Squeezed Graviton States

• Squeezed states of gravitons are nonclassical quantum states where squeezing operators redistribute uncertainty between conjugate field quadratures in linearized gravity.
• The topic covers production mechanisms from inflationary expansion to astrophysical sources like binary systems and axion clouds, yielding significant squeezing parameters that can reach values of 50–100.
• These states affect detector responses, interferometric noise, and quantum correlations, providing practical insights for probing quantum gravitational effects and modifications in effective field theories.

Squeezed states of gravitons are nonclassical quantum states of the linearized gravitational field in which quantum uncertainty is redistributed between conjugate field quadratures, or, in the cosmological two-mode formulation, between correlated modes of momenta $\mathbf{k}$ and $-\mathbf{k}$. They are generated by squeezing operators acting on the graviton vacuum or on displaced vacua, and they arise in several settings emphasized in the recent literature: inflationary expansion, reheating, time-dependent quadratic couplings in linearized gravity, binary motion, superradiant axion clouds around Kerr black holes, and strong-field shockwave scattering (Parikh et al., 2020, Kanno et al., 2021, Das et al., 23 Dec 2025, Dorlis et al., 14 May 2026, Staśto et al., 4 May 2026). Their importance lies in the fact that they modify graviton two-point functions, detector transition probabilities, interferometric strain noise, polarization correlations, and, in some analyses, even the short-distance structure of quantum field theory through light-cone smearing (Parikh et al., 2020, Trenggana et al., 28 Apr 2025, Matsui, 7 May 2026).

1. Formal definition and state structure

In linearized quantum gravity, the metric perturbation is expanded in harmonic-oscillator modes with annihilation and creation operators. In the single-mode description, a squeezed graviton state is generated by

$$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$

and the squeezed vacuum is

$|\zeta\rangle=S(\zeta)|0\rangle.$

For each mode one introduces quadratures

$$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$

with variances, after rotating by the squeeze angle $\phi$,

$$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$

Thus the amplitude $r$ controls exponential suppression in one quadrature and exponential enhancement in the conjugate one, while $\phi$ fixes the squeezed axis in phase space (Parikh et al., 2020).

In cosmology and in most detector calculations, the more natural object is the two-mode squeezed state of paired modes $\mathbf{k}$ and $-\mathbf{k}$0,

$-\mathbf{k}$1

Acting on the out-vacuum, this produces an entangled superposition of paired graviton number states, with coefficients proportional to $-\mathbf{k}$2 and an overall normalization $-\mathbf{k}$3 (Kanno et al., 2021).

Several recent works formulate the problem directly in a multimode language. The evolution operator then takes the generic form

$-\mathbf{k}$4

where $-\mathbf{k}$5 label momentum and polarization or helicity. In this representation the effective squeezing strength is often defined by a mode-sum such as $-\mathbf{k}$6, while the mean graviton number in the squeezed vacuum obeys $-\mathbf{k}$7 (Dorlis et al., 14 May 2026). A related Takagi-like decomposition diagonalizes the complex symmetric squeezing kernel into independent squeezed collective modes, each with quadrature variances $-\mathbf{k}$8 and $-\mathbf{k}$9 (Dorlis et al., 31 Jul 2025).

These constructions make clear that “squeezed graviton state” does not denote a single unique state. The literature uses single-mode, two-mode, and multimode formulations, but all share the same defining feature: a Bogoliubov mixing of annihilation and creation operators and an exponential quadrature asymmetry.

2. Cosmological origin: inflation, reheating, and large squeezing

The cosmological mechanism is the best developed. Expansion mixes positive- and negative-frequency graviton modes and unitarily generates two-mode squeezing of the $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$0 pair (Kanno et al., 2021). In de Sitter inflation the tensor mode equation takes the standard form with $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$1, and several papers derive large late-time squeezing for super-Hubble modes. One framework gives

$$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$2

with $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$3, so each Hubble time adds an $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$4 increment to the squeeze parameter (Das et al., 23 Dec 2025). Another analysis writes, in the super-horizon limit,

$$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$5

and reports $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$6 for sub-Hz to $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$7 modes exiting the horizon roughly $$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$8 e-folds before the end of inflation (Kanno et al., 2022).

The comparison between instantaneous and adiabatic vacua for massless gravitons shows essentially no qualitative difference in the squeezing amplitude. For a de Sitter phase joined to radiation domination, both schemes yield

$$S(\zeta)=\exp\!\left[\tfrac12\left(\zeta^* a^2-\zeta (a^\dagger)^2\right)\right], \qquad \zeta=r e^{i\phi},$$9

and for benchmark values $|\zeta\rangle=S(\zeta)|0\rangle.$0, $|\zeta\rangle=S(\zeta)|0\rangle.$1, and $|\zeta\rangle=S(\zeta)|0\rangle.$2, one obtains $|\zeta\rangle=S(\zeta)|0\rangle.$3 (Kanno et al., 2021). In the same analysis, introducing a finite-duration reheating stage enhances the result from approximately $|\zeta\rangle=S(\zeta)|0\rangle.$4 to approximately $|\zeta\rangle=S(\zeta)|0\rangle.$5, corresponding to an $|\zeta\rangle=S(\zeta)|0\rangle.$6 increase (Kanno et al., 2021).

The cosmological consequences are not limited to the power spectrum. The same two-mode squeezed vacua enhance Bell-inequality violation, with

$|\zeta\rangle=S(\zeta)|0\rangle.$7

and they amplify the anticommutator noise correlator by an $|\zeta\rangle=S(\zeta)|0\rangle.$8 term (Kanno et al., 2021). At the same time, cosmological magnetic fields appear to induce only weak decoherence of primordial graviton squeezing: one study finds $|\zeta\rangle=S(\zeta)|0\rangle.$9, while graviton-to-photon conversion during inflation remains at a few percent at most (Kanno et al., 2022).

The cosmological literature therefore treats graviton squeezing not as an exotic add-on but as a generic consequence of time-dependent backgrounds in the quadratic graviton action.

3. Source-driven and astrophysical production mechanisms

A broader framework states the point in general form: time-dependent couplings in the quadratic part of the linearized gravitational action generically produce squeezed states from the vacuum, so both time-dependent classical spacetimes and time-dependent classical matter typically produce squeezed states of gravity (Das et al., 23 Dec 2025). In this sense, inflation is a distinguished example rather than an isolated one.

For time-varying matter sources, the quadratic interaction Hamiltonian contains terms proportional to $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$0, and resonant growth appears when $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$1 (Das et al., 23 Dec 2025). In a circular Newtonian orbit, the dominant channel can exhibit linear-in-time growth,

$$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$2

which formalizes the statement that classical matter motion can squeeze graviton modes (Das et al., 23 Dec 2025). A related binary-source proposal argues that astrophysical binaries displace a primordial squeezed graviton vacuum into a squeezed-coherent state, so that binary gravitational waves can probe nonclassical initial graviton statistics rather than creating those statistics from scratch (Kanno et al., 27 Oct 2025).

The most developed astrophysical mechanism beyond binaries involves superradiant axion clouds around rotating black holes. In Kerr backgrounds, bosonic modes satisfy a superradiant instability condition

$$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$3

so long-lived quasi-bound clouds with large occupation numbers accumulate outside the horizon (Dorlis et al., 14 May 2026). Through the GR quartic interaction $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$4 and the Chern-Simons anomaly term $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$5, the cloud acts as a nonlinear pump and drives the graviton field into a squeezed multimode state (Dorlis et al., 14 May 2026, Dorlis et al., 2 Jul 2025).

Different benchmark analyses report different effective squeezing measures. One estimate gives $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$6 and $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$7 correlated quanta for astrophysical black-hole/axion-cloud systems (Dorlis et al., 14 May 2026). Closely related treatments report $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$8 with $$\hat X_1=\frac{\hat a+\hat a^\dagger}{\sqrt2},\qquad \hat X_2=\frac{\hat a-\hat a^\dagger}{\sqrt2\,i},$$9, or an effective $\phi$0 with resonant-mode occupations $\phi$1, under different benchmark assumptions and parameterizations (Mavromatos et al., 16 Dec 2025, Dorlis et al., 2 Jul 2025). All of these analyses agree, however, that the GR channel dominates the Chern-Simons channel by many orders of magnitude for realistic parameter choices (Dorlis et al., 2 Jul 2025).

Another proposed strong-field source comes from shockwave scattering in the Lipatov regime, where multi-graviton radiation is described by a generalized Susskind-Glogower squeezed coherent state and very large squeezing parameters $\phi$2 are argued to be feasible (Staśto et al., 4 May 2026).

4. Noise, geodesic deviation, and interferometric response

The most direct detector observable analyzed in the literature is the graviton-induced fluctuation in interferometer arm length. In linearized gravity, the geodesic separation operator is proportional to the metric perturbation, and, choosing the squeezed quadrature as the signal channel,

$\phi$3

If the detector aligns with the anti-squeezed quadrature, the result becomes

$\phi$4

In a more complete plane-wave treatment, the strain noise spectral density obeys

$\phi$5

which reduces to $\phi$6 for $\phi$7 (Parikh et al., 2020).

The stochastic-gravity and Heisenberg-picture approaches recast the same physics in terms of noise kernels. In the influence-functional formalism, the graviton Hadamard function generates a tensorial stochastic force in a Langevin equation for the geodesic separation of two masses, and the multimode squeezed state inserts explicit $\phi$8 and $\phi$9 factors into the noise kernel (Cho et al., 2021). In the Heisenberg-picture treatment, the nonrelativistic separation satisfies

$$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$0

and the squeezed-state symmetrized spectral density takes the form

$$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$1

in contrast to the thermal factor $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$2 (Haba, 2022).

The same squeezing-enhanced graviton noise has been proposed as a decohering environment for macroscopic quantum states of interferometer mirrors. For an equal-arm interferometer with $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$3 arms and $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$4 mirrors, the decoherence time induced by inflationary squeezed gravitons is estimated to be approximately $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$5 seconds (Kanno et al., 2021). This shifts the emphasis from direct graviton counting to indirect observation through entanglement loss and noise-induced dephasing.

Across these formulations, the common invariant claim is that squeezed graviton backgrounds do not merely renormalize a classical strain amplitude; they introduce phase-sensitive quantum noise with exponentially strong dependence on the squeeze parameter.

5. Quantum statistics, phase-sensitive signatures, and detector-specific probes

The phase sensitivity of squeezed gravitons is central experimentally. Vacuum and thermal graviton noise are phase-isotropic, whereas squeezed noise is anisotropic in quadrature space. One proposed strategy is to rotate the detector readout quadrature and search for a sinusoidal modulation of the noise variance proportional to

$$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$6

together with phase-locked modulation at $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$7 in cross-correlations between independent interferometers (Parikh et al., 2020). The same work emphasizes higher cumulants and non-Gaussian relations as additional discriminants between squeezed and purely Gaussian-thermal noise (Parikh et al., 2020).

A distinct detector model places the gravitational-wave detector in a one-dimensional harmonic trap and couples it through

$$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$8

Because the coupling is quadratic in $$(\Delta X_1)^2=\tfrac12 e^{-2r},\qquad (\Delta X_2)^2=\tfrac12 e^{+2r}.$$9, the relevant detector transitions are $r$0. In a squeezed vacuum, the detector can undergo $r$1 excitation even when $r$2, with probability proportional to $r$3 and resonance at $r$4 (Trenggana et al., 28 Apr 2025). The squeeze phase $r$5 controls interference between absorption and emission processes and can be detected through asymmetries in $r$6 versus $r$7 (Trenggana et al., 28 Apr 2025).

An important statistical clarification follows from binary-wave analyses. A pure squeezed vacuum is not sub-Poissonian; for a single squeezed mode,

$r$8

Sub-Poissonian statistics, and hence $r$9, arise only when the squeezed vacuum is also displaced by a coherent amplitude into a squeezed-coherent state (Kanno et al., 27 Oct 2025). This corrects a common simplification in discussions that identify “squeezed” with “sub-Poissonian” without specifying the state family.

For multimode axion-cloud sources, proposed signatures extend beyond strain variance to polarization-entanglement observables. One analysis predicts nonzero EPR-type correlators $\phi$0 together with higher-order cumulants that differ from thermal backgrounds (Dorlis et al., 14 May 2026). Closely related work states that the structure of the entangled state in a left-right polarization basis depends strongly on whether the production mechanism is the GR channel or the anomalous Chern-Simons channel (Dorlis et al., 31 Jul 2025).

6. Constraints, feasibility, and broader implications

Current direct bounds are modest but nontrivial. Using LIGO-Virgo data and the residual noise around GW150914 at $\phi$1, one analysis derives the constraint

$\phi$2

equivalently an upper bound on the squeezing parameter of the quantum gravitational-wave state (Hertzberg et al., 2021). Related axion-cloud studies cite the same LIGO/Virgo limit as a single-mode benchmark and note that null results can be converted into bounds on cloud lifetimes (Dorlis et al., 14 May 2026).

Feasibility estimates vary by detector concept. For aLIGO-class interferometers with $\phi$3 and strain sensitivity $\phi$4, one proposal states that effectively $\phi$5 is required for detection, while in an optimal two-detector cross-correlation experiment $\phi$6 could already be distinguished from vacuum at the $\phi$7 level, assuming months of integration (Parikh et al., 2020). In stochastic-background language, the squeezed-graviton quantum-noise spectrum can be written

$\phi$8

and comparisons with detector sensitivities suggest that LISA, Einstein Telescope, and future LIGO runs may enter relevant parameter regions for some cosmological spectra (Zhang et al., 2021).

At the conceptual level, the observational target is not unique. Single-mode interferometric bounds, multimode entangled sources, decoherence measurements, HBT-style intensity correlations, and harmonic-trap transition rates probe different state functionals of the graviton field. This suggests that translating one bound into another setup is model-dependent.

The implications also extend beyond detection. In one recent analysis, squeezed graviton fluctuations smear the light cone through the variance of the first-order correction to Synge’s world function, replacing the classical $\phi$9 singularity by a Gaussian of width $\mathbf{k}$0; the same smearing regularizes one-loop ultraviolet divergences in scalar self-energies and yields an inflationary squeezed-graviton correction of order $\mathbf{k}$1 to the one-loop self-energy (Matsui, 7 May 2026). This broadens the significance of squeezed graviton states from gravitational-wave phenomenology to the causal and short-distance structure of effective quantum field theory.

Taken together, the literature presents squeezed graviton states as a unifying language for quantum gravitational radiation in time-dependent backgrounds. Inflation supplies the canonical large-$\mathbf{k}$2 example; time-dependent matter, binaries, black-hole superradiance, and shockwave scattering provide source-driven realizations; and interferometers, harmonic traps, HBT correlations, decoherence experiments, and polarization cross-correlators provide complementary probes. The central technical fact remains the same across these settings: the squeeze parameter and squeeze phase enter observables exponentially and anisotropically, so that even modest changes in $\mathbf{k}$3 can qualitatively alter the phenomenology of graviton noise, coherence, and entanglement.