---
title: Squeezed Phonon Laser
url: https://www.emergentmind.com/topics/squeezed-phonon-laser
type: topic
---

# Squeezed Phonon Laser

Searching arXiv for the cited squeezed phonon laser and related papers to ground the article with current literature.
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{"query":"squeezed phonon laser","max_results":10}
Squeezed phonon lasing denotes a regime in which a vibrational mode undergoes self-sustained, coherent oscillation while quadrature fluctuations are modified by squeezing, so that gain, threshold behavior, linewidth narrowing, and phase coherence are treated together with reduced noise in a selected quadrature. In the recent literature, the term encompasses experimentally realized topological parametric phonon lasing in nonlinear nanoelectromechanical Dirac-vortex cavities, Floquet-engineered squeezed phonon lasing in solid-state defect platforms, bath-free squeezed lasing via intrinsic ion–phonon coupling in trapped ions, and nonreciprocal phonon lasing induced by directional optical squeezing in compound optomechanical systems [2107.11162] [2606.05083] [2601.05575] [2312.06896].

## 1. Definition and distinguishing characteristics

Phonon lasing—the acoustic analogue of an optical laser—occurs when driven gain overcomes mechanical damping, and a vibrational mode enters a regime of self-sustained, coherent oscillation. Generating squeezed phonon states—where one quadrature’s noise is suppressed below the zero-point level—opens the door to measurements below the standard quantum limit, offering enhanced force, mass or field sensing in nanomechanical platforms [2606.05083].

Within this framework, a squeezed phonon laser is not merely a phonon laser with large amplitude, and it is not merely a squeezed mechanical state. The defining feature is the coexistence of lasing-type gain and threshold behavior with quadrature-selective noise control. In the experimental topological realization, the squeezed-bosonic interaction provides phase-sensitive gain or deamplification depending on pump phase, and below threshold the quadrature ellipse narrows along one axis with a 40% reduction in noise variance; above threshold, a sharp peak at $\omega_0=\tfrac12\cdot \text{pump}$ appears once $V_{\rm pump}>2.09\ {\rm V}$ [2107.11162].

The literature also makes clear that squeezed phonon lasing is not tied to a single mechanism. One proposal enables a continuous transition from conventional lasing to phase-locked squeezed phonon lasing by Floquet engineering in a color-center platform [2606.05083]. Another realizes squeezed lasing in a trapped-ion system without relying on engineered baths or tailored dissipative reservoirs [2601.05575]. A distinct line of work uses directional optical squeezing to generate directional mechanical gain, leading to nonreciprocal phonon lasing with a well-tunable directional power threshold [2312.06896].

## 2. Dynamical mechanisms and threshold conditions

The principal mechanisms reported in the literature can be organized by the effective interaction that generates gain and by the form of the threshold condition.

| Platform | Effective interaction | Threshold condition |
|---|---|---|
| Dirac-vortex nanoelectromechanics | $(\hat a^{\dagger 2}+\hat a^2)$ squeezing term | $g=\gamma/2$ |
| Floquet-controlled solid-state defects | $B\equiv u\,b+v\,b^\dagger$ squeezed-mode coupling | $g_{\rm eff}^2/\gamma'_m>\gamma_m$ |
| Trapped ions | dual red- and blue-sideband ion–phonon coupling | $A_{\rm th}=B$ |
| Nonreciprocal optomechanics | directional optical squeezing modifies mechanical gain $G$ | $G=\gamma_m$ |

In the topological nanoelectromechanical system, the rotating-frame Hamiltonian of the Dirac-vortex mode is
\[
\hat H = \hbar\omega_0\,\hat a^\dagger\hat a
+\frac{\hbar B}{2}\,\hat a^\dagger\hat a^\dagger \hat a\,\hat a
+\frac{\hbar\,\alpha\,V_{\rm dc}\,V_{\rm pump}}{2}\,\Bigl(\hat a^{\dagger2} + \hat a^2\Bigr),
\]
with $g=\alpha V_{\rm dc}V_{\rm pump}/2$. The Heisenberg–Langevin analysis yields eigenvalues $\lambda_\pm=-\gamma/2\pm g-i\Delta$, so the onset of parametric instability and phonon lasing occurs when $g>\gamma/2$, equivalently
\[
V_{\rm pump}^{(\rm th)}=\frac{\gamma}{\alpha V_{\rm dc}}.
\]
The experiment finds $V_{\rm pump}^{(\rm th)}\approx 2.09\ {\rm V}$ for coherent pumping at optimal detuning $\Delta=0$ [2107.11162].

In the Floquet-controlled solid-state-defect proposal, periodic modulation at the primary red sideband resonance $\Omega=\Delta_1+\Delta_2+\omega_m$, $\nu=2\omega_m$ produces
\[
H_{\rm eff}^A=i\,g_{\rm eff}\bigl(u\,b^\dagger+v\,b\bigr)\sigma_1^+\sigma_2^+ + {\rm H.c.},
\]
where $u=J_0(\chi)/N$, $v=J_1(\chi)/N$, $N=\sqrt{|J_0^2(\chi)-J_1^2(\chi)|}$, and $g_{\rm eff}=(J\lambda_1/\omega_m)N$. After elimination of the ancilla sector, the semi-classical occupation obeys
\[
\frac{d\langle n\rangle}{dt}=2(g_{\rm eff}^2/\gamma'_m)(\langle n\rangle+1)-\gamma_m\langle n\rangle,
\]
leading to the threshold
\[
g_{\rm eff}^2/\gamma'_m>\gamma_m,
\qquad
J>J_{\rm th}=\omega_m\,\frac{\sqrt{\gamma_m\Gamma'}}{\lambda_1\,|J_0(\chi)^2-J_1(\chi)^2|^{1/2}}.
\]
Above threshold, $\langle n\rangle_\infty=(g_{\rm eff}^2/\gamma'_m\gamma_m)-1$ [2606.05083].

In the trapped-ion proposal, adiabatic elimination of the internal states yields an effective phonon-mode master equation
\[
\dot\rho_m=A\,\mathcal D[a^\dagger]\rho_m+B\,\mathcal D[a]\rho_m,
\]
with
\[
A=\frac{2|g_1|^2}{\gamma_1},
\qquad
B=\frac{2|g_2|^2}{\gamma_2},
\]
and the threshold is reached when
\[
A_{\rm th}=B
\quad\Longleftrightarrow\quad
\frac{|g_1|^2}{\gamma_1}=\frac{|g_2|^2}{\gamma_2}.
\]
This formulation keeps the gain–loss balance explicit while moving the squeezing into a unitary basis change [2601.05575].

In the nonreciprocal optomechanical scheme, the phonon-laser threshold is reached when $G=\gamma_m$, and the emitted-phonon number is $N_b=\exp[2(G-\gamma_m)/\gamma_m]$, so $N_b=1$ at threshold. The forward threshold power is
\[
P_{\rm th} \approx \frac{2\hbar\,\kappa_0\,\gamma_m\,\omega_l\left[(J_s^2+\kappa_0^2-\Delta_l\Delta_s)^2+\kappa_0^2(\Delta_s+\Delta_l)^2\right]}
{\kappa_1\,J_s\,g^2\,x_0^2\,\Delta_s},
\]
whereas in the backward direction one replaces $J_s\to J_0$ and $\Delta_s\to\Delta_l$ [2312.06896].

## 3. Topological Dirac-vortex parametric phonon laser

The experimentally realized topological platform is a 2D nanomechanical crystal that implements the Jackiw–Rossi model by imposing a vortex in the local Kekulé distortion, $d_0(r)=d_{\max}\tanh(|r|/R_0)$. The Dirac-vortex mode is a zero mode bound to the vortex core with spinor envelope
\[
\psi_0(r)\propto \frac{1}{[\cosh(|r|/R_0)]^s}\,W(r),
\qquad s\approx 0.52,
\]
where $W(r)$ is the Bloch profile in each unit cell and $R_0$ sets the modal area $S\propto R_0^2$. The device consists of a 2D hexagonal lattice of suspended silicon-nitride membranes of thickness $150\ {\rm nm}$ and $l_0=20.8\ \mu{\rm m}$, patterned with two radii of holes in each unit cell to implement a Kekulé modulation; after Al metallization ($40\ {\rm nm}$), the membrane array sits above a Si substrate that serves as ground [2107.11162].

In the $R_0/l_0=0.5$ device, the bare resonant frequency is $\omega_0/2\pi=45.238\ {\rm MHz}$. The Kerr coefficient is $B\approx 0.392\ {\rm kHz}\cdot{\rm nm}^{-2}$, the electromechanical tuning coefficient is $\alpha\approx 0.40\ {\rm kHz}\cdot{\rm V}^{-2}$, and the damping rate is $\gamma/2\pi\approx 7.9\ {\rm kHz}$. A combined bias $V_{\rm dc}$ and parametric pump $V_{\rm pump}\cos(2\omega_0 t)$ are applied between the Al electrode and substrate. The electrostatic force $F\sim (V_{\rm dc}+V_{\rm pump}\cos 2\omega_0 t)^2$ couples to the Dirac-vortex deflection $W(r,t)$, yielding both a linear tuning of $\omega_0$ and the two-phonon interaction [2107.11162].

The topological properties reported for this cavity differ sharply from those of conventional cavities. Unlike conventional cavities whose fundamental frequency shifts with size, with ${\rm FSR}\sim 1/S$, the Dirac-vortex frequency is topologically pinned to the bulk Dirac point. In measurements from $R_0/l_0=0.5\to 4$, $f_0$ stays within $\pm 0.05\%$ even though $S$ increases by over an order of magnitude. The spacing between the zero mode and its first excited bound state remains nearly constant, with $\Delta\omega/2\pi \simeq 1.5\ {\rm MHz}$ for $R_0/l_0$ from $0.5$ to $4$, defying the $1/S$ law. The mode profile decays algebraically as $\cosh^{-s}$ rather than as a simple Gaussian or sine-mode, and the sideband modes correspond to excited Jackiw–Rossi bound states rather than higher longitudinal or transverse Fabry–Pérot orders [2107.11162].

Experimentally, the quadratures were recorded via a laser interferometer or via the electrode readout demodulated at $\omega_0$. With $V_{\rm pump}$ below threshold, the quadrature ellipse narrows along one axis, with a 40% reduction in noise variance, directly evidencing squeezed vacuum fluctuations. Under a coherent pump with bandwidth $\ll\gamma$, a sharp peak at $\omega_0=\tfrac12\cdot \text{pump}$ appears once $V_{\rm pump}>2.09\ {\rm V}$, and the spectrum linewidth narrows from $\sim 8\ {\rm kHz}$ down to $\sim 1\ {\rm kHz}$. Under an incoherent white-noise pump with bandwidth $\gg\gamma$, lasing still occurs but with a slightly higher threshold, $2.75\ {\rm V}$, due to reduced spectral density near $2\omega_0$ [2107.11162].

These results were summarized in the source as the first squeezed, topological phonon laser, combining a Jackiw–Rossi Dirac-vortex cavity, strong second-order electromechanical nonlinearity, and observation of phonon lasing above threshold, phase-sensitive amplification, vacuum squeezing, and linewidth narrowing in a single integrated platform. A plausible implication is that the combination of frequency pinning and nearly constant mode spacing directly addresses the large-area single-mode problem in bosonic cavities.

## 4. Floquet-engineered squeezed phonon lasing in solid-state defects

A distinct route uses a single mechanical mode of frequency $\omega_m$ coupled dispersively to a principal pair of two-level defects embedded in a suspended hBN membrane, while an identical ancilla pair provides engineered dissipation. The time-dependent Hamiltonian is
\[
H(t)=\omega_m\,b^\dagger b+\sum_{j=1,2}\Delta_j\,\sigma_j^z/2-\lambda_1\,\sigma_1^z(b+b^\dagger)
+J\cos(\Omega t)\sigma_1^x\sigma_2^x+\epsilon\cos(\nu t)\sigma_2^z
\]
plus analogous primed terms for the ancilla spins. Under periodic driving and in the regime $\lambda_1\ll \omega_m$, a Magnus/Floquet expansion yields the effective squeezed-mode coupling of the principal sector and, under a different sideband resonance for the ancilla, an engineered squeezed-mode decay rate $\gamma'_m\sim g_{\rm eff}^{\prime 2}/\Gamma'$ [2606.05083].

The squeezed character is encoded by the Bogoliubov relation $|u^2-v^2|=1$, which identifies
\[
B\equiv u\,b+v\,b^\dagger
\]
as a squeezed-mode operator. Above threshold, the Wigner function of the mechanical mode in the squeezed basis $B$ becomes a ring of radius $\sqrt{\langle n\rangle_\infty+1/2}$ with reduced width along one quadrature. For canonical quadratures
\[
X_\theta=\frac{1}{2}(b\,e^{-i\theta}+b^\dagger e^{i\theta}),
\]
the steady-state variances satisfy
\[
{\rm Var}(X_{\min})=\frac14 e^{-2r},
\qquad
{\rm Var}(X_{\max})=\frac14 e^{2r},
\]
with squeezing parameter
\[
r=\operatorname{arctanh}(v/u).
\]
The same study analyzes the emission spectrum
\[
S(\omega)=\int_{-\infty}^{\infty} d\tau\, e^{-i\omega\tau}\langle b^\dagger(\tau)b(0)\rangle_\infty
\]
and the zero-delay second-order correlation
\[
g^{(2)}(0)=\frac{\langle b^\dagger b^\dagger b b\rangle}{\langle b^\dagger b\rangle^2}.
\]
The latter evolves from thermal values $g^{(2)}(0)\approx 2$ below threshold to Poissonian $g^{(2)}(0)\to 1$ for conventional lasing at $\chi=0$, and to super-Poissonian $g^{(2)}(0)\to 3/2$ in the squeezed-lasing regime at $\chi\approx 1$ [2606.05083].

The Floquet control protocol also distinguishes conventional, squeezed, and phase-locked squeezed lasing. At $\nu=2\omega_m$, one realizes squeezed amplification into $B$; for $\nu\gg\omega_m$, the usual beam-splitter-type coupling recovers a standard phonon laser. The dimensionless drive amplitude $\chi=2\epsilon/\nu$ controls the balance between the Stokes and anti-Stokes sidebands and tunes the squeezing strength continuously from zero to moderate values. Furthermore, choosing $\nu=\omega_m$ and $\Omega=\Delta_1+\Delta_2+\omega_m$ yields an extra term proportional to $J J_1(\chi)\sigma_1^+\sigma_2^+$ that explicitly breaks the $U(1)$ phase symmetry, leading to phase-locked squeezed lasing without any external injection tone [2606.05083].

The feasible parameter set used in the numerical simulations is given, in units of $\omega_m$, by $\lambda_1/\omega_m=0.04$, $J/\omega_m\approx 0.2$–$0.4$, $\gamma_m\approx 10^{-4}$, $\gamma'_m\approx 10\gamma_m$, $\Gamma_j\approx 0.03$, $\bar n_m\approx 10^{-3}$, and $\bar n_s\approx 10^{-5}$. These values correspond to hBN membranes of radius $R\approx 0.1\ \mu{\rm m}$, $\omega_m/2\pi\approx 350\ {\rm MHz}$, mechanical $Q\sim 10^6$, and magnetic field gradients $\partial_z B\gtrsim 10^8\ {\rm T/m}$ that yield $\lambda_1/\omega_m\sim 0.04$. The proposal states that a steady-state squeezed phonon laser enables force or mass sensors whose sensitivity improves by a factor $e^r$ over the standard quantum limit and that one expects a $\sqrt{1+2\langle n\rangle}$ enhancement in signal-to-noise ratio [2606.05083].

## 5. Bath-free squeezed phonon lasing in trapped ions

The trapped-ion realization uses two effective two-level ions coupled to a single common vibrational mode of frequency $\omega_m$, with each ion driven simultaneously on its red- and blue-sidebands. Under the Lamb–Dicke and rotating-wave approximations, the interaction Hamiltonian is
\[
H=\sum_{i=1}^2\Bigl[
g_{i,r}\,a\,\sigma_i^+ + g_{i,r}^*\,a^\dagger\,\sigma_i^-
+ g_{i,b}\,a^\dagger\,\sigma_i^+ + g_{i,b}^*\,a\,\sigma_i^-
\Bigr].
\]
Each ion undergoes spontaneous decay at rate $\gamma_i$ into an auxiliary level, and in the regime $\gamma_i\gg |g_{i,r}|,|g_{i,b}|$ the internal states can be adiabatically eliminated, yielding the effective phonon-mode master equation quoted above [2601.05575].

The squeezing is encoded through the unitary operator
\[
S(r,\theta)=\exp\!\Bigl[\frac{r}{2}\bigl(e^{-i\theta}a^2-e^{i\theta}a^{\dagger 2}\bigr)\Bigr].
\]
If the ratio of red and blue couplings satisfies
\[
\tanh(r)=\left|\frac{g_{1,r}}{g_{1,b}}\right|
=\left|\frac{g_{2,b}}{g_{2,r}}\right|<1,
\qquad
\theta=(\phi_{1,r}-\phi_{1,b})=(\phi_{2,r}-\phi_{2,b}),
\]
then with $b\equiv S a S^\dagger$ the Hamiltonian takes the standard phonon-laser form in the squeezed frame,
\[
H_{\rm sq}=g_1\,b^\dagger\,\sigma_1^+ + g_2\,b\,\sigma_2^+ + {\rm h.c.}
\]
The source explicitly characterizes this as a “bath-free” approach, because it uses only coherent sideband drives and standard optical pumping, embeds all squeezing into a unitary basis change, and avoids the technical overhead and extra noise of engineered squeezed reservoirs [2601.05575].

Well above threshold, the phonon mode reaches a displaced squeezed coherent state. Denoting by $\alpha_s\equiv \langle b\rangle$ the coherent amplitude, the laboratory-mode average phonon number is
\[
\langle n\rangle=\langle a^\dagger a\rangle
=|\alpha_s|^2\cosh(2r)+\sinh^2(r).
\]
The second-order correlation is given analytically by
\[
g^{(2)}(0)=\frac{\langle a^\dagger a^\dagger a a\rangle}{\langle a^\dagger a\rangle^2},
\]
with the full expression reported in the source, and the quadrature variances for
\[
X_\theta\equiv \tfrac12(ae^{-i\theta}+a^\dagger e^{i\theta})
\]
satisfy
\[
\Delta X_\theta^2=
\frac14\Bigl[e^{-2r}\cos^2(\theta-\varphi)+e^{2r}\sin^2(\theta-\varphi)\Bigr].
\]
The minimum variance is therefore $e^{-2r}/4<1/4$ [2601.05575].

Above threshold the lasing mode has a free, uniformly random phase, and the proposal introduces a weak resonant force
\[
H_d=\varepsilon(ae^{-i\phi_d}+a^\dagger e^{i\phi_d}),
\qquad \varepsilon\ll g_{1,2},
\]
to lock it. Without the drive, the phase-diffusion rate is
\[
D_\phi^{(0)}\approx \frac{B-A}{4\langle n\rangle}
=\frac{\Gamma}{4\langle n\rangle},
\]
where $\Gamma=B-A$ is the small decay above threshold. With the drive, the effective phase diffusion is suppressed as
\[
D_\phi^{(\rm locked)}\sim D_\phi^{(0)}\times e^{-2r}\Bigl(\frac{\varepsilon}{\Gamma}\Bigr)^2
\ll D_\phi^{(0)}.
\]
The locked quadrature $X_{\phi_d+\pi/2}$ is squeezed by roughly the same factor $e^{-2r}$ [2601.05575].

The proposal gives realistic mixed-species ion-chain parameters: trap axial frequency $\omega_m/2\pi\approx 1$–$5\ {\rm MHz}$, Lamb–Dicke parameter $\eta\sim 0.05$–$0.1$, sideband Rabi rates $|g_{i,r}|,|g_{i,b}|\sim 5$–$50\ {\rm kHz}$, and optical pumping rates $\gamma_i/2\pi\sim 50$–$300\ {\rm kHz}$. The example above threshold yields $A/2\pi\simeq 5\ {\rm kHz}$, $B/2\pi\simeq 4\ {\rm kHz}$, squeezing parameter $r\approx 0.85$, expected $\langle n\rangle\sim 10$–$20$, $g^{(2)}(0)\sim 1.2$–$1.5$, and quadrature-noise reduction to $\Delta X^2\simeq e^{-2r}/4\approx 0.06$ [2601.05575].

## 6. Directionality, phase locking, and application domains

The nonreciprocal variant is implemented with two evanescently coupled whispering-gallery-mode microresonators and two tapered waveguides. One resonator is a standard optomechanical resonator supporting an optical mode and a mechanical breathing mode coupled by radiation pressure. The other is a purely optical resonator made of a $\chi^{(2)}$ medium and pumped by a strong laser at $\omega_p\approx 2\omega_a$, phase-matched so that only the counter-clockwise optical mode is squeezed. In the forward case, the optomechanical resonator couples to the squeezed mode; in the backward case, it couples to the unsqueezed mode. This asymmetry produces direction-dependent mechanical gain and hence nonreciprocal phonon lasing [2312.06896].

The squeezed frame is introduced by the Bogoliubov transformation
\[
a_{s,\Leftarrow}=\cosh r\, a_{2,\Leftarrow}+e^{-i\theta}\sinh r\, a_{2,\Leftarrow}^\dagger,
\]
with
\[
r=\frac14\ln\!\left[\frac{\Delta_l-\Lambda}{\Delta_l+\Lambda}\right],
\qquad (|\Delta_l|>\Lambda \text{ for stability}),
\]
leading, under the rotating-wave approximation, to an effective forward Hamiltonian with renormalized parameters
\[
\Delta_s=(\Delta_l+\Lambda)e^{2r},
\qquad
J_s=J_0\cosh r.
\]
By varying the pump power $P_p$, which sets
\[
\Lambda=\sqrt{\frac{8g_d^2\kappa_2 P_p}{\hbar\omega_p\kappa_p^2}},
\]
one tunes $r$, $\Delta_s$, and $J_s$, so the forward threshold can be shifted over orders of magnitude while the backward threshold remains unchanged. The isolation parameter is
\[
\mathcal R=10\log_{10}\!\left[\frac{N_b(\Lambda\neq 0)}{N_b(\Lambda=0)}\right].
\]
The reported applications are nonreciprocal acoustic amplification, one-way phononic network elements, and back-scattering-immune force or sound sensing [2312.06896].

Across platforms, phase locking appears in more than one form. In the Floquet hBN proposal, explicit breaking of the $U(1)$ phase symmetry produces phase-locked squeezed lasing without any external injection tone [2606.05083]. In the trapped-ion proposal, phase coherence is stabilized by an external coherent drive that suppresses phase diffusion while preserving the squeezed quadrature structure [2601.05575]. In the topological nanoelectromechanical platform, phase-sensitive amplification arises directly from the $(\hat a^{\dagger 2}+\hat a^2)$ interaction under $2\omega_0$ pumping [2107.11162].

A common misconception is that squeezed phonon lasing is synonymous with a single parametric-cavity architecture. The available results instead span topological Dirac-vortex cavities, Floquet-controlled solid-state defects, trapped ions driven on red- and blue-sidebands, and compound optomechanical systems with directional optical squeezing. Another common misconception is that squeezing and lasing are necessarily competing regimes. The reported models and measurements treat them jointly: one quadrature can be squeezed while the mode still exhibits threshold behavior, linewidth narrowing, and coherent emission [2107.11162] [2606.05083].

The application space reported in these works includes large-area single-mode phonon and photon lasers with minimal sensitivity to fabrication imperfections, testbeds for nonlinear and non-Hermitian topological physics, topologically protected phononic logic circuits, sub-standard-quantum-limit sensing in solid-state devices, quantum metrology, phononic information processing, directional acoustic amplifiers, and one-way phononic networks [2107.11162] [2606.05083] [2312.06896]. A plausible implication is that squeezed phonon lasing has become less a single device concept than a family of gain-and-squeezing protocols whose unifying theme is the controlled coexistence of coherent phonon generation and quadrature engineering.

Source: https://www.emergentmind.com/topics/squeezed-phonon-laser