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Synchronization transition in the quantum dynamics of two spins

Y. Liu, Frédéric Piéchon, Jean-Noël Fuchs

Published Mar 22, 2013• 0 citations
OpenAlex

Abstract

Motivated by the spin self-rephasing recently observed in an atomic clock, we introduce a simple dynamical model to study the competition between dephasing and synchronization. Two spins $S$ are taken to be initially parallel and in the plane perpendicular to an inhomogeneous magnetic field $\Delta$ that tends to dephase them. In addition, the spins are coupled by exchange interaction $J$ that tries to keep them locked. The analytical solution of the classical dynamics shows that, there is a phase transition to a synchronized regime for sufficiently large exchange interaction $J>\Delta$ compared to the inhomogeneity. The quantum dynamics is solved analytically in four limits -- large/small $J/\Delta$ and large/small $S$ -- and numerically in between. In sharp contrast to the classical case, the quantum solution features very rich $S$-dependent multiscale dynamics. For any finite $S$, there is no synchronization but a crossover around $J=\Delta$ between two regimes. The synchronization transition is only recovered when $S\to \infty$, approaching the classical solution in a non-trivial way. Quantum effects therefore suppress the synchronization transition.

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Motivated by the spin self-rephasing recently observed in an atomic clock, we introduce a simple dynamical model to study the competition between dephasing and synchronization. Two spins SSS are taken to be initially parallel and in the plane perpendicular to an inhomogeneous magnetic field Δ\DeltaΔ that tends to dephase them. In addition, the spins are coupled by exchange interaction JJJ that tries to keep them locked. The analytical solution of the classical dynamics shows that, there is a phase transition to a synchronized regime for sufficiently large exchange interaction J>ΔJ>\DeltaJ>Δ compared to the inhomogeneity. The quantum dynamics is solved analytically in four limits -- large/small J/ΔJ/\DeltaJ/Δ and large/small SSS -- and numerically in between. In sharp contrast to the classical case, the quantum solution features very rich SSS-dependent multiscale dynamics. For any finite SSS, there is no synchronization but a crossover around J=ΔJ=\DeltaJ=Δ between two regimes. The synchronization transition is only recovered when S→∞S\to \inftyS→∞, approaching the classical solution in a non-trivial way. Quantum effects therefore suppress the synchronization transition.
Motivated by the spin self-rephasing recently observed in an atomic clock, we introduce a simple dynamical model to study the competition between dephasing and synchronization. Two spins S are taken to be initially parallel and in the plane perpendicular to an inhomogeneous magnetic field Δ that tends to dephase them. In addition, the spins are coupled by exchange interaction J that tries to keep them locked. The analytical solution of the classical dynamics shows that there is a phase transition to a synchronized regime for sufficiently large exchange interaction compared to the inhomogeneity. The quantum dynamics is solved analytically in four limits —large/small and large/small S — and numerically in between. In sharp contrast to the classical case, the quantum solution features very rich S -dependent multiscale dynamics. For any finite S , there is no synchronization but a crossover around between two regimes. The synchronization transition is only recovered when , approaching the classical solution in a non-trivial way. Quantum effects therefore suppress the synchronization transition.
We analyze the static and dynamic properties of two Ising-coupled quantum spins embedded in a common bosonic bath as an archetype of dissipative quantum mechanics. First, we elucidate the ground-state phase diagram for an Ohmic and a sub-Ohmic bath using a combination of bosonic numerical renormalization group (NRG), analytical techniques, and intuitive arguments. Second, by employing the time-dependent NRG we investigate the system's rich dynamical behavior arising from the complex interplay between spin-spin and spin-bath interactions. Interestingly, spin oscillations can synchronize due to the proximity of the common non-Markovian bath and the system displays highly entangled steady states for certain nonequilibrium initial preparations. We complement our nonperturbative numerical results by exact analytical solutions when available and provide quantitative limits on the applicability of the perturbative Bloch-Redfield approach at weak coupling.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
Can synchronization -- the widespread spontaneous emergence of coordinated dynamics in classical nonlinear systems -- also occur in the quantum regime? This question has recently sparked intense research into collective behavior and temporal self-organization in quantum systems. Typical quantum features such as the linearity of time evolution and the presence of quantum noise seem to hinder the appearance of synchrony at the microscopic level. At the same time, quantum coherence and quantum correlations may provide novel mechanisms for enhancing synchronization beyond its classical counterpart. We here survey recent theoretical and experimental advances in quantum synchronization, ranging from the characterization of synchronous oscillations and genuinely nonclassical forms of synchrony to many-body synchronization on quantum networks.
A system consisting of a number of trapped two-level atoms in the presence of external inhomogeneous magnetic field undergoes dephasing: classically, since each atom feels a different field along its trajectory, the pseudospin rotation rates differ; as a result the average spin decays. It has been demonstrated in recent years that in the case of systems initiated in the atomic coherent state such spin dephasing can be prevented by tuning the interaction between the atoms to induce so-called spin self-rephasing. While the effect has been studied theoretically from a semiclassical point of view, a quantum-mechanical description is limited. In this paper we provide a numerical simulation of an ab initio model and provide realistic examples of spin self-rephasing used to counteract the effect of inhomogeneous magnetic field. We found that the spin self-rephasing method can be also used to prevent atomic squeezed states from decoherence: while the dephasing inhibition is lower as compared with the case of coherent states, it is still comparatively large and may be useful in real interferometric scenarios.
We discuss the emergence of spontaneous synchronization for an open spin-pair system interacting only via a common environment. Under suitable conditions, and even in the presence of detuning between the natural precession frequencies of the two spins, they are shown to reach a long-lasting transient behavior where they oscillate in phase. We explore the connection between the emergence of such a behavior and the establishment of robust quantum correlations between the two spins, analyzing differences between dissipative and dephasing effects. In particular, in the regime in which synchronization occurs, quantum correlations are more robust for shorter synchronization times and this is related to a separation between system decay rates.