Introduction

Nonlinear interactions of modes with vastly different eigenfrequencies (VDE) are unique because, unlike standard internal resonances1,2,3,4,5, the eigenfrequencies need not be rationally related. In VDE modes, interactions occur between the envelope of the high-frequency (HF) carrier signal and the oscillations of the low-frequency (LF) carrier signal (Fig. 1). While these VDE modal interactions are peculiar, they are ubiquitous and occur in a wide range of fields of physics. Examples include (i) cavity optomechanics6,7,8,9,10,11,12,13 and plasmomechanics14,15,16,17,18, where these interactions are between HF optical modes and the LF mechanical modes; (ii) interactions between HF nano- and LF micro-mechanical modes19,20,21; (iii) certain classes of aeroelastic instabilities, such as stall flutter22 and transverse galloping23, where these interactions are between HF vortex modes of the turbulent wake (the so-called Kármán vortex street) and the LF modes of the elastic structure (Fig. 1); and many other systems24,25,26,27,28,29,30,31,32,33,34,35,36. These interactions have gained significant interest, particularly in cavity optomechanics37,38,39,40,41,42,43,44,45, since they offer novel means to generate engineered quantum states46,47,48,49 and practically unlimited bandwidth for enhanced sensing of acceleration50, mass51, force52, vibration53, chemical quantities54, and biological quantities55.

Fig. 1: Nonlinear interactions of modes with vastly different eigenfrequencies (VDE).
figure 1

a The VDE modal interactions are associated with a resonant interaction between the oscillating envelope of the high-frequency mode (blue) and the signal oscillations of the low-frequency mode (red). Examples of systems that exhibit VDE modal interactions include: b Cavity optomechanics, where the interaction is between optical (eigenfrequency ωopt) and mechanical (eigenfrequency ωmech) modes. c Plasmomechanical oscillators, where the interaction is between localized-gap plasmon (eigenfrequency ωLGP) and mechanical modes (eigenfrequency ωmech). d Interactions between nano-mechanical (eigenfrequency ωnano) and micro-mechanical (eigenfrequency ωmicro) modes. e Aeroelastic instabilities in which the high-speed upstream velocity U∞ generates a high-frequency turbulent vortex mode in the wake (eigenfrequency ωvortex) that interacts with a low-frequency mode of the mechanical structure (eigenfrequency ωstruct).

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Nonlinear VDE modal interactions have been of interest for decades56,57,58,59,60,61,62,63,64,65,66. However, to the best of our knowledge, no theory presents a simple model that captures the essential physics of these interactions and maps them onto a single generic (normal) form. In this paper, we develop such a theory. In particular, we consider the lowest-order (quadratic) nonlinear modal coupling, and show that VDE modal interactions can be mapped onto the normal form of a supercritical Hopf bifurcation described by the Stuart-Landau oscillator67,68. Moreover, we present a simple prototypical pendulums system that exhibits VDE modal interactions and offers a simple conceptual view of the generic characteristics of these interactions.

Results and Discussion

Minimalistic model

To derive a minimalistic model for VDE modal interactions, we consider a pair of driven vibration modes that, by the definition of eigenmodes, are linearly uncoupled. We denote their modal coordinates as (q0, q1) and their eigenfrequencies as (ω0, ω1), where ω0 ≪ ω1. The Hamiltonian of the system is given by H = H0 + H1 + Hint, where \({H}_{0,1}=({p}_{0,1}^{2}+{\omega }_{0,1}^{2}{q}_{0,1}^{2})/2-{q}_{0,1}{F}_{0,1}\cos ({\omega }_{{F}_{0,1}}t)\) are the Hamiltonians of the individual modes, p0,1 are the attendant momenta, F0,1 and \({\omega }_{{F}_{0,1}}\) are the amplitude and frequency of the modal drives, respectively, and Hint = Hint(q0, q1) is the interaction Hamiltonian, which necessarily couples the modes in a nonlinear way.

We restrict the analysis to the lowest order nonlinearity, and write the following single-term interaction Hamiltonian \({H}_{{{{{{{{\rm{int}}}}}}}}}=\alpha {q}_{0}{q}_{1}^{2}\), which is consistent with the interaction Hamiltonian in cavity optomechanics that generates the radiation-pressure force42. We note that the inclusion of a term \(\beta {q}_{0}^{2}{q}_{1}\) in the Hamiltonian is also possible; however, its contribution is negligible since it does not promote energy exchange during the interactions of interest. Therefore, with the inclusion of linear dissipation terms, we obtain the following dynamical system (Fig. 2)

$${\ddot{q}}_{0}+2{\Gamma }_{0}{\dot{q}}_{0}+{\omega }_{0}^{2}{q}_{0}+\alpha {q}_{1}^{2}={F}_{0}\cos ({\omega }_{{F}_{0}}t),$$
(1)
$${\ddot{q}}_{1}+2{\Gamma }_{1}{\dot{q}}_{1}+{\omega }_{1}^{2}{q}_{1}+2\alpha {q}_{0}{q}_{1}={F}_{1}\cos ({\omega }_{{F}_{1}}t).$$
(2)

where the Γ1,2 are the modal decay rates.

Fig. 2: The minimalistic nonlinear model for interactions between modes with vastly different eigenfrequencies.
figure 2

A pair of linear modes are nonlinearly coupled via the interaction Hamiltonian Hint. As described in our analysis, we consider the case in which the high-frequency mode (q1-blue) is driven in its blue sideband with \({\omega }_{{F}_{1}}={\omega }_{1}+{\omega }_{{F}_{0}}\), and its relaxation time is significantly shorter than the relaxation time of the low-frequency mode (q0-red) \({\Gamma }_{1}^{-1}\ll {\Gamma }_{0}^{-1}\). The nonlinear pendulums system has similar modal equations and offers a simple conceptual view of these nonlinear modal interactions, where the low-frequency/high-frequency mode corresponds to the symmetric/antisymmetric mode of the pendulums system. The dashed lines of the symmetric/antisymmetric mode represent a nonzero angle, which biases the system and breaks its symmetry. k and ks are the torsional stiffness of the shafts.

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Eqs. (1)-(2) are similar to the modal equations (see Supplementary Note 1) of a pair of biased pendulums that are coupled via stiff torsional spring (Fig. 2), and with the nonlinearities truncated at the quadratic order. Note that in the pendulums system, there is an additional term \(\alpha {q}_{0}^{2}\) in the equation of q0 (see Supplementary Note 1) that has negligible contribution to the leading order approximation. Therefore, we conceptually associate the LF mode with the symmetric mode of the pendulums \({q}_{0}=({\theta }_{L}+{\theta }_{R})/\sqrt{2}\), and the HF mode with the antisymmetric mode of the pendulums \({q}_{1}=({\theta }_{L}-{\theta }_{R})/\sqrt{2}\) (Fig. 2). We note that Eqs. (1)-(2) can be readily generalized to include multiple LF and HF modes, LF mode nonlinearities, and noise (Methods).

For weakly nonlinear modal interactions (\(| \alpha {q}_{1}^{2}/{\omega }_{0}^{2}{q}_{0}| \ll 1\), \(| \alpha {q}_{0}/{\omega }_{1}^{2}| \ll 1\)), light damping [\(| 2{\Gamma }_{0}\dot{q}/({\omega }_{0}^{2}{q}_{0})| \ll 1\), \(| 2{\Gamma }_{1}{\dot{q}}_{1}/({\omega }_{1}^{2}{q}_{1})| \ll 1\)], and weak external drives [\(| {F}_{1}/({\omega }_{1}^{2}{q}_{1})| \ll 1\), \(| {F}_{0}/({\omega }_{0}^{2}{q}_{0})| \ll 1\)] that operate at near-resonance conditions (\(| {\omega }_{{F}_{0}}-{\omega }_{0}| /{\omega }_{0}\ll 1\), \(| {\omega }_{{F}_{1}}-{\omega }_{1}| /{\omega }_{1}\ll 1\)), we make the following ansatz for the modal dynamics

$${q}_{0}(t) \, = \, {\bar{q}}_{0}\,+\,{A}_{0}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{0}}t}\,+\,{{{{{{{\rm{cc}}}}}}}},\,{\dot{q}}_{0}(t)={{{{{{{\rm{i}}}}}}}}{\omega }_{0}{A}_{0}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{0}}t}\,+\,{{{{{{{\rm{cc}}}}}}}},\\ {q}_{1}= \, {A}_{1}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{1}}t}+{{{{{{{\rm{cc}}}}}}}},\,{\dot{q}}_{1}(t)={{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{1}}{A}_{1}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{1}}t}+{{{{{{{\rm{cc}}}}}}}}.$$
(3)

Here cc denotes the complex-conjugate of the preceding term, \({\bar{q}}_{0}=-\alpha \langle {q}_{1}^{2}\rangle /{\omega }_{0}^{2}\) is a DC deflection that arises from the time-independent component of \({q}_{1}^{2}\), and A0,1 are the complex-amplitudes of the LF and HF modes.

Since \({\omega }_{{F}_{0}}\ll {\omega }_{{F}_{1}}\), we can treat q0 as a quasi-static variable in Eq. (2) and apply the method of averaging, or equivalently, the rotating wave approximation (RWA) to obtain the following complex-amplitude equations

$${\dot{A}}_{0}= -({\Gamma }_{0}+{{{{{{{\rm{i}}}}}}}}\Delta {\omega }_{0}){A}_{0}+\frac{{{{{{{{\rm{i}}}}}}}}\alpha }{2\pi }\int\nolimits_{t}^{t+\frac{2\pi }{{\omega }_{{F}_{0}}}}\,| {A}_{1}{| }^{2}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{0}}t}dt\\ -\frac{{{{{{{{\rm{i}}}}}}}}{F}_{0}}{4{\omega }_{{F}_{0}}},$$
(4)
$${\dot{A}}_{1}= -\left({\Gamma }_{1}+{{{{{{{\rm{i}}}}}}}}\Delta {\omega }_{1}\right){A}_{1}\,+\,\frac{{{{{{{{\rm{i}}}}}}}}\alpha }{{\omega }_{{F}_{1}}}({A}_{0}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{0}}t}\,+\,{A}_{0}^{* }{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{0}}t}){A}_{1}\\ -\frac{{{{{{{{\rm{i}}}}}}}}{F}_{1}}{4{\omega }_{{F}_{1}}},$$
(5)

where \(\Delta {\omega }_{0}={\omega }_{{F}_{0}}-{\omega }_{0}\) and \(\Delta {\omega }_{1}={\omega }_{{F}_{1}}-{\omega }_{1}-\alpha {\bar{q}}_{0}/(2{\omega }_{{F}_{1}})\) are the frequency detunings of the LF and HF modes.

A nonstandard feature of Eq (5) is that certain slowly varying excitation effects persist after the RWA, with frequency \({\omega }_{{F}_{0}}\). Specifically, we see that under certain conditions, in particular where A0 is constant, A1 can oscillate with a frequency \({\omega }_{{F}_{0}}\). Furthermore, since Eq. (5) is linear in A1, we can formally solve it in terms of the yet unknown complex amplitude of the LF mode \(({A}_{0}=| {A}_{0}| {{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\phi }_{0}})\), i.e., \({A}_{1}(t)={{{{{{{{\rm{e}}}}}}}}}^{-{\mathfrak{g}}(t)}\{{A}_{10}-[({{{{{{{\rm{i}}}}}}}}{F}_{1})/(4{\omega }_{{F}_{1}})]\int{{{{{{{{\rm{e}}}}}}}}}^{{\mathfrak{g}}(t)}dt\}\), where A10 is determined from the initial condition of A1 and \({\mathfrak{g}}(t)=({\Gamma }_{1}+{{{{{{{\rm{i}}}}}}}}\Delta {\omega }_{1})t-{{{{{{{\rm{i}}}}}}}}\alpha | {A}_{0}| \sin ({\omega }_{{F}_{0}}t+{\phi }_{0})/({\omega }_{{F}_{1}}{\omega }_{{F}_{0}})\). Consequently, for constant A0, we can use the Jacobi-Anger expansion to write an explicit formula for the evolution of A1, e.g., \({{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}\alpha | {A}_{0}| \sin ({\omega }_{{F}_{0}}t+{\phi }_{0})/({\omega }_{{F}_{1}}\Delta {\omega }_{1})}=\mathop{\sum }\nolimits_{n = -\infty }^{\infty }{J}_{n}(u){{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}n({\omega }_{{F}_{0}}t+{\phi }_{0})}\), where \(u=\alpha | {A}_{0}| /({\omega }_{{F}_{1}}{\omega }_{{F}_{0}})\), Jn is the nth Bessel function of the first kind, and integrate the resulting expansion term by term. Moreover, since the HF modes typically decay faster than the LF modes, we assume that Γ1 ≫ Γ0, and therefore, q1 adiabatically tracks q0 when \(t\gg {\Gamma }_{1}^{-1}\) (for more details about the adiabatic approximation see3). Thus, in the adiabatic tracking regime of q1, we find that

$$| {A}_{1}{| }^{2} \equiv \,f(u)={\left(\frac{{F}_{1}}{4{\omega }_{{F}_{1}}}\right)}^{2}{{{{{{{{\rm{e}}}}}}}}}^{-2{\Gamma }_{1}t}\int{{{{{{{{\rm{e}}}}}}}}}^{[{\Gamma }_{1}t+{{{{{{{\rm{i}}}}}}}}(\Delta {\omega }_{1}t-u\sin ({\omega }_{{F}_{0}}t+{\phi }_{0}))]}dt \\ \times\int{{{{{{{{\rm{e}}}}}}}}}^{[{\Gamma }_{1}t-{{{{{{{\rm{i}}}}}}}}(\Delta {\omega }_{1}t-u\sin ({\omega }_{{F}_{0}}t+{\phi }_{0}))]}dt.$$
(6)

For u ⩽ 1, we use a standard Taylor expansion truncated at cubic order to obtain the approximation \(| {A}_{1}{| }^{2}\approx f(0)+{f}^{{\prime} }(0)u+{f}^{{\prime\prime} }(0){u}^{2}/2+{f}^{{\prime\prime\prime} }(0){u}^{3}/6\), where \(f(0)={[{F}_{1}/(4{\omega }_{{F}_{1}}\sqrt{{\Gamma }_{1}^{2}+\Delta {\omega }_{1}^{2}})]}^{2}\) is used to evaluate the DC deflection \({\bar{q}}_{0}=-\alpha \langle {q}_{1}^{2}\rangle /{\omega }_{0}^{2}\approx -2\alpha f(0)/{\omega }_{0}\) near the onset of VDE modal interactions. By substitution of the truncated expansion of ∣A1∣2 into Eq. (4), we obtain the following Stuart–Landau oscillator67,68 for the LF mode

$${\dot{A}}_{0}=(\sigma -l| {A}_{0}{| }^{2}){A}_{0}-\frac{{{{{{{{\rm{i}}}}}}}}{F}_{0}}{4{\omega }_{{F}_{0}}},$$
(7)

where

$$\sigma = \, {g}_{1}\frac{\Delta {\omega }_{1}}{{\omega }_{{F}_{1}}}\left[2{\Gamma }_{1}+{{{{{{{\rm{i}}}}}}}}\left(\frac{{\Gamma }_{1}^{2}+\Delta {\omega }_{1}^{2}}{{\omega }_{{F}_{0}}}-{\omega }_{{F}_{0}}\right)\right]-{\Gamma }_{0}-{{{{{{{\rm{i}}}}}}}}\Delta {\omega }_{0},\\ \ell = \, {g}_{2}\frac{\Delta {\omega }_{1}}{{\omega }_{{F}_{1}}}\left[{\Gamma }_{1}(3{\Gamma }_{1}^{2}+8{\omega }_{{F}_{0}}^{2}-5\Delta {\omega }_{1}^{2})\right.\\ +\left.{{{{{{{\rm{i}}}}}}}}\left(\frac{{\Gamma }_{1}^{4}-\Delta {\omega }_{1}^{4}}{{\omega }_{{F}_{0}}}+{\omega }_{{F}_{0}}({\Gamma }_{1}^{2}+5\Delta {\omega }_{1}^{2}-4{\omega }_{{F}_{0}}^{2})\right)\right],$$

and g1,2 are non-negative quantities given by

$${g}_{1}={\left(\frac{\alpha {F}_{1}}{4{\omega }_{{F}_{1}}}\right)}^{2}\mathop{\prod }\limits_{n=-1}^{1}\frac{1}{{\Gamma }_{1}^{2}+{(\Delta {\omega }_{1}+n{\omega }_{{F}_{0}})}^{2}},$$
$${g}_{2}=\frac{3}{2}{\left(\frac{{\alpha }^{2}{F}_{1}}{4{\omega }_{{F}_{1}}^{2}}\right)}^{2}\mathop{\prod }\limits_{n=-2}^{2}\frac{1}{{\Gamma }_{1}^{2}+{(\Delta {\omega }_{1}+n{\omega }_{{F}_{0}})}^{2}}.$$

Therefore, in the adiabatic regime, the HF mode is functionally dependent on the LF mode. A geometric view of this process is that the faster-decaying dynamics end up on an invariant manifold on which the slower dynamics evolve. The retarded backaction from the HF mode completely modifies the properties of the LF mode. To be specific, we see that at leading order, the interaction between the two modes leads to the following changes in the LF mode: (i) The effective linear damping coefficient Γ0eff ≡ −ℜ{σ} = Γ0 − 2Γ1g1Δω1 can be markedly different from Γ0. In particular, we find that Γ0eff > Γ0 for red-detuned drive frequencies (i.e., negative detuning Δω1 < 0) of the HF mode, and Γ0eff < Γ0 for blue-detuned drive frequencies (i.e., positive detuning Δω1 > 0) of the HF mode. Moreover, for sufficiently large HF mode drive amplitude (F1), Γ0eff becomes negative, and self-induced oscillatory motion, i.e., lasing, is generated in the LF mode. (ii) The linear stiffness effect δω0A0 ≡ (ℑ{σ} + Δω0)A0, shifts the eigenfrequency of the LF mode. For δω0/ω0 ≪ 1, the shifted eigenfrequency can be approximated by \({\tilde{\omega }}_{0} \approx {\omega }_{0} + \delta {\omega }_{0}\). (iii) The (cubic) nonlinear damping effect ℜ{ℓ}∣A0∣2A0 introduces a new damping mechanism, which dominates at large amplitudes of the LF mode (\(| {A}_{0}| \gg \sqrt{| \Re \{\sigma \}/\Re \{\ell \}| }\)). And, (iv) the (cubic) nonlinear spring effect − ℑ{ℓ}∣A0∣2A0 introduces an additional Duffing nonlinearity69, which yields an amplitude-dependent frequency in the LF mode.

Eq. (7) reveals that the normal form of VDE modal interactions is a simple single-mode nonlinear oscillator, specifically the Stuart–Landau oscillator. Furthermore, Eq. (7) is consistent with the complex-amplitude equation one obtains from the following driven van der Pol–Duffing oscillator70

$$\frac{{d}^{2}v}{d{\tau }^{2}}-\epsilon (1-{v}^{2})\frac{dv}{d\tau }+v+\gamma {v}^{3}=F\cos ({\Omega }_{F}\tau ),$$
(8)

where \(v=({q}_{0}-{\bar{q}}_{0})/L\) is the non-dimensional displacement of the LF mode from its equilibrium position (\({\bar{q}}_{0}\)), and τ = t/T is the nondimensional time. All other parameters, including characteristic time (T) and length (L) scales, are specified in Supplementary Note 2.

From the foregoing analysis, we deduce that Eq. (8) and Eqs. (1)-(2) are dynamically equivalent when the adiabatic approximation holds. That is, the self-induced oscillations of the LF mode are a manifestation of nonlinear interaction with a blue-detuned HF mode, or alternatively, the leakage of energy from the blue-detuned HF mode generates negative linear damping in the LF mode, which results in self-induced oscillations. While Eq. (8) enables a considerably simpler view of VDE modal interactions, it still possesses an intricate bifurcation structure71 and can exhibit a wide range of dynamical responses, including chaotic attractors72. Consequently, in the remainder of this paper, we focus on a limited range of dynamical responses corresponding to injection locking and pulling of the LF mode. As shown below, injection locking and pulling of the LF mode generate tunable frequency combs in the HF and LF modes, respectively. These tunable frequency combs have potential use in a wide range of applications spanning from frequency metrology73 to molecular fingerprinting74.

Frequency combs generation

To explore the injection locking and pulling phenomena of the LF mode, we consider the scenario in which the drive frequency of the HF mode is blue-detuned \(\Delta {\omega }_{1}={\omega }_{{F}_{0}}\) and its amplitude (F1) is relatively large, such that Γ0eff < 0 (i.e., self-induced oscillations of the LF mode occur) and F0/F1 ≪ 1 (weak external harmonic injection to the LF mode). Using polar notation for the complex amplitude of the LF mode \({A}_{0}=-{{{{{{{\rm{i}}}}}}}}{a}_{0}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}[{\varphi }_{0}+\arg (\ell )]}/2\), we find from Eq. (7) that to leading order (Methods), the phase dynamics are governed by the Adler equation75

$$\frac{d{\varphi }_{0}}{ds}={\Omega }_{L}-\sin {\varphi }_{0},$$
(9)

where \(s=[{F}_{0}| \ell | /(4{\omega }_{{F}_{0}}\sqrt{\Re \{\sigma \}\Re \{\ell \}})]t\) is the non-dimensional time of the Adler equation, and \({\Omega }_{L}=4{\omega }_{{F}_{0}}\Im \{{\ell }^{* }\sigma \}\sqrt{\Re \{\sigma \}/\Re \{\ell \}}/({F}_{0}| \ell | )\) is the non-dimensional one-sided frequency-locking range (i.e., the overall locking range of the LF mode is ± ΩL around \({\tilde{\omega }}_{0}t/s\)).

Eq. (9) is a reduced-order, simplified, single-dimension dynamical system. To obtain this Adler equation, we effectively eliminate the dynamics of the HF mode (via adiabatic approximation) and then eliminate the amplitude dynamics under the assumption of weak injection (Methods) to achieve an equation for only the phase dynamics. We note that the assumption of weak injection is not mandatory, but it greatly simplifies the analysis. Without this condition, one needs to consider the generalized Adler equation76, which makes the analysis more complicated, especially when considering the amplitude dynamics.

To integrate Eq. (9), we set \(u(s)={{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\varphi }_{0}(s)}\) and obtain the equation du/ds = (1 + 2iΩLu − u2)/2, which can be readily solved to yield u(s) = [(u(0) − us)uus − (u(0) − uus)useλs]/[u(0) − us − (u(0) − uus)eλs], where us,us = iΩL ∓ λ and \(\lambda =\sqrt{1-{\Omega }_{L}^{2}}\). From Eq. (9) and u(s), we see that ∣ΩL∣ < 1 corresponds to injection locking of the LF mode, where for s ≫ λ−1, \(\sin {\varphi }_{0}={\Omega }_{L}\), u = us, and \({q}_{0}(t)={\bar{q}}_{0}+{a}_{0}\sin ({\omega }_{{F}_{0}}t+{\varphi }_{0}+\angle \ell )\). The condition ∣ΩL∣ < 1 can be viewed as a case in which the frequency of the external drive in Eq. (8) is close enough to the frequency of the unperturbed limit cycle (i.e., when F = 0) such that synchronization/injection-locking is achieved. The injection-locked LF mode, which has constant amplitude and phase, generates a periodically modulated complex-amplitude of the HF mode \({A}_{1}(t)={{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\varphi }_{1}(t)}{\sum }_{n}{a}_{1n}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}n({\omega }_{{F}_{0}}t+{\phi }_{0})}\), where \({\varphi }_{1}(t)={{{{{{{\rm{i}}}}}}}}\alpha {a}_{0}\sin ({\omega }_{{F}_{0}}t+{\varphi }_{0}+\angle \ell )/(2{\omega }_{{F}_{1}}{\omega }_{{F}_{0}})\), \({a}_{1n}={F}_{1}{J}_{n}[\alpha {a}_{0}/(2{\omega }_{{F}_{1}}{\omega }_{{F}_{0}})]/[4{\omega }_{{F}_{1}}({\omega }_{{F}_{0}}(1-n)-{{{{{{{\rm{i}}}}}}}}{\Gamma }_{1})]\), and Jn is the Bessel function of the first kind. These periodic modulations of A1 create a frequency comb around ω1, where the spacing between the spectral lines of the comb is \({\omega }_{{F}_{0}}\) (Fig. 3a and b). Therefore, by tuning the injected frequency \({\omega }_{{F}_{0}}\), we can control the spacing of the frequency comb of the HF mode. Injection pulling of the LF mode is associated with ∣ΩL∣ > 1 in which u(s), and therefore, φ0(s) are periodic functions with a period75 of \(2\pi /\sqrt{{\Omega }_{L}^{2}-1}\). Alternatively, we can view the condition ∣ΩL∣ > 1 as a case in which the frequency of the external drive in Eq. (8) is not sufficiently close to the frequency of the unperturbed limit cycle such that there are quasi-periodic oscillations of the LF mode. The non-uniform (highly non-harmonic) periodic modulations of φ0 create a frequency comb around \({\tilde{\omega }}_{0}\). The spacing between the spectral lines of the comb is \(\sqrt{{\Omega }_{L}^{2}-1}(s/t)\) (Fig. 3c and d). Hence, by tuning ΩL, we can control the spacing of the comb fingers in the spectrum of the LF mode.

Fig. 3: Injection locking and pulling of the low-frequency mode.
figure 3

a The constant amplitude and phase of the injection-locked low-frequency mode generate periodic modulations in the complex amplitude of the high-frequency mode, b which correspond to a frequency comb around ω1 with a spacing of \({\omega }_{{F}_{0}}\) in the power spectral density of q1. c The unlocked phase of the injection-pulled low-frequency mode is periodically modulated in a highly non-uniform rate with a frequency of \(\sqrt{{\Omega }_{L}^{2}-1}(s/t)\). Consequently, the temporal responses of the LF mode q0 are associated with distinct transitions from long calmer epochs to short windows of large modulations, d generating a frequency comb around \({\tilde{\omega }}_{0}\) with spacing of \(\sqrt{{\Omega }_{L}^{2}-1}(s/t)\) in the power spectral density of q0. All the shown results are obtained from the numerical integration of Eqs. (1)-(2), with \({\Gamma }_{0}=0.01,{\Gamma }_{1}=0.2,\,{\omega }_{0}=1,\,{\omega }_{1}=200,\,\alpha =100,\,{F}_{0}=0.2,\,{F}_{1}=50,\,{\omega }_{{F}_{1}}={\omega }_{1}+{\omega }_{{F}_{0}}\), and \({\omega }_{{F}_{0}}=\)0.98 (blue), 1(azure), 1.02 (cyan) in the injection locking regime and \({\omega }_{{F}_{0}}=\)1.021(red), 1.022(burgundy), 1.023(orange) in the injection pulling regime.

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Conclusions

We derived and analyzed a simple generic model for the intricate dynamics of VDE modal interactions that occur in a wide class of dynamical systems. We showed that the dynamics of VDE interactions can be mapped onto a single normal form, the Stuart-Landau oscillator, and can be conceptually viewed as the energy exchange between the symmetric and antisymmetric modes of a simple prototypical pendulums system. We studied in detail the phenomena of injection locking and pulling of the LF mode, which corresponds to a blue-detuned HF mode and a weakly driven LF mode. Our study reveales that injection locking and pulling can be exploited to generate tunable frequency combs in both the HF and the LF modes. Furthermore, these frequency combs are outcomes of the phase dynamics of the LF mode, which are governed by the well-known Adler equation; therefore, injection locking and pulling of the LF mode can be mapped onto the motion of an overdamped particle in a tilted washboard potential (Fig. 3, insets). The study of injection locking and pulling phenomena serves as a showcase for the capabilities of this simple model, which describes generic behavior of these systems and can be used to explore other phenomena, such as cooling and heating of several LF modes (with or without drive), in a straightforward manner.

Methods

A generalized model for VDE modal interactions

To generalize the model of Eqs. (1)-(2), we consider the Hamiltonian H = Hlf + Hhf + Hint, where \({H}_{{{{{{{{\rm{lf}}}}}}}}}=\mathop{\sum }\nolimits_{i = 1}^{n}{p}_{{L}_{i}}^{2}/2+{\omega }_{{L}_{i}}^{2}{q}_{{L}_{i}}^{2}/2+{\beta }_{i}{q}_{{L}_{i}}^{4}/4-{q}_{{L}_{i}}{F}_{{L}_{i}}\cos ({\omega }_{{F}_{{L}_{i}}}t)\) is the Hamiltonian of n low-frequency (LF) modes and each of these modes can have a Duffing nonlinearity (when βi ≠ 0), \({H}_{{{{{{{{\rm{hf}}}}}}}}}=\mathop{\sum }\nolimits_{i = 1}^{m}{p}_{{H}_{i}}^{2}/2+{\omega }_{{H}_{i}}^{2}{q}_{{H}_{i}}^{2}/2-{q}_{{H}_{i}}{F}_{{H}_{i}}\cos ({\omega }_{{F}_{{H}_{i}}}t)\) is the Hamiltonian of m high-frequency (HF) modes, and \({H}_{{{{{{{{\rm{int}}}}}}}}}={\sum }_{i,j,{j}^{{\prime} }}{\alpha }_{ij{j}^{{\prime} }}{q}_{{L}_{i}}{q}_{{H}_{j}}{q}_{{H}_{{j}^{{\prime} }}}\) is the interaction Hamiltonian and its coefficients \({\alpha }_{ij{j}^{{\prime} }}\) are symmetric with respect to j and \({j}^{{\prime} }\), i.e., \({\alpha }_{ij{j}^{{\prime} }}={\alpha }_{i{j}^{{\prime} }j}\). With the inclusion of linear dissipation and thermal noise terms (which are connected via the fluctuation-dissipation theorem77), we obtain the following dynamical system

$${\ddot{q}}_{{L}_{i}} + \,2{\Gamma }_{{L}_{i}}{\dot{q}}_{{L}_{i}}+{\omega }_{{L}_{i}}^{2}{q}_{{L}_{i}}+{\beta }_{i}{q}_{{L}_{i}}^{3}+\mathop{\sum}\limits_{j,{j}^{{\prime} }}{\alpha }_{ij{j}^{{\prime} }}{q}_{{H}_{j}}{q}_{{H}_{{j}^{{\prime} }}}\\ = \, {F}_{{L}_{i}}\cos ({\omega }_{{F}_{{L}_{i}}}t)+{\xi }_{{L}_{i}}(t),$$
(10)
$${\ddot{q}}_{{H}_{i}}+ 2{\Gamma }_{{H}_{i}}{\dot{q}}_{{H}_{i}}+{\omega }_{{H}_{i}}^{2}{q}_{{H}_{i}}+2{q}_{{H}_{i}}\mathop{\sum}\limits_{j}{\alpha }_{jii}{q}_{{L}_{j}}\\ + 2\mathop{\sum}\limits_{j,{j}^{{\prime} }\ne i}{\alpha }_{ji{j}^{{\prime} }}{q}_{{L}_{j}}{q}_{{H}_{{j}^{{\prime} }}}={F}_{{H}_{i}}\cos ({\omega }_{{F}_{{H}_{i}}}t)+{\xi }_{{H}_{i}}(t),$$
(11)

where the \({\Gamma }_{{L}_{i}}\) and \({\Gamma }_{{H}_{i}}\) are the modal decay rates, and \({\xi }_{{L}_{i}}\) and \({\xi }_{{H}_{i}}\) are zero-mean delta-correlated independent noise terms, so that \(\langle {\xi }_{{L}_{i}}(t) \rangle = \langle {\xi }_{{H}_{i}}(t) \rangle =0\), \(\langle {\xi }_{{L}_{i}}(t){\xi }_{{L}_{j}}(t+\tau ) \rangle =2{\delta }_{ij}\delta (\tau ){{{{{{{{\mathcal{D}}}}}}}}}_{{\xi }_{{L}_{i}}}\), and \(\langle {\xi }_{{H}_{i}}(t){\xi }_{{H}_{j}}(t+\tau ) \rangle =2{\delta }_{ij}\delta (\tau ){{{{{{{{\mathcal{D}}}}}}}}}_{{\xi }_{{H}_{i}}}\). The above idealization of the noises also applies to general non-Gaussian noises, as long as their correlation times are considerably shorter than the relaxation time of the modes78.

We make the ansatz

$${q}_{{L}_{i}}(t) \, = \, {\bar{q}}_{{L}_{i}}+{A}_{{L}_{i}}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{i}}}t}+{{{{{{{\rm{cc}}}}}}}},\\ {\dot{q}}_{{L}_{i}}(t) \, = \, {{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{i}}}{A}_{{L}_{i}}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{i}}}t}+{{{{{{{\rm{cc}}}}}}}},\\ {q}_{{H}_{i}}(t) \, = \, {A}_{{H}_{i}}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{H}_{i}}}t}+{{{{{{{\rm{cc}}}}}}}},\\ {\dot{q}}_{{H}_{i}}(t) \, = \, {{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{H}_{i}}}{A}_{{H}_{i}}(t){{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{H}_{i}}}t}+{{{{{{{\rm{cc}}}}}}}},$$
(12)

where \({\bar{q}}_{{L}_{i}}=-\langle {\sum }_{j,{j}^{{\prime} }}{\alpha }_{ij{j}^{{\prime} }}{q}_{{H}_{j}}{q}_{{H}_{{j}^{{\prime} }}} \! \rangle /{\omega }_{{L}_{i}}^{2}\) are the DC deflections of the LF modes that arise from the time-independent component of \({q}_{{H}_{j}}{q}_{{H}_{{j}^{{\prime} }}}\), and \({A}_{{L}_{i}}\,\)\(({A}_{{H}_{i}})\) are the complex-amplitudes of the LF (HF) modes. Treating the \({q}_{{L}_{i}}\) as quasi-static variables in Eq. (11) and applying the rotating wave approximation (RWA), we obtain the following complex-amplitude equations

$${\dot{A}}_{{L}_{i}}= -\left[{\Gamma }_{{L}_{i}}+{{{{{{{\rm{i}}}}}}}}\left(\Delta {\omega }_{{L}_{i}}-\frac{3{\beta }_{i}}{2{\omega }_{{F}_{{L}_{i}}}}| {A}_{{L}_{i}}{| }^{2}\right)\right]{A}_{{L}_{i}}\\ +\frac{{{{{{{{\rm{i}}}}}}}}}{2\pi }\mathop{\sum}\limits_{j,{j}^{{\prime} }}{\alpha }_{ij{j}^{{\prime} }}\int\nolimits_{0}^{\frac{2\pi }{{\omega }_{{F}_{{L}_{i}}}}}{A}_{{H}_{j}}{A}_{{H}_{{j}^{{\prime} }}}^{* }{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}({\omega }_{{F}_{{H}_{j}}}-{\omega }_{{F}_{{H}_{{j}^{{\prime} }}}})t}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{i}}}t}dt\\ -\frac{{{{{{{{\rm{i}}}}}}}}{F}_{{L}_{i}}}{4{\omega }_{{F}_{{L}_{i}}}}-\frac{{{{{{{{\rm{i}}}}}}}}}{2{\omega }_{{F}_{{L}_{i}}}}\langle {\xi }_{{L}_{i}}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{i}}}t}\rangle ,$$
(13)
$${\dot{A}}_{{H}_{i}}= -({\Gamma }_{{H}_{i}}+{{{{{{{\rm{i}}}}}}}}\Delta {\omega }_{{H}_{i}}){A}_{{H}_{i}}\\ +\frac{{{{{{{{\rm{i}}}}}}}}}{2{\omega }_{{F}_{{H}_{i}}}}\mathop{\sum}\limits_{j}{\alpha }_{jii}({A}_{{L}_{j}}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{j}}}t}+{A}_{{L}_{j}}^{* }{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{L}_{j}}}t}){A}_{{H}_{i}}\\ +\frac{{{{{{{{\rm{i}}}}}}}}}{2\pi }\mathop{\sum}\limits_{j,{j}^{{\prime} }\ne i}{\alpha }_{ji{j}^{{\prime} }}{A}_{{L}_{j}}^{* }{A}_{{H}_{{j}^{{\prime} }}}\int\nolimits_{0}^{\frac{2\pi }{{\omega }_{{F}_{{H}_{i}}}}}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}({\omega }_{{F}_{{H}_{j}^{{\prime} }}}-{\omega }_{{F}_{{L}_{j}}})t}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{H}_{i}}}t}dt\\ -\frac{{{{{{{{\rm{i}}}}}}}}{F}_{{H}_{i}}}{4{\omega }_{{F}_{{H}_{i}}}}-\frac{{{{{{{{\rm{i}}}}}}}}}{2{\omega }_{{F}_{{H}_{i}}}}\langle {\xi }_{{H}_{i}}{{{{{{{{\rm{e}}}}}}}}}^{-{{{{{{{\rm{i}}}}}}}}{\omega }_{{F}_{{H}_{i}}}t}\rangle ,$$
(14)

where \(\Delta {\omega }_{{L}_{i}}={\omega }_{{F}_{{L}_{i}}}-{\omega }_{{L}_{i}}\) and \(\Delta {\omega }_{{H}_{i}}={\omega }_{{F}_{{H}_{i}}}-{\omega }_{{H}_{i}}-{\sum }_{j}{\alpha }_{jii}{\bar{q}}_{{L}_{j}}/(2{\omega }_{{F}_{{H}_{i}}})\) are the frequency detunings of the LF and HF modes.

Similar to Eq. (5), Eq (14) contains certain slowly varying excitations after the RWA, with frequencies \({\omega }_{{F}_{{L}_{i}}}\). In particular, for constant \({A}_{{L}_{i}}\)’s, the \({A}_{{H}_{i}}\) oscillate with frequencies \({\omega }_{{F}_{{L}_{i}}}\). Moreover, from Eq. (14), we see that a pair of HF modes are coupled when they are separated in frequency by one of the LF mode’s frequencies, i.e., \(| {\omega }_{{H}_{j}}-{\omega }_{{H}_{j}^{{\prime} }}| \approx {\omega }_{{L}_{i}}\). Refs. 79,80 show experimental observations of this type of resonant coupling in cavity optomechanical systems. Consequently, Eq. (14) represents a set of linearly coupled equations in \({A}_{{H}_{i}}\). We can formally solve Eq. (14) in terms of the yet unknown complex amplitudes of the LF modes (\({A}_{{L}_{i}}\)). Then, in the adiabatic tracking regime of \({A}_{{H}_{i}}\) (under the assumption that \({\Gamma }_{{H}_{i}}\gg {\Gamma }_{{L}_{i}}\)), we can obtain a set coupled Stuart-Landau oscillators for the complex amplitude of the LF modes. While we leave the details of such an analysis for future study, it is worth noting that even in the case of a single HF mode (and multiple LF modes), its backaction leads to linear and nonlinear coupling between the LF modes. This type of LF modal coupling, which is mediated by the HF modes, has been experimentally observed in Refs. 12,81 in cavity optomechanics. Therefore, Eqs. (13)-(14), which account for the important effects of nonlinearity of the LF modes and noise8,82, can be viewed as a direct extension of the simplified model of Eqs. (1)-(2), which is clearly relevant to a wider class of systems.

The Adler equation

From Eq. (7), we find that the polar notation \({A}_{0}=-{{{{{{{\rm{i}}}}}}}}{a}_{0}{{{{{{{{\rm{e}}}}}}}}}^{{{{{{{{\rm{i}}}}}}}}[{\varphi }_{0}+\arg (\ell )]}/2\) yields the following pair of equations

$${\dot{a}}_{0}=\left(\Re \{\sigma \}-\Re \{\ell \}\frac{{a}_{0}^{2}}{4}\right){a}_{0}-\frac{\varepsilon {f}_{0}}{2{\omega }_{{F}_{0}}}\cos \theta ,$$
(15)
$${\dot{\varphi }}_{0}=\Im \{\sigma \}-\Im \{\ell \}\frac{{a}_{0}^{2}}{4}-\frac{\varepsilon {f}_{0}}{2{\omega }_{{F}_{0}}{a}_{0}}\sin \theta ,$$
(16)

where \(\theta ={\varphi }_{0}+\arg (\ell )\), and εf0 = F0, which is used to explicitly denote the smallness of F0 (ε ≪ 1). For ε = 0, the amplitude of the LF mode reaches the steady-state value \({a}_{0{{{{{{{\rm{s}}}}}}}}s}=2\sqrt{\Re \{\sigma \}/\Re \{\ell \}}\). Thus, in the presence of weak injection, we make the ansatz a0(t) = a0ss + εη(t), and obtain the following evolution equation to the perturbation \(\dot{\eta }=-2\Re \{\sigma \}\eta -[{f}_{0}/(2{\omega }_{{F}_{0}})]\cos \theta\). We see that the perturbation η is strongly damped; hence, for t ≫ 1/ℜ{σ}, we can set \(\dot{\eta }=0\) to obtain

$${a}_{0}=2\sqrt{\frac{\Re \{\sigma \}}{\Re \{\ell \}}}-\frac{\varepsilon {f}_{0}}{4{\omega }_{{F}_{0}}\Re \{\sigma \}}\cos \theta .$$
(17)

Substitution of Eq. (17) into Eq. (16) yields

$${\dot{\varphi }}_{0} \, = \, \Im \{\sigma \}-\Re \{\sigma \}\frac{\Im \{\ell \}}{\Re \{\ell \}}-\frac{\varepsilon {f}_{0}| \ell | }{4{\omega }_{{F}_{0}}\sqrt{\Re \{\sigma \}\Re \{\ell \}}}\sin {\varphi }_{0}\\ +\,O({\varepsilon }^{2}),$$
(18)

which is the well-known Adler equation.