Squeezed out-of-time-order correlator and cosmology

S Shajidul Haque,Bret Underwood

doi:10.1103/physrevd.103.023533

S Shajidul Haque, Bret Underwood

Open Access

https://doi.org/10.1103/physrevd.103.023533

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Abstract

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 are 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.

Highlights

Unlike classical systems, characterizing the nature of chaos in quantum mechanics and quantum many body systems can be challenging
We find that the of-time-order correlator (OTOC) for highly squeezed states is exponentially large, implying that highly squeezed states are “primed” for quantum chaos
Analogous to the classical case, an exponentially growing OTOC is commonly associated with quantum chaos

Summary

INTRODUCTION

Unlike classical systems, characterizing the nature of chaos in quantum mechanics and quantum many body systems can be challenging. A highly squeezed state with φ 1⁄4 nπ with this same linear growth in squeezing exhibits what appears to be the opposite of quantum chaos: an exponential decrease in the OTOC, as in (12) This quantum attractor– like behavior for a specific squeezing angle in the quantum phase space is an artifact of restricting ourselves to the single unequal-time commutator (11) and motivates a generalization of the OTOC. It is straightforward to check that the matrix Mhas a unit determinant for squeezed states; since it is a 2 × 2 matrix, it is necessarily symplectic, as its classical counterpart This formulation of the squared matrix of out-of-time-order correlators is preferable to focusing on the single OTOC constructed using 1⁄2qðηÞ; pðη0Þ, because it avoids a reliance in one particular direction in the phase space of canonical coordinates. OTOC (33), as expected, identifying the Lyapunov exponent λ ∼ k as before

SQUEEZED STATES IN QUANTUM FIELD THEORY AND THE OTOC

COSMOLOGICAL PERTURBATIONS AND THE OTOC

Expanding backgrounds

Contracting backgrounds

DISCUSSION