[Paper Review] Non-stationarity and Dissipative Time Crystals: Spectral Properties and Finite-Size Effects

Cameron Booker, Berislav Buča+1 more|arXiv (Cornell University)|May 11, 2020

Quantum many-body systems140 references73 citations

TL;DR

This paper introduces dissipative time crystals in open quantum many-body systems, demonstrating that persistent, experimentally observable time-periodic oscillations can emerge via engineered dissipation—specifically two-body loss and gain—leading to coherent dynamics between GHZ states. The key contribution is a framework linking commensurate purely imaginary eigenvalues of the Liouvillian to robust time-crystalline order, even in finite systems, with examples showing that dynamical symmetries and dark Hamiltonians can stabilize such behavior without full isolation.

ABSTRACT

We discuss the emergence of non-stationarity in open quantum many-body systems. This leads us to the definition of dissipative time crystals which display experimentally observable, persistent, time-periodic oscillations induced by noisy contact with an environment. We use the Loschmidt echo and local observables to indicate the presence of a finite sized dissipative time crystal. Starting from the closed Hubbard model we then provide examples of dissipation mechanisms that yield experimentally observable quantum periodic dynamics and allow analysis of the emergence of finite sized dissipative time crystals. For a disordered Hubbard model including two-particle loss and gain we find a dark Hamiltonian driving oscillations between GHZ states in the long-time limit. Finally, we discuss how the presented examples could be experimentally realized.

Motivation & Objective

• To define and characterize dissipative time crystals in open quantum systems where non-stationary, time-periodic dynamics persist due to environmental coupling.
• To identify the spectral conditions—specifically commensurate purely imaginary eigenvalues of the Liouvillian—under which such time-crystalline behavior emerges.
• To demonstrate that robust time-periodic oscillations can arise not only from dynamical symmetries but also from non-local effective Hamiltonians in dissipative systems.
• To provide experimentally realizable models based on the Hubbard model with two-body loss and gain, enabling observation of finite-size dissipative time crystals.
• To clarify the distinction between true dissipative time crystals and quasi-time-crystalline behavior with incommensurate frequencies, and to explore their thermodynamic limits.

Proposed method

• Uses the Lindblad master equation framework to model open quantum systems with Markovian dissipation, focusing on the Liouvillian superoperator L.
• Analyzes the spectral properties of L, particularly the structure of purely imaginary eigenvalues, to determine conditions for time-crystalline order.
• Employs the Loschmidt echo as a stringent, non-engineered probe of periodicity in generic initial states, capturing finite-size effects.
• Applies local observables and correlation functions to detect persistent oscillations in expectation values, even in the absence of exact symmetries.
• Constructs explicit models based on the Hubbard Hamiltonian with two-body loss and gain, and with inhomogeneous magnetic fields, to realize time-crystalline dynamics.
• Identifies dark states and effective dark Hamiltonians through spectral analysis, showing that coherent oscillations between GHZ states emerge at long times.

Experimental results

Research questions

• RQ1 What spectral conditions in the Liouvillian spectrum are necessary and sufficient for the emergence of persistent time-periodic oscillations in open quantum many-body systems?
• RQ2 Can dissipative time crystals exist without dynamical symmetries or dark states, and if so, how are they stabilized?
• RQ3 How do finite-size effects influence the observability of time-crystalline order, and can the Loschmidt echo serve as a robust, generic probe of such behavior?
• RQ4 To what extent can experimentally realistic dissipation mechanisms—such as two-body loss and gain—induce time-crystalline order in platforms like ultracold atoms?
• RQ5 How does the presence of a strong dynamical symmetry relate to the commensurability of eigenfrequencies and the stability of time-crystalline oscillations?

Key findings

• Robust time-periodic oscillations in open quantum systems require the purely imaginary eigenvalues of the Liouvillian to form a commensurate, nowhere-dense set; incommensurability leads to dephasing and loss of periodicity.
• In a disordered Hubbard model with two-body loss and gain, the long-time dynamics are governed by an effective dark Hamiltonian that drives coherent oscillations between two GHZ states.
• The Loschmidt echo of a generic initial state exhibits periodicity with a Poincaré recurrence time that scales exponentially with system size, confirming the presence of time-crystalline order in finite systems.
• A model with only two-body loss exhibits quasi-time-crystalline behavior with a few incommensurate frequencies, demonstrating a distinct class of dissipative time crystals without strong dynamical symmetries.
• Even without global dynamical symmetries, an inhomogeneous magnetic field can induce effective non-local dynamics between GHZ states, showing that time-crystalline order can emerge from local interactions and dissipation.
• The framework allows for the realization of time-crystalline order in experimentally accessible cold-atom platforms, with robustness to parameter variations, suggesting feasibility for future experimental observation.