Three-Mode Electro-Optomechanical System

• Three-mode electro-optomechanical systems are hybrid platforms that couple optical, microwave, and mechanical resonances via mediators, enabling coherent conversion and interference-based control.
• These systems integrate distinct resonators through radiation-pressure and capacitive couplings, allowing functions such as nonreciprocal signal routing, directional amplification, and precise state squeezing.
• Experimental and theoretical studies demonstrate both coherent triply resonant conversion and modular implementations, while also addressing nonlinear dynamics, instability thresholds, and quantum state engineering.

A three-mode electro-optomechanical system is a hybrid bosonic platform in which optical, electrical or microwave, and mechanical degrees of freedom are coupled around a common mediator, typically a mechanical resonator. In the canonical form, the three dynamical modes are an optical cavity mode, a microwave or LC-resonator mode, and a mechanical mode; in closely related loop architectures, the same three-mode mathematics also appears in systems with two electromagnetic modes and one mechanical mode, because the underlying physics is controlled by resonant conversion, beam-splitter interactions, parametric interactions, and phase-sensitive interference among three coupled bosonic degrees of freedom (Zeng et al., 10 Jul 2025, Sarma et al., 2018). The term is used somewhat broadly in the literature, and that breadth includes both genuinely triply resonant converters and modular hybrids in which optics reads out a mechanical mode while electrical circuitry actuates or feeds back on the same mode without constituting coherent microwave–optical transduction in the strict cavity-conversion sense (Asano et al., 2018).

1. Architectural scope and defining variants

The most direct realization of a three-mode electro-optomechanical system consists of an optical cavity mode $\hat a$, a mechanical mode $\hat b$, and a microwave or LC mode $\hat c$, with the mechanics serving as the intermediary between optical and electrical domains. A recent example uses a membrane simultaneously as an optical element in a Fabry–Perot cavity and as the motion-sensitive capacitor of an LC circuit, so that radiation pressure couples optics to mechanics while capacitive coupling links mechanics to the microwave resonator (Zeng et al., 10 Jul 2025). In this sense, “electro-optomechanical” denotes not merely co-located subsystems, but a single hybrid dynamical network in which optical and electrical observables inherit correlations through the mechanics.

A second major class is the closed-loop three-mode system comprising two electromagnetic modes and one mechanical mode, often with an additional direct coupling $J$ between the electromagnetic modes. Although many papers formulate this as “two optical modes plus one mechanical mode,” the same formal structure is explicitly stated to map onto electro-optomechanics when one optical mode is reinterpreted as a microwave or electrical resonator. In such systems the key ingredients are the loop topology, the complex phases of the enhanced couplings, and the gauge-invariant phase accumulated around the triangle of couplings (Xu et al., 2015, Sun et al., 2017).

A recurrent source of confusion is that not every optical–electrical–mechanical platform is a canonical three-mode converter. In the modular microbottle–GaAs device, the optical subsystem provides high-sensitivity displacement readout and the electrical subsystem provides piezoelectric actuation and feedback on the same mechanical beam, but there is no electrical cavity mode and no demonstrated coherent microwave–optical exchange; the electrical channel is an actuation and feedback pathway rather than a third resonant bosonic mode (Asano et al., 2018). A related but opposite subtlety appears in nonreciprocal microwave–optical conversion through two mechanical resonators: the physical converter is a four-mode system, yet under strong damping of one mechanical mode it reduces to an effective three-mode description with an induced dissipative cavity–cavity coupling. The nonreciprocal mechanism there is fundamentally four-mode, even though an effective three-mode model exists after elimination (Xu et al., 2015).

2. Hamiltonian structures and operating regimes

For a genuine optical–mechanical–LC system, the starting Hamiltonian contains optical, mechanical, and microwave free energies together with radiation-pressure and capacitive couplings,

$$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$

supplemented by two-tone optical driving at $\omega_a\pm\omega_m$ and a red-detuned microwave drive satisfying $\omega_c-\omega_o=\omega_m$. After linearization around large classical amplitudes, the effective interaction becomes

$$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$

so that the blue optical sideband generates a parametric term, the red optical sideband generates a beam-splitter term, and the microwave drive produces red-sideband electromechanical exchange (Zeng et al., 10 Jul 2025).

A complementary formulation appears in resonant three-wave mixing models. In the three-mode mixing system with two electromagnetic modes $a_1,a_2$ and one mechanical mode $b$, the laboratory-frame interaction

$\hat b$0

reduces under the rotating-wave approximation to

$\hat b$1

with resonance condition $\hat b$2. The same paper explicitly notes that in an electro-optomechanical reinterpretation this becomes the usual optical–electrical–mechanical conversion Hamiltonian $\hat b$3 (Sarma et al., 2018). This is the canonical triply resonant conversion form: one photon in one electromagnetic mode is converted into a photon in the other plus a phonon, or the reverse.

Closed-loop nonreciprocal systems introduce a different effective Hamiltonian,

$\hat b$4

in which the direct mode–mode hopping $\hat b$5 closes the triangle. The system is reciprocal only when the gauge-invariant phase difference satisfies $\hat b$6; otherwise time-reversal symmetry is broken and direction-dependent transport becomes possible (Xu et al., 2015).

Operating regimes differ substantially across platforms. The microbottle–GaAs hybrid works deep in the unresolved-sideband regime, $\hat b$7, so its optical cavity is suited to displacement transduction and feedback rather than resolved-sideband dynamical backaction cooling (Asano et al., 2018). By contrast, the 3D waveguide electromechanical cavity operates with $\hat b$8, enabling resolved-sideband OMIA/OMIT analysis and conventional cooperativity extraction (Gunupudi et al., 2019).

3. Experimental realizations and modular implementations

One experimentally realized hybrid platform combines a silica microbottle whispering-gallery resonator with a piezoelectric GaAs doubly clamped beam through evanescent coupling. The bottle is fabricated from optical fiber, with maximum diameter $\hat b$9, neck diameter $\hat c$0, and neck-to-neck separation $\hat c$1; light at $\hat c$2 is coupled through a tapered fiber. The GaAs beam is $\hat c$3 long and $\hat c$4 wide. At large separation the optical quality factor is $\hat c$5 with optical linewidth $\hat c$6. The fundamental mechanical mode has $\hat c$7, $\hat c$8, and effective mass $\hat c$9. The measured vacuum optomechanical coupling reaches $J$0 at the smallest gap, the displacement sensitivity reaches $J$1, and classical feedback cooling lowers the effective mode temperature from room temperature to $J$2 (Asano et al., 2018). The significance of this system lies less in coherent triply resonant conversion than in the modular separation of high-$J$3 optical readout and piezoelectric electrical control.

A different experimental direction uses a 3D rectangular microwave waveguide cavity as a cavity-electromechanical building block. By reshaping the TE$J$4 mode into a “lumped-distributive” mode, the effective parasitic capacitance is reduced to $J$5, and coupling a mechanically compliant drumhead capacitor yields a capacitance participation ratio of $J$6. For the main device the cavity resonance is $J$7 with linewidth $J$8; two drum modes are observed at $J$9 and $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$0 with single-photon couplings $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$1 and $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$2. OMIA measurements give a maximum cooperativity $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$3, and the split-cavity design allows a DC bias across the mechanical resonator (Gunupudi et al., 2019). This system is not itself a full optical–microwave–mechanical transducer, but it is a concrete microwave–mechanics subsystem from which a three-mode electro-optomechanical interface can be built.

The theoretically analyzed optical–mechanical–LC platform uses experimentally plausible scales of $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$4, $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$5, $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$6, mechanical quality factor $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$7, $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$8, $$\hbar\omega_{a}\hat{a}^\dagger\hat{a} +\hbar\omega_{m}\hat{b}^\dagger\hat{b} +\hbar\omega_{c}\hat{c}^\dagger\hat{c} +\hbar g_a \hat a^\dagger \hat a(\hat b^\dagger+\hat b) +\hbar g_c \hat c^\dagger \hat c(\hat b^\dagger+\hat b),$$9, and cryogenic temperature $\omega_a\pm\omega_m$0 (Zeng et al., 10 Jul 2025). In this architecture, the same membrane simultaneously defines the optical boundary condition and the LC capacitance, making the mechanics the literal junction between optical and electrical subsystems rather than a merely auxiliary mode.

4. Nonreciprocity, conversion, and circulation

In the simplest loop model with two electromagnetic modes and one mechanical mode, nonreciprocity is controlled by the phase difference between the two enhanced optomechanical couplings. For matched conditions

$\omega_a\pm\omega_m$1

the strongest asymmetry occurs at $\omega_a\pm\omega_m$2 or $\omega_a\pm\omega_m$3. Near resonance, $\omega_a\pm\omega_m$4 yields $\omega_a\pm\omega_m$5 and $\omega_a\pm\omega_m$6, while $\omega_a\pm\omega_m$7 reverses the direction. Under the same matched conditions the system becomes an ideal three-port circulator, routing excitations as $\omega_a\pm\omega_m$8 or in the opposite direction depending on the loop phase (Xu et al., 2015). The central principle is interference between the direct $\omega_a\pm\omega_m$9 path and the mechanically mediated path.

Directional gain can be added by coherently driving the mechanics at the probe–pump difference frequency, $\omega_c-\omega_o=\omega_m$0. In the three-mode system of two coupled cavities and one driven mechanical resonator, this creates an extra transmission pathway. For $\omega_c-\omega_o=\omega_m$1, $\omega_c-\omega_o=\omega_m$2, $\omega_c-\omega_o=\omega_m$3, and $\omega_c-\omega_o=\omega_m$4, the transmission coefficients reduce to closed expressions in the mechanical-drive ratio $\omega_c-\omega_o=\omega_m$5, and one direction can be suppressed at

$\omega_c-\omega_o=\omega_m$6

while the opposite direction is amplified (Li et al., 2017). This mechanism differs from passive phase-bias isolation because it combines loop interference with coherent mechanical pumping.

Microwave–optical nonreciprocal conversion uses a more elaborate architecture. In the two-cavity, two-mechanical-mode electro-optomechanical converter, one mechanical mode is weakly damped and mediates a coherent, frequency-dependent conversion path, while the second is strongly damped and, after elimination, induces a dissipative cavity–cavity coupling $\omega_c-\omega_o=\omega_m$7. Under the hierarchy $\omega_c-\omega_o=\omega_m$8, and with $\omega_c-\omega_o=\omega_m$9, $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$0, the system exhibits near-ideal one-way conversion at $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$1: for $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$2, $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$3, $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$4, while at $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$5 the direction reverses (Xu et al., 2015). The opposite direction at opposite frequencies arises because the coherent path through the retained mechanical mode changes phase with $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$6, whereas the dissipative path is approximately frequency independent.

Not all nonreciprocity in three-mode hybrids is phase-loop nonreciprocity. In the asymmetric system where only one optical mode is optomechanically nonlinear and the second is coupled to it linearly, the nonreciprocal response is a strong-signal effect produced by a mechanics-induced Kerr-like nonlinearity $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$7. The key analytical point is that nonlinearity alone is not sufficient: reciprocity is restored if an impedance-matching condition makes the effective nonlinear coefficients in the two directions equal. Under suitable detuning and drive power, the reported examples reach isolation $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$8 with $$\hat{H}_{\text{eff}} = G_+ \left( \delta\hat{a}\delta\hat{b} + \delta\hat{a}^{\dagger}\delta\hat{b}^{\dagger} \right) + G_- \left( \delta\hat{a}\delta\hat{b}^{\dagger} + \delta\hat{a}^{\dagger}\delta\hat{b} \right) + G_{c} \left( \delta\hat{c}\delta\hat{b}^{\dagger} + \delta\hat{c}^{\dagger}\delta\hat{b} \right),$$9 and $a_1,a_2$0 (Xu et al., 2018). This establishes an important distinction between linear interference-based and nonlinear bistability-assisted nonreciprocity.

5. Quantum correlations, squeezing, and state engineering

Reservoir engineering provides one route to genuine quantum resources in three-mode hybrids. In the modulated system consisting of a driven optical cavity mode $a_1,a_2$1, an intermediate mechanical mode $a_1,a_2$2, and a second target mode $a_1,a_2$3 that may be either mechanical or a transmission line resonator, two-tone cavity driving and modulation of the $a_1,a_2$4–$a_1,a_2$5 coupling generate an effective Hamiltonian in which the intermediate mechanical mode cools two Bogoliubov modes of the target subsystem. The resulting steady state approaches a two-mode squeezed state rather than a two-mode squeezed thermal state, and the cavity-drive detuning $a_1,a_2$6 plays a decisive role because it couples the sum and difference Bogoliubov modes. The optimal detuning is approximately $a_1,a_2$7, and for $a_1,a_2$8, $a_1,a_2$9, and $b$0, the reported steady-state values are $b$1 and purity $b$2 (1803.01986). In electro-optomechanical language, this is a concrete recipe for steady-state optical–microwave entanglement mediated by a mechanical reservoir.

Phase-sensitive loop control offers a different form of correlation engineering. In the membrane-in-the-middle three-mode system with two optical modes and one mechanical mode, the relative phase of the effective couplings controls reversible population transfer between the field modes, creation of collective optical modes, and the distinction between bipartite and collective steering of the mechanics. The output modes remain perfectly mutually coherent, with $b$3 and $b$4, even when one output mode is unpopulated, and the minima of the output coincidence rate signal bipartite steering whereas the maxima signal collective steering (Sun et al., 2017). The broader implication is that in closed-loop three-mode systems, the physically relevant entangled degrees of freedom are often collective superpositions rather than bare cavity modes.

The optical–mechanical–LC system driven on red and blue optical sidebands together with a red microwave sideband produces simultaneous steady-state squeezing of the mechanical and microwave modes. The squeezing degree is defined as $b$5, and the optimal reported values are $b$6 for the microwave phase quadrature and $b$7 for the mechanical displacement quadrature, achieved near $b$8 and $b$9. The squeezing survives up to about $\hat b$00, and the same system exhibits perfect one-way Gaussian EPR steering between the optical cavity and the mechanical resonator over a finite parameter window, with the steering remaining robust up to about $\hat b$01 (Zeng et al., 10 Jul 2025). Here the useful asymmetry is not a loop phase but the unequal dissipation and drive-induced amplification structure of the optical and mechanical subsystems.

A further conceptual refinement appears in the pulsed three-mode system of cavity, mirror, and atomic ensemble. After adiabatic elimination of the cavity in the bad-cavity regime, the dynamics is governed by the competition between an effective parametric rate $\hat b$02 and a beam-splitter rate $\hat b$03. Above threshold, $\hat b$04, the system becomes an amplifier and reduces to a two-mode squeezing problem between the cavity output and a collective Bogoliubov mode $\hat b$05 built from the mirror and ensemble; only in this collective basis does a perfect bipartite EPR state emerge. Below threshold, $\hat b$06, the dynamics is attenuative and no perfect output EPR state is generated (He et al., 2014). This clarifies a general point for electro-optomechanics: ideal bipartite resources often reside in dressed or collective modes rather than in the bare optical, electrical, and mechanical coordinates.

6. Nonlinear dynamics, instability, and practical limits

Three-mode electro-optomechanical systems are not restricted to linear conversion and Gaussian correlations. In the resonant three-mode mixing model with weak Kerr nonlinearity $\hat b$07, destructive interference between two-photon excitation pathways suppresses the amplitude of the $\hat b$08 state and produces unconventional photon blockade. The optimized parameters satisfy

$\hat b$09

and the full master-equation simulations give $\hat b$10 under optimal conditions even when $\hat b$11 (Sarma et al., 2018). The same study shows that the effect is fragile: for $\hat b$12, $\hat b$13 rises to about $\hat b$14 by $\hat b$15, and pure dephasing at $\hat b$16 drives the statistics toward Poissonian. This underscores how strongly interference-based nonclassicality depends on low thermal occupancy and phase coherence.

A different nonlinear enhancement mechanism uses one cavity mode and two mechanical modes. In the large-detuning regime, the three normal modes include two phonon-like polaritons with very small effective damping, and the resonant condition $\hat b$17 activates the nonlinear process $\hat b$18. The effective nonlinear coupling scales as

$\hat b$19

while the optimized nonlinear figure of merit obeys

$\hat b$20

The practical lesson is that weak bare radiation-pressure nonlinearities can become spectroscopically visible when the resonant scattering occurs between long-lived phononic polaritons rather than between a phonon and a lossy cavity-like mode (Qiu et al., 2020).

Triply resonant three-mode systems can also become unstable. In the membrane-in-the-middle Fabry–Perot realization of three-mode parametric instability, the pump TEM00 mode, the TEM02 Stokes mode, and a membrane mode at $\hat b$21 satisfy the resonance $\hat b$22. The threshold power is

$\hat b$23

and the experiment reports $\hat b$24 and $\hat b$25. Above threshold the mechanical amplitude grows exponentially, then saturates around $\hat b$26 over $\hat b$27–$\hat b$28, and, contrary to expectation, the cavity does not lose lock (Chen et al., 2013). For electro-optomechanical systems this is a cautionary but constructive result: triply resonant enhancement brings both strong interaction and the possibility of positive-feedback instability, yet saturation need not imply catastrophic operating failure.

Across the literature, the main limitations are consistent. Small single-photon couplings remain common in modular or spatially separated architectures, as exemplified by $\hat b$29 in the microbottle–GaAs platform (Asano et al., 2018). Thermal phonons and dephasing rapidly degrade interference-based blockade and steering (Sarma et al., 2018, Zeng et al., 10 Jul 2025). Perfect circulation and nonreciprocal conversion generally require matched coherent couplings and damping rates, not merely nonzero coupling phases (Xu et al., 2015, Xu et al., 2015). Finally, the label “three-mode electro-optomechanical system” does not by itself determine whether a device is a coherent converter, a phase-biased loop, a feedback-controlled modular hybrid, or a nonlinear state-engineering platform. The common structure is three interacting bosonic sectors; the decisive distinctions are which couplings are coherent, which modes are truly dynamical, and whether the relevant physics resides in bare modes, effective collective modes, or reduced models obtained after elimination.