Quantum Synchronization Blockade: Energy Quantization hinders Synchronization of Identical Oscillators

TL;DR

In the quantum regime, energy quantization prevents identical oscillators from synchronizing, and detuning can actually promote synchronization, contrasting classical expectations.

Contribution

This paper introduces the concept of quantum synchronization blockade, showing how energy quantization inhibits synchronization of identical oscillators.

Findings

Quantum synchronization blockade demonstrated in Kerr-type oscillators

Detuning can enhance synchronization in the quantum regime

Proposed implementations with superconducting circuits and trapped ions

Abstract

Classically, the tendency towards spontaneous synchronization is strongest if the natural frequencies of the self-oscillators are as close as possible. We show that this wisdom fails in the deep quantum regime, where the uncertainty of amplitude narrows down to the level of single quanta. Under these circumstances identical self-oscillators cannot synchronize and detuning their frequencies can actually help synchronization. The effect can be understood in a simple picture: Interaction requires an exchange of energy. In the quantum regime, the possible quanta of energy are discrete. If the extractable energy of one oscillator does not exactly match the amount the second oscillator may absorb, interaction, and thereby synchronization is blocked. We demon- strate this effect, which we coin quantum synchronization blockade, in the minimal example of two Kerr-type self-oscillators and predict consequences for small oscillator networks, where synchronization between blocked oscillators can be mediated via a detuned oscillator. We also propose concrete implementations with super- conducting circuits and trapped ions. This paves the way for investigations of new quantum synchronization phenomena in oscillator networks both theoretically and experimentally.

Equations

$$H=\omega a^{†}a-K(a^{†}{)}^2{a}^2$$

$${\mathcal{L}}^{+}=\frac{{\gamma }_{+}}{2}\sum _n{f}_{+}(n)\mathcal{D}\left[\sqrt{n}|n⟩⟨n-1|\right],$$

$${\mathcal{L}}^{-}=\frac{{\gamma }_{-}}{2}\sum _n{f}_{-}(n)\mathcal{D}\left[\sqrt{n}|n-1⟩⟨n|\right],$$

$$f_{\pm }(n)=\frac{{\sigma }_{\pm }^2}{(n-n_{\pm }{)}^2+{\sigma }_{\pm }^2},$$

$$\dot{\rho }=-i\left[\sum _j{H}_j+V,\rho \right]+\sum _j{\mathcal{L}}_j\rho ,$$

$${\dot{\alpha }}_j=-\left(i{\Omega }_j({\alpha }_j){\alpha }_j+\frac{{\Gamma }_j({\alpha }_j)}{2}\right){\alpha }_j-i{C}_{jk}{\alpha }_k+{\eta }_j$$

$$S=2\pi \underset{\varphi }{\max }[P(\varphi )]-1,$$

$$\Delta +2K\Delta n\pm 2K=0,$$

$${\lambda }_{-}$$

$$-\frac{{\gamma }_{1+}}{2}(m_1{f}_{1+}(m_1)+(m_1+1)f_{1+}(m_1+1))-\frac{{\gamma }_{1-}}{2}((m_1-1)f_{1-}(m_1-1)+m_1{f}_{1-}(m_1))$$

$$-\frac{{\gamma }_{2+}}{2}((m_2+2)f_{2+}(m_2+2)+(m_2+1)f_{2+}(m_2+1))-\frac{{\gamma }_{2-}}{2}((m_2+1)f_{2-}(m_2+1)+m_2{f}_{2-}(m_2)).$$

$${\rho }_{m_1-1,m_2+1;m_1{m}_2}^{(1)}=-iC\sqrt{m_1(m_2+1)}\frac{1}{{\lambda }_{-}}\left({\rho }_{m_1{m}_2;m_1{m}_2}^{(0)}-{\rho }_{m_1-1,m_2+1;m_1-1,m_2+1}^{(0)}\right)$$

$$S=2\left|\sum _{m_1=1,m_2=0}^{\infty }{\rho }_{m_1-1,m_2+1;m_1,m_2}^{(1)}\right|$$

Full text

Coupled self-oscillating systems can spontaneously synchronize, i.e., align their phase and frequency. This phenomenon Pikovsky et al. (2003); Balanov et al. (2009) is observed in a multitude of systems, ranging from the spontaneous blinking of fireflies in unison to the firing of neurons in the human brain, and technical applications such as lasers.

The laser is a well-known example of a quantum system that is described as a self-osillator. However, its steady state far above threshold settles into a coherent state, which is essentially classical Gardiner and Zoller (2004); Walls and Milburn (2008). Therefore, its synchronization behavior could so far be fully described within a semiclassical picture Schleich and Scully (1988); Schleich et al. (1988), which allows for efficient simulations. Along this line, powerful methods have been developed capable of describing large quantum oscillator arrays, such as complex lasing media Türeci et al. (2006); Malik et al. (2015), arrays of optomechanical systems Ludwig and Marquardt (2013); Weiss et al. (2016) and polariton condensates Eastham (2008); Wouters (2008); Khan and Türeci (2016); Sieberer et al. (2016).

The rapid experimental progress Leibfried et al. (2003); Koch et al. (2007); Schoelkopf and Girvin (2008); Dykman (2012) in the control of quantum oscillators and in the engineering of their dissipative reservoirs Poyatos et al. (1996); Myatt et al. (2000); Vahala et al. (2009); Kienzler et al. (2015); Holland et al. (2015); Roßnagel et al. (2016); Fitzpatrick et al. (2017) is opening the opportunity to study synchronization deep in the quantum regime, where only a few energy states are populated Mari et al. (2013); Lee and Sadeghpour (2013); Lee et al. (2014); Walter et al. (2014a, b); Xu et al. (2014); Xu and Holland (2015); Ameri et al. (2015); Lörch et al. (2016); Weiss et al. (2017); Galve et al. (2016); Giorgi et al. (2016). In this regime semiclassical methods can fail Lee and Sadeghpour (2013) and anharmonicity on the level of single quanta has been identified Lörch et al. (2016) as a crucial ingredient to demonstrate quantum effects in synchronization.

In this paper, we discuss a new class of effects in the synchronization of quantum self-oscillators: For the simplest case of two coupled self-oscillators we find that a finite frequency detuning between different oscillators may enable synchronization in the quantum regime, while synchronization between (nearly) identical self-oscillators is suppressed. Relatedly, two identical oscillators of different amplitude are found to synchronize better than oscillators of the same amplitude. These findings are in stark contrast to our classical expectation and elude any semiclassical model. The effect generalizes to oscillator networks: identical oscillators, while unable to synchronize directly, can synchronize via a third detuned oscillator. We propose possible implementations in a network of superconducting circuits Koch et al. (2007); Schoelkopf and Girvin (2008); Johansson et al. (2014) or using trapped ions Leibfried et al. (2003); de Clercq et al. (2016) to demonstrate the effect experimentally. Our study opens up a novel regime of synchronization with genuine quantum features that can be observed with state-of-the-art quantum hardware.

Quantum Model of the System.– We consider a network of anharmonic oscillators each described by the Hamiltonian

[TABLE]

where $a$ is a bosonic annihilation operator, $\omega$ is the natural frequency of the oscillator, and the Kerr parameter $K$ quantifies the anharmonicity. Crucially, the quantum oscillators are subject to dissipation which drives them into self-sustained oscillations (limit cycles). In the framework of open quantum systems this is modeled with a Lindblad operator $\mathcal{L}=\mathcal{L}^{(-)}+\mathcal{L}^{(+)}$ consisting of damping $\mathcal{L}^{(-)}$ and amplification $\mathcal{L}^{(+)}$.

To unravel quantum signatures most clearly, we aim for a narrow distribution of Fock states in steady state, ideally a single Fock state. One way to achieve Rips et al. (2012) this is with highly nonlinear dissipators

[TABLE]

where the individual terms induce transitions from Fock state $|n\rangle$ to Fock state $|n-1\rangle$. The transition rates $\propto f_{+}$ ($f_{-}$) are highly peaked just below (above) the desired Fock state 111Note that the dissipators in Eq. (Quantum Synchronization Blockade: Energy Quantization hinders Synchronization of Identical Oscillators) have no off-diagonal elements between each other in the Lindblad equation Rips et al. (2012). Depending on the concrete system and parameters these coherences may be present. The effects we want to show exist in both cases though, so we use this simpler model here., as illustrated in Fig. 1 (a). Our physical implementation described below results in

[TABLE]

where $n_{\pm}$ and $\sigma_{\pm}$ are mean and variance of the Lorentzian. Choosing $n_{+}$ near an integer $n_{0}$ and $n_{-}$ near $n_{0}+1$ stabilizes that particular Fock state $|n_{0}\rangle$, where a high fidelity is achieved if both $\sigma_{-},\sigma_{+}\ll 1$. For simplicity, we choose from here on $\sigma_{\pm}=\sigma,\gamma_{\pm}=\gamma$, and $n_{-}=n_{+}+1$.

This corresponds to the extreme quantum limit of self-oscillations, where the energy distribution is so sharp that only a single Fock state is populated. Therefore, due to the phase-number uncertainty, the phase must be in a superposition of all phases. In comparison, the state of an ordinary laser, as described by an incoherent mixture of coherent states, also has an undefined phase of classical uncertainty but not as a result of superposition.

The quantum master equation for the density matrix $\rho$ of a complete network of such self-oscillators (numbered with index $j$) that are reactively coupled, is given by

[TABLE]

where the commutator between $A$ and $B$ is denoted as $[A,B]$ and the coupling matrix associated with the interaction $V$ fulfills $C_{jk}=C_{jk}^{*}$ and $C_{jj}=0$.

Classical Model of the System.– We will now introduce the corresponding classical description to be able to compare the quantum system to its classical limit. The self-oscillators in Eqs. (4) can be described by the Langevin equations

[TABLE]

with the classical oscillator amplitudes $\alpha_{j}$. Here $\Omega_{j}(\alpha)=\omega_{j}-2K_{j}\left|\alpha\right|^{2}$ and $\Gamma_{j}(\alpha)=\gamma_{j-}f_{j-}(\left|\alpha\right|^{2})-\gamma_{j-}f_{j+}(\left|\alpha\right|^{2})$ are the amplitude-dependent frequency and damping rate of the $j$-th oscillator, and $C_{jk}$ is the coupling matrix from Eq. (4). Finally $\eta_{j}$ is a white-noise process with correlator $\langle\eta_{k}(t)\eta_{j}(t^{\prime})\rangle=\delta_{kj}\delta(t-t^{\prime})\gamma_{T}n_{T}$ where $n_{T}$ is the thermal bath occupation and $\gamma_{T}$ is the coupling rate to the bath. For conceptual clarity we adopt here the fully classical picture neglecting quantum noise induced by the damping terms $\propto\gamma_{+},\gamma_{-}$. Our main conclusions are not affected by this choice. A derivation of the semi-classical equations including quantum noise can be found in Ref. Grimm et al. (2016).

Synchronization Measures.– To quantify synchronization between two oscillators we consider the distribution $P(\phi)=\iint_{0}^{2\pi}\mathrm{d}\phi_{1}\mathrm{d}\phi_{2}\delta(\phi_{1}-\phi_{2}-\phi)p(\phi_{1},\phi_{2})$ of their relative phase $\phi$. For the quantum steady state $\rho_{ss}$, we define $p(\phi_{1},\phi_{2})=\langle\phi_{1},\phi_{2}|\rho_{ss}|\phi_{1},\phi_{2}\rangle$ with phase states $|\phi\rangle=\frac{1}{2\pi}\sum_{n=0}^{\infty}e^{in\phi}|n\rangle$ Barak and Ben-Aryeh (2005). For the classical case, we define $p(\phi_{1},\phi_{2})$ as the probability of $(\alpha_{1},\alpha_{2})$ to have phases $(\phi_{1},\phi_{2})$ in the steady state of Eq. (5). In both cases we choose the synchronization measure Hush et al. (2015); Davis-Tilley and Armour (2016)

[TABLE]

i.e. a scaled maximum of the relative phase distribution.

Quantum Synchronization Blockade.– We now consider the self-oscillator depicted in Fig. 1 (a) coupled to another such self-oscillator with all identical parameters, except for the natural frequencies which are detuned by $\Delta=\omega_{1}-\omega_{2}$. According to classical intuition, the strongest tendency to synchronize as a function of $\Delta$ as measured by (6) is always achieved at $\Delta=0$, where both oscillators are identical. This picture is confirmed by the numerical solution of Eq. (5), which is presented in Fig. 2 (a). It is consistent with analytical results obtained in a study of exciton-polariton condensates Wouters (2008), corresponding to the zero-temperature limit of Eq. (5).

The classical intuition is not valid in the quantum system described by Eqs. (4). We investigate the same setup with parameters deep in the quantum regime, where the limit cycle is essentially stabilized to a single Fock state $|n_{0}\rangle$. The numerical result depicted in Fig. 2 (b) converge with decreasing coupling strengths to an analytical perturbation theory derived in Sup . The phase synchronization measure is suppressed at $\Delta=0$, where $S$ has a local minimum. Instead, phase synchronization is now maximal at two peaks at $\Delta=\pm 2K$.

We call this phenomenon the quantum synchronization blockade, as it only occurs deep in the quantum regime, where almost all population is stabilized to a single Fock state. The transition from quantum to classical is visualized in Fig. 3 (a): For a narrow Fock distribution around $\sigma=0.2$ the two maxima of synchronization appear at $\Delta=\pm 2K$, as just discussed. With increasing width $\sigma$, the maxima merge to one broad resonance around $\Delta=0$, as classically expected.

In a second scenario we consider self-oscillators of identical frequency, now differing only in the amplitude $\bar{n}$ at which they are stabilized. Oscillator 1 is stabilized to an integer $\bar{n}=n_{0}$ as before, while the amplitude $\bar{n}$ of oscillator 2 is varied continuously. The result is shown in Fig. 3 (b): In the quantum regime of small $\sigma$ synchronization is maximal at $\bar{n}_{2}=\bar{n}_{1}\pm 1$, i.e. oscillators with a finite difference in amplitude amplitude are most likely to synchronize. Again the classical intuition, that maximal synchronization will be present for identical oscillators with $\bar{n}_{1}=\bar{n}_{2}$, is confirmed in the classical regime of larger $\sigma$.

Thus, in contrast to classical expectation, synchronization of two quantum oscillators can be enhanced by making the oscillators more heterogeneous via detuning their frequency or via a mismatch in their amplitude. The result can be explained as follows: For two oscillators to interact efficiently, the process $\propto a_{k}^{\dagger}a_{j}$ of exchanging one excitation must be resonant by conserving energy. For oscillator $j$ in state $|n_{j}\rangle$ to transfer an excitation to oscillator $k$ initially in state $|n_{k}\rangle$ it is required that $E(|n_{j},n_{k}\rangle)=E(|n_{j}-1,n_{k}+1\rangle)$. Writing the energy as $E(|n_{j},n_{k}\rangle)=\langle n_{j},n_{k}|H|n_{j},n_{k}\rangle=\omega_{j}n_{j}+\omega_{k}n_{k}-K(n_{j}^{2}+n_{k}^{2}-n_{j}-n_{k})$, this leads to the two resonances

[TABLE]

where $\Delta=\omega_{j}-\omega_{k}$ and $\Delta n=n_{j}-n_{k}$. This resonance condition is one of the main results of our paper and is illustrated in Fig. 3 (c), showing the synchronization measure as a function of both $\Delta$ and $\Delta n$ in the quantum regime. For an illustration of the resonance condition in the case of identical oscillators, see  Sup .

Equation (7) includes an offset of $2K$ stemming from the mismatch of energy in the exchange of a single quantum of energy described above. Classically, arbitrarily small quanta may be exchanged, so that the offset does not exist. For the oscillators to interact efficiently (and thereby to synchronize), an upward transition of oscillator $k$ must be resonant with a downward transition of oscillator $j$, or vice versa.

Fig. 3 (d) shows how the resonances may be resolved for increasing $K$. For $K=0$ we have the situation of resonant harmonic oscillators  Lee and Sadeghpour (2013), where $P(\phi)$ is a bimodal distribution at $K=0$. Increasing $K$ first leads to a suppression of the resonance and then to a splitting at $\Delta=\pm 2K$. In this regime only one maximum of $P(\phi)$ survives.

Oscillator Networks.–

Having established the quantum synchronization blockade, we now use this understanding to explore consequences for networks of oscillators. In the following, we focus on small networks, as these are easiest to implement experimentally. Consider the three-oscillator network depicted in the inset of Fig. 4 (c), where two identical oscillators $A$ and $B$ are coupled indirectly by connecting them with coupling strength $V_{1}$ to an oscillator $C$ which has a relative detuning $\Delta$ with respect to $A$ and $B$. We first look at the case where the direct coupling $V_{2}$ between the two identical oscillators is zero. As shown in Fig. 4 (a), resonances occur at $\Delta=\pm 2K$ for the synchronization between detuned oscillators, as expected from the two-oscillator case. Figure 4 (b) shows that identical oscillators can now synchronize via mediation of the detuned oscillator, again with resonances at $\Delta=\pm 2K$. In this way, the synchronization blockade can be lifted. This finding is also confirmed for the larger network of four oscillators, where two pairs of identical oscillators are connected in a ring in alternating order, see Sup .

Conversely, as shown in Figs. 4 (c) and (d), turning on a coupling $V_{2}$ between the identical oscillators can suppress synchronization. This effect is most pronounced for identical oscillators, for which a strip of suppressed synchronization appears along $V_{1}\propto V_{2}$.

Implementation.–

Nonlinear damping of the form (3) can be naturally achieved for an anharmonic oscillator mode $a$ coupled to a linear cavity mode $c$ by coupling the number $\propto c^{\dagger}c$ of the cavity to the quadrature $a+a^{\dagger}$. Driving the cavity on the red (blue) sideband will lead to a positive (negative) damping Rips et al. (2012); Grimm et al. (2016). Due to the anharmonic level structure the position of the sidebands depends on the oscillator amplitude. Therefore, in contrast to ordinary sideband cooling, the strength of both damping and amplification depends nonlinearly on the oscillation amplitude.

In a rotating frame of the cavity drive $E$ the Hamiltonian is given by $H_{c}=-\delta c^{\dagger}c+E(c+c^{\dagger})+g_{0}c^{\dagger}c(a+a^{\dagger}),$ where $\delta$ is the laser detuning and $g_{0}$ is the coupling rate. Defining $g=g_{0}\sqrt{\langle c^{\dagger}c\rangle}$ this can be linearized in the regime of large amplitudes (${\langle c^{\dagger}c\rangle}\gg 1$) as $H_{c}=-\delta c^{\dagger}c+g(a^{\dagger}+a)(c+c^{\dagger}).$ Assuming that the cavity decay rate $\kappa$ fulfills $g\ll\kappa$ such that the cavity can be adiabatically eliminated, the parameters of our dissipators (Quantum Synchronization Blockade: Energy Quantization hinders Synchronization of Identical Oscillators) are approximately given by Rips et al. (2012) $\gamma=4g^{2}/\kappa$, $\sigma=\kappa/8K$, and $n_{\pm}=\pm(\delta_{\pm}-\omega_{0})/2K$.

Thus, to achieve small $\sigma$ and thereby stabilize a Fock state, a large anharmonicity to cavity noise ratio $K/\kappa\gtrsim 1$ is required. As depicted in Fig. 3 (d) and reflected in the perturbation theory from Sup , also $K/\gamma\gtrsim 1$ is necessary. As $\gamma=4g^{2}/\kappa$ and $g\ll\kappa$, we have the hierarchy $\kappa>\gamma$ and therefore only $K/\kappa\gtrsim 1$ remains as the feasibility condition for our specific implementation. This condition is a challenging requirement on the experimental setup. For instance, optomechanical systems, while highly coherent, still lack strong enough anharmonicity. While this may be overcome in the future e.g. using auxiliary coupling to a Cooper pair transistor Rimberg et al. (2014), we propose an implementation using superconducting circuits and, alternatively, trapped ions. In both platforms a large anharmonicity $K\gg\kappa,\gamma$ can be achieved with state-of-the-art technology.

An implementation using superconducting circuits is schematically depicted in Fig. 1 (b) for the case of two capacitively coupled self-oscillators $a_{j}$. To implement larger networks, the array can be extended along the greyed out coupling capacitors. One choice of self-oscillators are transmon qubits Koch et al. (2007) which are sufficiently anharmonic, while offering a long enough coherence time. The auxiliary cavities for amplification ($b_{j}$) and damping ($c_{j}$), are coupled to the self-oscillator via an interaction of optomechanical form, $c^{\dagger}c(a+a^{\dagger})$. This can be brought about by embedding a SQUID in the auxiliary cavity Johansson et al. (2014). The particular Lorentzian form (3) was assumed as a concrete example, but the scheme is quite general, i.e. any other setup with both nonlinear damping and amplification could be used; any other means of Fock state stabilization such as Holland et al. (2015); Souquet and Clerk (2016); Ma et al. (2017) will be equally suitable for our purposes.

An implementation using trapped ions is depicted in Fig. 1 (c). Ions trapped in adjacent highly anharmonic potentials de Clercq et al. (2016) can become self-oscillators with dissipation engineered as follows: The roles of the cavities for amplification and damping are now played by the internal level structure of the ion, with one transition driven on the blue sideband and another transition on the red sideband. The use of two transitions is similar to the schemes Lee and Sadeghpour (2013); Hush et al. (2015) to implement self-oscillators with ions. The ions are naturally coupled via the Coulomb interaction Hush et al. (2015); Brown et al. (2011); Harlander et al. (2011).

We note that to observe the effect presented here each node of the network needs to have an anharmonic spectrum consisting of at least three levels, excluding arrays of harmonic oscillators or qubits Sup .

Conclusion and Outlook.–

To conclude, we have described a novel effect referred to as the quantum synchronization blockade, which prevents identical nonlinear oscillators from synchronizing deep in the quantum regime. This is in stark contrast to the classical regime, where oscillators synchronize best when on resonance. Complementarily we have demonstrated that detuned auxiliary oscillators can lift this blockade by indirectly mediating synchronization between identical oscillators. These effects will be observable in state-of-the-art quantum systems such as superconducting circuits and trapped ions, for which we have proposed concrete implementations. Our article thus opens a new perspective for the exploration of synchronization in Bose-Hubbard-van der Pol-type networks.

Acknowledgments.–

We would like to acknowledge helpful discussions with A. H. Safavi-Naeini and S. M. Girvin. This work was financially supported by the Swiss SNF and the NCCR Quantum Science and Technology. A.N. holds a University Research Fellowship from the Royal Society and acknowledges support from the Winton Programme for the Physics of Sustainability. All quantum steady-state solutions were obtained in QuTiP Johansson et al. (2013) and the classical Monte-Carlo trajectories were performed in Julia Bezanson et al. (2017). This work was supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 732894 (FET Proactive HOT).

Appendix A Analytical perturbation theory in the coupling

To gain further understanding of the synchronization behavior in the quantum regime, we rederive the synchronization condition (7) explicitly based on perturbation theory in the coupling. To this end we define the perturbation as $\mathcal{L}_{1}\rho=-i[V,\rho]$ and the unperturbed Lindbladian as $\mathcal{L}_{0}\rho=\sum_{j}-i[H_{0j},\rho]+L_{j}\rho$, where we use the definitions from Eq. (4) from the main text and $j=1,2$ indicates the oscillator. The steady-state density operator to first order in the coupling $V$ is then $\rho^{(1)}=-\mathcal{L}_{0}^{-1}\mathcal{L}_{1}\rho^{(0)}$, which we calculate term by term in the following.

The uncoupled density operator $\rho^{(0)}$ is diagonal and for the individual oscillator its (unnormalized) entries $\rho_{mm}^{(0)}\propto(\gamma_{+}/\gamma_{-})^{m}\prod_{i=1}^{m}f_{+}(i)/\prod_{i=1}^{m}f_{-}(i)$ are solved via a recursion relation. The superoperator $\mathcal{L}_{0}$ can be decomposed into a term coupling diagonal density matrix elements and a term coupling off-diagonal elements separately. In the off-diagonal subspace, without neglecting any terms, the inverse $\mathcal{L}_{0}^{-1}$ can be obtained by inverting the diagonal so that

$\mathcal{L}_{0}^{-1}|m_{1}-1,m_{2}+1\rangle\langle m_{1},m_{2}|=\lambda_{-}^{-1}|m_{1}-1,m_{2}+1\rangle\langle m_{1},m_{2}|$ with

[TABLE]

As $\mathcal{L}_{1}$ couples only neighboring Fock states, $\rho^{(1)}=-\mathcal{L}_{0}^{-1}\mathcal{L}_{1}\rho^{(0)}$ has nonzero elements only on the minor diagonals, so that the first-order correction for the steady state is

[TABLE]

and a similar expression for $\rho^{(1)}_{m_{1}+1,m_{2}-1;m_{1}m_{2}}$. The synchronization measure (6) from the main text can now be written as the sum

[TABLE]

of the off-diagonal elements (S2). The analytical expression (S3) for $S$ is a sum over terms (A) centered around $2K$ which have widths scaling with $\gamma_{\pm}$. As an approximate rule, we therefore need $K\gg\gamma_{\pm}$ to resolve the resonances, which is in agreement with the numerical simulation shown in Fig. 3 (d) from the main text.

Appendix B Network of four oscillators

We have also considered a network of four oscillators, where two pairs of identical oscillators are connected in a ring in alternating order. The two species again differ only in a relative detuning $\Delta$ of their frequencies $\omega$. As shown in Fig. 5, synchronization can be mediated via detuned oscillators, but is blocked for identical oscillators due to the quantum synchronization blockade.

Appendix C Illustration of the effect and minimal model

In future studies it will be interesting to investigate how the quantum synchronization blockade generalizes to larger networks. It is desirable to restrict the system to a small Hilbert space, to make these simulations numerically feasible. Therefore the individual components of the lattice would ideally be constituted by qubits. However, at least three energy levels per system are required to observe the quantum synchronization blockade effect introduced in this article.

This can be understood from Figure 6, where the origin of the effect is visualized. The crucial diagonal levels are marked in red. These levels are resonant with their neighbors (connected with red lines) for harmonic oscillators. The levels become off-resonant with their neighbors for anharmonic oscillators, resulting in the quantum synchronization blockade for identical oscillators.

A minimal model must keep at least one such red level and its neighbors. The lowest red level with neighbors is $|1,1\rangle$, which is coupled to $|0,2\rangle$ and $|2,0\rangle$. To include these levels, each system must contain at least three levels ($|0\rangle$, $|1\rangle$, $|2\rangle$). As a model must be anharmonic for the red levels to be off-resonant with their neighbors, an anharmonic oscillator with three levels is the minimum model for the effect to be observable. The minimal model is further illustrated in Figure 7. It is an interesting direction to investigate whether also spin systems with three or more levels could be a feasible platform for this effect as well.

The quantum synchronization blockade reported here is unrelated to the observation, that classical anharmonic oscillators with unbalanced losses (resulting in stabilization at different amplitudes) need a finite detuning to compensate for the disbalance to reach maximal synchronization. The suppression of synchronization reported here contradicts the classical expectation, as we have shown. It is also unrelated to the case of two coupled, initially aligned qubits with strongly unbalanced coupling to a thermal quantum bath reported in Ref. 37: Interestingly, the alignment of qubits during time evolution can probe the properties of the bath. Depending on its spectral density, a finite detuning of the unbalanced qubits may favor alignment after transient times before the system equilibrates with the bath.