Squeezed States in Gravity

TL;DR

The work establishes a systematic framework showing that time-dependent quadratic couplings in linearized quantum gravity generically produce squeezed graviton states from the vacuum. It develops and compares three methods—time evolution, Bogolyubov transformations, and Magnus expansion—to extract the squeezing parameter $r(t)$, and applies them to gravity in FRW, Kasner, and matter-sourced scenarios. The results show that background dynamics and time-varying matter sources can induce linear-in-time growth of squeezing in gravitational modes, with potential implications for primordial tensor fluctuations and gravitational-wave detector noise. Overall, the paper links concrete perturbative gravity effects to observable quantum features of the gravitational field, highlighting both cosmological and astrophysical contexts where squeezing can be significant.

Abstract

We present a general framework for the production of squeezed quantum states of the gravitational field in linearized quantum gravity. Time-dependent couplings in the quadratic part of the action generically produce squeezed states from the vacuum. Using the harmonic oscillator as an example, we describe three techniques to obtain the squeezing parameter from such quadratic terms. For gravity, the action to quadratic order in metric perturbations contains couplings both to background curvature as well as to matter sources. Thus, both time-dependent classical spacetimes and time-dependent classical matter typically produce squeezed states of gravity.

Figures (4)

• Figure 1: Plot of the function $r(t)$ which can be inferred by numerically solving the equations \ref{['eqsfull1']} with $F(t) = -\frac{A}{4}\sin (2 \omega t)$ with $A = 0.1$ and $\omega_0 =1$.
• Figure 2: Plot of the function $r(t)$ obtained by numerically solving the equations \ref{['eqsfull1']} with with $F(t) = - \frac{A}{4} \sin( 2 \omega_0 t)$ (in blue) and $F(t) = - \sum_{n = 0}^{10} \frac{A}{4} \sin(2 \omega_0 (1+ \sqrt{2} n) t)$ (in orange) with $A = 0.1$ and $\omega_0 =1$.
• Figure 3: Plot of the squeezing parameter $r(t)$ for the decoupled perturbation modes in a Kasner universe. Note the linear growth of the squeezing parameter with time.
• Figure 4: Phase space diagram for a coherent state $D(\alpha)\ket{0}$ (left) and a displaced squeezed state $D(\alpha) S(r, \varphi) \ket{0}$ (right). The angle $\varphi/2$ determines the direction of reduced uncertainty in phase space. The real and imaginary parts of $\alpha$ correspond to the expectation values of $X$ and $P$.