---
title: Polarized Nonlinear SU(1,1) Interferometers
url: https://www.emergentmind.com/topics/polarized-nonlinear-su-1-1-interferometers
type: topic
---

# Polarized Nonlinear SU(1,1) Interferometers

to=arxiv_search  彩神争霸大发快unction:search  qq彩票 早点加盟json_string
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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
{"query":"all:\"truncated SU(1,1) interferometer\" OR all:\"polarization-based\" OR all:\"active SU(1,1) atom interferometry\"", "max_results": 10, "sort_by": "relevance"}
Polarized nonlinear SU(1,1) interferometers are active interferometers in which the relevant bosonic modes are distinguished, coupled, or routed by polarization and the passive beam splitters of SU(2) 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 [1711.04552][1704.04261][1906.07213][2605.17798].

## 1. SU(1,1) structure and nonlinear mode transformations

An SU(1,1) 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,
$$
\hat S(\zeta)=\exp(\zeta \hat a^\dagger \hat b^\dagger-\zeta^* \hat a \hat b),
\qquad
\zeta=r e^{i\phi},
$$
with Bogoliubov input–output relations
$$
\hat a_{\mathrm{out}}=u\,\hat a_{\mathrm{in}}+v\,\hat b_{\mathrm{in}}^\dagger,\qquad
\hat b_{\mathrm{out}}=u\,\hat b_{\mathrm{in}}+v\,\hat a_{\mathrm{in}}^\dagger,
$$
where \(u=\cosh r\) and \(v=e^{i\phi}\sinh r\). For vacuum input, the first nonlinear stage produces a two-mode squeezed vacuum with \(\langle n_a\rangle=\langle n_b\rangle=\sinh^2 r\) and strong pair correlations [1711.04552][1704.04261].

The group-theoretic description is naturally expressed through the SU(1,1) generators
\(K_0=(a^\dagger a+b^\dagger b+1)/2\),
\(K_1=(a^\dagger b^\dagger+a b)/2\),
and
\(K_2=(a^\dagger b^\dagger-a b)/(2i)\),
whose algebra differs from SU(2) by the noncompact sign structure. In this language, the interferometric sequence is a nonlinear “boost,” then a phase evolution generated by \(K_z\), then either a second nonlinear boost or a direct measurement. The distinction is operationally important: in SU(2) interferometry the conserved quantity is typically total particle number and the observable phase is a difference phase, whereas in SU(1,1) the total number fluctuates and the interferometer is sensitive to a sum phase [1711.04552][2211.08867].

Several architectures coexist within this framework. A full SU(1,1) uses two nonlinear stages. A truncated SU(1,1) omits the second stage and relies on direct homodyne or intensity readout of the outputs of the first stage. An unbalanced SU(1,1) 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 [1906.07213][2605.17798][1704.04261].

## 2. Meanings of polarization in SU(1,1) interferometry

In the strongest sense, a polarized nonlinear SU(1,1) 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(1,1) atom interferometry, the two modes are the \(m_F=+1\) and \(m_F=-1\) spin-polarization side modes of \(^{87}\mathrm{Rb}\), populated pairwise from a pump mode in \(m_F=0\) 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(1,1) interferometer [1711.04552].

In optical birefringence sensing, polarization can appear twice: as the degree of freedom that labels parallel SU(1,1) 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 \(H\) and one for \(V\), 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 \(|\Phi^\pm\rangle\) and \(|\Psi^\pm\rangle\) across \(H\) and \(V\) [2603.08857].

A different usage appears in the polarization-based truncated interferometer in hot \(^{85}\mathrm{Rb}\) 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 [1906.07213].

A common misconception is that “polarized SU(1,1)” always implies orthogonally polarized signal and idler. The fiber unbalanced SU(1,1) interferometer based on \(\chi^{(3)}\) 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 \(\chi^{(3)}\) fiber FWM and that polarization in that implementation is a technical parameter for phase matching and stability, not a primary resource [2605.17798].

## 3. Polarization-based truncated optical interferometers and joint-quadrature readout

The most direct optical realization of the topic is the polarization-based truncated SU(1,1) interferometer demonstrated in hot \(^{85}\mathrm{Rb}\) vapor on the D1 line at \(794.9727\ \mathrm{nm}\). A pump and probe from the same amplified diode laser enter a \(25\ \mathrm{mm}\) cell at \(\approx 4\ \mathrm{mrad}\), with a Wollaston polarization beam displacer splitting them into two rails separated by \(\approx 4\ \mathrm{mm}\). The upper rail is seeded with a weak probe of \(\approx 30\ \mu\mathrm{W}\) and produces strong local oscillators of \(\approx 1\ \mathrm{mW}\) 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 \(1\ \mu\mathrm{W}\). Each rail carries \(\approx 160\ \mathrm{mW}\) of pump power, the pump polarization is rotated by \(90^\circ\) relative to the probe before the cell, and the cell is operated at \(106^\circ\mathrm{C}\) inside a three-layer magnetic shield [1906.07213].

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
\(\hat X_j=(\hat j+\hat j^\dagger)/\sqrt{2}\) and
\(\hat P_j=(\hat j-\hat j^\dagger)/(i\sqrt{2})\),
with joint observables
\(\hat X_\pm=\hat X_a\pm\hat X_b\) and
\(\hat P_\pm=\hat P_a\pm\hat P_b\).
For ideal two-mode squeezing,
\(V(\hat X_-)=V(\hat P_+)=e^{-2r}\)
and
\(V(\hat X_+)=V(\hat P_-)=e^{+2r}\).
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 [1906.07213].

This polarization architecture addresses a technical weakness of earlier truncated implementations: it produces intrinsic port symmetry, interference visibility above \(98\%\), and improved common-mode rejection without post-measurement balancing. The experiment observed up to \(-1.8\ \mathrm{dB}\) of squeezing below the shot-noise limit in both the intensity-difference \(X_-\) and phase-sum \(Y_+\) joint quadratures; the abstract reports \(\approx 2\ \mathrm{dB}\). The Duan–Simon inseparability criterion,
\(\mathcal I=V(\hat X_-)+V(\hat P_+)\le 2\),
gave \(\mathcal I=1.32\pm0.04<2\), while the best measured variances were
\(\langle\Delta X_-\rangle^2=\langle\Delta Y_+\rangle^2=0.66\pm0.03\).
The stricter EPR criterion was not met, with a minimum conditional-variance product of \(\approx 1.75\) [1906.07213].

The frequency response is a defining feature of the polarization-based design. Nearly flat squeezing was observed from \(1\ \mathrm{MHz}\) down to \(50\ \mathrm{kHz}\) on a spectrum analyzer, and oscilloscope acquisition with FFT showed squeezing down to \(\le 200\ \mathrm{Hz}\). 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 [1906.07213].

## 4. 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 \(N_0\approx 500\) in \(m_F=0\) drives resonant spin exchange that converts pairs of pump atoms into correlated atoms in \(m_F=+1\) and \(m_F=-1\). In the undepleted-pump approximation, the effective Hamiltonian is
$$
H=i\hbar\kappa\left(e^{i\phi_p}a^\dagger b^\dagger-e^{-i\phi_p}ab\right),
$$
or, in the \(K\)-representation, \(H=2\hbar\kappa K_x+2\hbar\delta K_z\). A first amplification stage of duration \(t_1\) creates the SU(1,1) coherent state from vacuum, the system then acquires a sum phase \(\phi_\uparrow+\phi_\downarrow\), and a second matched amplification stage maps that phase onto the mean populations of the two spin modes [1711.04552].

For balanced gains \(r_1=r_2=r\), the output population is
$$
N_{\mathrm{out}}(\Theta)=2\sinh^2(2r)\cos^2(\Theta/2),
$$
with \(\Theta=(\phi_1-\phi_2)+(\alpha+\beta)\). This expression makes the nonlinear time-reversal property explicit: \(N_{\mathrm{out}}\) is maximal at \(\Theta=0\) and vanishes at \(\Theta=\pi\). Experimentally, first-stage durations of \(6\)–\(8\ \mathrm{ms}\) with \(\kappa\approx 2\pi\times20\ \mathrm{Hz}\) generated side-mode populations \(\langle N\rangle=2\sinh^2(\kappa t_1)\), and residual minima of \(\approx 0.5\)–\(1.5\) 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
\(\Delta\phi_{HL}=[\langle N\rangle(\langle N\rangle+2)]^{-1}\) for the mean number of phase-sensing atoms inside the interferometer [1711.04552].

Optical birefringence sensing generalizes the polarized SU(1,1) 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 \(\theta\), a birefringent sample of retardance \(\phi\) and axis angle \(\delta\), internal phase controls, and internal loss channels coherently mix the \(H\) and \(V\) interferometers between state generation and measurement. The sensitivity is defined by
\(\delta\phi^2=[\Delta N/(\partial\langle N\rangle/\partial\phi|_{\phi_0})]^2\),
and the classical benchmark is \(\delta\phi_c=1/\sqrt{\langle N\rangle}\) [2603.08857].

This device predicts practical enhancement beyond the shot-noise limit by \(3\)–\(15\ \mathrm{dB}\), with the exact achievable enhancement governed solely by the loss. For a representative \(|\Phi^+\rangle\) setting around \(\phi=0\), the lossless improvements are reported as \(-5.44\ \mathrm{dB}\) at \(g=1\), \(-9.98\ \mathrm{dB}\) at \(g=1.5\), and \(-14.36\ \mathrm{dB}\) at \(g=2\). At realistic \(10\%\) internal loss and \(g=2\), the two-dimensional sensitivity maps versus \((\phi,\delta)\) show up to \(8.78\ \mathrm{dB}\) improvement below the shot-noise limit. The polarization settings \((\alpha,\beta,\theta)\) determine which Bell state is realized: \(|\Phi^+\rangle\) for \((0,0,0)\), \(|\Phi^-\rangle\) for \((\pi,0,0)\), \(|\Psi^+\rangle\) for \((\pi,0,\pi/8)\), and \(|\Psi^-\rangle\) for \((\pi,\pi,\pi/8)\) [2603.08857].

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(1,1) Mach–Zehnder configuration with two cascaded parametric interactions. Two retrieval methods were demonstrated: a phase-shift method and a visibility method, the latter reaching \(\pm0.002\pi\)-level accuracy and meeting current optical industry standards [1810.12498].

## 5. Unbalanced, multimode, and integrated polarized architectures

Time-domain multiplexed SU(1,1) interferometry extends the polarization question into multimode settings. The unbalanced fiber SU(1,1) interferometer uses two identical \(\chi^{(3)}\) FWM OPAs formed by \(300\ \mathrm{m}\) dispersion-shifted fibers at \(77\ \mathrm{K}\), pumped by a \(50\ \mathrm{MHz}\) mode-locked laser with \(\sim 9\ \mathrm{ps}\) pulses and a one-slot delay \(A_T=20\ \mathrm{ns}\) in the signal arm between OPA1 and OPA2. OPA2 then mixes delayed signal pulses with undelayed idler pulses from OPA1, 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 \(g^{(2)}(0s,0i)\approx1.73\), \(g^{(2)}(0s,\pm1s)\approx1.18\), and \(g^{(2)}(0s,\pm1i)\approx1.13\) [2605.17798].

The joint-intensity variance over \(M\) measured modes is reported as
\(R(M)\equiv \mathrm{Var}(N_M)/\mathrm{Var}_{SNL}\),
with high-gain scaling
\(R(M)\approx (8/p^4)(1/M)\)
for \(\mu_1\approx\mu_2\) and same-slot joint measurement. Experimentally, power gains reached \(p^2\approx38\), measurements addressed up to \(M=30\) modes, and \(R(M)\approx-0.9\ \mathrm{dB}\) below SNL was achieved for \(M\gtrsim10\), with the noise remaining below SNL up to \(M=30\). 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 \(\chi^{(2)}\) type-II OPAs using PBSs and wave plates while retaining the joint-intensity framework and covariance structure [2605.17798].

Integrated SU(1,1) 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 \(L=8\ \mathrm{mm}\), are separated by \(l=10\ \mathrm{mm}\), use a pump wavelength \(\lambda_p=766\ \mathrm{nm}\), and a poling period \(\Lambda=126\ \mu\mathrm{m}\). 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, \(\Delta\tilde\beta\approx-\Delta\beta\). The resulting joint spectral amplitude acquires a \(\cos(\phi/2)\) modulation, allowing almost perfect destructive interference at \(\phi\approx(2m+1)\pi\) when the polarization conversion is ideal and the pump is CW [2012.03751].

The integrated theory emphasizes a structural difference between single-mode and highly multimode SU(1,1) interferometers. In multimode operation, the Schmidt spectrum changes with phase, residual background remains at the dark fringe because higher-order dispersion prevents exact \(\Delta\tilde\beta=-\Delta\beta\) away from the central frequency, and the normalized phase sensitivity degrades with gain near the dark fringe according to
\(\Delta\phi/\Delta\phi_{SNL}\approx\sinh(\gamma/2)/\gamma\).
Spectral filtering around \(\omega_p/2\) 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 [2012.03751].

Spectral engineering by nonlinear interference provides a further route to polarized implementations. In pulsed fiber FWM SU(1,1) 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 \(\chi^{(2)}\) down-conversion, polarization-maintaining fibers, and birefringent dispersive media. In the telecom example with \(50\ \mathrm{m}\) DSF sections, \(7\ \mathrm{m}\) SMF dispersive medium, and \(\lambda_{p0}=1548.5\ \mathrm{nm}\), the two-stage device yields \(\xi_s\approx98\%,97\%,96\%\) for \(m=1,2,3\) islands with \(\bar g_s^{(2)}\gtrsim1.9\), while multistage designs with \(N=3\)–\(5\) give \(\bar g_s^{(2)}>1.95\) and \(\xi_s>95\%\) simultaneously [1811.07646].

## 6. 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(1,1) can saturate the quantum Cramér–Rao bound. For the joint-quadrature observable \(M_{\lambda Q}\), choosing \(\lambda=\tanh(2r)\) yields QCRB saturation for all gains in the lossless bright-seeded case, and with vacuum seeding the full SU(1,1) with intensity detection reaches the optimum \(\Delta^2\phi=\mathrm{csch}^2(2r)\), exactly saturating the QCRB. By contrast, vacuum-seeded homodyne noise-power readout is worse by a factor of \(2\), though still sub-SQL [1704.04261]. Including loss, the optimized weighted homodyne remains polarization-agnostic and directly transferable to polarization-encoded modes, with
\[
\lambda_{\mathrm{opt}}=
\frac{\sqrt{\eta_s}\sqrt{\eta_i}\sinh(2r)}
{1-\eta_i+\eta_i\cosh(2r)} ,
\]
and the phase quadratures on both homodynes still optimal [1802.04314].

Another misconception is that full SU(1,1) 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 \(\eta=1/2\), the phase sensitivity becomes
\(\Delta\phi_{\mathrm{opt}}=1/(\sqrt{2}\,e^{2g}|\alpha|)\),
which approaches the QCRB in the bright, high-gain regime. The same work gives the direct mapping \(a\rightarrow H\), \(b\rightarrow V\) for polarized implementations using type-II interactions and polarization-resolved homodyne [2012.04236].

The role of loss is platform dependent but systematic. In the hot-vapor truncated interferometer, optical loss after the cell of about \(30\%\), pump leakage, thermal jitter, and residual LO phase fluctuations reduced the observed squeezing; a beam-splitter loss model inferred \(\approx-2.9\ \mathrm{dB}\) underlying joint-quadrature squeezing for \(\eta\approx0.7\) [1906.07213]. In the hyper-entangled birefringence sensor, the exact achievable enhancement is governed solely by the loss [2603.08857]. In the fiber USUI, the total detection efficiency is \(\approx63\%\), while seed-induced technical noise and residual phase jitter limit the observed \(-0.9\ \mathrm{dB}\) multimode squeezing [2605.17798]. In the integrated KTP interferometer, imperfect polarization conversion and higher-order dispersion limit the depth of the dark fringe [2012.03751]. In a Kerr-enhanced SU(1,1), 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 [2103.07844].

The phase-sum character of SU(1,1) 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(1,1) substitution gives a null signal because the relevant phase sum vanishes. The proposed remedy is a nested SU(1,1)-in-Sagnac geometry in which the entangled pair in each SU(1,1) co-propagates, while a complementary SU(1,1) runs in the opposite direction. In that case the dark-port SNR scales as
\(\mathrm{SNR}\propto (G+g)^2 I_{ps}\Delta^2\),
yielding a quantum enhancement factor \((G+g)^2\) at fixed probe power when the classical loop area is minimized [2211.08867].

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 [1906.07213]. Nonlinear-interference polarimetry uses visible-light detection to measure infrared retardation and is relevant to material research, optical inspection, and quality control [1810.12498]. Spectral-domain OCT with nonlinear interferometers uses idler probing and signal-wavelength detection; in that literature, polarization-sensitive SU(1,1)-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 [2108.05998]. Time-multiplexed and integrated platforms connect polarized SU(1,1) physics to quantum networking, on-chip interferometry, and scalable multimode resource generation [2605.17798][2012.03751].

Polarized nonlinear SU(1,1) 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.

Source: https://www.emergentmind.com/topics/polarized-nonlinear-su-1-1-interferometers