Introduction

Noise is widely regarded as a nuisance that limits the transmission and processing of information1. The adverse effect of random fluctuations is even more dramatic in quantum physics, since quantum coherence and quantum correlations, two essential resources of quantum technologies2, are highly susceptible to external disturbances3. Surprisingly, the nontrivial interplay between noise and nonlinear dynamics may induce order and organization4,5,6,7. Classical noise-induced phenomena, such as pattern formation, noise-induced transport and stochastic resonance, occur in a great variety of contexts, from physics and chemistry to biology and engineering4,5,6,7. However, observing the constructive role of noise in quantum systems is far more challenging8,9,10,11,12,13.

Noise-induced synchronization is another counterintuitive consequence of random fluctuations14,15,16,17,18,19,20,21,22,23; it is, for instance, thought to be relevant for collective neuronal synchronization in the brain21,22,23. Synchronization is a general concept in classical24,25,26,27,28,29,30 and quantum31,32,33,34,35,36,37,38,39,40 physics: synchronous motion usually arises when coupled nonlinear oscillators adjust their internal rhythms, and oscillate in unison24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40. Synchronization phenomena have lately found interesting applications in communication systems41,42 and in complex networks43. In noise-induced synchronization, collective oscillations arise through the constructive influence of a noise source. This classical effect has been observed in lasers19,20 and in sensory neurons21. In the quantum regime, noise-induced synchronization has been predicted to occur in many-body systems and to lead to entangled collective oscillations40. So far, synchronous oscillations with stronger than classical, long-distance correlations have not been observed.

We here report the experimental realization of noise-induced quantum synchronization in a linear chain of superconducting transmon qubits with nearest-neighbor interactions44. We observe the occurrence of stable, synchronized oscillations of the magnetizations of the edge qubits when Gaussian noise is applied to a single qubit. We further show that the corresponding synchronized state is not only entangled, with nonzero concurrence45, but that it is given by a maximally entangled mixed state which exhibits the maximum obtainable amount of entanglement for a given degree of mixedness46,47,48,49,50,51. Such states are regarded as direct generalizations of maximally entangled pure Bell states46,47,48,49,50,51, and play an important role in mixed-state quantum information processing52,53,54,55. We additionally confirm the robustness of the observed quantum synchronization phenomenon to detuning of the natural frequencies of the qubits. We concretely obtain Arnold-tongue-like patterns24,25,26,27,28,29,30, for both synchronization and entanglement, as a function of detuning and noise strength.

Results

Experimental system

Our experiments are implemented on a superconducting quantum processor, comprising a one-dimensional array of tunably coupled transmon qubits44 (Fig. 1a, b). The qubits act as artificial spins, where the jth qubit frequency ωj/(2π) can be controlled, in the range from  ~3.2 GHz to  ~4.6 GHz, by applying an external flux through the dedicated Z line of the corresponding qubit and coupler56. Likewise, the coupling constants (Jj = J)/(2π) between the qubits is set to  ~10 MHz by applying an external flux on the associated C line56 (Supplementary Information). Each qubit can be individually addressed and driven into the excited state by applying a microwave pulse through its XY control line. The lattice model of the experiments can be described by the Hamiltonian of a one-dimensional quantum XY chain of N spins in a transverse field57

$${H}_{0}=\frac{\hslash J}{2}{\sum}_{j=1}^{N-1}\left({\sigma }_{j}^{x}{\sigma }_{j+1}^{x}+{\sigma }_{j}^{y}{\sigma }_{j+1}^{y}\right)+{\sum}_{j=1}^{N}\hslash {\omega }_{j}{\sigma }_{j}^{z},$$
(1)

where \({\sigma }_{j}^{\, x,y,z}\) are the local Pauli operators acting on site j. We apply Gaussian white noise ξ(t) with zero mean and autocorrelation \(\langle \xi (t)\xi ({t}^{{\prime} })\rangle=\Gamma \delta (t-{t}^{{\prime} })\), with noise strength Γ, by locally modulating the natural frequencies of the individual qubits on the desired sites58 (Supplementary Information) (Fig. 1c, d). This stochastic contribution corresponds to the addition of the Hermitian operator \(\xi (t){\sigma }_{u}^{z}\) (acting locally on site u) to the Hamiltonian (1). Stable synchronization of the entire quantum chain is predicted to occur, for a homogeneous chain with constant frequencies, ωj = ω, and noise applied to a single site u, when the two conditions, N = 5 + 3m and u = 3n (\(m,n\in {\mathbb{N}}\)), are satisfied40. This leads to a decoherence-free subspace, that is decoupled from the surroundings59, with only a single eigenmode, whose frequency determines the synchronization frequency. In the following, we analyze the occurrence of stable noise-induced quantum synchronization in a chain of N = 5 qubits (corresponding to m = 0) when the noise acts on the third qubit (corresponding to n = 1) in the middle of the chain (Fig. 1c). This is the smallest spin chain in which noise-induced quantum synchronization is expected to appear. Additional cases with larger number of spins (N = 8 and 11), as well as with violations of the synchronization conditions are presented in the Supplementary Information.

Fig. 1: Experimental system.
figure 1

a The chip of a transmon qubit chain is designed with a flip-chip technique with the top layer consisting of tunable-frequency qubits (with 5, 8 and 11 qubits) and tunable-frequency couplers, that facilitate interactions between qubits. b Other elements of the chip sit in the bottom layer including the resonators, the qubit control and flux lines (XYZ lines) and neighboring coupler flux lines (C lines). c A one-dimensional XY chain of five transmon qubits, with coupling constant J and edge-spin frequencies detuned by Δ, is subjected to d Gaussian white noise ξ(t) with effective amplitude γ.

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Noise-induced quantum synchronization

We initially prepare the quantum spin chain in the separable state \(| \Psi (0)\left.\right\rangle=| 1\left.\right\rangle \otimes {| 0\left.\right\rangle }^{\otimes 4}\), where \(| 1\left.\right\rangle\) and \(| 0\left.\right\rangle\) denote the respective excited and ground states of the qubits (Fig. 2a, b). The state of each qubit is determined by measuring the state-dependent transmission of a dedicated readout resonator, with frequency centered around 6.1 GHz, coupled to each qubit using a dispersive readout scheme (Supplementary Information).

Fig. 2: State preparation.
figure 2

a The system is initially prepared in a separable state where qubit Q1 is prepared in the excited state while all the other qubits Qj of the quantum chain are in the ground state; external noise ξ(t) is applied to the central spin Q3. b Quantum circuits used in the tomographic measurement of the two-qubit density matrix ρ15 of the two end spins. c The measured real part of the density matrix ρ15 of the synchronized edge qubits at time Jt = 3π corresponds to a maximally entangled mixed state.

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Figure 3 displays the temporal evolution of the measured local z-polarization \(\langle {\sigma }_{j}^{z}\rangle\) of the individual qubits, without (Fig. 3a) and with (Fig. 3b) Gaussian noise applied to the middle of the chain (the reduced noise strength is γ = Γ/J ≈ 1.3). In the absence of noise, the initial excitation travels through the chain in a wave-like manner, is reflected at the open boundaries, and bounces back and forth between the edges of the chain, without collective coordination between the individual qubits (Fig. 3a). By contrast, in the presence of Gaussian noise, qubits 1 and 5 (as well as qubits 2 and 4) oscillate in phase at a single frequency of  ~20 MHz, in agreement with the predicted value 2J, after some transient (Fig. 3b, c). The synchronization frequency is set by the coupling constant of the quantum chain and not by the eigenfrequencies of the qubits. The faint lines in the background of Fig. 3c, d represent the measured unsynchronized, quasi-unitary dynamics of the magnetizations, when the external Gaussian noise is switched off.

Fig. 3: Noise-induced quantum synchronization.
figure 3

a The local z-polarizations \(\langle {\sigma }_{j}^{z}\rangle\) of a chain of five qubits do not synchronize during noise-free unitary dynamics. b Synchronized oscillations, with frequency  ~ 2J = 20 MHz, between c qubits 1–5 and d qubits 2–4 occur, when Gaussian white noise with strength γ = 1.3 is applied to the third spin. The associated unsynchronized unitary evolution is shown in the background. e Pearson correlation coefficients C15 and C24 of qubits 1–5 and qubits 2–4 (symbols) converge to one, indicating perfect correlation between the corresponding qubits. The synchronized regime, Cij ≥ 0.9, is attained for Jt ≥ 3π (red arrow). Good agreement with theory (solid lines) is obtained.

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In order to quantitatively characterize the synchronized oscillations of the respective polarizations, we use the Pearson correlation coefficient, a standard measure of quantum synchronization38,39,40, defined as the ratio of the covariance and the respective standard deviations, \({C}_{ij}={{{\rm{cov}}}}\left(\langle {\sigma }_{i}^{z}\rangle,\langle {\sigma }_{j}^{z}\rangle \right)/\sqrt{{{{\rm{var}}}}(\langle {\sigma }_{i}^{z}\rangle ){{{\rm{var}}}}(\langle {\sigma }_{j}^{z}\rangle )}\)60. The latter quantity provides a measure of the degree of linear correlation between observables; it ranges from  −1 (corresponding to anticorrelated oscillations) to  +1 (indicating correlated evolution). Figure 3e shows the Pearson correlation coefficients C15 and C24 of the measured z-polarizations of qubits 1–5 and qubits 2–4, as a function of time (symbols). Both quickly converge to one, demonstrating almost perfectly correlated oscillations over the entire duration of the experiment, in good agreement with the theoretical simulations (solid lines) (Supplementary Information). We specifically consider two qubits to be synchronized for Pearson coefficients Cij ≥ 0.9, which happens for times Jt ≥ 3π (red arrow). We also mention that synchronized oscillations occur for arbitrary initial states as long as they have a nonzero overlap with the synchronized mode (Supplementary Information).

The synchronization condition ensures that the noise dynamically suppresses all the eigenmodes of the system except one, as seen in the measured Fourier spectrum of the magnetizations (Fig. 4a, b). The frequency  ~20 MHz of the surviving mode determines the synchronization frequency (see below). This synchronization effect may thus be regarded as a quantum generalization of the classical synchronization mechanism known as “suppression of natural dynamics"28,29,30. It is worth mentioning that dissipation is necessary to induce stable synchronization in a quantum many-body system, since the required permutation invariance of local observables, such as \({\sigma }_{j}^{z}\), is not guaranteed for purely unitary evolution39 (Supplementary Information).

Fig. 4: Eigenmode suppression.
figure 4

a Modulus of the Fourier transform of the magnetizations after being synchronized for time Jt = 3π without, and b with noise, showing the suppression of all the system eigenmodes, except one.

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We next analyze the robustness of the synchronous oscillations. To that end, we introduce a variable detuning Δ between the natural frequencies of the synchronized end qubits via a term \({H}_{1}=(\hslash \Delta /2)({\sigma }_{1}^{z}-{\sigma }_{5}^{z})\) added to the system Hamiltonian (1) (Fig. 1c). Figure 5a displays the measured Pearson correlation coefficient C15 at the onset of the synchronization regime Jt = 3π, when both the reduced noise amplitude γ and the detuning Δ are varied. We recognize a structure which is reminiscent of an Arnold tongue which defines the synchronized domain of classical synchronization phenomena24,25,26,27,28,29,30. Noise-induced quantum synchronization appears to be a robust effect that occurs in a wide range of parameters of the system. We note that the synchronization region is enlarged when the detuning is reduced and when the noise strength is increased; it is thus easier to synchronize identical spins with strong noise (provided the noise amplitude remains below the noise-induced quantum Zeno regime61,62). We again obtain good agreement with theoretical simulations (Fig. 5b).

Fig. 5: Stability regions and entanglement.
figure 5

Arnold tongue of synchronization. Experimental data (a) and numerical simulation (b) for the Pearson correlation coefficients C15, extracted at a given time Jt = 3π (red arrow), as a function of noise amplitude γ and detuning Δ between the two end spins. Larger values of detuning are detrimental, but the system can still be synchronized by increasing the noise strength. Concurrence \({{{\mathcal{C}}}}({\rho }_{15})\) (c) and fidelity F(M, ρ15) (d) of the predicted maximally entangled mixed state (M) and the measured two-qubit density operator (ρ15) as a function of time. Both quantities display non-vanishing steady oscillations for Jt ≥ 2π, showing that the synchronized two-qubit state is entangled Good agreement between data (dots) and theory (lines) is observed. Entanglement tongue. Experimental data (e) and numerical simulation (f) for the concurrence \({{{\mathcal{C}}}}({\rho }_{15})\) at a given time Jt = 3π as a function of noise amplitude γ and the detuning Δ. Increasing both the detuning and the noise strength diminishes the amount of entanglement in the system.

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Maximally entangled mixed states

Entanglement is a fundamental resource in quantum information science; mechanisms creating entangled states are hence of great importance2. There seems, however, to be no direct relationship between quantum synchronization and quantum correlations in general63,64,65. In view of the detrimental influence of noise on quantum properties3, it is therefore all the more remarkable that noise-induced synchronization is expected to give rise to entangled synchronized edge qubits40. In order to test this feature, we tomographically reconstruct the two-qubit state of the two end spins66 as illustrated in Fig. 2c and evaluate the concurrence, \({{{\mathcal{C}}}}(\,{\rho }_{15})=\max \left(0,\sqrt{{\kappa }_{1}}-\sqrt{{\kappa }_{2}}-\sqrt{{\kappa }_{3}}-\sqrt{{\kappa }_{3}}\right)\); the operator ρ15 is here the reduced density matrix of the two edge qubits and κn are the ordered eigenvalues of the product \({\rho }_{15}{\widetilde{\rho }}_{15}\), with \({\widetilde{\rho }}_{15}\) the spin flipped state45. Figure 5c shows that \({{{\mathcal{C}}}}({\rho }_{15})\) exhibits nonzero steady oscillations, clearly indicating the presence of synchronized entangled edge qubits. This observation reveals an intriguing connection between collective quantum behavior and nonclassical correlations. The concurrence reaches a steady state for Jt ≥ 2π, thus slightly before the two edge qubits are fully synchronized (Jt ≥ 3π).

The reconstructed two-qubit state ρ15 at time Jt = 3π is explicitly given in Fig. 2c: it has the form of a maximally entangled mixed state which can be parametrized as \(\rho \,=\, {p}_{1}| {\Psi }^{-}\left.\right\rangle \left\langle \right.{\Psi }^{-}|+{p}_{2}\,| 00\left.\right\rangle \left\langle \right.00|+{p}_{3}| {\Psi }^{+}\left.\right\rangle \left\langle \right.{\Psi }^{+}|+{p}_{4}| 11\left.\right\rangle \left\langle \right.11|\) with p1  ≥  p2  ≥  p3  ≥  p446, where \(| {\Psi }^{-}\left.\right\rangle=(| 01\left.\right\rangle -| 10\left.\right\rangle )/\sqrt{2}\) and \(| {\Psi }^{+}\left.\right\rangle=(| 01\left.\right\rangle+| 10\left.\right\rangle )/\sqrt{2}\) are the usual Bell states2 and ∑kpk = 1. The concurrence of the maximally entangled mixed state can be analytically determined as \({{{\mathcal{C}}}}={p}_{1}-{p}_{3}-2\sqrt{{p}_{2}{p}_{4}}\)46. The fidelity of the measured state ρ15 and the theoretical maximally entangled mixed state M given, for the parameters of the experiment, by

$$M=\left(\begin{array}{cccc}0&0&0&0\\ 0&1/3&-1/6&0\\ 0&-1/6&1/3&0\\ 0&0&0&1/3\end{array}\right),$$
(2)

is \(F(M,{\rho }_{15})=\,{{\mbox{Tr}}}\,\left[\scriptstyle\sqrt{\sqrt{M}{\rho }_{15}\sqrt{M \, }}\right]=99.3\%\) (Supplementary Information). The maximally entangled mixed state is hence already created at the beginning of the synchronized regime (the fidelity between measured and theoretical states at Jt = 4π is F = 99.6%). The time evolution of the fidelity of the measured two-qubit state and the theoretical maximally entangled mixed state is shown in Fig. 5d. It exhibits a behavior similar to that of the concurrence (Fig. 5c). In particular, it displays steady oscillations for Jt ≥ 2π. Maximally entangled mixed states define a class of quantum states for which no more entanglement can be created by any global unitary operations46,47,48,49,50,51. They have the interesting property that they are more entangled than Werner states with the same purity46,47,48,49,50,51. So far, maximally entangled mixed states have only been generated in optical systems49,50,51.

Like the Pearson correlation coefficient C15, the concurrence \({{{\mathcal{C}}}}({\rho }_{15})\) is robust to detuning of the edge qubit frequencies. It appears in a large parameter domain (Fig. 5e), and exhibits an (inverted) Arnold-tongue-like structure that results from the competition of two different mechanisms: on the one hand, entanglement between the initially separable end spins is created through the unitary evolution of the system67,68,69,70; on the other hand, noise, which drives the quantum synchronization process, destroys quantum correlations. The value of the effective noise strength γ, which controls the synchronization time40, sets the maximal amount of entanglement that the synchronized state can have once it has reached the (quasi)-stationary state in the decoherence-free subspace. The concurrence of the maximally entangled mixed state thus decreases when the noise amplitude or the detuning are increased. The entanglement tongue seen in Fig. 5e is often regarded as a quantum generalization of the classical Arnold tongue71. As before, good agreement with theoretical simulations is found (Fig. 5f).

Discussion

Classical synchronization gives rise to fascinating collective oscillation phenomena24,25,26,27,28,29,30. On the other hand, entanglement has been recognized as a powerful resource for quantum applications2. Both are ubiquitous in current science and technology. We have taken a first step towards merging these two fields by demonstrating the occurrence of entangled, quantum synchronization in a chain of transmon qubits. Applying Gaussian white noise to one site of the chain, perfectly correlated in-phase oscillations at a frequency set by the coupling constant of the chain, with a Pearson coefficient close to one, have been observed. These findings provide a unique illustration of the nontrivial interplay between noise and unitary dynamics in a quantum many-body system, leading to collective behavior, and at the same time, to the creation of distant quantum correlations with nonzero concurrence. In view of their generality, we expect these results to be important for future studies of quantum-enhanced synchronization, including synchronization-based quantum communication72,73, complex quantum networks74,75 and quantum metrology76.