The term “synchronous” is composed of two Greek words, “syn” and “kronous” which mean “together” and “time”, respectively. Therefore, synchronous events refer to the phenomena that occur simultaneously. Scientifically, synchronization is a remarkable ubiquitous phenomenon where mutually interacting objects behave in unison, ranging from neural oscillations in the human brain, to synchronous flash of fireflies, to the spread of the epidemic disease, to the wobbly motion of London’s Millennium bridge, and a set of coupled pendula. The first study of synchronization dates back to 1673 when Huygens observed that pendula clocks hanging from the same bar had the same period and phase [1].  

One aspect of synchronization in physics can be thought of as a phase transition when by changing the interaction strength in a coupled system the behavior spontaneously changes from an incoherent, and sometimes chaotic phase, to a synchronized one. An important requirement for having such synchronized dynamics is that a system has more than one stable state. This feature is called multi-stability and it can potentially allow for long-time dynamics. In recent years, the study of synchronization and chaos in the context of dynamical systems has become an active area of research. Semi-classical synchronization, aka limit cycles, has been examined in the mean-field limit of various quantum many-body systems where the quantum correlations can be ignored [2]. For strongly-interacting many-body systems or the ones with finite-dimensional Hilbert spaces, however, such as an array of finite qubits or electrons the quantum correlation plays an important role hence rendering the mean-field treatments invalid [3]. Very recently such systems have been the subject of intense studies to seek an answer to the unsettled question of “quantum synchronization” which asks whether the semi-classically predicted synchronous phase would survive the quantum correlation noise (e.g. [4]).

In this work, we study the emergence of true quantum synchronization in an open quantum system of interaction fermions, i.e. elementary quantum particles that all materials are made of. The system must be open (dissipative) as the typical dense and incommensurate spectrum of a many-body quantum system leads to rapid dephasing and an effective stationary state, a phenomenon known as the eigenstate thermalization hypothesis (ETH). Using recent pioneering results from 2019 [5], by identifying a dynamical symmetry operator of the conserved dynamics of our system, we devise Lindblad operators that would not be “seen” by the system, hence leaving its dynamics intact, independent of the system size. That means several observables would have a non-vanishing oscillatory behavior despite the system being lossy. Further, we examine the response of the system in the presence of other Lindblad operators and show the oscillations taper off in small systems, but they persist as the system size increases and approaches the thermodynamic limit. Finally, we show the validity of our theoretical predictions through several numerical examples. As these oscillatory temporal behaviors are reminiscent of spatially-periodic behaviors in electronic crystals they are called “time crystals”. Our model is of the recently proposed class of time crystals induced and stabilized by dissipation (opening the system), called “dissipative time crystals” (distinguished from time crystals that have been proposed about a decade ago in periodically-driven closed systems called “discrete time crystals”). Due to the aforementioned stability and the emergence of the time crystal in the thermodynamic limit only, our model satisfies the criteria for a boundary time crystal and constitutes the first example of a boundary time crystal in a genuine quantum many-body system to the best of our knowledge. The model is both a dissipative and boundary time crystal.

Quantum synchronization is a promising burgeoning field of research with several technical applications, for example in creating coherent time-dependent magnetic fields from synchronized spins to improve the resolution of MRI images, improving the security of quantum key distribution (QKD) protocols for communications, or devising more precise measurements of time. Our results presented in this work pave the way toward studying such interesting phenomenon in controllable platforms such as cold atoms, superconducting qubits, and quantum dots.

[1] C. Huygens and R. J. Blackwell. Christiaan Huygens’ The Pendulum Clock, or, Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks. Iowa State
University Press, Iowa, USA, ISBN 978-0-8138-0933-5 (1986).

[2] S. Sonar, M. Hajdušek, M. Mukherjee, R. Fazio, V. Vedral, S. Vinjanampathy, and L.-C. Kwek. Squeezing Enhances Quantum Synchronization. Phys. Rev. Lett. 120, 163601 (2018).

[3] B. Buca, C. Booker, D. Jaksch. Algebraic theory of quantum synchronization and limit cycles under dissipation. SciPost Phys. 12, 097 (2022).

[4] L.-C. Kwek. No Synchronization for Qubits. Physics 11, 75 (2018).

[5] B. Buca, J. Tindall and D. Jaksch. Non-stationary coherent quantum many-body dynamics through dissipation. Nat. Commun. 10, 1730 (2019).