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arXiv:1904.11763 · Physical Review A

Synchronization in two-level quantum systems

Álvaro Parra-López, Joakim Bergli

Published Jun 4, 2020• 43 citations
arXivHTMLPDF

Abstract

Recently, it was shown that dissipative quantum systems with three or more levels are able to synchronize to an external signal. Some researchers have stated that this is not possible for two-level systems, as those systems lack a stable limit cycle in unperturbed dynamics, but others have demonstrated, under a different definition of synchronization, that it is possible in qubits in models which include other elements. We show how a quantum two-level system can be understood as containing a valid limit cycle as the starting point of synchronization and that it can synchronize its dynamics to an external weak signal. This is demonstrated by analytically solving the Lindblad equation of a two-level system coupled to an environment, determining the steady state. This is a mixed state with contributions from many pure states, each of which provides a valid limit cycle. We show that this is sufficient to phase lock the dynamics to a weak external signal, hence clarifying synchronization in two-level systems. We use the Husimi $Q$ representation to analyze the synchronization region, defining a synchronization measure which characterizes the strength of the phase locking. Also, we study the stability of the limit cycle and its deformation with the strength of the signal in terms of the components of the Bloch vector of the system. Finally, we generalize a model of the three-level system to illustrate how the stationary fixed point of that model can be changed into a limit cycle similar to the one that we describe for the two-level system.

DOI
10.1103/physreva.101.062104

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Recently, it was shown that dissipative quantum systems with three or more levels are able to synchronize to an external signal. Some researchers have stated that this is not possible for two-level systems, as those systems lack a stable limit cycle in unperturbed dynamics, but others have demonstrated, under a different definition of synchronization, that it is possible in qubits in models which include other elements. We show how a quantum two-level system can be understood as containing a valid limit cycle as the starting point of synchronization and that it can synchronize its dynamics to an external weak signal. This is demonstrated by analytically solving the Lindblad equation of a two-level system coupled to an environment, determining the steady state. This is a mixed state with contributions from many pure states, each of which provides a valid limit cycle. We show that this is sufficient to phase lock the dynamics to a weak external signal, hence clarifying synchronization in two-level systems. We use the Husimi QQQ representation to analyze the synchronization region, defining a synchronization measure which characterizes the strength of the phase locking. Also, we study the stability of the limit cycle and its deformation with the strength of the signal in terms of the components of the Bloch vector of the system. Finally, we generalize a model of the three-level system to illustrate how the stationary fixed point of that model can be changed into a limit cycle similar to the one that we describe for the two-level system.
It is well known that a system with two or more levels exists a limit cycle and can be synchronized with an external drive when the system and the drive are directly coupled. One might wonder if a system can synchronize with the external drive when they are not coupled directly. In this paper, we examine this case by considering a composite system consisting of two coupled two-level quantum systems, one of which is driven by an external field, while another couples to the driven one. Due to the decoherence caused by environments, the composite system would stay in a mixed state, and an effective limit cycle is formed, so phase locking could occur. We find the phase locking phenomenon in the phase diagram characterized by Husimi QQQ function, and the synchronization can be generated consequently that we will refer to indirect synchronization. The SSS function defined in the earlier study can also be used to measure the strength of synchronization. We claim that indirect synchronization is possible. This result provides us with a method to synchronize a quantum system that coupled to its neighbour without interacting with external drive directly.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
We investigate the minimal Hilbert-space dimension for a system to be synchronized. We first show that qubits cannot be synchronized due to the lack of a limit cycle. Moving to larger spin values, we demonstrate that a single spin 1 can be phase locked to a weak external signal of similar frequency and exhibits all the standard features of the theory of synchronization. Our findings rely on the Husimi Q representation based on spin coherent states which we propose as a tool to obtain a phase portrait.
Synchronizing a few-level quantum system is of fundamental importance to the understanding of synchronization in the deep quantum regime. We investigate quantum phase synchronization of a two-level system (qubit) driven by a semiclassical laser field, in the presence of a general non-Markovian dissipative environment. The phase preference of the qubit is demonstrated through Husimi Q-function, and the existence of a limit cycle is also shown in our system. Synchronization of the qubit is quantified using the shifted phase distribution. The signature of quantum phase synchronization viz the Arnold tongue is obtained from the maximal value of the shifted phase distribution. Two distinct types of qubit dynamics is considered depending on the reservoir correlation time being very short and a situation when bath correlation time is finite. In the Markov regime of the environment, the phase preference of the qubit goes away in the long time limit, whereas the long-time phase localization persists in the non-Markovian regime. We also plot the maximum of the shifted phase distribution in two ways: (a) by varying the detuning and laser driving strength, and (b) by varying the system-bath coupling and laser driving strength. Various system-environment parameters determine the synchronization regions and the qubit phase synchronization is shown to be enhanced in the non-Markov regime.
Quantum synchronization has been a subject of intensive research in the last decade. In this Letter, we propose a quantum Liénard system whose classical equivalent features two limit cycles to one of which the system will converge. In the quantum case, both limit cycles coexist in a single steady state. Each of these limit cycles localizes to a distinct phase if coupled to an external drive: one quantum state can thus be assigned two phases. Furthermore, coupling two such oscillators leads to the simultaneous appearance of synchronization and a synchronization blockade. To shed light on this apparent paradoxical result, we introduce finer measures of quantum synchronization.
Spontaneous macroscopic quantum synchronization is an emergent phenomenon where an ensemble of quantum oscillators achieves global phase coherence through the interplay of interaction and dissipation. To illuminate this phenomenon, we study an ensemble of two-level systems (TLS) and establish its associated nonlinear quantum master equation, for which self-consistent analytical solutions of quantum synchronization can be obtained. The trajectories on the Bloch sphere vividly illustrate how dissipation and interaction drive the system toward a synchronized state. We present a phase diagram for macroscopic synchronization as a function of interaction strength and the gain-to-damping ratio. Furthermore, we demonstrate full synchronization and partial synchronization between two groups of TLS with different natural frequencies. This work establishes ensemble of TLS as a remarkable system for understanding spontaneous quantum synchronization.