Krylov Complexity in early universe

TL;DR

This paper applies Krylov complexity analysis to the early universe, revealing differences between open and closed systems and showing how dissipative effects influence quantum states during various cosmological phases.

Contribution

It introduces a novel application of Krylov complexity to cosmology, including the construction of open two-mode squeezed states and derivation of evolution equations for squeezing parameters.

Findings

Krylov complexity distinguishes open and closed quantum system behaviors in cosmology.

Dissipative effects cause rapid decoherence, with the universe behaving as strongly dissipative during inflation.

Krylov complexity and entropy evolve similarly across different inflationary potentials during RD and MD.

Abstract

The Lanczos algorithm offers a framework for constructing wave functions in closed and open quantum systems from their Hamiltonians. Since the early universe is inherently an open system, we employ this algorithm to investigate Krylov complexity across various cosmological phases: inflation, radiation domination (RD), and matter domination (MD). Our results highlight a clear distinction in Krylov complexity between the closed- and open-system methodologies. To accurately capture the influence of potentials during RD and MD, we examine a set of inflationary potentials, including the Higgs potential, $R^2$ inflation, and chaotic inflation, while incorporating violations of slow-roll conditions. This study is conducted in conformal time through the preheating stage. Numerically, we find that the evolution of Krylov complexity and Krylov entropy shows remarkable similarity across different potentials during RD and MD. Furthermore, we rigorously construct an open two-mode squeezed state using the second kind of Meixner polynomial. Based on this construction, we derive for the first time the evolution equations for the squeezing parameter $r_k$ and phase $\phi_k$ in terms of the scale factor. Our analysis indicates that dissipative effects lead to rapid decoherence-like behavior. In addition, we observe that the inflationary universe behaves as a strongly dissipative system, whereas during the RD and MD epochs the universe exhibits weak dissipative characteristics. This work opens new perspectives for studying the universe from a quantum-informational viewpoint.