., 2019). This suggests that high-gain SU(1,1) interferometry is not restricted to single-mode metrology but can be adapted to parallel spatial metrology if mode matching and phase flatness are maintained.

Undetected-photon spectroscopy and tomography

High-gain SU(1,1) interferometers are especially effective when the probe wavelength is difficult to detect. In Fourier-transform infrared spectroscopy with undetected photons, a high-gain single-crystal double-pass SU(1,1) interferometer produces visible output powers measurable by a standard Si power meter while keeping the MIR idler power on the sample very low (Hashimoto et al., 2024). At 400 $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$4W pump power, a fit

$a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$5

with $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$6, $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$7 implies $a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$8 (Hashimoto et al., 2024). The same work reports high-gain visibility

$a_s^{(o)} = G a_s^{(i)} + g a_i^{\dagger(i)}$9

which remains high even for significant idler loss, unlike the low-gain visibility

$g=\sinh r$00

(Hashimoto et al., 2024).

In nonlinear optical coherence tomography, high-gain parametric down-conversion yields $g=\sinh r$01 idler photons per pulse probing the sample and $g=\sinh r$02 detected signal photons per pulse, so standard CCDs or spectrometers suffice (Machado et al., 2020). In the apKTP-based low-coherence interferometer, the ratio of detected signal power after the second pass to probing idler power after the first pass is approximately

$g=\sinh r$03

which at high gain scales as $g=\sinh r$04 (Zotti et al., 3 Sep 2025). Experimentally, amplification factors up to about 208 are reported, with SNR up to 40 dB and axial resolution down to $g=\sinh r$05 after optimizing the aperiodic poling range (Zotti et al., 3 Sep 2025).

Generalizations beyond SU(1,1)

The SU(1,2) interferometer, built from four four-wave mixers and three modes, is a direct nonlinear extension of SU(1,1). With vacuum inputs it reaches the Heisenberg limit in terms of internal photon number and improves absolute phase accuracy relative to SU(1,1) because the total intensity inside the interferometer is higher (Wu et al., 2015). Although this is not an SU(1,1) device, it serves as evidence that multi-stage active interferometry can be interpreted as effective gain multiplication. A plausible implication is that some performance benefits associated with “high gain” can also be achieved architecturally, by increasing the number of active stages and modes rather than driving a single stage deeper into a non-ideal regime.

7. Historical development, misconceptions, and outlook

Historically, SU(1,1) interferometry was first formulated as a nonlinear alternative to the Mach–Zehnder, but many contemporary developments have shifted emphasis from abstract phase sensitivity to system-level engineering under realistic constraints. The literature now spans degenerate and non-degenerate OPAs, bulk crystals, hot-vapor FWM, double-pass single-crystal devices, integrated waveguides, multimode spatial systems, and hybrid nested architectures (Anderson et al., 2017, Manceau et al., 2017, Frascella et al., 2019, Du et al., 2020, Ferreri et al., 2022).

One common misconception is that increasing gain always improves sensitivity without qualification. Several results contradict this. In stochastic phase estimation, there is a finite optimal gain because the measurement noise depends self-consistently on the tracking error (Zheng et al., 2020). In integrated multimode devices, high-order dispersion destroys the quantum advantage beyond a certain gain even without loss (Ferreri et al., 2022). In FWM-based platforms, excess noise in the second amplifier can dominate at high gain (Du et al., 2020). Thus the statement “high-gain SU(1,1) interferometry is exponentially better” is only valid within an idealized model.

A second misconception is that the main benefit of SU(1,1) comes purely from squeezing. The general homodyne analysis shows that improvement can originate from noise reduction and/or signal amplification, depending on operating point, gain distribution, and detection scheme (Jana et al., 21 May 2026). In some dark-fringe single-output configurations, enhancement is primarily due to noiseless signal amplification at nearly unchanged output noise; in bright-fringe or joint-homodyne schemes, squeezing of the measured quadrature plays a larger role (Jana et al., 21 May 2026).

A third misconception is that balanced gain is generically optimal. Both theory and experiment indicate otherwise. Gain-unbalanced interferometers with a boosted second amplifier can be markedly more tolerant to external loss (Manceau et al., 2016, Manceau et al., 2017), while equal-gain single-port homodyne can be most robust to internal loss (Jana et al., 21 May 2026). Optimality is therefore contingent on the loss budget and the measurement strategy.

Current directions suggested by the literature include stronger dispersion engineering in integrated platforms (Ferreri et al., 2022), broader-bandwidth and lower-chirp aperiodic poling for MIR interferometry (Zotti et al., 3 Sep 2025), further development of parametric homodyne for detector-loss immunity (Natan et al., 4 Feb 2026), and application of nested active interferometry to high-power precision instruments such as gravitational-wave detectors (Du et al., 2020). Across these strands, the central design lesson is consistent: the practical figure of merit for a high-parametric-gain SU(1,1) interferometer is not gain in isolation, but gain deployed in the right stage, with the right readout, under a controlled internal-loss and mode-matching budget.