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Journal of Physics A Mathematical and Theoretical

Quantum synchronization of the Schrödinger–Lohe model

Sun-Ho Choi, Seung‐Yeal Ha

Published Aug 18, 2014• 31 citations
Publisher

Abstract

We present a quantum synchronization estimate of the Schrödinger–Lohe (S–L) model introduced by Lohe (2010 J. Phys. A: Math. Theor. 43 465301 ). The S–L model describes the dynamics of quantum oscillators on the nodes of a quantum network. When the coupling strength is positive and the maximal L 2 distances between normalized initial wave functions are smaller than , we show that the L 2 distances between wave functions converge to zero exponentially fast. Our result generalizes earlier work by Chi et al (2014 J. Math. Phys. 55 052703 ) for the Lohe model.

DOI
10.1088/1751-8113/47/35/355104

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We present a quantum synchronization estimate of the Schrödinger–Lohe (S–L) model introduced by Lohe (2010 J. Phys. A: Math. Theor. 43 465301 ). The S–L model describes the dynamics of quantum oscillators on the nodes of a quantum network. When the coupling strength is positive and the maximal L 2 distances between normalized initial wave functions are smaller than , we show that the L 2 distances between wave functions converge to zero exponentially fast. Our result generalizes earlier work by Chi et al (2014 J. Math. Phys. 55 052703 ) for the Lohe model.
Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.
We consider a network of quantum oscillators in which quantum states are distributed among connected nodes by means of unitary transformations. The distributed states interact with each local state according to a time-dependent interaction Hamiltonian, which is modeled by a Hermitian operator constructed in terms of the states themselves, thereby introducing nonlinear network interactions. For qubit nodes of differing natural frequencies, we show numerically that for a sufficiently large coupling constant, synchronization of quantum nodes occurs in which the spins of all qubits are mutually aligned with a common frequency of oscillation, following initial transient configurations. We discuss the significance of quantum synchronization as a means to create copies of unknown quantum states.
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.
We show the emergence of spontaneous synchronization between a pair of detuned quantum oscillators within a harmonic network. Our model does not involve any nonlinearity, driving, or external dissipation, thus providing the simplest scenario for the occurrence of local coherent dynamics in an extended harmonic system. A sufficient condition for synchronization is established by building upon the Rayleigh normal mode approach to vibrational systems. Our results show that mechanisms favoring synchronization, even between oscillators that are not directly coupled to each other, are transient energy depletion and crosstalk. We also address the possible buildup of quantum correlations during synchronization and show that indeed entanglement may be generated in detuned systems, starting from uncorrelated states and without any direct coupling between the two oscillators.
Natural and engineered classical systems are replete with examples of synchronization, understood as the adjustment of rhythms of physical systems. Such synchronization is at the heart of the stability of several classical technologies, such as mechanical bridges and electrical networks. Given the advent of quantum simulation and computation technologies, it is natural to study a quantum analogue of synchronization and explore novel applications. This review surveys synchronization in few and many-body quantum systems, measures that quantify them, and their applications to quantum technologies.