Squeezed Graviton States in Quantum Gravity

• Squeezed graviton states are nonclassical quantum states where correlations between graviton modes are enhanced or suppressed relative to the vacuum.
• They are constructed via multimode squeezing operators and Bogoliubov transformations, yielding measurable reductions in quantum noise and distinct entanglement features.
• These states offer insights into the quantum-to-classical transition in cosmology and provide potential observational signatures for quantum gravity in gravitational wave detectors.

Squeezed graviton states are nonclassical quantum states of the linearized gravitational field in which selected correlations between pairs of graviton modes (typically of opposite momentum or polarization) are enhanced or suppressed relative to the vacuum, producing a redistribution of fluctuations among gravitational field observables. These states arise naturally in quantum field theory treatments of gravity subject to cosmological expansion, black hole dynamics, or nonlinear gravitational interactions. They are central for both theoretical investigations of the quantum-to-classical transition in cosmology, emergent semiclassical geometry, and potential signatures of quantum gravity in current and future gravitational wave detectors.

1. Theoretical Construction and Mathematical Structure

The prototypical squeezed state for a single bosonic mode is generated by the unitary squeezing operator

$$S(r, \theta) = \exp\left\{\frac{1}{2}\left(re^{-i\theta}b^2 - re^{i\theta}b^{\dagger 2}\right)\right\}$$

where $b, b^\dagger$ are annihilation and creation operators, $r$ is the squeezing amplitude, and $\theta$ the squeezing phase. Extended to the gravitational field, the quantized metric perturbations $\hat{h}_{\mu\nu}$ are expanded in Fourier modes, and one constructs a (generally multimode) squeezed vacuum by acting on the Fock vacuum with a multimode analog of $S$.

For two-mode squeezing typical of cosmological and black hole contexts, the state takes the form

$$|\Psi\rangle = N \sum_{n=0}^{\infty} \exp\left[i n (\theta_k + \phi_k)\right] \left(\tanh r_k \right)^n |n_k\rangle \otimes |n_{-k}\rangle$$

where $r_k$ is the squeezing parameter for mode $k$, $N$ is normalization, and $\phi_k$ is an effective phase (Kanno et al., 2021).

In loop quantum gravity (LQG), squeezed vacua are constructed in a bosonic Hilbert space of oscillators associated with seeds of a graph $\Gamma$, and projected to gauge-invariant subspaces. Squeezed vacua in this context are parameterized by a complex, symmetric matrix $Y$ in the Siegel unit disk $$D = \{Y \in \mathrm{Mat}(4L, \mathbb{C})\,|\, Y=Y^T, 1-Y^\dagger Y > 0\}$$, leading to states

$$|Y) = \exp\left(\frac{1}{2}Y_{AB} F^{AB\,\dagger}\right)|0)$$

and, after imposing gauge constraints, physical squeezed states are $|\Gamma, Y) = P_R |Y)$ (Bianchi et al., 2016).

2. Physical Origins: Cosmology, Black Holes, and Gravitational Nonlinearity

Cosmology: During inflation, quantum fluctuations of the metric are exponentially stretched by the background expansion, leading to highly squeezed two-mode states for tensor perturbations. The squeezing parameter $r_k$ grows with the number of e-folds after horizon exit, and particle creation is described by Bogoliubov transformations: $$a_k(\eta) = \alpha_k(\eta)a_k(\eta_0) + \beta_k(\eta)a_{-k}^\dagger(\eta_0),\quad r_k(\eta) = \sinh^{-1}|\beta_k(\eta)| [2112.14496].$$ Both the "instantaneous vacuum" (field-diagonal) and "adiabatic vacuum" (positive-frequency) approaches yield nearly identical squeezing for massless gravitons, though the effective squeezing phase can differ.

Black Holes: Quasi-normal modes (QNMs) in perturbed black holes function as multimode gravitational squeezers (Su et al., 2017). The dynamical interaction Hamiltonian for a quantum field in the QNM background takes a quadratic form in creation/annihilation operators and drives the system into a multimode squeezed vacuum. The peak squeezing amplitude is inversely proportional to the cube of the imaginary part of the QNM frequency, and long-lived QNMs (with small damping rates) strongly amplify quantum field fluctuations—even Hawking radiation.

Nonlinear Gravitational Interactions: In the effective field theory of general relativity, nonlinear terms in the Einstein-Hilbert action couple gravitational wave modes, leading to self-interactions that are analogous to three-wave mixing in nonlinear optics (Guerreiro, 28 Jan 2025). The resulting interaction Hamiltonian generates nonclassical states, including squeezing and entanglement between GW modes. The core interaction is of the form

$$L_I = -\frac{1}{64\pi G} h^{L\,ij} (\partial_i h_+^H)(\partial_j h_+^H)$$

and, upon mode decomposition, directly induces squeezing among high-frequency GW modes in the presence of a low-frequency background.

Astrophysically, rotating black holes with axion clouds (grown by superradiance) seed multimode graviton squeezing via coherent axion annihilation ($2a \rightarrow 2g$), with the squeezing parameter and number of excited gravitons scaling favorably with the large axion occupation and the long cloud lifetime (Dorlis et al., 2 Jul 2025, Dorlis et al., 31 Jul 2025).

3. Quantum Properties and Observable Consequences

Reduced Uncertainty and Quantum Correlations: Squeezed graviton states manifest reduced noise in one field quadrature (e.g., the metric fluctuation) and enhanced noise in the conjugate (momentum) quadrature. The variance of a measured quadrature in a squeezed state is given by

$$\Delta Y'^2 = \frac{1}{2}\left[\cos^2(gt) + e^{-2r}\sin^2(gt)\right] [2408.00094]$$

where $r$ is the squeezing amplitude and $g$ denotes detector coupling. Sub-vacuum noise (variance below the standard "vacuum" level) can occur transiently.

Entanglement: Two-mode and multimode squeezing generate entanglement between graviton modes of opposite momentum, or between left–right (polarization) sectors. In the cosmological context, the squeezed state violates Bell-type inequalities with an expectation value

$$\langle\mathcal{B}\rangle = \sqrt{1+\tanh^2(2r_k)} \to \sqrt{2}\;\;\text{as}\ r_k \to \infty [2112.14496].$$

Axion-induced squeezing in black holes results in polarization-dependent entangled states, with the symmetry (Bell, antisymmetric) depending on whether the source interaction is conventional GR or anomaly-driven (Dorlis et al., 31 Jul 2025).

Quantum Noise and Graviton Detection: Squeezed gravitons generated during inflation or from astrophysical sources enhance quantum noise spectra (e.g., strain noise in interferometers) (Haba, 2022, Kanno et al., 2022). The statistical structure of the noise—such as strong two-point correlations and sub-vacuum features—provides an avenue for indirect measurement, though the observation of such quantum noise is not, by itself, definitive evidence for gravitational field quantization due to classical mimics (Carney, 31 Jul 2024).

Stimulated Emission and Quantum Coherence: Squeezed vacuum states are susceptible to stimulated emission when interacting with thermal backgrounds. The full quantum dynamics, including coherence between number eigenstates, modulate the graviton number evolution and can lead to resonant growth, requiring regularization procedures for IR modes in cosmological settings (Ota et al., 9 Apr 2025).

4. Mathematical and Group-Theoretic Frameworks

Bogoliubov Transformations and Squeezing Operators: Squeezing transformations are fundamentally linked to Bogoliubov transformations in both quantum mechanics and field theory (Zhou et al., 2019). The generic squeeze operator for fields is constructed as

$$S = \exp\left\{\int \frac{dp\,dq}{(2\pi)^2} f(p, q)[a(p)a(-q) - a^\dagger(p)a^\dagger(-q)]\right\}$$

with the function $f$ determined by matching two Hamiltonians or sectors.

Group Representation and Siegel Domain: In many-body or geometric settings, squeezed vacua can be classified by their parametrization in the Siegel unit disk (complex, symmetric matrices) and associated symplectic symmetry groups (e.g., Sp(4L, ℝ) for LQG), enabling a systematic paper of their expectation values, correlations, and completeness (Bianchi et al., 2016).

Takagi/Bloch-Messiah Decomposition: Multimode squeezing operators with complex symmetric kernels are diagonalized using Takagi (Schmidt) decomposition, reducing the multimode case to a product of single-mode squeezing operators, which facilitates the analysis of entanglement and occupation statistics (Dorlis et al., 31 Jul 2025).

5. Squeezed Graviton States in Graviton–Photon and Graviton–Matter Conversion

The possibility of enhancing the conversion probability between photons and gravitons using squeezed states has been established by explicit calculation in a quantum field theoretic framework (Ikeda et al., 2 Jul 2025). Placing incoming photons in a squeezed coherent state and the background graviton field in a squeezed vacuum (as from inflation) increases the photon–graviton conversion probability by enhancement factors such as $\cosh^2 r$ (photon squeezing) and $\cosh^2 z$ (graviton background squeezing), achieving factors of order $10^4$ at frequencies around 100 MHz. Moreover, the conversion process swaps or generates entanglement between the photon and graviton sectors, which is strictly forbidden in purely classical models, offering a direct operational signature of quantization.

In gravitational wave detectors or hypothetical quantum sensors within a harmonic trap, the presence of squeezed or squeezed coherent graviton states enables unique quantum processes, including simultaneous graviton annihilation (e.g., via $a^2$ terms) and energy transitions quantized in units of $2\hbar\omega$, which would not occur for classical metric fluctuations or for initial Fock states $|n_G\rangle$ (Trenggana et al., 28 Apr 2025).

6. Physical and Observational Implications

Connection to Semiclassical Geometry: Squeezed graviton states interpolate between highly quantum regimes (with strong pair correlations and quantum noise) and semiclassical spacetimes where expectation values of geometric operators follow classical trajectories but quantum correlations survive. In certain loop quantum gravity formulations, squeezed vacua allow explicit control of long-range correlations mimicking those expected from the Minkowski vacuum graviton propagator (Bianchi et al., 2016).

Limits on Proof of Quantum Gravity: While the observation of quantum noise (or even of graviton "clicks") in a detector responding to a squeezed graviton background would be a milestone, such phenomena can be reproduced by classical stochastic models under realistic detector couplings. True evidence for quantization requires observation of nonclassical features—such as negative quasi-probabilities or entanglement not reproducible with classical field statistics (Carney, 31 Jul 2024, Ikeda et al., 2 Jul 2025).

Astrophysical Prospects: Superradiant axionic clouds near rotating black holes may act as powerful natural "graviton squeezers", generating observable squeezed graviton radiation depending on the cloud lifetime and axion occupation (Dorlis et al., 2 Jul 2025, Dorlis et al., 31 Jul 2025). Graviton squeezing contributes to modified quantum noise signatures and potentially to observable nonclassical correlations in gravitational wave data.

7. Experimental and Conceptual Frontiers

Squeezed graviton states represent a natural setting to test the operational distinctions between classical and quantum gravity by leveraging advances in quantum optics, squeezed-state engineering, and quantum measurement theory. They are at the interface of quantum cosmology, quantum field theory in curved spacetime, and quantum information applied to gravitational systems. Ongoing efforts focus on the precise quantification of squeezing and entanglement in realistic astrophysical and cosmological scenarios, as well as the development of detection strategies robust against classical stochastic mimics.

Further theoretical developments include systematic group-theoretic classification of squeezing in gauge-invariant sectors, extension of the formalism to curved and highly dynamical spacetime backgrounds, and exploration of new quantum signatures emergent from the interplay of nonlinearity, topology, and quantum coherence among graviton modes.