Hidden Squeezing in Quantum Systems

• Hidden squeezing is a quantum phenomenon where noise reduction occurs in nonstandard observables that conventional measurements, such as standard quadratures or Stokes parameters, fail to capture.
• It emerges when complex, frequency-dependent covariance elements are revealed through optimized measurement bases and mode transformations, surpassing limitations of standard homodyne detection.
• Advanced schemes like synodyne detection and resource-theoretic optimizations are essential for uncovering and exploiting hidden squeezing in multimode and integrated quantum systems.

Searching arXiv for recent and foundational papers on hidden squeezing. Hidden squeezing denotes forms of squeezing that are physically present in a quantum or stochastic optical, optomechanical, atomic, or interferometric system but are not revealed by the most conventional observables or measurement bases. Across the literature, the term has been used in several technically distinct senses: noise reduction below the vacuum or coherent-state benchmark in hidden optical-polarization variables rather than ordinary Stokes observables (Singh et al., 2010); squeezing encoded in complex, frequency-dependent covariance elements that standard homodyne detection cannot access (Buchmann et al., 2016, Ockeloen-Korppi et al., 2018, Gouzien et al., 2022, Nello et al., 24 Jun 2026); squeezing temporarily concealed in a transformed basis to mitigate decoherence and later recovered by inverse operations (Brewster et al., 2018); and, more broadly, squeezing that becomes manifest only after selecting an appropriate generalized observable, transformed mode basis, or resource benchmark (Bräuer et al., 12 Mar 2025). The unifying feature is basis dependence: the noise reduction exists, but naive quadrature, polarization, or mode-resolved measurements can miss it.

1. Conceptual scope and definitions

The most restrictive usage of the term arises in hidden optical-polarization states (HOPS), where the relevant order parameter is not the ordinary polarization relation based on amplitude ratio and phase difference, but a hidden relation based on amplitude ratio and phase sum (Singh et al., 2010). In that setting, hidden squeezing means noise reduction below the vacuum or coherent-state level in the hidden optical-polarization variables, not in the usual Stokes parameters (Singh et al., 2010).

A broader usage appears in continuous-variable and multimode settings. There, hidden squeezing refers to variance reduction that is present in the full covariance structure of the field but inaccessible to standard homodyne detection because the relevant amplitude–phase or sideband correlations are complex-valued, frequency dependent, multimode, or encoded in transformed observables (Buchmann et al., 2016, Gouzien et al., 2022, Nello et al., 24 Jun 2026). In this sense, squeezing is hidden whenever the optimal measurement basis is not a single real quadrature selected by a monochromatic local oscillator.

A further extension is resource-theoretic. "Generalized squeezing as a witness" treats squeezing as variance suppression below the best value attainable by a prescribed free class of states and free operations (Bräuer et al., 12 Mar 2025). This suggests that hidden squeezing need not refer to a single physical mechanism; it can denote any resource that becomes visible only after optimization over observables, nonlinear transformations, or free operations.

2. Hidden optical-polarization squeezing

In the HOPS framework, a monochromatic bi-modal field with orthogonally polarized collinear modes is characterized not by the ordinary index of polarization $N_x,N_y$0 but by the hidden index $N_x,N_y$1 The defining feature is that the ratio of real amplitudes and the sum of phases are non-random, whereas the ordinary phase-difference criterion fails (Singh et al., 2010). In quantum form, the hidden-polarization criterion is $N_x,N_y$2 which replaces the usual polarized-light relation $\hat a_y=p\,\hat a_x$ (Singh et al., 2010).

The reason these states are termed hidden is that ordinary Stokes analysis can classify them as apparently unpolarized. In the linear basis, the Stokes parameters evaluate to $N_x,N_y$3 even though the field possesses nontrivial hidden polarization structure (Singh et al., 2010). The appropriate observables are instead the hidden optical-polarization parameters $N_x,N_y$4 and their operator counterparts $N_x,N_y$5 $N_x,N_y$6 $N_x,N_y$7 These satisfy nontrivial commutators, including $N_x,N_y$8 so the hidden variables obey uncertainty relations and admit a squeezing notion (Singh et al., 2010).

For a monochromatic bi-modal chaotic optical field incident on a $\chi^{(2)}$ crystal and undergoing degenerate parametric amplification, the Hamiltonian is $N_x,N_y$9 The exact Heisenberg solutions are $Sq(kt,\Delta_h)$0 with $Sq(kt,\Delta_h)$1 Squeezing is then identified through a variance criterion such as $Sq(kt,\Delta_h)$2 which becomes $Sq(kt,\Delta_h)$3 The associated squeezing function is $Sq(kt,\Delta_h)$4 The onset condition is $Sq(kt,\Delta_h)$5 Numerically, no squeezing occurs before $kt \approx 0.22$ s; squeezing begins at the onset time $Sq(kt,\Delta_h)$6 and then grows with interaction time for both equal and unequal mode intensities (Singh et al., 2010). The dependence on interaction time is critical, while the dependence on $N_x,N_y$ is described as meager (Singh et al., 2010).

A closely related treatment of HOPS in degenerate parametric amplification uses a monochromatic double-mode coherent input and introduces a squeezing function $Sq(kt,\Delta_h)$ together with a "degree of Hidden Optical-Polarization" $H(t)$ as a non-classicality measure (Gupta et al., 2010). In that account, squeezing occurs when $Sq(kt,\Delta_h)$7 and non-classical hidden polarization is identified by $Sq(kt,\Delta_h)$8 This use of the term remains specific to hidden polarization observables rather than general quadrature squeezing (Gupta et al., 2010).

3. Complex covariance and the failure of standard homodyne detection

A major modern meaning of hidden squeezing is tied to continuous fields with frequency-dependent covariance matrices. "Complex Squeezing and Force Measurement Beyond the Standard Quantum Limit" shows that homodyne detectors are blind to squeezing spectra in which the correlation between amplitude and phase fluctuations is complex (Buchmann et al., 2016). For a continuous field, standard homodyne measures $Sq(kt,\Delta_h)$9 so only $\mathrm{Re}([\mathcal C(\omega)]_{12})$ contributes (Buchmann et al., 2016). If the relevant amplitude–phase covariance is purely imaginary, the field can be strongly correlated yet appear unsqueezed in homodyne readout.

The optomechanical example is paradigmatic. With linearized Hamiltonian $H(t)$0 the optical output quadratures obey $H(t)$1 $H(t)$2 with $H(t)$3 At $\omega=\omega_m$, the susceptibility $\chi_m(\omega_m)$ is purely imaginary, making the induced amplitude–phase correlations purely imaginary as well. The strongest squeezing is therefore hidden from homodyne at resonance (Buchmann et al., 2016).

This diagnosis was experimentally corroborated in microwave electromechanics. "Revealing hidden quantum correlations in an electromechanical measurement" observes ponderomotive squeezing in a superconducting microwave resonator–drumhead system and shows that standard balanced homodyne cannot access the full complex correlation matrix (Ockeloen-Korppi et al., 2018). The measured quadrature noise with a single local oscillator is $H(t)$4 again discarding $\mathrm{Im}[S_{XY}]$ (Ockeloen-Korppi et al., 2018). Using ordinary homodyne detection, the experiment observed a maximum squeezing of about 1.1 dB near $\chi^{(2)}$0 (Ockeloen-Korppi et al., 2018), but a bi-chromatic local oscillator recovered a stronger reduction corresponding to the smallest eigenvalue of the full correlation matrix.

The same structural issue appears in semiconductor lasers. "Complex frequency-dependent quadrature squeezing in semiconductor lasers" computes the frequency-resolved squeezing map from a fully quantum Langevin model and identifies hidden or complex squeezing in the output field (Nello et al., 24 Jun 2026). Standard homodyne accesses $H(t)$5 whereas the full covariance matrix contains the complex cross term $\chi^{(2)}$1 (Nello et al., 24 Jun 2026). The paper attributes the emergence of frequency-dependent and hidden squeezing in large part to the Henry linewidth enhancement factor $\chi^{(2)}$2, which mixes amplitude and phase noise through carrier dynamics (Nello et al., 24 Jun 2026).

4. Measurement schemes that reveal hidden squeezing

Because hidden squeezing is often a measurement-basis problem, several works focus on readout architectures that recover the inaccessible covariance components. The foundational proposal is synodyne detection, introduced for optomechanics as a two-tone local-oscillator scheme $H(t)$6 With this choice, the detected noise at $\chi^{(2)}$3 becomes $H(t)$7 so the complex phase of $\chi^{(2)}$4 can be exploited, and the noise can approach the smallest eigenvalue $\chi^{(2)}$5 (Buchmann et al., 2016). The same work connects this to force sensing beyond the standard quantum limit, since the local oscillator can combine AM back-action information and PM signal information with the phase needed for back-action accounting (Buchmann et al., 2016).

The electromechanical experiment implemented a closely related bi-chromatic local oscillator $H(t)$8 leading to a zero-frequency rotating-frame spectrum $H(t)$9 By tuning amplitudes and phases, the experiment selected the eigenvectors of the quadrature correlation matrix and accessed the smallest and largest eigenvalues directly (Ockeloen-Korppi et al., 2018).

In multimode integrated photonics, the measurement problem is more severe because the optimal mode itself changes with analysis frequency. "Hidden and detectable squeezing from micro-resonators" studies silicon and silicon nitride micro-resonators in a pulsed synchronously pumped regime and diagonalizes the transfer matrix via Bloch-Messiah decomposition, $\mathrm{Re}([\mathcal C(\omega)]_{12})$0 In the morphing-supermode basis, $\mathrm{Re}([\mathcal C(\omega)]_{12})$1 the independent squeezed modes have singular values $\chi^{(2)}$6 (Gouzien et al., 2022). Hidden squeezing is the gap between these ideal squeezing levels and what can be observed by standard homodyne with a physically realizable real local oscillator profile. The obstruction is twofold: the optimal supermode column $\chi^{(2)}$7 is typically frequency dependent and often complex-valued (Gouzien et al., 2022). The paper suggests pump engineering and dispersion engineering as routes to reduce the hidden fraction and notes that a fully general recovery would require an interferometer with memory effects capable of implementing a complex frequency-dependent local oscillator (Gouzien et al., 2022).

A plausible implication is that hidden squeezing should be regarded not only as a property of a state but also as a joint property of state, observable, and mode-matching constraint. This interpretation is explicit in the generalized witness framework, where one minimizes the variance over free transformations, $\mathrm{Re}([\mathcal C(\omega)]_{12})$2 and thereby treats squeezing as a resource certified only after basis optimization (Bräuer et al., 12 Mar 2025).

5. Multimode, coupled, and distributed manifestations

Hidden squeezing frequently appears in systems where the nonclassical resource is distributed across multiple modes, subsystems, or transformed degrees of freedom rather than concentrated in a single obvious quadrature.

A clear example is the concurrent generation of atomic spin squeezing and optical squeezing in a hot $\chi^{(2)}$8 ensemble under a stroboscopically engineered symmetric atom–light interaction (Jin et al., 2023). Using the Holstein–Primakoff mapping, $\mathrm{Re}([\mathcal C(\omega)]_{12})$3 $\mathrm{Re}([\mathcal C(\omega)]_{12})$4 the effective Hamiltonian couples the atomic mode to optical sidebands, $\mathrm{Re}([\mathcal C(\omega)]_{12})$5 and can be rewritten as a beam-splitter interaction with a squeezed Bogoliubov optical mode $\mathrm{Re}([\mathcal C(\omega)]_{12})$6 The experiment reported proof-of-principle concurrent squeezing at the same setting $\chi^{(2)}$9, $kt \approx 0.22$0: $kt \approx 0.22$1 for spin and $kt \approx 0.22$2 for light (Jin et al., 2023). The optical squeezing resides in multiple odd-frequency sidebands $kt \approx 0.22$3, $kt \approx 0.22$4, of a single spatial mode (Jin et al., 2023). This suggests a multimode notion of hidden squeezing in which the nonclassicality is encoded in sideband structure and atom–light normal modes rather than in a single carrier quadrature.

Another distributed manifestation appears in ultrastrong cavity QED. "Output Field-Quadrature Measurements and Squeezing in Ultrastrong Cavity-QED" shows that squeezing can be present in dressed intracavity variables while remaining absent from the propagating output field if the system is in its ground state (Stassi et al., 2015). The correct input-output relation is $\mathrm{Re}([\mathcal C(\omega)]_{12})$7 where $kt \approx 0.22$5 are positive and negative frequency parts defined in the dressed eigenbasis of the full interacting Hamiltonian (Stassi et al., 2015). Since $kt \approx 0.22$6 for the ground state, no output squeezing appears even if the dressed ground state has bare-photon squeezing-like features (Stassi et al., 2015). Observable squeezing arises only after appropriate driving prepares superpositions such as $kt \approx 0.22$7 or $kt \approx 0.22$8 (Stassi et al., 2015). In this usage, the squeezing is hidden in the dressed-state structure and virtual-photon content.

The idea can even be formulated beyond quadrature and polarization. "Coherence squeezing in optical interference" introduces Hermitian slit-coherence operators $\mathrm{Re}([\mathcal C(\omega)]_{12})$8 with $\mathrm{Re}([\mathcal C(\omega)]_{12})$9 and uncertainty relation $\omega=\omega_m$0 Coherence squeezing is defined by $\omega=\omega_m$1 relative to a two-mode coherent-state benchmark (Hanhisalo et al., 26 Feb 2026). The squeezing is hidden because it is not directly a field quadrature; it appears in the coherence sector and manifests indirectly through uncertainties of fringe magnitude and fringe position in double-slit interference (Hanhisalo et al., 26 Feb 2026).

6. Hidden squeezing as concealment, protection, or internal processing

Not all uses of the term refer to inaccessible observables. In some work, squeezing is hidden because the state is deliberately transformed into a less vulnerable basis and later restored.

"Reduced decoherence using squeezing, amplification, and anti-squeezing" studies non-Gaussian optical states, especially Schrödinger cat states, sent through a lossy channel (Brewster et al., 2018). The strategy is to squeeze the signal before transmission, pass it through loss, then deterministically amplify and anti-squeeze it. The single-mode squeezing operator is $\omega=\omega_m$2 with inverse $kt \approx 0.22$9 (Brewster et al., 2018). By reducing the separation of the cat components along the transmission-sensitive quadrature, the protocol lowers environmental which-path information during both loss and amplification (Brewster et al., 2018). The amplifier-induced idler photon number $\omega=\omega_m$3 is correspondingly reduced for a suitably squeezed input (Brewster et al., 2018). Here hidden squeezing means that part of the signal is temporarily concealed in a squeezed quadrature and recovered later by anti-squeezing. The squeezing is operationally useful even though it is not the final observable output state.

A related but more internalized concept appears in gravitational-wave interferometry. "Bidirectional Internal Squeezing for Gravitational-Wave Detectors" places two optical parametric amplifiers inside the signal-extraction cavity so that counter-propagating intracavity fields are squeezed on the inward pass and amplified on the outward pass (Vermeulen et al., 15 May 2026). The scheme does not inject an external squeezed vacuum; instead, the interferometer’s internal optical network reshapes the noise before readout. In the ideal balanced high-gain limit, the signal-referred quantum shot noise reaches the internal-dissipation Callen–Welton bound (Vermeulen et al., 15 May 2026). This suggests a form of hidden squeezing embedded in the internal cavity dynamics rather than visible as an externally supplied squeezed state.

Mechanical systems furnish an additional variant. In detuned parametric amplification with weak continuous measurement, the mechanical state can have stronger squeezing than is immediately visible in a fixed measured quadrature because the parametric drive creates nonzero covariance $\omega=\omega_m$4 Weak measurement can then exploit those correlations to infer the squeezed quadrature more precisely, surpassing the traditional steady-state 3 dB limit (Szorkovszky et al., 2011). Likewise, in nonlinear atomic force microscopy, a cantilever can exhibit strong squeezing in the covariance matrix of fluctuations while the mean trajectory appears ordinary; the squeezing is hidden in the phase-space distribution rather than in mean motion (Marzlin et al., 2021).

7. Unifying themes, misconceptions, and research directions

A common misconception is that hidden squeezing is a single, standardized concept. The literature shows instead that it is a family of related ideas organized around three recurring motifs.

First, hidden squeezing often reflects an observable mismatch. Standard Stokes parameters miss HOPS (Singh et al., 2010); standard homodyne misses imaginary covariance terms in optomechanics and semiconductor lasers (Buchmann et al., 2016, Ockeloen-Korppi et al., 2018, Nello et al., 24 Jun 2026); real fixed local oscillators miss complex morphing supermodes in micro-resonators (Gouzien et al., 2022). In these cases the squeezing is present in the covariance structure but absent from naive readout.

Second, hidden squeezing can reflect a mode or basis transformation. Bogoliubov optical sideband modes in atom–light interfaces (Jin et al., 2023), dressed positive-frequency operators in ultrastrong cavity QED (Stassi et al., 2015), and coherence operators in interference theory (Hanhisalo et al., 26 Feb 2026) all define non-obvious variables in which squeezing becomes the natural descriptor.

Third, hidden squeezing can be operational rather than merely descriptive. Squeezing can be used as a resource witness after optimization over allowed transformations (Bräuer et al., 12 Mar 2025), as a method for preserving non-Gaussian coherence during transmission (Brewster et al., 2018), or as an internally generated noise-shaping mechanism in interferometers (Vermeulen et al., 15 May 2026).

The principal technical challenge across these settings is not always state generation but state access. The recurring requirement is mode-matched, basis-adapted, or transformation-aware measurement. This suggests that future progress will depend at least as much on generalized detection architectures, covariance-based diagnostics, and resource-aware benchmarks as on stronger nonlinearities alone. A plausible implication is that the practical boundary between detectable squeezing and hidden squeezing will increasingly be determined by experimental control of measurement basis, temporal mode, sideband structure, and admissible transformations rather than by the intrinsic nonclassicality of the source itself.