Krylov Complexity in early universe

Ke-Hong Zhai, Lei-Hua Liu

Published 2024-11-27Version 1

The Lanczos algorithm offers a method for constructing wave functions for both closed and open systems based on their Hamiltonians. Given that the entire early universe is fundamentally an open system, we apply the Lanczos algorithm to investigate Krylov complexity across different phases of the early universe, including inflation, the radiation dominated period (RD), and the matter dominated period (MD). Notably, we find that Krylov complexity differs between the closed and open system approaches. To effectively capture the impact of potentials during the RD and MD phases, we analyze various inflationary potentials, including the Higgs potential, the $R^2$ inflationary potential, and chaotic inflationary potential, taking into account the violations of slow roll conditions. This analysis is conducted in terms of conformal time through the preheating process. Our numerical results indicate that the evolution of Krylov complexity and Krylov entropy is remarkably similar across both methods, regardless of the potential under consideration. Additionally, we rigorously construct what is referred to as an open two-mode squeezed state, utilizing the second kind of Meixner polynomials. Based on this construction, we are the first to calculate the evolution equations for $r_k$ and $\phi_k$ as they relate to the scale factor. Our findings suggest that dissipative effects lead to a rapid decoherence like behavior. Moreover, our results indicate that inflation behaves as a strongly dissipative system, while both the RD and MD phases exhibit characteristics of weak dissipation. This research provides new insights into exploring the universe from the perspective of quantum information.