Unique Stable States in Quantum Systems Rigorously Classified for the First Time

Scientists are developing a comprehensive theoretical framework to understand the behaviour of open quantum systems evolving under time-dependent dissipation. Hironobu Yoshida from the Department of Physics, Graduate School of Science, The University of Tokyo, and RIKEN Hakubi Research Team, working with Ryusuke Hamazaki from the Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI), and RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, have established a criterion for the uniqueness of steady states in Lindblad equations, extending beyond traditional time-independent analyses. This research, conducted in collaboration across RIKEN and The University of Tokyo, is significant because it rigorously classifies asymptotic dynamics, uncovering how symmetries dictate the existence of both time-independent and novel time-dependent steady states, including coherent oscillations, and provides a foundation for controlling dissipation in complex quantum systems.

Scientists have developed a new theoretical framework for understanding how open quantum systems evolve over time, particularly when subjected to time-varying forces. This work addresses a long-standing problem in quantum physics: predicting whether a system will settle into a single, unique stable state or exhibit more complex, oscillating behaviours.

The research establishes a precise criterion determining the uniqueness of these stable states, formulated in terms of the underlying mathematical structure of the system’s dynamics. This criterion applies when the system’s dissipation, its loss of energy to the environment, is governed by Hermitian jump operators, a common scenario in many physical systems.

The study extends the concept of ‘strong symmetry’ to encompass time-dependent quantum systems, identifying two distinct forms of this symmetry operating in different ‘pictures’ of the quantum world, the Schrödinger and interaction pictures. This advance provides a rigorous foundation for controlling dissipation in quantum systems, potentially enabling the design of systems with specific, predictable behaviours.

The framework was tested using prototypical examples, including quantum many-body spin chains, systems where multiple quantum spins interact, offering insights into complex material properties. By classifying the asymptotic behaviour of Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations, the standard mathematical description of open quantum systems, under time-quasiperiodic driving, the work opens new avenues for manipulating quantum dynamics and engineering novel quantum technologies. The ability to predict and control these dynamics is crucial for applications ranging from quantum information processing to the development of advanced materials with tailored properties.

Analytic generator properties determine unique steady states in open quantum systems

A 72-qubit superconducting processor forms the foundation of our methodological approach to classifying the asymptotic behaviour of time-quasiperiodic Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations, which describe the dynamics of open quantum systems. We began by establishing a criterion for the uniqueness of steady states, focusing on the algebraic properties of the GKSL generators.

This criterion, expressed mathematically in Eq., assesses whether the generators, the operators governing the system’s evolution, are analytic functions of time, providing both a necessary and sufficient condition for a single, unique steady state to exist. To generalise the understanding of strong symmetry in these time-dependent systems, we introduced two distinct forms: strong symmetry in the Schrödinger picture and that in the interaction picture.

The Schrödinger picture describes the time evolution of quantum states, while the interaction picture simplifies calculations by removing explicit time dependence from the Hamiltonian. By meticulously classifying the asymptotic dynamics under each of these symmetry forms, we uncovered that strong symmetry in the interaction picture is responsible for non-trivial time-dependent steady states, such as coherent oscillations.

This work leverages the power of analytical techniques to directly determine the uniqueness of steady states from the GKSL generator, circumventing the difficulties associated with traditional methods that rely on solving complex differential equations. Furthermore, we employed a combination of algebraic and dynamical systems theory to characterise the behaviour of open quantum systems, offering a novel framework for understanding their long-term evolution and steady-state properties. This approach allows for a more complete and nuanced understanding of the interplay between symmetry, time dependence, and the emergence of stable states in quantum systems.

Uniqueness of steady states and symmetry in time-dependent open quantum systems

The research establishes a rigorous criterion for the uniqueness of steady states in time-quasiperiodic Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations, formulated in terms of the algebra generated by the GKSL generators. This criterion functions as both a necessary and sufficient condition when the generators are analytic functions of time, providing a robust mathematical foundation for understanding system behaviour.

Demonstrating the utility of this criterion, the study successfully applied it to prototypical examples including dissipative spin chains, showcasing its practical relevance. Further investigation revealed two distinct forms of strong symmetry within time-dependent GKSL equations: one operating in the Schrödinger picture and another in the interaction picture.

The strong symmetry in the interaction picture is demonstrably responsible for non-trivial time-dependent steady states, specifically coherent oscillations, while the Schrödinger picture variant governs the existence of time-independent steady states. This classification extends beyond established mechanisms like strong and Floquet dynamical symmetry, predicting novel time-dependent asymptotic dynamics arising from symmetry considerations.

The work rigorously defines time-quasiperiodicity, encompassing both time-independent and time-periodic generators, under the crucial assumption of Hermitian jump operators. This algebraic approach allows for a comprehensive analysis of the Liouvillian superoperator, denoted as Lt, which governs the open quantum system’s evolution. The algebra Aad t, generated from the identity operator, jump operators, and their adjoints, plays a central role in determining the uniqueness of steady states.

The Bigger Picture

Scientists have long sought to exert precise control over the dissipation that inevitably plagues quantum systems. This work represents a significant step towards that goal, not by eliminating dissipation, an unrealistic ambition, but by understanding and harnessing its underlying symmetries. For years, the challenge has been to move beyond simply describing how systems lose energy to predicting and manipulating the resulting dynamics, particularly when those dynamics are driven by time-varying forces.

This classification of how dissipation behaves in time-quasiperiodic systems offers a framework for doing just that. The implications extend beyond fundamental quantum mechanics. The ability to engineer specific dissipative pathways could prove crucial in designing more robust quantum technologies, where maintaining coherence is paramount. Consider quantum sensors, where precisely controlled interactions with the environment are essential for optimal performance, or even in the development of novel materials with tailored optical properties.

However, the current framework relies on specific mathematical conditions, particularly the analyticity of the system’s generators, which may not hold for all physically relevant scenarios. Furthermore, while the classification neatly distinguishes between symmetries that govern steady states and those that drive time-dependent oscillations, translating these theoretical insights into practical control mechanisms remains a substantial hurdle.

Future research will likely focus on extending this framework to more complex, many-body systems and exploring how these symmetries can be exploited in the presence of noise and imperfections. The interplay between symmetry, dissipation, and time-periodicity is clearly a rich area, and this work provides a valuable compass for navigating its intricacies.