Steady states and oscillation modes of two driven dissipative oscillators with non-Hermitian coupling

Abstract: We analyze the dynamics of two harmonic oscillators with nonlinear, non-Hermitian coupling between them. Specifically, we consider a synthetic antiferromagnet composed of two ferromagnetic layers with antiferromagnetic interactions between the two layers, leading to the emergence of two oscillation modes of the combined system. We analyze the response of this system to external driving fields. We determine the allowed steady states under various driving conditions. We then analyze the dynamics of small deviations away from the different steady states, which reveals the normal modes that can be probed using spectroscopic techniques in experiment. As expected, the nonlinear system exhibits multistability, in which multiple steady states can exist for the same driving conditions. We find that the dynamical response can be drastically different depending on which one of the two modes is driven and on the strength of the driving. Specifically, we can obtain level-repulsion or level-attraction patterns in the spectrum. In addition to the rich variety of spectra, we find that some steady states exhibit normal modes that contain multiple frequencies. Our results explain recent experiments on synthetic antiferromagnets and provide guidance for designing experiments that can explore previously unseen phenomena in these systems.

• The paper develops a nonlinear model of two driven dissipative oscillators and shows that asymmetric non-Hermitian coupling produces level repulsion under acoustic-mode driving and level attraction under optical-mode driving.
• The analysis predicts drive-dependent multistability, including up to three coexisting steady states, threshold-controlled spectral switching, and experimentally testable stability changes in synthetic antiferromagnets.
• The paper demonstrates that perturbations around nontrivial steady states require multi-frequency normal modes, while strong-drive regimes may reveal exotic spectra not captured by conventional single-frequency coupled-oscillator theory.

This paper presents a theoretical analysis of two harmonically coupled, driven, dissipative oscillators with nonlinear, non-Hermitian inter-mode coupling, motivated by recent experiments on synthetic antiferromagnets (SAFs). The central result is that the asymmetric structure of the nonlinear coupling produces qualitatively distinct spectral responses depending on which collective mode—the acoustic or the optical—is driven: driving near the acoustic mode yields level-repulsion patterns, while driving near the optical mode yields level-attraction patterns, in direct correspondence with two recent experimental observations (2608.12799). The analysis further predicts multistability, drive-strength-dependent spectral switching, and normal modes containing multiple frequencies, none of which have yet been observed experimentally.

Model and equations of motion

The system consists of two ferromagnetic layers coupled antiferromagnetically, whose collective magnetizations hybridize into a low-frequency acoustic mode and a high-frequency optical mode. The dynamics are described by two coupled equations for the complex mode amplitudes $b_{ac}$ and $b_{op}$:

$$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$

$$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$

The coupling is parametric: the optical mode is driven by the square of the acoustic amplitude, while the acoustic mode couples to $b_{ac}^* b_{op}$. The system is assumed biased near the degeneracy condition $\omega_{op} \approx 2\omega_{ac}$, so that both modes participate significantly in the dynamics. The complex variables represent the two-dimensional magnetization vectors of the two layers. The authors note that the sign structure of $g_3$ matters: for imaginary $g_3$ the equations describe an intrinsically unstable system, although the $c_{ac}=0$ steady state can remain stable and is analyzed separately. The analysis is anchored to two experiments: one driving near the acoustic mode frequency, which exhibited level repulsion, and one driving near the optical mode with different coupling, which exhibited level attraction.

Steady states under optical-mode driving

For driving near the optical mode ($\omega_d \approx \omega_{op}$), the steady-state ansatz is $b_{op}$0 and $b_{op}$1. The steady-state condition reduces to a nonlinear matrix equation in which the off-diagonal element $b_{op}$2 depends on the unknown amplitude itself. Two families of solutions emerge:

• The trivial solution $b_{op}$3, in which the optical mode reaches the usual driven-damped steady state while the acoustic mode remains in its static equilibrium. This solution always exists.
• Nontrivial solutions with $b_{op}$4, obtained in closed form with an explicit expression for $b_{op}$5 and a phase $b_{op}$6 fixed by $b_{op}$7.

The nontrivial solutions exist only under threshold conditions on the drive amplitude. In the strong-damping regime (detunings smaller than decay rates), a nontrivial solution requires $b_{op}$8, i.e., the drive must exceed a minimum threshold. In the weak-damping regime, an additional solution exists only below a maximum drive strength, $b_{op}$9. The consequence is a drive-dependent multistability: one solution for weak driving, two for strong driving, and up to three in an intermediate window. Numerical integration shows that, among the three coexisting solutions in a representative parameter set ($$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$0 GHz, $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$1 GHz, $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$2 GHz, $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$3 GHz, $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$4 MHz), only one is dynamically stable; the other two are unstable against small perturbations. This multistability is the nonlinear analogue of the switching behavior exploited in bifurcation-based sensing.

Normal modes and multi-frequency perturbation dynamics

Linearizing around a steady state with $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$5 reveals a structurally important result: single-frequency perturbation ansätze of the standard form do not admit physical solutions. The authors show that a single-term ansatz leads to a determinant condition whose roots are generically complex (unphysical), appearing in complex-conjugate pairs. The correct description requires a two-term ansatz, constrained by frequency-matching conditions $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$6 and $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$7, with all components sharing a common decay rate $$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$8. Each normal mode therefore contains two frequencies.

For a representative strongly driven case ($$i \frac{db_{ac}}{dt} = (\omega_{ac} - i\kappa_{ac}) b_{ac} + 2 i g_3 b_{ac}^* b_{op} + \tau_{ac},$$9 GHz, $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$0 GHz, $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$1 GHz, $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$2 GHz), the determinant condition yields four $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$3 pairs: (5.886, 0.79), (2.216, 0.79), (2.23, 0.311), and (5.872, 0.311), organized into two decay-rate branches with complementary frequencies summing to $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$4. A four-term fitting function constructed from these values reproduces the numerically integrated perturbation dynamics with excellent agreement, confirming the multi-frequency normal-mode picture. This result is a direct, testable prediction: spectroscopy of the driven SAF should reveal modes oscillating at multiple frequencies simultaneously, a feature absent in linear coupled-oscillator theory.

For the trivial steady state $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$5, the perturbation equations decouple. The optical deviations oscillate at $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$6 with decay $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$7, while the acoustic deviations satisfy a self-coupled equation involving $$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$8. Solving the determinant condition yields two branches, one of which is always physical: either

$$i \frac{db_{op}}{dt} = (\omega_{op} - i\kappa_{op}) b_{op} - i g_3 b_{ac}^2 + \tau_{op}.$$9

or a degenerate-frequency branch with drive-dependent decay rate $b_{ac}^* b_{op}$0. Exactly one of the two square roots is real for given parameters, so the spectrum is unambiguous.

Spectra and the origin of level attraction versus repulsion

Combining these results produces normal-mode spectra as a function of the tunable acoustic frequency. Three regimes appear under optical-mode driving. For weak driving ($b_{ac}^* b_{op}$1 GHz), only the $b_{ac}^* b_{op}$2 solution exists and the spectrum shows level attraction. For intermediate driving ($b_{ac}^* b_{op}$3 GHz), the mode frequencies asymptote to $b_{ac}^* b_{op}$4 and $b_{ac}^* b_{op}$5, with exceptional points where branches meet. For strong driving ($b_{ac}^* b_{op}$6 GHz) near $b_{ac}^* b_{op}$7, the stable steady state switches to the $b_{ac}^* b_{op}$8 branch and the spectrum acquires an exotic multi-branch structure. The authors also show that for imaginary $b_{ac}^* b_{op}$9 the $\omega_{op} \approx 2\omega_{ac}$0 spectrum is formally identical to the real-$\omega_{op} \approx 2\omega_{ac}$1 case, so level attraction persists provided that state is dynamically stable—though the underlying equations admit diverging trajectories for large amplitudes, a limitation of the imaginary-coupling model.

Under acoustic-mode driving ($\omega_{op} \approx 2\omega_{ac}$2), the steady-state equation changes character: neither $\omega_{op} \approx 2\omega_{ac}$3 nor $\omega_{op} \approx 2\omega_{ac}$4 solutions exist for nonzero drive, and no closed-form solution is available; the authors proceed perturbatively, with $\omega_{op} \approx 2\omega_{ac}$5 and $\omega_{op} \approx 2\omega_{ac}$6 for weak driving. The weak-driving spectrum exhibits a clean level-repulsion (avoided-crossing) pattern, consistent with the earlier experiment, because the dominant perturbation coupling between $\omega_{op} \approx 2\omega_{ac}$7 and $\omega_{op} \approx 2\omega_{ac}$8 is effectively Hermitian. For strong acoustic driving ($\omega_{op} \approx 2\omega_{ac}$9 GHz), the repulsion pattern disappears, replaced by level-attraction features—including an asymmetric pair near $g_3$0 GHz where branches merge smoothly on one side and bifurcate abruptly on the other.

The paper identifies the microscopic origin of the attraction/repulsion dichotomy: the asymmetric nonlinear interaction terms generate, in the linearized equations, either a $g_3$1–$g_3$2 coupling term (which produces level attraction, in the optically driven $g_3$3 state) or an approximately Hermitian $g_3$4–$g_3$5 coupling (which produces level repulsion, in the weakly acoustically driven state). The two appendices reinforce this diagnosis: a model of two linearly coupled oscillators shows that real coupling gives repulsion while imaginary (dissipative) coupling gives attraction, and a single-variable model with an $g_3$6 coupling term demonstrates how multi-frequency modes arise generically when conjugate variables couple.

Limitations and open questions

Several limitations are stated explicitly. The linearized normal-mode analysis is perturbative and assumes small deviations; it does not address switching dynamics between coexisting steady states or the basins of attraction that determine which state is reached experimentally. The claim that single-frequency perturbation ansätze admit no physical solutions rests on numerical evaluation of the determinant condition across sampled parameter combinations rather than an analytical proof. The strong-driving spectra under acoustic driving are described qualitatively—the authors attribute the obscured level-repulsion pattern to the complicated dependence of $g_3$7 on $g_3$8 without a closed-form account. The predicted exotic spectra, including multi-frequency normal modes and the strong-drive level-attraction features, require drive amplitudes beyond current experimental capability, so their observation remains an open experimental question. Finally, the analysis assumes the rotating-wave-like restriction to a single circular drive component and the near-degeneracy condition $g_3$9; behavior outside these regimes is not treated.

Conclusion

This work provides a unified theoretical framework for the driven dynamics of two nonlinearly, non-Hermitianly coupled oscillators modeling a synthetic antiferromagnet. By solving for steady states and linearizing around them with appropriately generalized multi-frequency ansätze, the authors explain the experimentally observed contrast between level repulsion (acoustic driving) and level attraction (optical driving) as a consequence of the asymmetric parametric coupling, and they predict multistability, drive-threshold-dependent spectral switching, and normal modes with multiple oscillation frequencies. The results supply concrete parameter targets—drive amplitude, bias field, and drive frequency—for experiments seeking to observe these regimes.