⚛️ high-energy theory

Krylov complexity and oscillatory quantum dynamics in two-field inflation

This paper investigates the Krylov complexity of primordial perturbations in two-field inflation, revealing that coupled curvature and entropy modes exhibit distinct, oscillatory dynamics and squeezing histories after horizon exit, driven by relative phases and nonlinear channel interactions.

Original authors: Tao Li, Lei-Hua Liu, Hai-Bing Fu

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tao Li, Lei-Hua Liu, Hai-Bing Fu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, a period of rapid expansion known as inflation smoothed out the cosmos and planted the seeds for everything we see today, from galaxies to stars. While this expansion is well understood as a physical process, the quantum nature of the tiny fluctuations that grew into these structures remains a deep mystery. Scientists have long studied how these fluctuations behave, but a newer approach looks at them through the lens of quantum information. Instead of just measuring how strong a fluctuation is, researchers now ask how complex the quantum state becomes as it evolves. This complexity, a measure of how much a simple starting point spreads out into a vast web of possibilities, offers a fresh way to track the history of the early universe.

A team of physicists has now applied this idea to a specific, more complicated version of the early universe model. While many theories assume the universe was driven by a single field of energy, others suggest that two different fields interacted during inflation. This paper investigates what happens to the quantum complexity when these two fields are coupled together. The researchers found that the interaction between the fields creates two distinct pathways for quantum information to spread. These pathways do not behave the same way; one responds quickly to the cosmic expansion, while the other lags behind before surging forward. This difference creates a unique, oscillating pattern in the complexity of the universe's quantum state, a signature that would be invisible if the universe had only one field.

The study began by modeling the universe with two interacting fields: one that drives the expansion and another that acts as a sidekick, or entropy field. In the simplest models, the quantum state of the universe is described as a single pair of linked vibrations. However, when two fields are present, the system becomes a four-part structure, involving two pairs of linked vibrations that can exchange energy. The researchers solved the equations governing how these vibrations evolve from the very beginning of inflation, starting from a calm, empty state and moving forward in time. They tracked how the "squeezing" of these quantum states changed, a process where the uncertainty in the position and momentum of the particles becomes distorted as the universe expands.

As the simulations progressed, a clear story emerged. The two fields did not squeeze in unison. The second channel of interaction began to grow and distort the quantum state almost immediately after the fluctuations moved beyond the reach of the cosmic horizon, the boundary where light can no longer travel fast enough to keep up with expansion. This second channel dominated the intermediate phase of the evolution. The first channel, however, waited. It remained quiet for a while before suddenly growing strong and taking over the later stages of the timeline. This delay and the subsequent rapid growth created a two-stage history for the quantum complexity of the universe.

The researchers measured this complexity by calculating how far the quantum state had spread from its original, simple form. They found that the total complexity was simply the sum of the contributions from these two separate channels. Because the state of the universe is highly sensitive to the timing and phase of these interactions, the complexity did not just grow smoothly. Instead, it developed a wavy, oscillating pattern. These ripples in the complexity were caused by the relative timing of the two fields. When the universe was small and the fluctuations were still inside the horizon, the waves were small and subtle. But once the fluctuations crossed the horizon and the squeezing became intense, these small timing differences were amplified dramatically, turning into large, visible fluctuations in the complexity.

This oscillating behavior is a direct result of the two fields talking to each other. In a universe with only one field, the complexity would grow in a steady, predictable manner. The presence of the second field introduces a new layer of interaction that causes the growth to speed up and slow down in a rhythmic way. The researchers also looked at the "entropy" of this state, which measures how spread out the probability of finding the system in different configurations is. Like the complexity, the entropy showed a two-stage growth pattern, rising first in one channel and then in the other, confirming that the information in the universe was being distributed in a specific, non-uniform way.

The study provides a detailed map of how quantum information spreads in a multi-field universe. It shows that the history of the early universe is not just a story of expansion, but also a story of how different quantum channels interact and compete. The fact that one channel dominates early and another later suggests that the specific details of how the fields are connected leave a permanent mark on the quantum structure of the cosmos. This work extends our understanding beyond the simplest models, offering a new tool to distinguish between different theories of inflation. By looking at the oscillations in quantum complexity, scientists may one day be able to tell if the early universe was driven by a single force or a complex dance of multiple fields, even if those fields are no longer present today.

Technical Summary: Krylov Complexity and Oscillatory Quantum Dynamics in Two-Field Inflation

Problem Statement
While Krylov complexity has recently been applied to inflationary cosmology to study the spreading of quantum information and operator growth, existing literature has predominantly focused on single-field or effectively single-field descriptions. In these scenarios, the quantum state is typically modeled as a two-mode squeezed state associated with a single scalar degree of freedom. However, this framework cannot capture quantum information transfer between different scalar degrees of freedom, a feature intrinsic to multi-field inflation. The authors aim to extend the Krylov complexity framework to two-field inflation, where curvature and entropy perturbations are coupled. The central problem is to determine how this curvature–entropy coupling modifies Krylov-space spreading, specifically across the horizon crossing, and to identify distinct dynamical signatures that differentiate multi-field dynamics from single-field limits.

Methodology
The authors construct a quantitative framework for analyzing the Krylov-space dynamics of primordial perturbations in a two-field inflationary model:

Hamiltonian Construction: Starting from the quadratic action for coupled curvature ( $\xi$ ) and entropy ( $\psi$ ) perturbations, the authors derive the equations of motion and the corresponding quadratic Hamiltonian. The system is treated in the Bunch–Davies basis, with numerical solutions for the mode functions obtained for specific slow-roll parameters and coupling strengths.
State Representation: The resulting Gaussian quantum state is described as a four-mode squeezed state belonging to the symplectic group $Sp(4, \mathbb{R})$ . Through the Bloch–Messiah decomposition, the authors identify two independent diagonal squeezing channels (eigenchannels) rather than a single ladder. The state is expressed as a product of two two-mode squeeze operators acting on the vacuum.
Evolution Equations: The authors derive coupled differential equations for the squeezing strengths ( $r_{1k}, r_{2k}$ ) and squeezing angles ( $\phi_{1k}, \phi_{2k}$ ) for both channels. These equations depend on the Hamiltonian coefficients ( $A_{ii}, B_{ii}$ ), which encode the curvature–entropy coupling via the mode functions.
Krylov Complexity and Entropy: Using the probability distribution of the four-mode squeezed state in the occupation number basis, the authors calculate the Krylov complexity ( $K_k$ ) and Krylov entropy ( $S_K$ ). They demonstrate that the total complexity separates into the sum of contributions from the two independent channels ( $K_k = K_{1k} + K_{2k}$ ).
Lanczos Coefficients: The authors derive channel-resolved Lanczos coefficients ( $b_n^{(1)}, b_m^{(2)}$ ) corresponding to the hopping amplitudes along the two independent directions of the two-dimensional Krylov lattice.

Key Results
The numerical analysis reveals distinct dynamical behaviors for the two squeezing channels compared to the single-field case:

Asymmetric Squeezing Histories: The two channels exhibit different onset times. The second channel develops strong squeezing earlier and dominates the intermediate evolution, whereas the first channel shows a delayed but rapid enhancement. This asymmetry is a direct dynamical consequence of the curvature–entropy coupling.
Oscillatory Dynamics: Both channels display pronounced oscillatory structures in their Krylov complexities, particularly in the superhorizon regime. These oscillations originate from the relative phases entering the squeezing equations (specifically the term $\sin(2\phi_{ik} - \arg A_{ii})$ ).
Amplification Mechanism: The nonlinear relation $K_{ik} = \sinh^2 r_{ik}$ acts as a strong amplifier. While phase modulations exist in the subhorizon regime, they are significantly amplified in the strongly squeezed superhorizon regime, leading to large fluctuations in the Krylov complexity.
Channel-Resolved Observables: The Krylov entropy and Lanczos coefficients also reflect this two-channel structure. The entropy grows in a two-stage pattern corresponding to the sequential onset of squeezing in the two channels. The Lanczos coefficients for the second channel become larger than those for the first, consistent with the earlier and stronger growth of the second channel's squeezing strength.
Consistency Check: The authors verify that in the limit where the entropy sector is removed (reducing to a single-field model), the formalism correctly recovers the standard single-field Krylov complexity ( $K_k = \sinh^2 r_k$ ) and the one-dimensional Krylov chain.

Significance and Claims
The paper claims that Krylov complexity serves as a sensitive probe of genuine multi-field dynamics that is complementary to traditional observables like the power spectrum, isocurvature fraction, and non-Gaussianity.

Distinguishing Multi-Field Dynamics: The study demonstrates that the two-field system contains two squeezing eigenchannels with distinct dynamical responses, unlike the single ladder of the two-mode squeezed state in single-field inflation. The separation of background-induced spreading from interaction-induced spreading is proposed as a key capability of this framework.
Oscillatory Signatures: The authors highlight that the oscillatory structures in Krylov complexity across horizon crossing are a specific signature of the phase-sensitive squeezing dynamics in coupled systems. These features are absent in single-channel descriptions.
Theoretical Diagnostic: By probing the dynamical evolution of the quantum state rather than just final correlation functions, Krylov complexity offers a new theoretical diagnostic for multi-field interactions, potentially revealing information about the underlying field-space geometry and trajectory turning that is not accessible through standard statistical measures.

The authors conclude that this work extends the squeezing-state Krylov description from effectively single-field systems to the simplest coupled multi-field setting, providing a foundation for future extensions to general $N$ -field inflation.

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