Hard Mode in Three-Mode Optomechanical Systems

• Hard mode is a bistable regime in three-mode optomechanics where the transition to high-intensity generation occurs abruptly rather than continuously.
• The system utilizes two optical modes and one mechanical mode obeying a Brillouin frequency relation to enable seed-controlled switching via stability modification.
• Numerical analysis shows that a weak resonant seed significantly lowers the pump power threshold, paving the way for optical transistor and memory applications.

A three-mode optomechanical system can exhibit a hard excitation mode in which phonon generation and excitation of a lower-frequency optical mode begin through a discontinuous, hysteretic jump rather than a continuous threshold crossing. In the configuration studied in "A control of threshold of hard excitaion mode of optomechanical system by a low-intensity seed wave" (Mukhamedyanov et al., 19 Jun 2025), two optical cavity modes and one mechanical mode satisfy a Brillouin-type frequency relation, one optical mode is driven by an off-resonant pump, and the lower-frequency optical mode is resonantly driven by a weak seed. The central result is that a low-intensity seed wave can significantly reduce the threshold of the hard excitation mode by changing the stability of solutions in the bistability region, so that switching from a non-generating state to a generating one occurs at lower pump intensity (Mukhamedyanov et al., 19 Jun 2025).

1. Three-mode optomechanical configuration

The system consists of a high-frequency optical mode of frequency $\omega_1$ with annihilation operator $\hat a_1$, a low-frequency optical mode of frequency $\omega_2$ with annihilation operator $\hat a_2$, and a mechanical phonon mode of frequency $\omega_b$ with annihilation operator $\hat b$. These are related by the Brillouin-type condition

$\omega_1 - \omega_2 = \omega_b .$

Two coherent external fields act on the cavity: an off-resonant pump of frequency $\omega$ drives the higher optical mode, while a seed wave at frequency $\omega_2$ resonantly drives the lower optical mode (Mukhamedyanov et al., 19 Jun 2025).

Under the rotating-wave approximation, the Hamiltonian is

$$\begin{aligned} \hat H &= \hbar \omega_1 \hat a_1^\dagger \hat a_1 + \hbar \omega_2 \hat a_2^\dagger \hat a_2 + \hbar \omega_b \hat b^\dagger \hat b \ &\quad + \hbar g \left( \hat a_1^\dagger \hat a_2 \hat b + \hat a_1 \hat a_2^\dagger \hat b^\dagger \right) \ &\quad + \hbar \Omega_1 \left( \hat a_1^\dagger e^{-i\omega t} + \hat a_1 e^{+i\omega t} \right) \ &\quad + \hbar \Omega_2 \left( \hat a_2^\dagger e^{-i\omega_2 t} + \hat a_2 e^{+i\omega_2 t} \right). \end{aligned}$$

The three-wave mixing term describes conversion between the optical modes via the phonon mode, $\hat a_1$0, with coupling strength $\hat a_1$1 (Mukhamedyanov et al., 19 Jun 2025).

Including dissipation with rates $\hat a_1$2, $\hat a_1$3, and $\hat a_1$4, and neglecting noise, the semiclassical equations of motion for the mode amplitudes are

$\hat a_1$5

$\hat a_1$6

$\hat a_1$7

The analysis assumes $\hat a_1$8 and both coherent drives switched on at $\hat a_1$9 (Mukhamedyanov et al., 19 Jun 2025).

2. Bistability and the hard excitation mode

In this system, soft excitation denotes a continuous onset of generation above threshold, with a unique stable steady state for each pump value. By contrast, hard excitation mode denotes a bistable regime in which a low-intensity non-generating state and a high-intensity generating state coexist, producing a jump-like switch and hysteresis as the pump is varied (Mukhamedyanov et al., 19 Jun 2025).

After transforming to rotating frames,

$\omega_2$0

the stationary equations become

$\omega_2$1

$\omega_2$2

$\omega_2$3

Here the pump detuning is $\omega_2$4, the seed is resonant so $\omega_2$5, and $\omega_2$6 is the effective mechanical detuning (Mukhamedyanov et al., 19 Jun 2025).

For the unseeded case, $\omega_2$7, one steady-state branch is the trivial non-generating solution

$\omega_2$8

A nontrivial generating solution with finite $\omega_2$9 and $\hat a_2$0 can coexist with it (Mukhamedyanov et al., 19 Jun 2025).

The possibility of hard excitation is governed by the inequality

$\hat a_2$1

When this holds, there is a bistable interval

$\hat a_2$2

with analytic boundaries

$\hat a_2$3

and

$\hat a_2$4

Within this interval, the low- and high-intensity branches are both stable; above $\hat a_2$5, the low-intensity branch loses stability and the system must jump to the generating state (Mukhamedyanov et al., 19 Jun 2025).

3. Seed-wave control of the threshold

The lower-frequency optical mode is directly seeded by the resonant drive $\hat a_2$6. Once $\hat a_2$7, the strictly zero solution

$\hat a_2$8

no longer exists. Instead, two qualitatively distinct steady-state branches remain: a low-intensity branch continuously connected to the unseeded trivial state as $\hat a_2$9, and a high-intensity branch corresponding to the generating state (Mukhamedyanov et al., 19 Jun 2025).

Numerical solutions of the dynamical equations show that the hard excitation persists in the seeded system: at low $\omega_b$0 the system follows the low-intensity branch, and with increasing pump it still undergoes an abrupt jump to the high-intensity branch. The decisive modification is that the jump occurs at a smaller pump amplitude, and the effective switching point moves downward as $\omega_b$1 increases (Mukhamedyanov et al., 19 Jun 2025).

The paper states that, with seed present, no closed analytic threshold formula is given; the threshold is obtained numerically. The principal trend is that increasing $\omega_b$2 shifts the effective switching point from $\omega_b$3 toward $\omega_b$4. The total optical power required for switching, counted as pump plus seed, can drop by a factor of several relative to the single-pump threshold (Mukhamedyanov et al., 19 Jun 2025).

This threshold reduction is specific to the hard excitation regime. In the soft excitation regime, where the bistability condition is not satisfied, the seed only slightly shifts the generation threshold and does not produce a dramatic reduction or a discontinuous jump (Mukhamedyanov et al., 19 Jun 2025).

4. Stability mechanism and threshold reduction

The stability analysis is based on linearization around a steady solution,

$\omega_b$5

leading to a linear system

$\omega_b$6

where $\omega_b$7 is the Jacobian. Stability is determined by the eigenvalues $\omega_b$8: the steady state is stable when all $\omega_b$9 and unstable when at least one eigenvalue has positive real part (Mukhamedyanov et al., 19 Jun 2025).

Without seed, at $\hat b$0 one real part crosses zero, so the non-generating solution loses stability and the system switches to the generating state. With seed, the eigenvalue spectrum of the low-intensity branch changes in two linked ways. First, at fixed seed amplitude, increasing $\hat b$1 makes the least damped eigenvalue less negative. Second, at fixed pump amplitude, increasing $\hat b$2 likewise reduces the magnitude of the most weakly damped eigenvalue (Mukhamedyanov et al., 19 Jun 2025).

The paper interprets this as a reduction of the stability of the low-intensity branch inside the bistable window. Because the high-intensity branch already exists there, weakening the stability of the low-intensity solution makes switching to the generating branch occur at lower pump intensity. The stated mechanism is therefore not the appearance of a new branch, but a seed-induced change in the stability landscape of the bistability region (Mukhamedyanov et al., 19 Jun 2025).

A plausible implication is that the seed acts as a controlled perturbation of the gain-loss balance in the coupled $\hat b$3 subsystem. The paper explicitly notes that the seed provides an additional coherent input to $\hat b$4, enhances the effective couplings $\hat b$5 and $\hat b$6 in the linearized dynamics, and thereby pushes an eigenvalue toward zero at a lower pump value (Mukhamedyanov et al., 19 Jun 2025).

5. Numerical regime and representative parameters

For the hard excitation regime discussed in the main numerical examples, the paper uses

$\hat b$7

$\hat b$8

and

$\hat b$9

For these parameters, the one-pump hard excitation threshold is

$\omega_1 - \omega_2 = \omega_b .$0

and the hard excitation condition is satisfied (Mukhamedyanov et al., 19 Jun 2025).

The plotted examples compare the unseeded case with seeded cases such as

$\omega_1 - \omega_2 = \omega_b .$1

In the unseeded case, the intensities remain very low until $\omega_1 - \omega_2 = \omega_b .$2 reaches $\omega_1 - \omega_2 = \omega_b .$3, where they jump abruptly to high values. With seed, the intensities no longer vanish at low pump, and the jump to the high-intensity branch occurs at smaller $\omega_1 - \omega_2 = \omega_b .$4, between $\omega_1 - \omega_2 = \omega_b .$5 and $\omega_1 - \omega_2 = \omega_b .$6. Increasing $\omega_1 - \omega_2 = \omega_b .$7 moves the jump point toward $\omega_1 - \omega_2 = \omega_b .$8 and reduces the unstable region (Mukhamedyanov et al., 19 Jun 2025).

In the soft-excitation example, the paper states that parameters are chosen so that the bistability inequality is not satisfied, and quotes

$\omega_1 - \omega_2 = \omega_b .$9

In that regime all intensity curves are smooth, with only a slight threshold shift under seeding (Mukhamedyanov et al., 19 Jun 2025).

These numerical results support a sharp distinction between two operating regimes. In the bistable regime, a weak resonant seed can move the switching boundary substantially. Outside bistability, the same seed acts only as a modest perturbation.

6. Functional interpretation and broader context

The paper interprets the seeded hard excitation regime as a possible basis for all-optical transistors and logical elements. In this mapping, the output is the intensity of the generated low-frequency optical mode or of the phonon mode; the pump amplitude $\omega$0 acts as a main input; and the seed amplitude $\omega$1 acts as a control input. A low-intensity output corresponds to a logical “0,” while the high-intensity generating state corresponds to a logical “1” (Mukhamedyanov et al., 19 Jun 2025).

The proposed transistor-like mode of operation is to bias the pump near the original hard excitation threshold. With no seed, the system remains in the low-intensity state. Turning on a weak seed lowers the effective threshold below the fixed pump value and triggers the jump to the high-intensity state. Because the output intensity change is large, the paper identifies high on/off contrast and lower pump requirements as potential advantages of this mechanism (Mukhamedyanov et al., 19 Jun 2025).

The same bistability also suggests memory-like behavior. Since two stable states coexist over the same pump interval, seed pulses can in principle trigger transitions between them, and hysteresis can maintain the chosen state after the control input is removed. This suggests optical memory and logic functionality, although the paper presents this as an implication rather than a device-level implementation (Mukhamedyanov et al., 19 Jun 2025).

Within the broader nonlinear-optical context, the work is compared to optical bistability in Kerr cavities or saturable absorbers, injection seeding and injection locking in lasers, and stimulated Brillouin scattering with a Stokes seed. The specific feature emphasized here is that the nonlinearity is mediated by the optomechanical three-wave interaction, so phonons play an explicit role in the switching dynamics (Mukhamedyanov et al., 19 Jun 2025).

The main limitation identified is that hard excitation requires a specific detuning condition and a narrow bistable operating window. The paper also notes sensitivity to noise, the challenge of realizing two optical modes with frequency separation $\omega$2 and sufficient coupling $\omega$3 in an integrated platform, and the absence of a closed analytic threshold formula when the seed is present (Mukhamedyanov et al., 19 Jun 2025). Within those constraints, the study establishes seed-controlled destabilization of the low-intensity branch as a concrete mechanism for reducing the hard excitation threshold in a three-mode optomechanical system.