Abstract
Synchronization and entanglement constitute fundamental collective phenomena in multiunit classical and quantum systems, respectively, both equally implying coordinated system states. Here, we present a direct link for a class of isolated quantum manybody systems, demonstrating that synchronization emerges as an intrinsic system feature. Intriguingly, quantum coherence and entanglement arise persistently through the same transition as synchronization. This direct link between classical and quantum cooperative phenomena may further our understanding of strongly correlated quantum systems and can be readily observed in stateoftheart experiments, for example, with ultracold atoms.
Introduction
Understanding collective dynamical phenomena constitutes a topical challenge across physics and beyond, with distinct implications for the classical and quantum realms. How collective phenomena in classical and quantum worlds are linked is largely unknown. Synchronization constitutes one of the most basic cooperative dynamics in classical systems. It indicates the locking of states of coupled classical units and governs the dynamics of physical, chemical, and biological systems^{1,2,3,4,5,6,7,8}. Entanglement constitutes the most fundamental phenomenon in manybody quantum systems and indicates correlations that are genuinely quantum mechanical. Two quantum particles are entangled if they cannot be described by independent singleparticle states. Such entanglement thereby determines the quantum systems’ inherent complexity^{9,10} and unique computational power^{11,12}.
In this article, we present a direct link between classical synchronization and quantum entanglement. We investigate a paradigmatic class of isolated nonlinearly coupled quantum systems combining the classical theory of synchronization with simulations of quantum dynamics and meanfield as well as higherorder analysis. We reveal that and how synchronization phenomena impact entanglement. Intriguingly, transient squeezing and number fluctuations indicating genuine entanglement emerge through and exactly at the transition to classical synchronization. Moreover, the dynamics of classical phase locking quantitatively predicts the growth of quantum number fluctuations, and for large system sizes becomes an exact indicator of the growth. As the quantum system is isolated, synchronization is not externally induced but emerges through selforganized dynamics. We demonstrate how this quantumclassical link on the level of collective phenomena may be experimentally verified, for example with ultracold atoms^{13,14,15,16,17}. For a paradigmatic and experimentally relevant class of systems, these results thus indicate that the substantial parts of the emergence of entanglement—a genuine quantum feature—can be traced back to a classical synchronization process.
Results
Signatures of synchronization
Consider the dynamics of a quantum manybody system described by Schrödinger’s equation with the Hamiltonian
describing L spatially localized modes j∈{1, …, L} (ref. 18) with onsite twobody interactions of energy scale U. denotes the annihilation and the creation operator for the jth mode and is the number operator.
The quantum manybody system (1) exhibits a sharp transition from a weakly to a strongly correlated regime. When the coupling strengths exceed a critical value, correlations emerge dynamically and persist independent of the initial state. This transition (Fig. 1) becomes apparent already for systems with just two modes, which arise in the longitudinal LipkinMeshkowGlick model (see Supplementary Note 1). Figure 1a,b illustrate the different dynamical regimes for a coherent initial state Ψ(0)〉=z, Δφ〉, that is, a state which is maximally localized in phase space (see Methods section).
For small coupling strengths , correlations remain negligible and the modes gradually dephase, so the phase coherence α_{12}(t) defined by decays rapidly. The Husimi function Q(t), representing the quantum phase space density (see Methods section), spreads in the phase direction, such that the relative phase of the two modes becomes undefined.
In contrast, for sufficiently large coupling strengths , we observe transient squeezing of the quantum state (Fig. 1c): The phase space density Q(t) is compressed in the phase direction and the uncertainty of the relative phase decreases below the standard uncertainty limit, indicating manybody entanglement^{13,19}. Moreover, a strong phase coherence prevails in the long term (Fig. 1d). The reduction of phase fluctuations is accompanied by the emergence of number entanglement: The number fluctuations exceed the maximum possible for any separable (nonentangled) quantum state, indicated by the entanglement parameter W_{12}>0 in Fig. 1e (see Methods section). Strikingly, this type of entanglement is persistent.
Squeezing and entanglement systematically emerge for coupling strengths above some critical value, whereas they are absent below, see Fig. 2. The observed transition indicates a quantum analogue of the classical synchronization transition^{1,2,3,4}. To see this, consider the meanfield limit and derive the equations of motion for the amplitudes from Heisenberg’s equation, neglecting quantum fluctuations and approximating . We obtain
(see Supplementary Note 3 for more details). Given a total number N of excitations, the manifold of the phase space defined by for all is invariant under the dynamics such that in time (see Supplementary Note 3 and ref. 20). With initial conditions on this manifold denoted by the dynamics (2) reduces to
The intrinsic frequencies and rescaled coupling strengths become and , respectively.
This mean field limit constitutes a system of Kuramoto oscillators^{2}—a paradigmatic model of classical nonlinear dynamics characterizing synchronization and other collective phenomena^{3,4}. For two modes, the dynamics is fully characterized by the phase difference Δφ=φ_{2}−φ_{1} and the population imbalance z=(c_{2}^{2}−c_{1}^{2})/(c_{2}^{2}+c_{1}^{2}) as the total number of excitations is conserved. The phase dynamics on the invariant manifold z=0 becomes
with ω=ω_{2}−ω_{1}. This Kuramoto system bifurcates at K_{c}=ω/2, precisely indicating the quantum transition point, see Fig. 2. Below K_{c} no steady states exist and the phases are unlocked. For K>K_{c} phase locking emerges such that Δφ(t) tends to the fixed point Δφ*=arcsin(ω/2K), which shapes the corresponding quantum dynamics (Fig. 1a versus b): The Husimi function is contracted at the fixed point such that phase squeezing emerges. Simultaneously, the dynamics is unstable in the zdirection, indicating the growth of number fluctuations. The classical Kuramoto dynamics can thus be seen as a skeleton of the full quantum dynamics^{21} and the onset of classical synchronization as a skeleton for the emergence of quantum correlations.
The correspondence of classical phase locking and the growth of quantum fluctuations becomes analytically exact for large populations of globally coupled oscillators, that is, with large (Fig. 3a–d). We define the Kuramoto order parameter^{1,3,4}
In the generic case, the magnitude r relaxes to a fixed value measuring the degree of phase order and γ oscillates with the mean frequency +Uν, ν=N/L being the density of atoms or excitations per mode. Transforming to a corotating frame of reference, γ becomes constant and the classical equations of motion (3) simplify to
A bifurcation occurs when the coupling K increases: For KK_{c} all oscillators drift independently such that r=0. For K>K_{c} the oscillators with get phaselocked, such that r>0.
To describe quantum fluctuations beyond meanfield, we decompose the annihilation operators into the condensate mode and the quantum fluctuations , and insert this ansatz into the Heisenberg equations of motion (see Supplementary Note 4 and Supplementary Fig. 4 for more details). To linear order in this yields the Bogoliubovde Gennes equations^{22}
On the Kuramoto manifold the Bogoliubovde Gennes operator is given by
where the coefficients are given by
in the rotating frame. In the limit N, L→∞ with fixed ν = N/L, all terms with vanish as L^{−1} such that the operator describes whether quantum fluctuations grow. For r=0 all eigenvalues of are real, implying that fluctuations do not grow. Once synchronization sets in and r>0, the eigenvalues of the phaselocked oscillators are purely imaginary,
such that quantum fluctuations grow exponentially as . The maximum growth rate becomes
This growth rate scales as the classical synchronization order parameter (Fig. 3b). Drifting oscillators typically have real eigenvalues, except for the ones in the immediate vicinity of the phaselocked region (compare Fig. 3c versus d). Hence the classical synchronization transition^{1,2,3,4,5} has a direct quantum counterpart.
Potential experimental realizations
Our predictions are observable in experiments, for example, with BoseEinstein condensates (BECs) in optical lattices^{13,14,15,16,17} or modulated photonic lattices^{23,24}. In a tilted or accelerated lattice, the eigenmodes are localized (Fig. 4a) and the energies are arranged in a ladder, (ref. 25). The atomic interactions induce the nonlinear coupling of the neighbouring modes with and otherwise. A detailed discussion of how these parameters depend on the experimental setting is provided in the Supplementary Note 2 and the Supplementary Figs 1–3.
Synchronization is detected from the momentum density ρ(k, t) measured in a timeofflight image (Fig. 4b). For weak coupling , the modes dephase^{15} such that the coherences vanish. No relative phase is defined and ρ(k, t) delocalizes over the Brillouin zone (Fig. 4b,c). Strong coupling induces synchronization such that the coherences are partly preserved and ρ(k, t) shows a localized peak which does not blur (Fig. 4e,f). The peak remains steady in the center of the Brillouin zone as the phases are locked at a constant value (Fig. 4d). Momentum space localization thus provides a robust experimental quantum signature of synchronization. Synchronization implies number fluctuations signalling entanglement (Fig. 4g), as above for two modes. This entanglement is persistent and emerges for all pure BECs and Fock states with homogeneous density.
Another possibility to experimentally realize our predictions is given by Floquet engineered optical lattices^{16,17}.
Robustness to dissipation
In many models studied so far, quantum signatures of synchronization are induced externally, via dissipation or a common driving^{26,27,28,29,30,31,32,33,34,35}. In contrast, the coupling to the environment is not the cause of synchronization in the quantum manybody system (1), which directly bears the Kuramoto model in the meanfield limit. Indeed, this classical synchronization model qualitatively predicts that the quantum system relaxes to different states depending on the coupling strength. The phase coherence , which is most easily accessible in experiments, converges to a nonzero value up to some small residual fluctuations (Figs 1 and 4). But how robust is this intrinsic form of quantum synchronization to perturbations from the environment?
We are reporting a particular destructive case of quantum dissipation, where independent phase noise couples to all modes of the system. Such a noise source arises in experiments with ultracold atoms in optical lattices due to incoherent scattering of photons from the lattice beams^{36} or collisions with the background gas^{37}. The noisy dynamics can be well captured using a quantum master equation in Lindblad form
where is the density operator and κ the noise rate. Numerical simulations of the master equation (12) for two modes indicate the influence of phase noise on the evolution of phase coherence. Without the synchronization coupling (for K=0), already weak noise completely destroys the phase coherence α_{12}(t) within a few periods T as shown in Fig. 5. In contrast, the decay of α_{12}(t) is slower by orders of magnitude in the presence of the coupling (for K>0). Hence, the effects described in the present paper should be experimentally observable also in the presence of noise.
The robustness of the phase coherence depends crucially on the coupling strength K. For supercritical coupling K>K_{c} the quantum state tends to a highly entangled superposition of atoms being localized in one of the wells, making it more susceptible to noise. Phase coherence decays in time, but the decay is still much slower than without coupling. The modecoupling through the Hamiltonian induces phase coherence also for 0<KK_{c}, but without strong number fluctuations. This form of coherence is remarkably robust. After an initial drop, the phase coherence α_{12}(t) remains almost constant in time for the subcritical coupling K=0.1 (Fig. 5a) and the final value of α_{12}(t) is almost independent of the noise strength κ (Fig. 5b). For different sources of noise or dissipation (less uncorrelated for instance) we expect an even better robustness in all mentioned regimes.
Discussion
In summary, we have unearthed the manifestation of classical synchronization in a class of quantum manybody systems, providing a direct link between collective classical and quantum dynamics. So far, synchronization and entanglement have been mostly studied as two separate phenomena in the classical and quantum worlds, respectively. Recent previous works considered aspects of synchronization in open quantum model systems, where the coupling to the environment is crucial. The interaction with a common thermal bath can incude synchronization of qubits^{26} or harmonic oscillators^{27}, as well as a common classical driving field^{28}. Synchronization has also been studied for quantum van der Pol oscillators^{29,30,31,32} and other driven dissipative oscillators^{33,34}. In all these cases, dissipation and external driving play a crucial role, for instance the selfsustained oscillations of the van der Pol oscillators are entirely driven by the exchange of excitations with the bath. It has also been shown that quantum effects can prevent the occurrence of synchronization of coupled spins^{38}. The Kuramoto model itself has been recovered in the semiclassical limit of different quantum system, in particular particles moving in tilted washboard potential^{35}, coupled optomechanical oscillators^{39} or Josephson junction arrays^{40}. In contrast, we here analysed a class of isolated quantum systems, demonstrating that synchronization emerges as an intrinsic system feature. We recover the celebrated Kuramoto model in the meanfield limit, which can be seen as a skeleton for the fullquantum manybody dynamics.
Indeed, the transition to synchronization clearly indicates squeezing, longterm coherence and persistent entanglement. Moreover, the dynamics of phase locking in the synchronization process indicates the growth of number fluctuations, becoming exact in the limit of large system sizes. Our findings can be directly verified by stateoftheart experiments, for instance with ultracold atoms in accelerated or driven optical lattices^{13,14,15,16,17} or modulated photonic lattices^{23,24}. These experiments are facilitated by the observed robustness with respect to dissipation. They offer a unique control over the system parameters such that various distributions of natural frequencies can be realized which can give rise to different types of synchronization phase transitions^{41,42}. Advanced imaging techniques allow to observe the global phase coherence as well as number distributions with single site resolution. These results thus offer a novel perspective on a correspondence between classical and quantum dynamics, on the level of collective phenomena.
Methods
Coherent states
Spin coherent states are defined as . They are maximally localized in phase space and thus provide a natural link to the classical meanfield dynamics^{18}. The projection of a quantum state on a coherent states defines the Husimi function Q(z, Δφ; t)≡〈z, ΔφΨ(t)〉^{2}. It carries all information about the quantum state and shares properties of a classical phase space density^{43}.
Squeezing
To quantify squeezing one defines the collective operators which form an angular momentum algebra. The squeezing parameter is then defined as where is a rotation of the vector operator . Spectroscopic squeezing with ξ^{2}<1 is only possible for entangled states and enables highprecision quantum metrology^{13,19}.
Number entanglement
The variance of the number difference between two modes is bounded for every pure separable state as (ref. 44). If the entanglement parameter exceeds zero for a pure state, this unambiguously proves the entanglement of the modes. States with large W_{jk} are used in precision quantum metrology^{45,46,47}.
Tilted optical lattices
To study the quantum dynamics in tilted or accelerated optical lattices, we expand the bosonic field operator in the singleparticle eigenmodes assuming that tunneling to excited Bloch bands is negligible. The eigenmodes are arranged in ladders with equidistant eigenenergies , where ω_{B} is the Bloch frequency^{25}. Depending on the tilting, the eigenmodes are strongly localized in real space (Fig. 4a), such that they couple only to the nearest neighbours. The momentum space density is given by
In the noninteracting case U=K=0 the coherences are constant in magnitude and the phases evolve according to . The atoms show a periodic dynamics, referred to a Bloch oscillations, with the Bloch period T_{B}=2π/ω_{B} (refs 25, 48). Pure onsite interactions lead to dephasing such that the coherences vanish, →0 for and the momentum density reads , which is extended over the entire first Brillouin zone^{15,49}.
Data availability
The data that support the findings of this study (in particular simulation source code and figure raw data) are available from the corresponding author upon request.
Additional information
How to cite this article: Witthaut, D. et al. Classical synchronization indicates persistent entanglement in isolated quantum systems. Nat. Commun. 8, 14829 doi: 10.1038/ncomms14829 (2017).
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Acknowledgements
We thank Matteo di Volo and Alessandro Vezzani for helpful discussions and gratefully acknowledge support from the Helmholtz association (grant no. VHNG1025 to D.W.), the FIL 2014 program of Parma University, and the Max Planck Society (grant to M.T).
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Affiliations
Forschungszentrum Jülich, Institute for Energy and Climate Research (IEKSTE), 52428 Jülich, Germany
 Dirk Witthaut
Institute for Theoretical Physics, University of Cologne, Zuelpicher Str. 77, 50937 Köln, Germany
 Dirk Witthaut
Network Dynamics, Max Planck Institute for Dynamics and SelfOrganization (MPIDS), Am Faßberg, 37077 Göttingen, Germany
 Dirk Witthaut
 & Marc Timme
Dipartimento di Scienze Matematiche, Fisiche ed Informatiche, Universitá di Parma, Via G.P. Usberti 7/a, 43124 Parma, Italy
 Sandro Wimberger
 & Raffaella Burioni
INFN, Sezione di Milano Bicocca, Gruppo Collegato di Parma, Parco Area delle Scienze, 7/A, 43124 Parma, Italy
 Sandro Wimberger
 & Raffaella Burioni
Department of Physics, University of Darmstadt, 64289 Darmstadt, Germany
 Marc Timme
Institute for Theoretical Physics, Technical University of Dresden, 01062 Dresden, Germany
 Marc Timme
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Contributions
D.W. and M.T. designed and conceived the research. D.W. worked out the theory and carried out the numerical experiments. D.W. and S.W. worked out the possible experimental realizations. All authors contributed ideas, analysis tools and to the writing of the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Dirk Witthaut.
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