Quantum synchronization effects induced by strong nonlinearities

Introduction.—Mathematical modelling has shown us how the immense variety and beauty of nature can be governed by nonlinear differential equations May (2019); Edelstein-Keshet (2005); Jordan and Smith (2007); Debnath (2012). Such equations, owing to their nonlinearity, are difficult to analyze and their application to physical processes has come to be known as nonlinear science Campbell (1987); Scott (2006).

How nonlinear effects play out in quantum-mechanical systems is a subject of intense investigation. By far, chaos has attracted the most attention Stöckmann (2000); Haake (1991); Waltner (2012); Wimberger (2014). Nonetheless, noise effects, such as stochastic resonance Gammaitoni et al. (1998); Wellens et al. (2003); Ludwig (2019); Wagner et al. (2019) and coherence resonance Wellens et al. (2003); Kato and Nakao (2021) have also been prominent, while a recent example is Ref. Schmolke and Lutz (2023). Another widely recognized effect is noise-induced transitions Horsthemke and Lefever (2006). Recent work has extended this effect to quantum mechanics by using a quantum stochastic differential equation and its correspondence to nonlinear dissipation Chia et al. (see also Ref. Chia et al. (2020a)). There are also promising applications of nonlinear dissipation such as stabilizing bosonic qubits for fault-tolerant quantum computing Mirrahimi et al. (2014); Leghtas et al. (2015); Chamberland et al. (2022), and enhancing the sensitivity of quantum sensors Dutta and Cooper (2019); Sánchez Muñoz et al. (2021).

A relative newcomer to the study of nonlinear effects in quantum systems is synchronization Rosenblum and Pikovsky (2003). Its most elementary form consists of applying a sinusoidal force, say with amplitude $f$, and frequency $\Omega_{\rm d}$, to a self-sustained oscillator. Synchronization is then the modification of the oscillator frequency to $\Omega_{\rm d}$. A prototypical model is the driven van der Pol (vdP) oscillator Strogatz (2018), defined by phase-space coordinates $(x,y)$ satisfying

where primes denote differentiation with respect to the argument (in this case $t$, representing time). In the absence of forcing ($f=0$) the oscillator is characterized by $\omega_{0}$, and a nonlinearity parameter $\mu$ which controls how much the oscillator is damped towards an amplitude of order $|q|$. An important feature of the undriven vdP system is the existence of a supercritical Hopf bifurcation at $\mu=0$, via which a stable limit cycle appears for $\mu>0$ Strogatz (2018).

At $\mu=0$, (1) is entirely linear. This motivates one to consider the quasilinear limit of (1), defined by $\mu\longrightarrow 0^{+}$. In this limit the vdP oscillator is well approximated by the Stuart–Landau (SL) oscillator, the steady state of which is rotationally symmetric in phase space (a circular limit cycle). This makes the SL oscillator much simpler to analyze, and has thus served as a starting point in the literature on quantum synchronization for continuous-variable systems, e.g. Refs. Lee and Sadeghpour (2013); Walter et al. (2014); Sonar et al. (2018); Mok et al. (2020); Lee et al. (2014); Bastidas et al. (2015); Weiss et al. (2017); Walter et al. (2015); Morgan and Hinrichsen (2015); Ishibashi and Kanamoto (2017); Amitai et al. (2018); Bandyopadhyay et al. (2020, 2021); Kato et al. (2019); Kato and Nakao (2020); Bandyopadhyay and Banerjee (2021). The trade-off of course, is that effects taking place at finite values of $\mu$ are excluded. A well known example of this is relaxation oscillations in the undriven vdP oscillator¹¹1In fact, vdP intended for (1) to model relaxation oscillations in an electrical circuit Ginoux and Letellier (2012). To observe relaxation oscillations in quantum theory one needs to quantize the exact vdP model, and it is only relatively recently that such efforts have been made Shah et al. (2015); Chia et al. (2020b); Ben Arosh et al. (2021). Jordan and Smith (2007); Strogatz (2018); Jenkins (2013). More effects start to appear if driving is included, such as quasiperiodicity and chaos Broer and Takens (2011); Lakshmanan and Rajaseekar (2012); Moon (2008), both of which are absent in the driven SL oscillator.

In this work, we investigate the effects of nonlinearity in quantum oscillators by considering a more general model based on the classical Duffing–van der Pol (DvdP) oscillator. This adds $\zeta\,x^{3}$ to $y^{\prime}$ where $\zeta$ is another nonlinearity parameter. To overcome the inadequacy of the SL model we propose a quantum DvdP oscillator in which the vdP and Duffing nonlinearities (respectively $\mu$ and $\zeta$) are nonvanishing, but also not arbitrarily large. Our model is accurate up to order $(\mu/\omega_{0})^{2}$, at which the distinct signatures of strong nonlinearity appear, such as relaxation oscillations Chia et al. (2020b). Our approach has the benefit of capturing novel nonlinear effects while evading the large computational cost of simulating quantum systems with very strong nonlinear dissipations.

We show that for a single oscillator with periodic forcing there exists a critical Duffing nonlinearity, above which further increases in $\zeta$ enlarges the synchronization bandwidth (the amount of detuning the forcing can tolerate from the oscillator and still entrain it). This result is similar to the synchronization enhancement from the classical literature Antonio et al. (2015), but now generalized to quantum oscillators.²²2It is also worth mentioning that nonlinear oscillators are of interest to quantum information too, when they are coupled to qubits. In this context the Duffing nonlinearity has been shown to both increase and stabilize the oscillator-qubit entanglement Montenegro et al. (2014). In contrast, the vdP nonlinearity activates genuine quantum effects. Coupling two vdP oscillators dissipatively may lead to either amplitude death (the cessation of oscillations), or mutual synchronization. Classically, amplitude death occurs only when the two oscillators are sufficiently detuned Saxena et al. (2012). Interestingly, we find this need not be the case for quantum oscillators. We show that two quantum vdP oscillators possessing relatively small limit cycles and nonvanishing nonlinearities can exhibit amplitude death even with zero detuning. Larger limit cycles on the other hand can mutually synchronize from a state of amplitude death if their nonlinearity is increased.

We also consider reactively coupled oscillators. Two such SL oscillators cannot develop positional correlations, and hence do not synchronize. This is true regardless of whether the oscillators are classical or quantum. We show here that at finitely large nonlinearity, position correlations behave rather differently between the classical and quantum oscillators: Two reactively coupled quantum vdP oscillators can undergo nonlinearity-induced correlations whereby their position correlation increases as they become more nonlinear. In contrast, we find that making the analogous classical oscillators more nonlinear monotonically reduces their position correlation. The nonlinearity-induced correlations in the quantum vdP oscillators are thus a consequence of both their quantum nature and strong nonlinearity.

Model.—For simplicity we consider here a dimensionless DvdP model in terms of the nonlinearity parameters $\lambda\equiv\mu q^{2}/\omega_{0}r^{2}$ and $\beta\equiv\zeta q^{2}/\omega_{0}^{2}r^{2}$ in which $r$ is a dimensionless scale parameter:

Note that $\tilde{x}\equiv xr/q$ is now a function of $\tilde{t}=\omega_{0}t$, and we have also included a dimensionless external force parameterized by $F=fr/\omega_{0}^{2}q$ and $\omega_{\rm d}=\Omega_{\rm d}/\omega_{0}$. From the approximate analysis of (2), the leading contribution to the oscillator frequency is quadratic in $\lambda$, and linear in $\beta$ (see Appendix), given by $\omega\approx 1+r^{2}(3\beta/2-\lambda^{2}r^{2}/16)$. This motivates a Bogoliubov–Krylov time-average of the equations of motion up to these orders, giving Sanders et al. (2007)

where $\alpha=(\tilde{x}+i\,\tilde{y})/2$. For $F=0$, (3) predicts a limit-cycle amplitude of $2|\alpha|=2r$ with the expected frequency shifts due to $\lambda$ and $\beta$. Additionally, note the first-order averaging in $\lambda$ for $\beta=0$ yields the SL equation. Our approximate model captures the effects of strong vdP nonlinearity of order $\lambda^{2}$. We seek a quantum master equation $\rho^{\prime}={\cal L}\rho$ such that $\braket{\hat{a}}^{\prime}={\rm Tr}[\hat{a}\,{\cal L}\rho]$ (with $[\hat{a},\hat{a}^{\dagger}]=\hat{1}$) agrees with (3) in the mean-field limit Chia et al. (2020b). It can then be shown that this is satisfied by the Lindbladian Lindblad (1976); Gorini et al. (1976); Breuer and Petruccione (2002)

where

We have also defined ${\cal D}[\hat{c}]\equiv\hat{c}\,\text{\Large$\cdot$}\,\hat{c}^{\dagger}-(\hat{c}^{\dagger}\hat{c}\,\text{\Large$\cdot$}+\text{\Large$\cdot$}\,\hat{c}^{\dagger}\hat{c})/2$ for any $\hat{c}$, and a dot denotes the position of $\rho$ when acted upon by a superoperator. Detailed derivation can be found in the Appendix. We remark that both the higher-order Kerr terms and the nonlinear two-photon dissipation in our proposed model can be implemented in circuit QED Hillmann and Quijandría (2022); Leghtas et al. (2015). The tunability of the limit cycle radius $r$ allows us to access different parameter regimes of the quantum oscillator, in particular the quantum ($r\ll 1$), and semiclassical ($r\approx 1$) regimes. We have included the second-order contributions in $\lambda$ in our model for its nonlinearity-tuning capability, since the terms linear in $\lambda$ neither affect the limit-cycle amplitude nor phase dynamics. The Duffing nonlinearity translates to a Kerr term in ${\cal L}$. This model can be considered as an alternative to the quantum SL oscillator, but with flexibility in tuning the nonlinearity. All our numerical results for a given parameter set are obtained with a sufficiently large truncation of the Hilbert space by ensuring the corresponding steady-state power spectrum converge.

Nonlinearity-enhanced synchronization.—We study first the frequency locking of the approximate quantum DvdP oscillator to a periodic force [(4) and (5)]. The synchronization bandwidth is the range of $\omega_{\rm d}$ for which the oscillator frequency is locked to the driving frequency at steady state. This is achieved when $|\omega_{\rm d}-\tilde{\omega}|=0$, where $\tilde{\omega}$ the observed frequency of the driven oscillator, obtained from the peak of its spectrum averaged over one period of the drive Chia et al. (2020b).

Here we find the Duffing nonlinearity to enhance quantum synchronization: For a range of $\lambda$ and a fixed $r$, increasing $\beta$ past a critical value widens the synchronization bandwidth linearly. This is illustrated in Fig. 1(a) where the synchronization bandwidth is plotted as a contour against $\bar{\beta}\equiv\beta r^{2}$ and $F/r$. The critical value of $\bar{\beta}$ is indicated by the red dashed line, where the bandwidth is equal to its corresponding value at $\bar{\beta}=0$. However, this enhancement does not occur for all values of the vdP nonlinearity. In Fig. 1(b) we see that an increase of $\bar{\lambda}\equiv\lambda r^{2}$ from its value in Fig. 1(a) can ruin the gain in synchronization bandwidth due to $\bar{\beta}$. Noting that the dissipative terms in ${\cal L}$ are all proportional to $\lambda$, this effect can be qualitatively attributed to the phase diffusion due to quantum noise, which is known to inhibit synchronization Walter et al. (2014); Lee et al. (2014); Lee and Sadeghpour (2013). We can develop some understanding of the quantum DvdP by examining its classical analog. Using the method of harmonic balance on $x$, we are able to derive the conditions for nonlinearity-enhanced synchronization analytically for the classical DvdP oscillator (see derivation in Appendix), given by

This shows clearly the existence of a critical value of $\bar{\beta}$, and a finite interval of $\bar{\lambda}$ over which the synchronization enhancement occurs. These results are consistent with Fig. 1 except for the fact that quantum noise makes the range of $\lambda$ for synchronization enhancement in the quantum DvdP oscillator smaller compared to the classical range as seen in Fig. 1(b).

We have also shown in Appendix that increasing the vdP nonlinearity in the quantum model can only reduce the synchronization bandwidth. One can thus be certain that the synchronization enhancement seen in our model is induced by the Duffing nonlinearity. We shall see next that the vdP nonlinearity can induce synchronization in coupled oscillators from a state of amplitude death.

Nonlinearity-induced synchronization and amplitude death.—Two dissipatively coupled vdP oscillators (i.e. no Duffing nonlinearity) can be described by the Lindbladian

where ${\cal L}_{k}$ ($k=1,2$) is the Lindbladian for oscillator $k$, defined by setting $\hat{a}$ to $\hat{a}_{k}$ and $\beta=F=0$ in (4) and (5). We have assumed the oscillators to be identical (i.e. same $\lambda$ and $r$) except for an initial detuning of $\Delta$, and denoted their coupling strength by $\eta$.

In this case, frequency locking occurs when the observed frequencies of the two oscillators become identical at steady state. As before, the observed frequency of oscillator $k$ is given by the location at which its spectrum is peaked, i.e. where $S_{k}(\omega)=\int_{-\infty}^{\infty}dt\,\exp(i\omega t)\langle\hat{a}^{\dagger}_{k}(t)\hat{a}_{k}(0)\rangle$ is maximized, and where the two-oscillator steady state is to be used. For a fixed $\eta$, we define the synchronization bandwidth to be the range of $\Delta$ for which the two oscillators lock frequencies. It will also be interesting to look at position correlations in the two oscillators at steady state, defined by

where $\hat{x}_{k}=\hat{a}_{k}+\hat{a}^{\dagger}_{k}$. Note that frequency locking implies a nonzero $\Sigma$, but not vice versa.

In addition to frequency locking, dissipatively coupled oscillators can also cease to oscillate. If the oscillators are classical, then this may happen for a range of $\eta$ provided that $\Delta$ is sufficiently large. And if both oscillators stabilize to the same phase-space point, which may be taken to be the origin without loss of generality, then the effect is termed amplitude death Saxena et al. (2012); Koseska et al. (2013). To define amplitude death in quantum oscillators we generalize the notion of P-bifurcations from classical stochastic systems to the steady-state Wigner function of a reduced state (see Ref. Chia et al. and other references therein). In this case, amplitude death is said to occur if the single-oscillator Wigner functions peak at the origin in quantum phase space. This approach is consistent with previous studies on amplitude death in coupled quantum oscillators Bandyopadhyay et al. (2020); Amitai et al. (2018); Bandyopadhyay et al. (2021); Ishibashi and Kanamoto (2017).

In Fig. 2 we work out regions of frequency locking and amplitude death in the $(\eta,\Delta)$ parameter space for (7), along with $\Sigma$, shown as a contour in Fig. 2(a) and (c). Two especially interesting scenarios are—when the limit cycles are relatively large compared to quantum noise [Fig. 2(a)]; and when they become small, being more susceptible to quantum noise [Fig. 2(c)].

In Fig. 2(a) we have indicated the boundary between frequency locking and amplitude death by a dash-dotted line, while no identifiable phenomenon occurs to the left of the solid line. Note the dash-dotted line is a Hopf-bifurcation curve because the transition from amplitude death to frequency locking is facilitated by a Hopf bifurcation. Clearly, $\Sigma$ is larger inside the frequency-locking region. Especially significant here is the effect of the vdP nonlinearity on synchronization. Whereas in the single-oscillator case the vdP nonlinearity had only detrimental effects, it now has a constructive role by enlarging the (mutual) synchronization bandwidth. We illustrate this in Fig. 2(b) where additional frequency-locking boundaries for different values of vdP nonlinearity $\lambda$ are plotted. It can be seen that increasing $\lambda$ enlarges the frequency-locking region (see also Appendix for some classical analysis). This means that two oscillators in a state of amplitude death with an $(\eta,\Delta)$ lying above the Hopf-bifurcation curve in Fig. 2(a) will transit suddenly to a state of synchronized oscillations when $\lambda$ is sufficiently increased. This may be appropriately called nonlinearity-induced mutual synchronization.

Turning now to the case of a small limit cycle $(r\ll 1)$ in Fig. 2(c), we find frequency locking to be absent while position correlations become negligible. The blue solid line delineates the boundary of amplitude death. Most striking is the persistence of amplitude death at zero detuning (i.e. $\Delta=0$). Classically, some frequency mismatch between the two oscillators must be present in order for amplitude death to occur Saxena et al. (2012). This signifies a clear distinction between classical and quantum dynamics. A loss of amplitude death at $\Delta=0$ may then be expected if we increased either $r$ or $\lambda$ (while holding the other constant). This is indeed what we find, as illustrated in Fig. 2(d), where such a quantum-to-semiclassical transition is captured by increasing $\bar{\lambda}\equiv\lambda r^{2}$.

Since we have focused exclusively on the vdP nonlinearity here, we note that incorporating a Duffing nonlinearity into our model will not in fact change the frequency-locking boundary. We can understand this from the classical coupled equations of motion, where we see that $\beta$ appears only in the phase dynamics of the oscillators, and that such terms vanish at steady state for identical oscillators (equal limit cycle radii) Aronson et al. (1990).

Nonlinearity-induced correlations.—It is known that two reactively coupled SL oscillators cannot synchronize nor share position correlations. This is true even in the quantum case (see Appendix). However, we show here that positional correlations between two reactively coupled quantum vdP oscillators do develop. Here we must use the exact vdP Lindbladian Chia et al. (2020b), because under reactive coupling, the approximate model does not produce any off-diagonal elements in the steady state, and hence cannot generate correlations in the two oscillators. As with the dissipatively coupled system, two reactively coupled vdP oscillators can be modelled by considering two uncoupled vdP oscillators with annihilation operators $\hat{a}_{1}$ and $\hat{a}_{2}$, coupled by the Hamiltonian $g(\hat{a}_{1}\hat{a}^{\dagger}_{2}+\hat{a}^{\dagger}_{1}\hat{a}_{2})$, where $g$ is the reactive coupling strength. As before, we assume that both oscillators have the same nonlinearity $\lambda$.

In Fig. 3(a), we generate a contour plot of $\Sigma$ as a function of $\lambda$ and $g$ at $r=0.3$ and $\Delta=0.05$. From this we see that for a fixed $g$, increasing the oscillator nonlinearity beyond the SL regime leads to stronger correlations. We illustrate this more clearly in Fig. 3(b) by showing how $\Sigma$ varies as a function of $\lambda$ for $g=0.2$, 0.4, and 0.8, which are marked in Fig. 3(a) by vertical dashed lines. Note this also shows the existence of an optimal $\lambda$ which maximizes $\Sigma$. Such nonlinearity-induced correlations are absent in the corresponding classical model. At a given coupling strength, the position correlation in two reactively coupled classical vdP oscillators decreases monotonically as $\lambda$ increases (see Appendix).

Conclusion.— Our work goes beyond the well-studied paradigm of weak nonlinearity in quantum synchronization, and provides the first systematic study of quantum synchronization effects for strong nonlinearity. We introduced a new quantum oscillator model which captures intriguing effects induced by two strong nonlinearities. We showed that a strong Duffing nonlinearity leads to a linear enhancement of the synchronization bandwidth in driven oscillators. We also reported genuine quantum synchronization effects exclusive to strong nonlinearity which are not observed previously: Increasing the vdP nonlinearity enhances the synchronization bandwidth, and revives synchronization between dissipatively-coupled oscillators in amplitude death. For reactively-coupled vdP oscillators on the other hand, we find that strong nonlinearity induces position correlations which are impossible in the weakly-nonlinear limit. Our model provides a new paradigm for studying other strongly nonlinear effects such as chaos Stöckmann (2000); Strogatz (2018); Haake (2000); Braun (2001).