Lindbladian Spectra Fail to Reliably Diagnose Chaos in Open Quantum Systems

Caio B. Naves of the Stockholm University and colleagues have identified a key flaw in the common practice of applying level statistics, typically used for Hamiltonian systems, to analyse the behaviour of open quantum systems. Their research reveals that the Lindbladian spectrum, which describes the evolution of these systems, behaves fundamentally differently from traditional Hamiltonian spectra, meaning it cannot reliably indicate chaotic dynamics. The team show that level statistics can be manipulated independently of the underlying system dynamics, potentially leading to misleading interpretations of chaos, and link this to the emergence of a non-Hermitian skin effect. These findings underscore the necessity for developing a new set of tools to diagnose quantum chaos in open systems where non-normality is prevalent.

Lindbladian spectral analysis reveals non-normality’s influence on open quantum systems

Numerical diagonalisation of the Lindbladian superoperator proved central to revealing this unexpected behaviour. It involves finding the eigenvalues and eigenvectors of the Lindbladian, dictating the time evolution of the open quantum system and functioning as a fingerprint of energy levels representing how the system changes over time. By carefully calculating these spectra for both a driven harmonic oscillator and a tight-binding model, both subject to incoherent loss and gain, statistical distributions of the energy levels could then be analysed.

Finite-sized systems with open boundary conditions allowed isolation of the impact of non-normality, a mathematical property where small changes can cause large effects, on the resulting spectral patterns. The Lindbladian spectrum of open quantum systems was analysed using numerical diagonalisation, focusing on a driven harmonic oscillator and a one-dimensional tight-binding model. Both systems were subjected to incoherent loss and gain, simulating interaction with a thermal bath; the tight-binding model also incorporated incoherent tunneling. Calculations were performed on finite-sized systems using open boundary conditions to isolate the effects of non-normality on spectral patterns, with a bosonic truncation cutoff introduced for the harmonic oscillator and a fixed system size used for the tight-binding model to enable analysis of spectral statistics and the Uhlmann fidelity.

Ginibre repulsion in open quantum systems independent of dynamical chaos

A driven harmonic oscillator and tight-binding model have demonstrated Ginibre level repulsion despite the complete absence of chaotic dynamics, a result previously considered impossible. Prior methods assumed a direct link between spectral statistics and chaos, mirroring the behaviour of Hamiltonian systems; however, spectral statistics can be tuned almost arbitrarily, with spectral correlations manipulated independently of the underlying system dynamics, challenging the Grobe-Haake-Sommers conjecture regarding chaotic systems exhibiting level repulsion. This decoupling arises from the unique behaviour of eigenvalues with small real parts and the strong non-normality inherent in Lindbladian systems, necessitating new diagnostic tools for identifying genuine quantum chaos.

Ginibre level repulsion, a statistical property of eigenvalue distribution, can occur in a driven harmonic oscillator and a tight-binding model even without any chaotic behaviour present. This contradicts the long-held Grobe-Haake-Sommers conjecture which predicted level repulsion specifically in systems mirroring classically chaotic counterparts, as spectral correlations can be manipulated independently of the system’s dynamics. Further analysis revealed this decoupling is linked to the behaviour of eigenvalues with small real parts and the strong non-normality typical of Lindbladian systems, a type of mathematical operator describing open quantum systems. The emergence of a non-Hermitian skin effect, relating boundary-induced eigenvector localization to spectral instability, was also observed. While these findings establish a new understanding of spectral statistics, they do not yet clarify how to reliably identify genuine quantum chaos in complex, real-world open quantum systems; a strong diagnostic tool remains elusive.

Lindbladian spectra challenge established methods for detecting quantum chaos

Diagnosing quantum chaos has long relied on identifying patterns in a system’s energy levels, a technique borrowed from the study of isolated quantum systems. However, the statistical fingerprints of chaos can appear in open quantum systems even when the underlying dynamics are perfectly orderly. This decoupling arises because the behaviour of Lindbladian spectra, which describe how these systems evolve, is fundamentally different from that of their closed counterparts.

This discovery does not invalidate years of research into open quantum systems; instead, it refines our understanding of how to interpret spectral data. Identifying chaotic signatures via energy levels remains valuable for isolated systems, but applying this directly to those experiencing environmental interaction, known as open quantum systems, requires caution. Researchers must now focus on developing new diagnostic tools less susceptible to these spectral illusions and better suited to accurately characterise dynamics in these complex scenarios.

The Lindbladian spectrum, representing patterns in a system’s energy levels, can be misleading when applied to open quantum systems. Unlike their isolated counterparts, these systems exhibit spectral patterns that falsely suggest disorder even when the underlying physics is orderly. Confirming that established methods for diagnosing quantum chaos can fail when applied to open quantum systems represents a significant advance in understanding how these systems behave. This decoupling arises because the behaviour of Lindbladian spectra differs fundamentally from that of closed quantum systems, focusing dynamics on eigenvalues with small real parts and strong non-normality; patterns in a system’s energy levels can occur independently of the actual dynamics within the system, challenging the long-held Grobe-Haake-Sommers conjecture.

Researchers found that the statistical patterns in the Lindbladian spectrum do not reliably indicate quantum chaos in open quantum systems. This matters because previous methods for identifying chaotic behaviour relied on these spectral patterns, potentially leading to misinterpretations of system dynamics. The study demonstrates that these patterns can arise even when the underlying physics is not chaotic, due to the unique behaviour of Lindbladian spectra and the influence of eigenvalues with small real parts. The authors highlight the need for alternative diagnostic tools to accurately characterise open quantum systems.