---
title: 'Squeezed Light: Quantum Noise and Applications'
url: https://www.emergentmind.com/topics/squeezed-light
type: topic
---

# Squeezed Light: Quantum Noise and Applications

Squeezed light is a nonclassical electromagnetic-field state in which quantum fluctuations are redistributed between conjugate field quadratures so that one selected quadrature has variance below the vacuum or shot-noise level, while the orthogonal quadrature is anti-squeezed. For a bosonic mode with annihilation operator $\hat a$, quadratures may be written as $\hat X=(\hat a+\hat a^\dagger)/\sqrt{2}$ and $\hat P=(\hat a-\hat a^\dagger)/(i\sqrt{2})$, with $[\hat X,\hat P]=i$. A phase-rotated quadrature is $\hat X_\theta=(e^{-i\theta}\hat a+e^{i\theta}\hat a^\dagger)/\sqrt{2}$; squeezing occurs when $\Delta X_\theta^2<1/2$ in this normalization. The uncertainty principle is not violated: reduced fluctuations in one quadrature are accompanied by increased fluctuations in its conjugate. Squeezed states were theoretically described in the early 1970s, first observed in the mid-1980s, and developed into an operational technology for quantum metrology, continuous-variable quantum information, interferometry, integrated photonics, and precision microscopy [1401.4118] [2307.08394].

## 1. Physical definition and quantum-state structure

For vacuum and coherent states, both quadratures have equal variance. A pure squeezed state has an elliptical Gaussian Wigner function rather than the circular Gaussian associated with vacuum. If the squeeze parameter is $r$, an appropriate orientation gives

$$
\Delta X_\theta^2=\frac{1}{2}e^{-2r},\qquad
\Delta X_{\theta+\pi/2}^2=\frac{1}{2}e^{2r}.
$$

The product of the two variances remains at the minimum allowed by the uncertainty relation for a pure state. A squeezed vacuum has zero coherent displacement, whereas a displaced squeezed state, or bright squeezed light, combines a coherent amplitude with reduced fluctuations in a selected quadrature. Amplitude squeezing reduces amplitude or photon-number fluctuations; phase squeezing reduces phase fluctuations.

Squeezing is commonly expressed in decibels relative to the vacuum or shot-noise variance:

$$
S_{\mathrm{dB}}=10\log_{10}\left(\frac{V}{V_{\mathrm{SQL}}}\right).
$$

Negative values denote noise below the reference. A reduction of $3~\mathrm{dB}$ corresponds to a variance one-half of the shot-noise variance; $10~\mathrm{dB}$ corresponds to a variance one-tenth of the reference. The decibel convention concerns variance or noise power, not optical power or field amplitude.

The single-mode squeeze operator is

$$
\hat S(\zeta)=
\exp\left[\frac{1}{2}
\left(\zeta\hat a^2-\zeta^*\hat a^{\dagger2}\right)\right],
$$

where $\zeta=re^{i\phi}$. Its Bogoliubov transformation mixes annihilation and creation operators. An ideal squeezed vacuum contains only even photon numbers, reflecting pairwise photon generation. Displacement, loss, and detector inefficiency introduce odd-photon contributions and generally make the state mixed.

Squeezing is nonclassical because its Glauber–Sudarshan $P$-representation cannot be interpreted as a positive classical probability distribution over coherent states. A Gaussian squeezed state can nevertheless possess a positive Wigner function. Wigner-function negativity is therefore not required for quadrature squeezing. Sub-Poissonian photon statistics, characterized by a Fano factor below unity, commonly accompany amplitude squeezing in bright fields.

## 2. Single-mode, two-mode, and multimode squeezing

Single-mode squeezing reduces fluctuations of one quadrature of one optical mode and increases those of the conjugate quadrature. Two-mode squeezing instead acts on two distinct modes, conventionally $a$ and $b$. The two-mode squeezing operator is

$$
\hat S_2(\zeta)=
\exp\left(-\zeta\hat a\hat b+\zeta^*\hat a^\dagger\hat b^\dagger\right).
$$

The two-mode squeezed vacuum has the form

$$
|\mathrm{TMSV}\rangle
=
\frac{1}{\cosh r}
\sum_{n=0}^{\infty}
(\tanh r)^n|n\rangle_a|n\rangle_b.
$$

Its individual modes are thermal and are not locally squeezed. The nonclassicality appears in joint observables such as $X_a-X_b$ and $P_a+P_b$, whose variances can fall below the corresponding vacuum or separable-state bounds. These EPR-type correlations constitute an entangled continuous-variable resource.

A balanced beam splitter converts a two-mode squeezed vacuum into two single-mode squeezed vacua with orthogonal squeezing axes. Conversely, two single-mode squeezed states with orthogonal squeeze angles can be combined to form a two-mode squeezed state. In frequency-domain measurements, a single temporal mode is often equivalent to two-mode squeezing between symmetric sidebands at $\Omega+\nu$ and $\Omega-\nu$.

The term multimode may refer to temporal, spectral, polarization, or transverse spatial modes. In long-pulse or continuous-wave squeezed light, photons can have strong correlations in time and frequency even when the instantaneous photon density is small. A Schmidt or Takagi decomposition diagonalizes the squeezing kernel into independent global modes, but those modes can extend over the entire pulse. A Whittaker–Shannon construction instead samples an approximately band-limited joint temporal amplitude on a grid with spacing comparable to the coherence time, yielding localized temporal packets and a generally off-diagonal squeezing matrix. This representation makes local pair correlations and correlations between neighboring temporal cells explicit [2310.10919].

Spatial multimode correlations have also been transported through plasmonic structures. Four-wave mixing in a cavity-free $^{85}\mathrm{Rb}$ vapor source generated correlated probe and conjugate beams, and a silver nanohole array transmitted the probe through localized surface plasmon resonances. At approximately $36\%$ extraordinary optical transmission, $1.28~\mathrm{dB}$ of intensity-difference squeezing remained measurable, while a structured cross image generated with a spatial light modulator remained intact. The result was consistent with coherent photon–localized-surface-plasmon–photon transduction through a lossy linear channel [1209.3754].

## 3. Generation mechanisms

### Parametric down-conversion and optical parametric oscillation

The principal conventional source is second-order nonlinear parametric down-conversion. A pump photon is converted into signal and idler photons according to $\omega_p=\omega_s+\omega_i$. Degenerate operation produces single-mode squeezed vacuum; nondegenerate operation produces two-mode squeezed vacuum. Optical parametric amplifiers and optical parametric oscillators enhance the interaction by placing the nonlinear medium in a resonator and operating below threshold.

Thin-film lithium-niobate devices have integrated second-harmonic generation, optical-parametric amplification, local-oscillator routing, phase control, and balanced homodyne detection on approximately $1~\mathrm{cm^2}$ chips. A monolithic thin-film lithium-niobate OPO generated and characterized degenerate squeezed vacuum using $20~\mathrm{mW}$ of input power, with $0.55~\mathrm{dB}$ measured squeezing and $1.55~\mathrm{dB}$ anti-squeezing [2310.12954]. A modal-phase-matched TFLN ring, avoiding periodic poling by matching a $1550~\mathrm{nm}$ $\mathrm{TE}_0$ mode to a $775~\mathrm{nm}$ $\mathrm{TM}_2$ mode, produced $0.46~\mathrm{dB}$ measured squeezing and an inferred $1.5~\mathrm{dB}$ on-chip value [2406.07425].

Bright pulsed squeezing is useful where the signal is encoded in optical intensity. A periodically poled MgO-doped lithium-niobate waveguide produced $-3.2~\mathrm{dB}$ bright amplitude squeezing in approximately $6$-ps pulses at an optical power of approximately $3.2~\mathrm{mW}$, alongside $-3.6~\mathrm{dB}$ vacuum squeezing. A loss-corrected estimate gave $-15.4^{+2.7}_{-8.7}~\mathrm{dB}$ generated in the waveguide [2601.15565].

### Four-wave mixing and Kerr interactions

Third-order four-wave mixing annihilates two pump photons and creates signal–idler pairs. Atomic four-wave mixing in Rb vapor generated approximately $4.5~\mathrm{dB}$ of intensity-difference squeezing before transmission through a plasmonic array [1209.3754]. Fiber Kerr squeezing arises from the intensity-dependent refractive index $n=n_0+n_2I$, which shears the optical state in phase space.

Integrated silicon-nitride resonators provide strong field enhancement and $\chi^{(3)}$ nonlinear interaction. A photonic molecule consisting of two coupled silicon-nitride microrings selectively hybridized resonances associated with parasitic spontaneous and Bragg-scattering four-wave mixing while preserving the desired pump and signal resonances. It generated $8(1)~\mathrm{dB}$ inferred on-chip degenerate squeezing and $1.65(1)~\mathrm{dB}$ directly measured squeezing [2001.09474].

A silicon-nitride microring has also generated bright single-mode coherent-squeezed light through degenerate four-wave mixing at the pump frequency. The measured detector noise reduction was $-1.219~\mathrm{dB}$, corresponding to approximately $-4.7~\mathrm{dB}$ inferred inside the chip [2502.16278]. In this configuration the same resonator stores the pump, generates the nonlinear correlations, and supplies the coherent carrier.

### Optomechanical and matter-mediated mechanisms

Ponderomotive squeezing results when radiation-pressure shot noise drives a mechanical resonator and the resulting displacement is written back onto the optical field as a phase fluctuation. A silicon micromechanical resonator coupled to a nanophotonic cavity produced $4.5\pm0.5\%$ noise reduction below shot noise near a $28~\mathrm{MHz}$ mechanical resonance, despite a mechanical occupation of approximately $10^4$ phonons [1302.6179]. A membrane in a Fabry–Perot cavity produced $1.7\pm0.2~\mathrm{dB}$ detected optomechanical squeezing, with the orthogonal quadrature reaching approximately $25~\mathrm{dB}$ above shot noise [1306.1268].

Other proposed or demonstrated matter-mediated mechanisms include magnetostrictive cavity magnomechanics, exciton–phonon cavity QED, quantum-dot microcavities, and phase-engineered two-level emitters. A magnomechanical proposal transfers squeezing generated by magnetostrictive magnon–phonon interaction to a microwave cavity and predicts approximately $5.2~\mathrm{dB}$ stationary microwave squeezing for representative parameters [2101.02796]. An exciton–phonon microcavity model predicts approximately $7~\mathrm{dB}$ optical output squeezing, potentially approaching $10~\mathrm{dB}$ for reduced exciton loss; room-temperature operation requires materials with high exciton binding energy [2408.09323]. A microscopic quantum-dot cavity-QED model predicts up to approximately $5~\mathrm{dB}$ amplitude squeezing near the onset of stimulated emission using an incoherent pump, coherent seed, and four-wave-mixing correlations [2508.15114].

A theoretical cavity containing two coherently driven two-level qubits has been proposed to produce squeezing without an externally supplied nonlinear optical medium. Opposite cavity couplings, $g_1=-g_2$, cancel selected single-photon excitation pathways while preserving multiphoton pathways differing by two photons. The resulting even–odd photon-number structure and anomalous correlations produce quadrature squeezing in a hyperradiant regime [2205.06752].

## 4. Detection, characterization, and loss

Balanced homodyne detection combines the signal with a strong local oscillator on a balanced beam splitter. The difference photocurrent is proportional to a selected quadrature,

$$
\hat i_-(t)\propto
\alpha_{\mathrm{LO}}
\left(
\hat a e^{-i\theta}
+\hat a^\dagger e^{i\theta}
\right).
$$

Scanning the local-oscillator phase identifies the squeezed and anti-squeezed axes. A single photodiode generally measures amplitude fluctuations, whereas homodyne detection can measure arbitrary quadratures and is required for complete quadrature tomography. Frequency-domain spectrum analysis measures noise spectra and sideband correlations but does not by itself provide complete state tomography.

For a Gaussian state, the covariance matrix contains the state information apart from mean displacement. Its eigenvalues give the minimum and maximum quadrature variances, and its eigenvectors determine the squeeze angle. Homodyne distributions for all local-oscillator phases can be used for inverse-Radon reconstruction of the Wigner function.

Loss is equivalent to mixing the state with vacuum on a beam splitter. In normalized shot-noise units,

$$
V_{\mathrm{det}}=\eta V_{\mathrm{out}}+(1-\eta).
$$

Consequently, loss always moves a squeezed variance toward unity. The same relation applies to propagation loss, cavity intrinsic loss, imperfect mode matching, detector inefficiency, fiber coupling, and incomplete escape from a resonator. Large anti-squeezing makes phase noise particularly damaging because random rotation of the squeezing ellipse mixes the anti-squeezed quadrature into the measured quadrature.

The difference between directly observed and inferred intrinsic squeezing is therefore essential. The silicon-nitride photonic molecule produced $1.65(1)~\mathrm{dB}$ directly measured and $8(1)~\mathrm{dB}$ inferred on-chip squeezing [2001.09474]. The all-fiber entanglement-assisted source produced $7.5\pm0.1~\mathrm{dB}$ directly measured squeezing for a coherent mode, $5.1~\mathrm{dB}$ for a partially coherent mode, and $1.1~\mathrm{dB}$ for a chaotic amplified-spontaneous-emission mode [2406.19991]. The differing values reflect mode overlap, classical noise, propagation loss, polarization rotation, and detection efficiency rather than distinct definitions of squeezing.

## 5. Applications and demonstrated systems

### Interferometry and gravitational-wave detection

In an interferometer, phase-quadrature fluctuations generate shot noise, while amplitude fluctuations generate radiation-pressure back-action. Increasing optical power reduces shot noise but increases radiation-pressure noise. Appropriately rotated, frequency-dependent squeezing can correlate these noise sources and reduce both over different frequency bands.

Squeezed vacuum is injected into the dark port of gravitational-wave interferometers. GEO600 measured up to approximately $3.5$–$3.7~\mathrm{dB}$ improvement in quantum-noise-limited sensitivity, corresponding to a noise variance of approximately $0.45$ of the unsqueezed value [1611.03986]. Squeezed light has been used in GEO600, Advanced LIGO, and Advanced Virgo; LIGO reported up to approximately $2.15$–$2.2~\mathrm{dB}$ sensitivity improvement above approximately $150~\mathrm{Hz}$ [2307.08394]. Frequency-dependent squeeze-angle rotation, often implemented with filter cavities, is required when radiation-pressure noise becomes important at low frequencies.

Two-mode squeezing has a different role in quantum-dense metrology. One mode can be injected into an interferometer while its entangled partner is retained as a reference. Joint measurements can distinguish genuine phase signals from arbitrary phase-space disturbances such as back-scattered light.

### Quantum information and communications

Two-mode squeezed vacuum is a central resource for continuous-variable teleportation, dense coding, continuous-variable quantum key distribution, measurement-based quantum computing, and cluster-state generation. Squeezed states also support Gaussian operations, frequency-multiplexed processing, entanglement distribution, and non-Gaussian state engineering through photon subtraction or conditional detection.

An integrated silicon-nitride platform has demonstrated a complete on-chip path comprising Kerr microresonator, tunable ring filter, multimode-interference beam splitter, and balanced modified uni-traveling-carrier photodiodes. The device measured approximately $3~\mathrm{dB}$ squeezing in a two-mode squeezed quantum microcomb containing $34$ frequency-bin modes organized as $17$ symmetric pairs. Its post-squeezer circuit had $1.1~\mathrm{dB}$ optical loss and approximately $72\%$ on-chip quantum efficiency [2608.13218].

Topological photonic structures provide another route to robust multimode sources. An SSH-like fused-silica waveguide array localized pump, signal, and idler fields at protected edge and interface states. The protected channels retained signal–idler cross-correlations approximately five times larger than an unprotected trivial channel over the demonstrated propagation distances [2106.07425].

### Sensing, microscopy, and spectroscopy

Squeezing improves measurements when the signal is encoded in the low-noise quadrature. Applications include phase, force, displacement, spectroscopy, biological measurements, and nonlinear microscopy. In stimulated Raman scattering microscopy, bright amplitude squeezing can improve signal-to-noise ratio at fixed illumination power or provide the precision of a brighter classical beam with lower optical dose. The pulsed lithium-niobate waveguide source was developed specifically for this photodamage-limited regime [2601.15565].

Squeezed light has also been generated with ultrafast degenerate four-wave mixing in MgO. Strong-field modulation of the dielectric response controlled the effective $\chi^{(3)}$ on sub-cycle timescales, switching the output between amplitude and phase squeezing. Frequency-resolved balanced homodyne detection reconstructed multimode covariance and coherency matrices, with approximately $5~\mathrm{dB}$ squeezing reported [2512.17046].

### Plasmonic and nanoscale photonics

The survival of two-mode intensity correlations through an Ag localized-surface-plasmon extraordinary-optical-transmission array demonstrated that plasmonic resonances need not automatically destroy nonclassical correlations. Such structures may interface freely propagating quantum fields with nanoscale optical circuitry, although low-transmission regimes exhibited deviations from a simple attenuation model and possible mode-dependent degradation [1209.3754].

## 6. Limitations, engineering requirements, and directions

The principal limitation across squeezed-light platforms is loss. Even a source with strong intrinsic squeezing exhibits modest detector squeezing when escape, propagation, coupling, mode matching, and detector efficiencies are not close to unity. A $10$-dB state can be substantially degraded by relatively small loss because the anti-squeezed quadrature amplifies phase fluctuations and mode mismatch.

Other recurring limitations include finite nonlinear bandwidth, pump fluctuations, phase drift, cavity detuning, thermal noise, parasitic nonlinear processes, scattering, photorefractive resonance shifts, mechanical decoherence, spontaneous emission, and incomplete characterization of multimode transfer functions. In optomechanical and exciton–phonon systems, thermal motion adds uncorrelated noise rather than generating the squeezing itself. In integrated resonators, parasitic spontaneous and Bragg-scattering four-wave mixing can contaminate the desired mode. Engineered mode hybridization, normal dispersion, topological localization, and multimode spectral filtering are approaches to controlling these effects.

Integrated photonics seeks to reduce alignment sensitivity, footprint, pump power, and control complexity while preserving quantum correlations. Thin-film lithium niobate provides $\chi^{(2)}$ nonlinearity and fast electro-optic modulation; silicon nitride provides low-loss waveguides, high-$Q$ resonators, and $\chi^{(3)}$ four-wave mixing; heterogeneous integration combines low-loss transport with absorbing photodetectors on a common chip [2608.13218]. Modal phase matching avoids periodic poling but uses a higher-order pump mode and the weaker $d_{31}$ interaction [2406.07425]. Photonic molecules and topological waveguide arrays engineer the mode landscape to suppress unwanted processes or propagation crosstalk.

Future directions include higher escape efficiency, lower-loss couplers, improved detector quantum efficiency, active phase and resonance stabilization, frequency-dependent squeezing, larger multimode sources, quantum frequency conversion for mode isolation, and integrated non-Gaussian detection. Proposed mechanisms based on exciton–phonon interactions, quantum-dot four-wave mixing, magnetostriction, hyperradiant interference, and attosecond nonlinear-response control expand the material and dynamical regimes in which squeezing may be produced. These proposals remain subject to stability, thermal, dephasing, fabrication, and loss constraints.

Squeezed light therefore comprises a family of Gaussian and non-Gaussian quantum states rather than a single source technology. Its defining property is sub-vacuum noise in a selected quadrature or joint observable; its practical value depends on preserving that reduction through generation, mode selection, transmission, phase control, and detection. The progression from bulk nonlinear optics to plasmonic interfaces, mechanical mediators, fiber systems, integrated resonators, topological waveguides, and heterogeneous source-and-detector chips reflects an increasing emphasis on controllable multimode structure, low loss, and application-specific quadrature engineering.

Source: https://www.emergentmind.com/topics/squeezed-light