Polarized Nonlinear SU(1,1) Interferometers

• Polarized nonlinear SU(1,1) interferometers are active devices replacing passive beam splitters with nonlinear amplifiers that generate two-mode squeezed states.
• They use polarization either as a mode label or as a routing mechanism to achieve robust phase-sum detection and improved interference visibility.
• Experimental realizations in optical, atomic, and integrated platforms demonstrate enhanced sensitivity, with significant noise suppression below the shot-noise limit.

to=arxiv_search 彩神争霸大发快unction:search qq彩票 早点加盟json_string {"^^^^^^^^9query9 SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^^^^^", "^^^^^^^^9max_results9 ^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^, "^^^^^^^^9sort_by9 "^^^^^^^^9relevance9 to=arxiv_search 的天天中彩票function:search 微信上的天天中彩票 夫妻性生活影片json_string {"^^^^^^^^9query9 SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer\" OR ^^^^^^^^9all:\9 OR ^^^^^^^^9all:\9 SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) atom interferometry\"", "^^^^^^^^9max_results9 ^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^, "^^^^^^^^9sort_by9 "^^^^^^^^9relevance9 Polarized nonlinear SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers are active interferometers in which the relevant bosonic modes are distinguished, coupled, or routed by polarization and the passive beam splitters of SU(^^^^9max_results9^^^^) interferometry are replaced by nonlinear beam splitters such as optical parametric amplifiers, four-wave-mixing stages, or spin-exchange amplifiers. Their defining resource is two-mode squeezing, so they generate nonclassical probe states inside the interferometer and, in the standard formulation, respond to a phase sum rather than a phase difference. In practice, “polarized” can denote several distinct situations: orthogonally polarized optical signal and idler modes, spin-polarization side modes in an atomic system, polarization-selective routing and joint detection of frequency-distinct modes, or polarization management that stabilizes a nonlinear interferometer without serving as the encoded degree of freedom (&&&^^^^9query9^^^^&&&, &&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^&&&, &&&^^^^9max_results9^^^^&&&, &&&^^^^9sort_by9^^^^&&&).

^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^. SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) structure and nonlinear mode transformations

An SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer replaces the two passive beam splitters of a Mach–Zehnder interferometer with phase-sensitive parametric amplifiers. In the two-mode optical description, the nonlinear stage is a two-mode squeezer,

PRESERVED_PLACEHOLDER_^^^^9query9^^^^

with Bogoliubov input–output relations

PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^

where PRESERVED_PLACEHOLDER_^^^^9max_results9^^^^ and PRESERVED_PLACEHOLDER_^^^^9sort_by9^^^^. For vacuum input, the first nonlinear stage produces a two-mode squeezed vacuum with PRESERVED_PLACEHOLDER_^^^^9relevance9^^^^ and strong pair correlations (&&&^^^^9query9^^^^&&&, &&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^&&&).

The group-theoretic description is naturally expressed through the SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) generators PRESERVED_PLACEHOLDER_^^^^9query9^^^^, PRESERVED_PLACEHOLDER_^^^^9all:\9^^^^, and PRESERVED_PLACEHOLDER_^^^^9 OR all:\9^^^^, whose algebra differs from SU(^^^^9max_results9^^^^) by the noncompact sign structure. In this language, the interferometric sequence is a nonlinear “boost,” then a phase evolution generated by PRESERVED_PLACEHOLDER_^^^^9 OR all:\9^^^^, then either a second nonlinear boost or a direct measurement. The distinction is operationally important: in SU(^^^^9max_results9^^^^) interferometry the conserved quantity is typically total particle number and the observable phase is a difference phase, whereas in SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) the total number fluctuates and the interferometer is sensitive to a sum phase (&&&^^^^9query9^^^^&&&, &&&^^^^9 OR all:\9^^^^&&&).

Several architectures coexist within this framework. A full SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) uses two nonlinear stages. A truncated SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) omits the second stage and relies on direct homodyne or intensity readout of the outputs of the first stage. An unbalanced SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) has $r_1\neq r_2$ and/or an intentional path delay between the two nonlinear stages. These variants are not merely implementation choices: they determine whether the second stage acts as nonlinear readout, phase-sensitive preamplifier, multimode mixer, or is absent altogether (&&&^^^^9max_results9^^^^&&&, &&&^^^^9sort_by9^^^^&&&, &&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^&&&).

^^^^9max_results9^^^^. Meanings of polarization in SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometry

In the strongest sense, a polarized nonlinear SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer uses polarization itself as the mode label. This occurs in type-II or crossed-crystal optical designs and in atomic realizations where the two amplified modes are spin polarizations. In the spinor-BEC implementation of active SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) atom interferometry, the two modes are the PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ spin-polarization side modes of PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^, populated pairwise from a pump mode in PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9^^^^ by magnetization-conserving spin exchange. Because the interferometer modes are spin polarizations of the same hyperfine manifold, the device is explicitly a polarized nonlinear SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer (&&&^^^^9query9^^^^&&&).

In optical birefringence sensing, polarization can appear twice: as the degree of freedom that labels parallel SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers and as the physical quantity that couples them. The hyper-entangled birefringence sensor employs two polarization-selective generator OPAs and two polarization-selective measurement OPAs, one pair for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9^^^^ and one for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^, with a birefringent sample and quarter-wave plates coherently mixing the two polarization channels between them. Within each OPA, the signal and idler share the same polarization, but the crossed-crystal geometry and pump polarization generate Bell-state superpositions such as PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9 OR all:\9^^^^ across PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9 OR all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)99^^^^ (&&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^&&&).

A different usage appears in the polarization-based truncated interferometer in hot PRESERVED_PLACEHOLDER_^^^^9max_results9query9^^^^ vapor. There the physically squeezed modes are frequency-distinct probe and Stokes fields generated by four-wave mixing, but polarization optics define dual rails, generate bright local oscillators in one rail, recombine probe and Stokes from both rails, and send the joint field to a single balanced homodyne detector. Here polarization is not the quantum mode label of the squeezed pair; it is the routing and balancing resource that makes single-detector joint-quadrature readout possible (&&&^^^^9max_results9^^^^&&&).

A common misconception is that “polarized SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^)” always implies orthogonally polarized signal and idler. The fiber unbalanced SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer based on PRESERVED_PLACEHOLDER_^^^^9max_results9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ four-wave mixing instead operates with co-polarized pump, signal, and idler to maximize nonlinear coupling. In that system polarization is managed by fiber polarization controllers, wave plates, and a PBS for alignment and stability, but it is not the encoded degree of freedom. The same work explicitly notes that the type-I versus type-II distinction is not applicable to PRESERVED_PLACEHOLDER_^^^^9max_results9max_results9^^^^ fiber FWM and that polarization in that implementation is a technical parameter for phase matching and stability, not a primary resource (&&&^^^^9sort_by9^^^^&&&).

^^^^9sort_by9^^^^. Polarization-based truncated optical interferometers and joint-quadrature readout

The most direct optical realization of the topic is the polarization-based truncated SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer demonstrated in hot PRESERVED_PLACEHOLDER_^^^^9max_results9sort_by9^^^^ vapor on the D^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ line at PRESERVED_PLACEHOLDER_^^^^9max_results9relevance9^^^^. A pump and probe from the same amplified diode laser enter a PRESERVED_PLACEHOLDER_^^^^9max_results9query9^^^^ cell at PRESERVED_PLACEHOLDER_^^^^9max_results9all:\9^^^^, with a Wollaston polarization beam displacer splitting them into two rails separated by PRESERVED_PLACEHOLDER_^^^^9max_results9 OR all:\9^^^^. The upper rail is seeded with a weak probe of PRESERVED_PLACEHOLDER_^^^^9max_results9 OR all:\9^^^^ and produces strong local oscillators of PRESERVED_PLACEHOLDER_^^^^9max_results99^^^^ total probe plus Stokes after the cell; the lower rail is the quantum channel and is seeded with vacuum or a very weak probe below PRESERVED_PLACEHOLDER_^^^^9sort_by9query9^^^^. Each rail carries PRESERVED_PLACEHOLDER_^^^^9sort_by9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ of pump power, the pump polarization is rotated by PRESERVED_PLACEHOLDER_^^^^9sort_by9max_results9^^^^ relative to the probe before the cell, and the cell is operated at PRESERVED_PLACEHOLDER_^^^^9sort_by9sort_by9^^^^ inside a three-layer magnetic shield (&&&^^^^9max_results9^^^^&&&).

After the cell, the pump is blocked, the second beam displacer recombines the rails, and a half-wave plate plus PBS mix the probe and Stokes polarizations evenly on a single balanced photodetector. This arrangement implements a joint homodyne measurement. The measured quadratures are the standard single-mode quadratures PRESERVED_PLACEHOLDER_^^^^9sort_by9relevance9^^^^ and PRESERVED_PLACEHOLDER_^^^^9sort_by9query9^^^^, with joint observables PRESERVED_PLACEHOLDER_^^^^9sort_by9all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9sort_by9 OR all:\9^^^^. For ideal two-mode squeezing,

PRESERVED_PLACEHOLDER_^^^^9sort_by9 OR all:\9^^^^

and PRESERVED_PLACEHOLDER_^^^^9sort_by99^^^^. Experimentally, the relative phase of the two local oscillators determines whether the single detector reads out difference quadratures or sum quadratures, while the common LO phase selects amplitude-like versus phase-like joint quadratures (&&&^^^^9max_results9^^^^&&&).

This polarization architecture addresses a technical weakness of earlier truncated implementations: it produces intrinsic port symmetry, interference visibility above PRESERVED_PLACEHOLDER_^^^^9relevance9query9^^^^, and improved common-mode rejection without post-measurement balancing. The experiment observed up to PRESERVED_PLACEHOLDER_^^^^9relevance9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ of squeezing below the shot-noise limit in both the intensity-difference PRESERVED_PLACEHOLDER_^^^^9relevance9max_results9^^^^ and phase-sum PRESERVED_PLACEHOLDER_^^^^9relevance9sort_by9^^^^ joint quadratures; the abstract reports PRESERVED_PLACEHOLDER_^^^^9relevance9relevance9^^^^. The Duan–Simon inseparability criterion, PRESERVED_PLACEHOLDER_^^^^9relevance9query9^^^^, gave PRESERVED_PLACEHOLDER_^^^^9relevance9all:\9^^^^, while the best measured variances were PRESERVED_PLACEHOLDER_^^^^9relevance9 OR all:\9^^^^. The stricter EPR criterion was not met, with a minimum conditional-variance product of PRESERVED_PLACEHOLDER_^^^^9relevance9 OR all:\9^^^^ (&&&^^^^9max_results9^^^^&&&).

The frequency response is a defining feature of the polarization-based design. Nearly flat squeezing was observed from PRESERVED_PLACEHOLDER_^^^^9relevance99^^^^ down to PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^ on a spectrum analyzer, and oscilloscope acquisition with FFT showed squeezing down to PRESERVED_PLACEHOLDER_^^^^9query9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^. The upper cutoff was limited by photodiode gain-bandwidth and the lower cutoff by electronic noise, including power-line pickup. The same work attributes this unusually broad flat band to polarization-based balancing and the elimination of electronic signal subtraction between separate homodyne channels (&&&^^^^9max_results9^^^^&&&).

^^^^9relevance9^^^^. Spin-polarized, hyper-entangled, and birefringence-coupled implementations

The atomic realization in a spinor Bose–Einstein condensate makes the polarization degree of freedom completely explicit. A large pump population PRESERVED_PLACEHOLDER_^^^^9query9max_results9^^^^ in PRESERVED_PLACEHOLDER_^^^^9query9sort_by9^^^^ drives resonant spin exchange that converts pairs of pump atoms into correlated atoms in PRESERVED_PLACEHOLDER_^^^^9query9relevance9^^^^ and PRESERVED_PLACEHOLDER_^^^^9query9query9^^^^. In the undepleted-pump approximation, the effective Hamiltonian is

PRESERVED_PLACEHOLDER_^^^^9query9all:\9^^^^

or, in the PRESERVED_PLACEHOLDER_^^^^9query9 OR all:\9^^^^-representation, PRESERVED_PLACEHOLDER_^^^^9query9 OR all:\9^^^^. A first amplification stage of duration PRESERVED_PLACEHOLDER_^^^^9query99^^^^ creates the SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) coherent state from vacuum, the system then acquires a sum phase PRESERVED_PLACEHOLDER_^^^^9all:\9query9^^^^, and a second matched amplification stage maps that phase onto the mean populations of the two spin modes (&&&^^^^9query9^^^^&&&).

For balanced gains PRESERVED_PLACEHOLDER_^^^^9all:\9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, the output population is

PRESERVED_PLACEHOLDER_^^^^9all:\9max_results9^^^^

with PRESERVED_PLACEHOLDER_^^^^9all:\9sort_by9^^^^. This expression makes the nonlinear time-reversal property explicit: PRESERVED_PLACEHOLDER_^^^^9all:\9relevance9^^^^ is maximal at PRESERVED_PLACEHOLDER_^^^^9all:\9query9^^^^ and vanishes at PRESERVED_PLACEHOLDER_^^^^9all:\9all:\9^^^^. Experimentally, first-stage durations of PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^–PRESERVED_PLACEHOLDER_^^^^9all:\9 OR all:\9^^^^ with PRESERVED_PLACEHOLDER_^^^^9all:\99^^^^ generated side-mode populations PRESERVED_PLACEHOLDER_^^^^9 OR all:\9query9^^^^, and residual minima of PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\9max_results9^^^^ atoms per mode were observed near deamplification. The same system realizes phase-to-first-moment mapping, so enhanced slopes and suppressed fluctuations near the dark fringe yield the Heisenberg-like bound PRESERVED_PLACEHOLDER_^^^^9 OR all:\9sort_by9^^^^ for the mean number of phase-sensing atoms inside the interferometer (&&&^^^^9query9^^^^&&&).

Optical birefringence sensing generalizes the polarized SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) concept by coupling two polarization-selective nonlinear interferometers. The hyper-entangled architecture uses two generator OPAs and two measurement OPAs with crossed polarization axes. Quarter-wave plates at angle PRESERVED_PLACEHOLDER_^^^^9 OR all:\9relevance9^^^^, a birefringent sample of retardance PRESERVED_PLACEHOLDER_^^^^9 OR all:\9query9^^^^ and axis angle PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^, internal phase controls, and internal loss channels coherently mix the PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ and PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ interferometers between state generation and measurement. The sensitivity is defined by PRESERVED_PLACEHOLDER_^^^^9 OR all:\99^^^^, and the classical benchmark is PRESERVED_PLACEHOLDER_^^^^9 OR all:\9query9^^^^ (&&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^&&&).

This device predicts practical enhancement beyond the shot-noise limit by PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^–PRESERVED_PLACEHOLDER_^^^^9 OR all:\9max_results9^^^^, with the exact achievable enhancement governed solely by the loss. For a representative PRESERVED_PLACEHOLDER_^^^^9 OR all:\9sort_by9^^^^ setting around PRESERVED_PLACEHOLDER_^^^^9 OR all:\9relevance9^^^^, the lossless improvements are reported as PRESERVED_PLACEHOLDER_^^^^9 OR all:\9query9^^^^ at PRESERVED_PLACEHOLDER_^^^^9 OR all:\9all:\9^^^^, PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^ at PRESERVED_PLACEHOLDER_^^^^9 OR all:\9 OR all:\9^^^^, and PRESERVED_PLACEHOLDER_^^^^9 OR all:\99^^^^ at $r_1\neq r_2$^^^^9query9^^^^. At realistic $r_1\neq r_2$^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ internal loss and $r_1\neq r_2$^^^^9max_results9^^^^, the two-dimensional sensitivity maps versus $r_1\neq r_2$^^^^9sort_by9^^^^ show up to $r_1\neq r_2$^^^^9relevance9^^^^ improvement below the shot-noise limit. The polarization settings $r_1\neq r_2$^^^^9query9^^^^ determine which Bell state is realized: $r_1\neq r_2$^^^^9all:\9^^^^ for $r_1\neq r_2$^^^^9 OR all:\9^^^^, $r_1\neq r_2$^^^^9 OR all:\9^^^^ for $r_1\neq r_2$9, PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9query9^^^^ for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9max_results9^^^^ for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9sort_by9^^^^ (&&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^&&&).

A related polarization-sensitive nonlinear-interference geometry is the nonlinear Michelson interferometer used for infrared polarimetry. In that system, the two SPDC interactions occur in forward and return passes through a single PPLN crystal, and the signal interference depends on the polarization transformation applied only to the undetected infrared idler. The setup is described as functionally equivalent, for phase sensing, to an SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) Mach–Zehnder configuration with two cascaded parametric interactions. Two retrieval methods were demonstrated: a phase-shift method and a visibility method, the latter reaching PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9relevance9^^^^-level accuracy and meeting current optical industry standards (&&&^^^^9max_results9sort_by9^^^^&&&).

^^^^9query9^^^^. Unbalanced, multimode, and integrated polarized architectures

Time-domain multiplexed SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometry extends the polarization question into multimode settings. The unbalanced fiber SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometer uses two identical PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9query9^^^^ FWM OPAs formed by PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9all:\9^^^^ dispersion-shifted fibers at PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9 OR all:\9^^^^, pumped by a PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9 OR all:\9^^^^ mode-locked laser with PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query99^^^^ pulses and a one-slot delay PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^ in the signal arm between OPA^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ and OPA^^^^9max_results9^^^^. OPA^^^^9max_results9^^^^ then mixes delayed signal pulses with undelayed idler pulses from OPA^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, creating a global multimode covariance structure in which each mode is correlated with its partner in the same slot and with the signal and idler of the two adjacent slots. Measured second-order correlations include PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^, and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9^^^^ (&&&^^^^9sort_by9^^^^&&&).

The joint-intensity variance over PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9^^^^ measured modes is reported as PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9^^^^, with high-gain scaling

PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:\9^^^^

for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9 OR all:\9^^^^ and same-slot joint measurement. Experimentally, power gains reached PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9 OR all:\9^^^^, measurements addressed up to PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)99^^^^ modes, and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9query9^^^^ below SNL was achieved for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, with the noise remaining below SNL up to PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9max_results9^^^^. The polarization point is explicit: the system operates with co-polarized pump, signal, and idler, and polarization is managed for phase matching and stability rather than encoded as the resource. The same work nevertheless describes a direct adaptation to polarized PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9sort_by9^^^^ type-II OPAs using PBSs and wave plates while retaining the joint-intensity framework and covariance structure (&&&^^^^9sort_by9^^^^&&&).

Integrated SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers make polarization indispensable because type-II waveguide PDC suffers pronounced polarization walk-off and group-velocity mismatch. The multimode integrated KTP design addresses this by inserting a polarization converter between two periodically poled type-II PDC sections. In the example analyzed, the two sections have length PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9relevance9^^^^, are separated by PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9query9^^^^, use a pump wavelength PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9all:\9^^^^, and a poling period PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9 OR all:\9^^^^. The polarization converter swaps the ordinary and extraordinary roles of signal and idler so that, in the CW limit and to first order around degeneracy, PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9 OR all:\9^^^^. The resulting joint spectral amplitude acquires a PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results99^^^^ modulation, allowing almost perfect destructive interference at PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9query9^^^^ when the polarization conversion is ideal and the pump is CW (&&&^^^^9max_results9all:\9^^^^&&&).

The integrated theory emphasizes a structural difference between single-mode and highly multimode SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers. In multimode operation, the Schmidt spectrum changes with phase, residual background remains at the dark fringe because higher-order dispersion prevents exact PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^ away from the central frequency, and the normalized phase sensitivity degrades with gain near the dark fringe according to PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9max_results9^^^^. Spectral filtering around PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9sort_by9^^^^ suppresses side lobes and broadens the supersensitive band, whereas seeding localized to one mode or polarization breaks signal–idler balance and eliminates supersensitivity in this architecture (&&&^^^^9max_results9all:\9^^^^&&&).

Spectral engineering by nonlinear interference provides a further route to polarized implementations. In pulsed fiber FWM SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers with dispersive phase control, the joint spectral amplitude is sculpted into nearly factorable “islands” by a tunable internal phase. The paper is scalar in its baseline treatment, but it explicitly extends the strategy to type-II PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9relevance9^^^^ down-conversion, polarization-maintaining fibers, and birefringent dispersive media. In the telecom example with PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9query9^^^^ DSF sections, PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9all:\9^^^^ SMF dispersive medium, and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9 OR all:\9^^^^, the two-stage device yields PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by9 OR all:\9^^^^ for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9sort_by99^^^^ islands with PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9query9^^^^, while multistage designs with PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^–PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9max_results9^^^^ give PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9sort_by9^^^^ and PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9relevance9^^^^ simultaneously (&&&^^^^9max_results9 OR all:\9^^^^&&&).

^^^^9all:\9^^^^. Metrological limits, misconceptions, and application domains

A central theoretical result is that the second nonlinear stage is not always required to attain the fundamental sensitivity of the state prepared by the first one. In the bright-seeded case, homodyne detection on a truncated SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) can saturate the quantum Cramér–Rao bound. For the joint-quadrature observable PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9query9^^^^, choosing PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9all:\9^^^^ yields QCRB saturation for all gains in the lossless bright-seeded case, and with vacuum seeding the full SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) with intensity detection reaches the optimum PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9 OR all:\9^^^^, exactly saturating the QCRB. By contrast, vacuum-seeded homodyne noise-power readout is worse by a factor of PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance9 OR all:\9^^^^, though still sub-SQL (&&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^&&&). Including loss, the optimized weighted homodyne remains polarization-agnostic and directly transferable to polarization-encoded modes, with

PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9relevance99^^^^

and the phase quadratures on both homodynes still optimal (&&&^^^^9sort_by9query9^^^^&&&).

Another misconception is that full SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) recombination is always preferable. A modified architecture that replaces the second OPA with a passive beam splitter preserves all phase-bearing photons stimulated by the first OPA and avoids parametric annihilation back into the pump. With equal coherent seeding, balanced homodyne on one output, and optimal transmissivity near PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9query9^^^^, the phase sensitivity becomes PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^, which approaches the QCRB in the bright, high-gain regime. The same work gives the direct mapping PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9max_results9^^^^, PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9sort_by9^^^^ for polarized implementations using type-II interactions and polarization-resolved homodyne (&&&^^^^9sort_by9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^&&&).

The role of loss is platform dependent but systematic. In the hot-vapor truncated interferometer, optical loss after the cell of about PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9relevance9^^^^, pump leakage, thermal jitter, and residual LO phase fluctuations reduced the observed squeezing; a beam-splitter loss model inferred PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9query9^^^^ underlying joint-quadrature squeezing for PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9all:\9^^^^ (&&&^^^^9max_results9^^^^&&&). In the hyper-entangled birefringence sensor, the exact achievable enhancement is governed solely by the loss (&&&^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9max_results9^^^^&&&). In the fiber USUI, the total detection efficiency is PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9 OR all:\9^^^^, while seed-induced technical noise and residual phase jitter limit the observed PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query9 OR all:\9^^^^ multimode squeezing (&&&^^^^9sort_by9^^^^&&&). In the integrated KTP interferometer, imperfect polarization conversion and higher-order dispersion limit the depth of the dark fringe (&&&^^^^9max_results9all:\9^^^^&&&). In a Kerr-enhanced SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^), internal losses have a greater influence on the phase sensitivity than external ones, although the nonlinear phase element suppresses the loss-induced degradation relative to the linear case (&&&^^^^9sort_by9all:\9^^^^&&&).

The phase-sum character of SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) also creates a geometric pitfall. In a traditional Sagnac loop, counter-propagating waves acquire equal-magnitude, opposite-sign phases under rotation, so a straightforward SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) substitution gives a null signal because the relevant phase sum vanishes. The proposed remedy is a nested SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^)-in-Sagnac geometry in which the entangled pair in each SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) co-propagates, while a complementary SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) runs in the opposite direction. In that case the dark-port SNR scales as PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9query99^^^^, yielding a quantum enhancement factor PRESERVED_PLACEHOLDER_^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9all:\9query9^^^^ at fixed probe power when the classical loop area is minimized (&&&^^^^9 OR all:\9^^^^&&&).

The application space is correspondingly broad. The polarization-based hot-vapor truncated interferometer targets quantum-enhanced phase and absorption measurements at audio-band frequencies, low-light low-frequency quantum imaging, and magnetometry (&&&^^^^9max_results9^^^^&&&). Nonlinear-interference polarimetry uses visible-light detection to measure infrared retardation and is relevant to material research, optical inspection, and quality control (&&&^^^^9max_results9sort_by9^^^^&&&). Spectral-domain OCT with nonlinear interferometers uses idler probing and signal-wavelength detection; in that literature, polarization-sensitive SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^)-OCT is presented as an inferred extension rather than an explicitly modeled degree of freedom, suggesting a route to polarization-diversity detection and Jones/Mueller reconstruction with undetected photons (&&&^^^^9relevance9query9^^^^&&&). Time-multiplexed and integrated platforms connect polarized SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) physics to quantum networking, on-chip interferometry, and scalable multimode resource generation (&&&^^^^9sort_by9^^^^&&&, &&&^^^^9max_results9all:\9^^^^&&&).

Polarized nonlinear SU(^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^,^^^^9all:(polarization SU(1,1) interferometer) OR all:(polarized nonlinear SU(1,1) interferometer)9^^^^) interferometers therefore form not a single architecture but a family of active interferometers unified by two-mode squeezing and differentiated by the role of polarization. In some platforms polarization is the interferometric mode itself; in others it is the mechanism for routing, balancing, or stabilizing the nonlinear process. Across these realizations, the central trade-offs remain the same: gain versus loss, multimode richness versus mode selectivity, and nonlinear readout versus truncated measurement.