Phase-Locked Squeezed Phonon Lasing

• Phase-locked squeezed phonon lasing is a regime where self-oscillating mechanical modes exhibit stabilized amplitude with controlled phase and reduced noise through quadrature squeezing.
• Experimental platforms such as levitated nanoparticles and optomechanical crystals demonstrate phase locking, while theoretical proposals extend these concepts to include squeezed quadrature dynamics.
• Key methodologies include nonlinear feedback, PLL implementations, and Bogoliubov transformations that together improve force sensitivity and coherence for advanced metrological applications.

Phase-locked squeezed phonon lasing denotes a regime in which a self-oscillating mechanical mode combines laser-like phonon emission, an explicitly stabilized phase reference, and quadrature-selective fluctuation suppression in a squeezed basis. In the current literature, these ingredients appear with different degrees of completeness across platforms. Optically levitated nanoparticles and multimode optomechanical crystals have experimentally established phase-locked phonon lasing, long coherence, and phase-noise reduction, whereas controlled squeezed phonon lasing and its phase-locked variants have been proposed or theoretically analyzed in coupled-cavity optomechanics, trapped-ion systems, and Floquet-engineered solid-state spin-mechanical platforms (Zheng et al., 8 Apr 2026, Mercadé et al., 2021, Molinares et al., 3 Jun 2026).

1. Conceptual definition and scope

A phase-locked phonon laser is a self-oscillating mechanical mode whose amplitude is stabilized by nonlinear feedback and whose phase and frequency are actively locked to a reference. In the levitated-nanoparticle implementation, parametric feedback sets an energy-dependent damping so the center-of-mass mode self-oscillates, and a phase-locking loop suppresses phase diffusion and frequency drift. This converts the oscillator into a stable, coherent carrier whose amplitude or energy can be modulated by an ultra-weak external force and read out over long averaging times (Zheng et al., 8 Apr 2026).

In this context, squeezing refers to quadrature-selective noise redistribution. For a mechanical mode with squeezing parameter $r>0$, the quadrature variances are commonly written as

$$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$

so that one quadrature is squeezed and the conjugate quadrature is anti-squeezed. The solid-state Floquet proposal and trapped-ion squeezed-basis models make this quadrature structure explicit, either through Bogoliubov-mode amplification or through red- and blue-sideband engineering (Molinares et al., 3 Jun 2026, Lee et al., 9 Jan 2026).

A recurrent misconception is that phase locking and squeezing are interchangeable. They are not. The levitated-nanoparticle force-sensing work demonstrates phase-locked operation but explicitly states that no quadrature squeezing is implemented or analyzed. Likewise, the multimode Floquet optomechanical-crystal study reports phase locking and stability enhancement, but no phonon quadrature squeezing is reported (Zheng et al., 8 Apr 2026, Mercadé et al., 2021).

2. Gain, saturation, and threshold structure

Across platforms, phonon lasing is formulated as a gain-loss instability that saturates at finite amplitude. In the levitated-nanoparticle system, the cooling-state dynamics on the sensing axis are written as

$$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$

with mechanical susceptibility

$$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$

In the phonon-laser regime, the feedback-defined nonlinear damping law is

$$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$

and the effective amplitude damping is

$$\Gamma_{\mathrm{eff}}(E)=\Gamma_0+\delta\Gamma(E)=\Gamma_0+\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a.$$

Self-oscillation occurs when $\Gamma_{\mathrm{eff}}(E)<0$ at small amplitude, while the oscillation saturates at the stabilized mean energy $|E|=\gamma_a\hbar\Omega_0/\gamma_c$ (Zheng et al., 8 Apr 2026).

In squeezed-basis trapped-ion models, the same threshold logic is expressed through a Bogoliubov mode. The bath-free two-ion proposal defines

$$A=\cosh r \cdot a + e^{i\theta}\sinh r \cdot a^\dagger,$$

so that lasing occurs in the squeezed mode $A$ rather than in the bare phonon mode. The threshold is

$$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$0

and the minimal quadrature variance is

$$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$1

In the mixed-species and single-ion quantum theory, the threshold retains the same gain-loss form,

$$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$2

even when the lasing dynamics are moved into a squeezed basis (Lee et al., 9 Jan 2026, Baur et al., 20 Apr 2026).

This suggests a common structural principle: phase locking modifies the phase dynamics and linewidth, while squeezing modifies the mode basis and the fluctuation tensor; neither replaces the need for threshold crossing or nonlinear saturation.

3. Phase-locking mechanisms

In the levitated-nanoparticle architecture, phase locking is implemented with a PLL-like digital loop. A 50 MHz master clock is frequency-divided to obtain a square wave; a Kalman filter synthesizes a pure sinusoid to avoid DDS-induced phase-noise broadening; the particle’s phonon-laser signal is compared to the reference to produce frequency-error and phase-error signals; these are scaled, summed, passed to a digital integrator, and the integrator updates the AOM DC-bias once per oscillation cycle. The final DC-bias modulates trapping laser power to adjust phonon-laser frequency and phase. Under this architecture, phase locking extends the coherence time of the phonon-laser carrier to 12,500 s, while stable levitation is maintained down to $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$3 mW trapping power at $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$4 mbar (Zheng et al., 8 Apr 2026).

In Floquet phonon lasing in a multimode silicon optomechanical crystal cavity, phase locking is mediated by sideband-induced intermodal gain rather than by a direct external PLL. Temporal modulation at the difference frequency of two nearly degenerate GHz mechanical modes creates optical sidebands that seed coherent multimode emission. Experimentally, the phase-locked multimode lasing state exhibits a beatnote with sub-Hz linewidth, improved phase noise, and improved long-term frequency stability compared to single-mode lasing. The reported RMS jitter decreases from $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$5 ps to $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$6 ps, and phase noise improves by $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$7 dB at 10 kHz offset (Mercadé et al., 2021).

In the trapped-ion phonon laser operating close to the quantum ground state, phase locking is realized by an additional resonant mechanical drive. The phase dynamics follow an Adler-type equation,

$$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$8

with locking condition $$V_X=\frac{1}{2}e^{-2r}, \qquad V_P=\frac{1}{2}e^{2r},$$9. The experiment observes phase locking of the oscillator to an additional resonant drive and reconstructs phase diffusion through characteristic-function tomography, so that free-running lasing yields an annular Wigner function whereas injection produces azimuthal localization (Behrle et al., 2023).

Theoretical proposals extend these mechanisms. In the hBN Floquet scheme, switching to $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$0 produces a phase-referencing term that explicitly breaks $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$1 phase symmetry and locks the mechanical phase to the Floquet reference. In the phase-controlled coupled-cavity proposal of Zhang et al., the lasing phase is set by the complex three-mode coupling $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$2 carrying the inter-cavity OPA phase $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$3, and the phase obeys an Adler-type equation with locking range $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$4 (Molinares et al., 3 Jun 2026, 1706.02097).

4. Routes to squeezing and squeezed lasing

The clearest separation in the literature is between demonstrated phase locking and proposed squeezing. The levitated-nanoparticle force-sensing paper states unambiguously that it demonstrates phase-locked operation and ultra-weak force sensing, but does not report squeezed phonon-laser quadratures. It nevertheless identifies three feasible paths on that platform: parametric modulation at $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$5, reservoir engineering by tailoring feedback filters and LUTs to implement quadrature-selective damping, and measurement-based squeezing or backaction-evading readout. The strong, phase-stable carrier produced by phase locking is presented as an ideal phase reference for fixing the squeezing axis and suppressing phase diffusion (Zheng et al., 8 Apr 2026).

In trapped ions, squeezed phonon lasing is formulated directly in a Bogoliubov basis. The bath-free model uses simultaneous red- and blue-sideband driving to realize

$$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$6

with matching condition

$$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$7

The interaction then becomes a standard dual-channel phonon laser Hamiltonian in the squeezed mode $$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$8. The model predicts a displaced squeezed state with controllable quadrature squeezing and phase locking, all achieved without engineered reservoirs (Lee et al., 9 Jan 2026).

The mixed-species and single-ion quantum theory likewise treats squeezed-basis lasing through

$$\frac{d^2x}{dt^2}+(\Gamma_0+\delta\Gamma)\frac{dx}{dt}+\Omega_0^2x=\frac{F_{\mathrm{sto}}(t)+F_{\mathrm{weak}}(t)}{m},$$9

and couples heating and cooling sidebands directly to $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$0. That framework further analyzes a sensing protocol based on squeezed states using experimentally feasible parameters and reports a sensitivity enhancement of up to two orders of magnitude (Baur et al., 20 Apr 2026).

In the Floquet-engineered hBN membrane proposal, squeezing is intrinsic to the effective gain channel. The principal-spin pair generates the effective Hamiltonian

$$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$1

with

$$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$2

For $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$3, the proposal gives $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$4, $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$5, and $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$6, implying variance reduction $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$7, or $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$8 dB squeezing below vacuum in the ideal noiseless limit. The same work proposes a continuous transition from conventional lasing to phase-locked squeezed phonon lasing and shows that $$\chi(\omega)=\frac{1}{m(\Omega_m^2-\omega^2-i\Gamma_m\omega)}.$$9 in squeezed lasing without phase locking, while $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$0 in the phase-locked regime (Molinares et al., 3 Jun 2026).

In coupled-cavity optomechanics with tunable OPAs, Zhang et al. propose that the phase difference $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$1 between the parametric drives controls whether coherent hopping or two-mode squeezing dominates. In the $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$2 regime, the OPAs generate optically mediated two-phonon terms that yield an effective mechanical Hamiltonian

$$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$3

so that the same phase control that minimizes threshold also fixes the squeezed quadrature orientation (1706.02097).

5. Platforms and present research status

The literature is best read as a layered development rather than a single unified experimental record.

Platform and paper	Phase-locking status	Squeezing status
Optically levitated nanoparticle (Zheng et al., 8 Apr 2026)	Phase-locked phonon laser; coherence time 12,500 s; operation at $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$4 mW; ultra-weak force readout	No quadrature squeezing is implemented or analyzed
Silicon optomechanical crystal cavity (Mercadé et al., 2021)	Phase-locked multimode phonon lasing; sub-Hz beatnote; reduced jitter and Allan deviation	No phonon quadrature squeezing is reported
Trapped-ion phonon laser near the quantum regime, including the ETH Zürich implementation (Behrle et al., 2023)	Injection locking to an additional resonant drive; phase diffusion reconstructed from characteristic-function tomography	Squeezing is not observed in the standard single-sideband implementation
Coupled-cavity optomechanics with OPAs proposed by Zhang et al. (1706.02097)	Phase-controlled phonon laser with Adler-type locking to the optical reference is proposed	Mechanically squeezed phonon lasing is proposed
hBN membrane with Floquet-controlled solid-state defects (Molinares et al., 3 Jun 2026)	Floquet phase locking is proposed as part of the same effective model	Continuous transition to phase-locked squeezed phonon lasing is proposed
Two-ion and single-ion trapped-ion squeezed-basis models (Lee et al., 9 Jan 2026, Baur et al., 20 Apr 2026)	External coherent drives can stabilize phase coherence	Squeezed lasing in a Bogoliubov basis is developed theoretically

This distribution of results is significant. Experimental work has already established that phase locking can strongly suppress phase diffusion, narrow spectral features, and stabilize phonon-laser carriers over long averaging times. Theoretical work then uses that stabilized carrier as the missing ingredient for long-lived quadrature squeezing. A plausible implication is that fully realized phase-locked squeezed phonon lasing is now less a question of basic mechanism than of platform-specific noise engineering.

6. Metrology, coherence, and unresolved distinctions

The metrological importance of phase-locked phonon lasing is already explicit in levitated optomechanics. In the phase-locked levitated-nanoparticle experiment, stable and high-amplitude oscillation under low trap power reduces the force noise to $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$5. Under a loaded force, the system achieves a measurement resolution of $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$6 with a sensitivity of $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$7. Force extraction is performed from the cooling-state displacement PSD or from the phonon-laser energy-domain line at detuning $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$8, depending on operating mode (Zheng et al., 8 Apr 2026).

The squeezed-basis trapped-ion theory extends this metrological logic from low phase noise to low quadrature noise. Its sensing protocol based on squeezed states reports a sensitivity enhancement of up to two orders of magnitude. The gain factor is maximized when the squeezing angle aligns with the signal phase, and the theory explicitly notes the trade-off that squeezing increases effective heating and can challenge the Lamb–Dicke regime at large $$\delta\Gamma(E)=\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a,$$9 (Baur et al., 20 Apr 2026).

Three limitations recur across the literature. First, residual technical noise remains platform-dependent: in levitated systems, calibration is dominated by mass uncertainty from the Epstein model, while optical detection efficiency, AOM linearity, loop delay, and photon recoil bound performance; in trapped ions, spontaneous emission, motional heating, and drive phase noise degrade both linewidth and squeezing; in solid-state Floquet platforms, spin dephasing and thermal phonons raise threshold and broaden the spectrum (Zheng et al., 8 Apr 2026, Lee et al., 9 Jan 2026, Molinares et al., 3 Jun 2026). Second, phase locking alone does not guarantee nonclassicality. The multimode Floquet nanocavity explicitly treats its phase-noise reduction and beatnote locking as classical stabilization phenomena (Mercadé et al., 2021). Third, squeezing without phase stabilization is vulnerable to quadrature-axis diffusion; several proposals therefore treat phase locking not as an optional refinement but as the condition that preserves squeezed quadrature alignment over long averaging times (Zheng et al., 8 Apr 2026, Molinares et al., 3 Jun 2026).

The field is therefore defined by a precise asymmetry. Phase-locked phonon lasing is experimentally mature enough to support ultra-weak-force sensing, sub-Hz intermode beatnotes, and quantum-regime injection locking. Squeezed phonon lasing is theoretically mature enough to specify threshold conditions, Bogoliubov-mode dynamics, quadrature variances, $$\Gamma_{\mathrm{eff}}(E)=\Gamma_0+\delta\Gamma(E)=\Gamma_0+\left(\frac{\gamma_c}{\hbar\Omega_0}\right)E-\gamma_a.$$0 signatures, and sensing gains. Phase-locked squeezed phonon lasing, as a single experimentally consolidated regime, is best understood as the convergence point of these two lines of development rather than as a completed single-platform result.