Dual-Squeezing Mach-Zehnder Interferometer

• Dual-Squeezing Mach-Zehnder Interferometer is a quantum architecture that injects squeezed states at multiple stages to enhance phase resolution beyond classical limits.
• It encompasses variations such as dual-input squeezing, paired squeezer configurations, and cascaded optical parametric amplifiers for robust metrological performance.
• Practical designs focus on optimizing photon budgets and minimizing losses to realize Heisenberg scaling and improve measurement precision.

Searching arXiv for the cited papers and closely related recent work on dual-squeezing and Mach–Zehnder interferometry. Searching for “Dual-Squeezing Mach-Zehnder interferometer” and related interferometric squeezing architectures. A Dual-Squeezing Mach–Zehnder Interferometer is a Mach–Zehnder interferometric architecture in which squeezing is introduced in more than one interferometrically relevant stage rather than in only a single input or a single internal element. Across the literature, the term encompasses several closely related realizations: injection of squeezed states into both input ports, the use of two-mode squeezed light together with a coherent probe in a conventional SU(2) Mach–Zehnder interferometer, paired single-mode squeezers placed before and after the interferometer, and dual-arm or cascaded optical parametric amplifiers in thin-film lithium-niobate platforms. A common theme is that the interferometer’s phase sensitivity is improved when the photon budget is dominated by squeezing-generated photons and when the readout is matched to the squeezed quadrature or to a squeezing-assisted interaction-based readout (Müller et al., 2022, Tritschler et al., 2023, Gu et al., 1 May 2026).

1. Definition and interferometric variants

In the standard SU(2) Mach–Zehnder representation with input modes $a,b$, phase encoding is commonly written as

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$

while a single-arm encoding uses

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$

For pure probes, the quantum Fisher information is

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$

with the quantum Cramér–Rao bound

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$

These relations underlie the dual-squeezing analyses in both SU(2) and SU(1,1)-related settings (Müller et al., 2022).

One dual-squeezing definition uses squeezed states in both input ports of a conventional SU(2) Mach–Zehnder interferometer. The input can be two single-mode squeezed vacua,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$

more general squeezed-coherent inputs,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$

or a two-mode squeezed vacuum,

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$

In this sense, “dual squeezing” refers to nonclassical resources entering both interferometric channels (Müller et al., 2022).

A second usage places one single-mode squeezer at the input and a second, phase-locked single-mode squeezer immediately before detection on one output port. In that paired configuration, the full evolution is

$$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$

The balanced case sets $r_d=r_i$ and $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$0, while the unbalanced case permits $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$1 (Gu et al., 1 May 2026).

A third realization uses dual-arm or cascaded $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$2 optical parametric amplifiers. In the thin-film lithium-niobate ring–Mach–Zehnder architecture, the cited work explicitly distinguishes a conventional SU(2) arrangement with squeezers in both arms from an SU(1,1) configuration in which optical parametric amplifiers replace the beam splitters (Kundu et al., 2024).

A fourth route is indirect: a non-linear kicked single-arm Mach–Zehnder interferometer dynamically generates squeezing inside a loop or cavity. The cited analysis states that, in terms of performance, the kicked Mach–Zehnder “reproduces the squeezed-dominated scaling of dual-squeezed inputs when $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$3 dominates the photon budget” (Müller et al., 2022). This does not make the device literally dual-input squeezed, but it establishes a direct metrological correspondence.

2. Core metrological principle

For two independent single-mode squeezed vacua with equal squeezing $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$4, the mean photon number in each mode is $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$5, the total is $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$6, and the photon-number variance of one single-mode squeezed vacuum is

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$7

Since $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$8 and the modes are independent,

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$9

hence

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$0

For $$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$1, this gives

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$2

which yields Heisenberg scaling with

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$3

With coherent displacements, the QFI acquires a shot-noise term linear in the coherent photon numbers, but the optimal scaling remains governed by squeezing for large $$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$4 (Müller et al., 2022).

The same squeezing-dominated criterion appears in the kicked-interferometer analysis. Defining the photon number generated by intra-loop squeezing as $$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$5 and the injected coherent photon number as $$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$6, the squeezing fraction is

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$7

Heisenberg-limited scaling arises when

$$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$8

Using an effective squeezing parameter $$U_\phi(\phi)=\exp(-i\phi\,\hat{n}_a),\qquad \hat{n}_a=\hat{a}^\dagger\hat{a}.$$9,

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$0

and, in the near-aligned regime,

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$1

A sufficient condition is therefore

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$2

or approximately $F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$3 (Müller et al., 2022).

The paired-squeezer readout scheme addresses a different metrological bottleneck. For coherent plus squeezed-vacuum inputs, the quantum Fisher information is

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$4

so the ultimate bound is

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$5

At optimal resource allocation, $F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$6, giving $F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$7 (Gu et al., 1 May 2026). The cited work emphasizes that this Heisenberg scaling is theoretically available in the equal-intensity regime, but direct photon-number-difference detection fails in the single-squeezer Mach–Zehnder because the mean signal vanishes there (Gu et al., 1 May 2026). The second, paired squeezer restores a nonzero, phase-sensitive intensity-difference signal.

This suggests that dual squeezing has two distinct technical roles in the literature. One is resource generation, in which the interferometer is fed by more than one squeezed channel. The other is readout engineering, in which a second squeezer converts otherwise inaccessible phase information into a directly measurable observable.

3. Canonical architectures

The two-mode-squeezed integrated-photonics implementation uses a microring source governed by a four-wave-mixing Hamiltonian

$F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$8

Below threshold, with $F_Q(\varphi)=4\,\mathrm{Var}(H_\varphi),$9, this linearizes to

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$0

where

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$1

The conventional SU(2) Mach–Zehnder then mixes a coherent probe at one input and ring-generated two-mode squeezed light at the other, with output intensity-difference detection

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$2

The paper states that the optimum operating point is $\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$3 (Tritschler et al., 2023).

In thin-film lithium niobate, the ring–Mach–Zehnder interferometer integrates a periodically poled lithium-niobate waveguide inside a symmetric ring–Mach–Zehnder coupler. The nonlinear interaction is

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$4

with $\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$5. In the pump-depletion-free doubly resonant regime, the output quadrature spectral densities are

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$6

At resonance and ideal detection,

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$7

A convenient squeezing parameter is

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$8

and the measured variance under loss is

$\Delta\varphi\ge\frac{1}{\sqrt{F_Q(\varphi)}}.$9

The same platform description then adapts naturally to two PPLN squeezing sections, one in each arm of an SU(2) interferometer, or to two cascaded optical parametric amplifiers in an SU(1,1) arrangement (Kundu et al., 2024).

The paired-squeezer Mach–Zehnder uses balanced beam splitters and a differential phase shift

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$0

For a convenient optimal phase convention with $$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$1 real and $$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$2, the balanced dual-squeezer output operators are

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$3

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$4

The mean number difference is then

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$5

This replaces the single-squeezer expression

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$6

which vanishes throughout the equal-intensity regime and causes the direct-detection divergence (Gu et al., 1 May 2026).

A non-Gaussian dual-squeezing route begins from a two-mode squeezed vacuum,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$7

which a balanced beam splitter converts into two single-mode squeezed vacua with opposite squeeze phases. Weak tapping and photon-number-resolving detection in auxiliary modes herald two continuous-variable states of definite parity, one for each interferometer arm. The resulting input is

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$8

The paper describes this as a dual-squeezing configuration with photon subtraction in both arms (Podoshvedov et al., 2024).

4. Sensitivity formulas and scaling regimes

For the microring-based two-mode-squeezed SU(2) Mach–Zehnder, the phase sensitivity is evaluated by error propagation,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)\,|0\rangle_a\otimes S_b(r_2)\,|0\rangle_b,$$9

With coherent-state input only,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$0

With two-mode squeezed light plus a coherent probe, at $$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$1,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$2

In the ideal low-loss limit with a large coherent probe,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$3

The same work defines the comparison shot-noise limit, including pump photons used to drive the ring, as

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$4

(Tritschler et al., 2023).

For dual single-mode squeezed vacua, the relevant asymptotic statement is the quadratic QFI scaling

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$5

with

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$6

under the useful symmetric-loss approximation

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$7

The kicked-interferometer analysis compares its squeezing-dominated scaling to this benchmark and finds $$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$8 with a prefactor slightly below the NOON bound when photons originate predominantly from squeezing (Müller et al., 2022).

For the paired-squeezer scheme, direct photon-number-difference detection becomes operationally compatible with Heisenberg scaling. In the balanced case,

$$|\psi_{\mathrm{in}}\rangle=S_a(r_1)D_a(\alpha)\,|0\rangle_a\otimes S_b(r_2)D_b(\beta)\,|0\rangle_b,$$9

and

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$0

Using

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$1

the cited work gives

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$2

with $|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$3 tending to $|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$4 near the optimal working point (Gu et al., 1 May 2026). The optimum phase in the balanced equal-intensity regime is

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$5

For the photon-subtracted dual-squeezing interferometer, the QFI of the dual-arm heralded state is

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$6

with

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$7

In the symmetric case,

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$8

The cited interpretation is that the leading terms provide Heisenberg-like behavior as subtraction order increases (Podoshvedov et al., 2024).

5. Measurement strategies and the direct-detection problem

A recurring issue in squeezed-light Mach–Zehnder interferometry is that optimal quantum scaling need not be accessible with the most convenient observable. The paired-squeezer analysis makes this point explicitly: in the original coherent-plus-squeezed-vacuum Mach–Zehnder, direct photon-number-difference detection diverges exactly at the equal-intensity regime because

$|\psi_{\mathrm{in}}\rangle=S_{ab}(r)\,|0,0\rangle.$9

so at $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$0 the signal and its slope vanish for all $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$1 (Gu et al., 1 May 2026). The added detection-stage squeezer breaks this output symmetry and restores a finite slope.

The cited work further models imperfect detection efficiency $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$2 through

$$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$3

and with additive Gaussian readout noise of variance $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$4,

$$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$5

The same paper states that the dual-squeezer scheme remains robust against detection noise and that, numerically, $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$6 and $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$7 produce curves almost indistinguishable from $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$8 for moderate-to-large $$U_{\mathrm{DS\text{-}MZI}}=S_2(r_d,\theta_d)\cdot BS2\cdot U_\phi\cdot BS1\cdot S_1(r_i,\theta_i).$$9 (Gu et al., 1 May 2026).

In the non-Gaussian photon-subtraction scheme, intensity-difference detection has a different limitation. The Mach–Zehnder unitary is

$r_d=r_i$0

with

$r_d=r_i$1

For definite-parity inputs, $r_d=r_i$2, giving

$r_d=r_i$3

The error-propagation slope is therefore nonzero near $r_d=r_i$4 only when $r_d=r_i$5, which requires asymmetric heralding or slightly different auxiliary transmissions. The paper states that symmetric subtraction $r_d=r_i$6 yields superior QFI or QCR behavior but makes intensity-difference detection phase-blind; in that case parity detection would be the optimal choice (Podoshvedov et al., 2024).

The kicked-interferometer treatment emphasizes feasible measurements rather than a single optimal one. It lists projective measurements onto the SLD eigenbasis in principle, and then photon-number difference at the output ports, parity detection at one output port, and homodyne detection optimized for the squeezed-quadrature angle as practical approximations. In its single-arm model, homodyne detection using the second arm as a phase reference is explicitly mentioned as viable (Müller et al., 2022).

In a distinct traveling-wave optomechanical realization, the difference current of the Mach–Zehnder output is shown to be homodyne-equivalent. The output relations are

$r_d=r_i$7

and the difference-current spectrum takes the form

$r_d=r_i$8

The paper states that this measures the spectrum of squeezing, with

$r_d=r_i$9

(Barchielli et al., 2021). Although this is not itself a standard dual-input squeezed Mach–Zehnder, it clarifies the readout logic behind squeezing-sensitive difference-current measurements.

6. Loss, integrated implementations, and practical performance

Loss sensitivity is a central constraint for all dual-squeezing architectures. In the kicked-interferometer Gaussian-channel model, one pass of loss acts as

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$00

For coherent-state benchmarks,

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$01

while for squeezing-dominated kicked states,

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$02

so that

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$03

The same paper states that non-linear kicks can considerably increase sensitivity as measured by the QFI per unit time under small to moderate loss, with gains up to $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$04 at $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$05 and $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$06 (Müller et al., 2022).

In the chip-integrated two-mode-squeezing analysis, loss is represented by arm transmissivity $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$07 and also by the physical sensor length via

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$08

The paper identifies a critical length

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$09

corresponding to $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$10, beyond which the squeezing advantage disappears. It further states that for quantum advantage $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$11, $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$12, and that the main usage is therefore for small and compact devices (Tritschler et al., 2023).

The same work reports an integrated Si$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$13N$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$14 ring example with $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$15, $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$16, $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$17, and near-threshold pump $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$18, for which the squeezed-plus-coherent scheme surpasses both the coherent-only case and the SNL, while the improvement factor

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$19

can exceed $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$20 for compact devices at high decay ratio and stays $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$21 when $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$22 (Tritschler et al., 2023).

The thin-film lithium-niobate ring–Mach–Zehnder platform gives explicit coupling-controlled squeezing and threshold trade-offs. The OPO threshold is

$$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$23

which is minimized at $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$24. The reported performance includes: squeezing $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$25 at $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$26, $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$27, $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$28 with bandwidth $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$29; up to $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$30 at $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$31 with $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$32 and bandwidth $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$33; anti-squeezing up to $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$34; and heralded-photon purity $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$35 with heralding efficiency $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$36 using $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$37 pulses (Kundu et al., 2024).

The non-Gaussian photon-subtraction proposal focuses on practical low-squeezing regimes. It states that dual-arm subtraction can demonstrate gain sensitivity more than $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$38 relative to the initial single-mode squeezed vacuum inputs and can follow Heisenberg scaling for practical values of the single-mode squeezing $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$39. Using only one measurement-induced continuous-variable state together with a single-mode squeezed vacuum shows an increase of no more than $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$40. The same work states that intensity-difference measurement of two measurement-induced continuous-variable states in the two arms can surpass the QCR boundary of the original two single-mode squeezed vacuum states in the practical range of input squeezing $$U_\varphi=\exp(-i\,\varphi\,\hat{J}_z),\qquad \hat{J}_z=\tfrac{1}{2}\big(\hat{a}^\dagger\hat{a}-\hat{b}^\dagger\hat{b}\big),$$41 (Podoshvedov et al., 2024).

Taken together, these results support a consistent design rule already stated explicitly in several of the cited analyses: squeezing must dominate the useful photon budget, the operating point must be chosen so that the readout slope does not vanish, and losses must remain low enough that the nonclassical variance reduction is not washed out. A plausible implication is that dual-squeezing Mach–Zehnder interferometers are best viewed not as a single device class, but as a family of architectures that use multiple squeezing resources to align state preparation, interferometric evolution, and readout with the same metrological quadrature (Müller et al., 2022, Tritschler et al., 2023, Kundu et al., 2024, Gu et al., 1 May 2026).