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arXiv:2403.09769 · Physical Review Letters

Engineering Nonequilibrium Steady States through Floquet Liouvillians

Weijian Chen, Maryam Abbasi, Serra Erdamar, Jacob Muldoon, Yogesh N. Joglekar, Kater W. Murch

Published Mar 7, 2025• 6 citations
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Abstract

We experimentally study the transient dynamics of a dissipative superconducting qubit under periodic drive toward its nonequilibrium steady states. The corresponding stroboscopic evolution, given by the qubit states at times equal to integer multiples of the drive period, is determined by a (generically non-Hermitian) Floquet Liouvillian. The drive period controls both the transients across its non-Hermitian degeneracies and the resulting nonequilibrium steady states. These steady states can exhibit higher purity compared to those achieved with a constant drive. We further study the dependence of the steady states on the direction of parameter variation and relate these findings to the recent studies of dynamically encircling exceptional points. Our Letter provides a new approach to control non-Hermiticity in dissipative quantum systems and presents a new paradigm in quantum state preparation and stabilization.

DOI
10.1103/physrevlett.134.090402

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We experimentally study the transient dynamics of a dissipative superconducting qubit under periodic drive toward its nonequilibrium steady states. The corresponding stroboscopic evolution, given by the qubit states at times equal to integer multiples of the drive period, is determined by a (generically non-Hermitian) Floquet Liouvillian. The drive period controls both the transients across its non-Hermitian degeneracies and the resulting nonequilibrium steady states. These steady states can exhibit higher purity compared to those achieved with a constant drive. We further study the dependence of the steady states on the direction of parameter variation and relate these findings to the recent studies of dynamically encircling exceptional points. Our Letter provides a new approach to control non-Hermiticity in dissipative quantum systems and presents a new paradigm in quantum state preparation and stabilization.
The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system's density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly quantum domain, driven by the enhanced sensitivity and topological features EPs have shown in their classical realizations. Here, we present Floquet analysis of a prototypical qubit whose drive or dissipator strengths are varied periodically. We consider models with a single dissipator that generate global loss (phase damping) or mode-selective loss (spontaneous emission). In all cases, we find that periodic modulations lead to EP lines at small dissipator strengths and a rich EP structure in the parameter space. Our analytical and numerical results show that extending Lindblad Liouvillians to the Floquet domain is a potentially preferred route to accessing exceptional points in the transient dynamics towards the Lindblad steady state.
Open quantum systems interacting with an environment exhibit dynamics described by the combination of dissipation and coherent Hamiltonian evolution. Taken together, these effects are captured by a Liouvillian superoperator. The degeneracies of the (generically non-Hermitian) Liouvillian are exceptional points, which are associated with critical dynamics as the system approaches steady state. We use a superconducting transmon circuit coupled to an engineered environment to observe two different types of Liouvillian exceptional points that arise either from the interplay of energy loss and decoherence or purely due to decoherence. By dynamically tuning the Liouvillian superoperators in real time we observe a non-Hermiticity-induced chiral state transfer. Our study motivates a new look at open quantum system dynamics from the vantage of Liouvillian exceptional points, enabling applications of non-Hermitian dynamics in the understanding and control of open quantum systems.
We study the quantum evolution of a non-Hermitian qubit realized as a sub-manifold of a dissipative superconducting transmon circuit. Real-time tuning of the system parameters results in non-reciprocal quantum state transfer associated with proximity to the exceptional points of the effective Floquet Hamiltonian. We observe chiral geometric phases accumulated under state transport, verifying the quantum coherent nature of the evolution in the complex energy landscape and distinguishing between coherent and incoherent effects associated with exceptional point encircling. Our work demonstrates an entirely new method for control over quantum state vectors, highlighting new facets of quantum bath engineering enabled through time-periodic (Floquet) non-Hermitian control.
Laser technology has developed and accelerated photo-induced nonequilibrium physics, from both the scientific and engineering viewpoints. Floquet engineering, i.e., controlling material properties and functionalities by time-periodic drives, is at the forefront of quantum physics of light-matter interaction. However, it is limited to ideal dissipationless systems. Extending Floquet engineering to various materials requires understanding of the quantum states emerging in a balance of the periodic drive and energy dissipation. Here, we derive a general description for nonequilibrium steady states (NESSs) in periodically driven dissipative systems by focusing on systems under high-frequency drive and time-independent Lindblad-type dissipation. Our formula correctly describes the time average, fluctuation, and symmetry properties of the NESS, and can be computed efficiently in numerical calculations. This approach will play fundamental roles in Floquet engineering in a broad class of dissipative quantum systems from atoms and molecules to mesoscopic systems, and condensed matter.
In this paper, we investigate periodically driven open quantum systems within the framework of Floquet-Lindblad master equations. Specifically, we discuss Lindblad master equations in the presence of a coherent, time-periodic driving and establish their general spectral features. We also clarify the notions of transient and nondecaying solutions from this spectral perspective, and then prove that any physical system described by a Floquet-Lindblad equation must have at least one physical nonequilibrium steady state (NESS), corresponding to an eigenoperator of the Floquet-Lindblad evolution superoperator UF{\mathcal{U}}_{F}UF​ with unit eigenvalue. Since the Floquet-Lindblad formalism encapsulates the entire information regarding the NESS, it in principle enables us to obtain nonlinear effects to all orders at once. The Floquet-Lindblad formalism thus provides a powerful tool for studying driven-dissipative solid-state systems, which we illustrate by deriving the nonlinear optical response of a simple two-band model of an insulating solid and comparing it with prior results established through Keldysh techniques.