Squeezed out-of-time-order correlator and cosmology

Exponential growth in the out-of-time-order correlator (OTOC) is an important potential signature of quantum chaos. The OTOC is quite simple to calculate for squeezed states, whose applications are frequently found in quantum optics and cosmology. We find that the OTOC for a generic highly squeezed quantum state is exponentially large, suggesting that highly squeezed states are primed for quantum chaos. A quantum generalization of the classical symplectic phase space matrix can be used to extract the quantum Lyapunov spectrum, and we find this better captures the exponential growth of squeezed states for all squeezing angles compared to any single OTOC. By describing cosmological perturbations in the squeezed state language, we arc able to apply our calculations of the OTOC to arbitrary expanding and contracting backgrounds with fixed equation of state. We find that only expanding de Sitter backgrounds support an exponentially growing OTOC at late times, with a putative Lyapunov exponent consistent with other calculations. While the late-time behavior of the OTOC for other cosmological backgrounds appears to change depending on the equation of state, we find that the quantum Lyapunov spectrum shows some universal behavior: the OTOC grows proportional to the scale factor for perturbation wavelengths larger than the cosmological Hubble horizon.

Automated summary

Exponential growth in the out-of-time-order correlator (OTOC) is a key signature of quantum chaos, particularly for highly squeezed quantum states. By using a quantum generalization of the classical symplectic phase space matrix, the quantum Lyapunov spectrum can be extracted, providing a better understanding of the exponential growth of squeezed states. This allows for the application of OTOC calculations to various expanding and contracting cosmological backgrounds, with expanding de Sitter backgrounds showing consistent exponential growth at late times.