Mechanical Squeezed-Fock Qubit Gravimeter

• The paper demonstrates that engineering squeezed Fock states via dissipative stabilization and nonlinear drives forms mechanical qubits that transduce gravity through displacement, qubit rotation, or frequency shifts.
• It compares multiple architectures—from cavity optomechanics to levitated Duffing oscillators—each exploiting distinct squeezing regimes to improve sensitivity and maintain favorable mass scaling.
• The study emphasizes that while squeezing can enhance gravitational sensitivity, its benefits depend critically on proper alignment and control of noise, as excessive or misaligned squeezing may degrade performance.

A mechanical squeezed-Fock qubit gravimeter is a gravimetric architecture in which a mechanical mode is prepared in squeezed Fock states, squeezed-Fock-like states, or a squeezed-Fock two-level subspace, and gravity is inferred from the resulting displacement, qubit rotation, or qubit-frequency shift. In the current literature, this concept appears in several complementary forms: dissipatively stabilized displaced mechanical Fock states in cavity optomechanics, parametrically driven Kerr or Duffing oscillators whose eigenstates are squeezed Fock states, and levitated mechanical qubits in which gravity couples directly to the center-of-mass motion and can be aligned with an anti-squeezed quadrature (Brunelli et al., 2018, Qiao et al., 17 Jul 2025, Yousefjani et al., 27 May 2026).

1. Conceptual framework

The defining state family is the squeezed Fock ladder

$$|n\rangle_S \equiv \hat S(r)|n\rangle,\qquad \hat S(r)=\exp\!\Big[\frac{r}{2}(\hat a^2-\hat a^{\dagger 2})\Big],$$

or, in phase-sensitive form, $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$. In parametrically driven nonlinear oscillators, these states arise naturally after a Bogoliubov transformation $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$, and the lowest two states $\{|0\rangle_S,|1\rangle_S\}$ define the mechanical squeezed-Fock qubit (Qiao et al., 17 Jul 2025).

A distinct but closely related construction is dissipative stabilization of displaced finite Fock superpositions. In the optomechanical reservoir-engineering protocol based on mixed linear and quadratic couplings, the mechanical steady state is a displaced finite superposition of the first $n+1$ Fock states,

$$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$

which approaches the displaced Fock state $D(-\sqrt{n+1/2})|n\rangle$ in the large-squeezing limit (Brunelli et al., 2018). This establishes a non-Gaussian steady-state route to a mechanical qubit resource without requiring coherent number-state preparation.

In gravimetry, the crucial interaction is the force term

$H_g=-mgx,$

or, for a harmonically bound oscillator, the equivalent static displacement of the equilibrium position. In levitated implementations this is a direct center-of-mass coupling, and the force-to-quadrature coupling scales as $mgx_{\mathrm{zpf}}\propto g\sqrt m$. The literature emphasizes this as the key advantage over hybrid schemes in which coupling to an auxiliary nonlinear element scales as $1/\sqrt m$ and cancels the mass benefit (Yousefjani et al., 27 May 2026).

The phrase “qubit gravimeter” does not denote a single measurement modality. Depending on the architecture, gravity may be read out as a transverse qubit drive, a longitudinal qubit-frequency shift, a displacement estimated through quantum Fisher information, or a change in phonon-number statistics. This diversity is a central feature of the field rather than a disagreement in terminology.

2. State synthesis and qubit formation

One route to mechanical squeezed-Fock resources is dissipative optomechanical stabilization. The cavity-optomechanical proposal with three control lasers uses drives at $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$0 and $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$1 together with linear and quadratic dispersive couplings,

$$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$2

with squeezing parameter $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$3. After adiabatic elimination of the cavity, the effective mechanical master equation is

$$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$4

and the unique pure steady state is the null space of the engineered jump operator $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$5 when intrinsic damping is neglected. For the stable finite-squeezing protocol, the resonant coupling must satisfy

$$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$6

with $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$7 for stability (Brunelli et al., 2018).

A second route uses a parametrically driven nonlinear oscillator. In the Kerr formulation,

$$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$8

with $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$9. In the squeezed basis, the effective Hamiltonian is diagonal in $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$0-number states,

$$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$1

and the anharmonicity

$$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$2

is exponentially enhanced with $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$3 (Qiao et al., 17 Jul 2025). The Duffing version used for levitated particles has the same structural consequence: the squeezed-Fock ladder becomes spectrally isolated, and the lowest transition forms a controllable qubit provided the drive amplitudes and dissipative broadenings are much smaller than $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$4 (Yousefjani et al., 27 May 2026).

Experimental control over squeezed-Fock bases has been demonstrated outside gravimetry. In trapped ions, engineered Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonians in a squeezed Fock basis reproduced the $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$5 scaling of the matrix elements, created states up to excitations of $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$6, and reached $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$7 of squeezing (Kienzler et al., 2016). In a superconducting qubit–mechanical system, a 5.023 GHz HBAR mode was turned into a squeezed Kerr oscillator with $$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$8 of noise reduction below the ground-state variance, tunable Kerr nonlinearity, Wigner-function negativities, and high quantum Fisher information (Marti et al., 2023). These demonstrations do not by themselves realize a gravimeter, but they establish the state-engineering primitives that a gravimeter would require.

3. Gravity transduction mechanisms

The most elementary gravimetric channel is static force to static displacement. For a harmonically bound oscillator, a constant acceleration gives

$$\hat b=\cosh r\,\hat a+e^{i\theta}\sinh r\,\hat a^\dagger$$9

so gravity enters as a phase-space displacement of the mechanical state (Brunelli et al., 2018). This viewpoint underlies continuous-variable gravimetry with squeezed or squeezed-Fock-like probes.

In the levitated squeezed-Fock qubit proposal, the transduction is instead a transverse qubit drive. Choosing $\{|0\rangle_S,|1\rangle_S\}$0 aligns gravity with the anti-squeezed $\{|0\rangle_S,|1\rangle_S\}$1-quadrature, so that

$\{|0\rangle_S,|1\rangle_S\}$2

and, in the qubit subspace,

$\{|0\rangle_S,|1\rangle_S\}$3

Starting in $\{|0\rangle_S,|1\rangle_S\}$4, the excited-state probability is

$\{|0\rangle_S,|1\rangle_S\}$5

This architecture uses the anti-squeezed quadrature to enhance the gravity-induced transition rate while preserving the direct mass scaling of the mechanical force coupling (Yousefjani et al., 27 May 2026).

A related but not identical direct-mechanical-qubit framework uses the lowest two Fock states of a Duffing oscillator as the sensor. There, projection into the qubit subspace gives

$\{|0\rangle_S,|1\rangle_S\}$6

while the squeezed-Fock extension yields

$\{|0\rangle_S,|1\rangle_S\}$7

The explicit consequence is that $\{|0\rangle_S,|1\rangle_S\}$8-squeezing reduces the gravity–qubit coupling and worsens the Rabi-based sensitivity by $\{|0\rangle_S,|1\rangle_S\}$9, whereas anti-squeezing the $n+1$0-quadrature improves the sensitivity by the factor $n+1$1 (Huo et al., 16 Apr 2026). This point is often obscured when “squeezing enhancement” is stated without specifying the coupling quadrature.

A third transduction channel is quadratic rather than linear. In the squeezed-Fock qubit model based on a Kerr oscillator, direct linear-force coupling in the squeezed basis gives a transverse drive amplitude

$n+1$2

so the linear-force responsivity decreases with increasing $n+1$3. The same work therefore emphasizes a spring-constant signal,

$n+1$4

for which the sensitivity improves with squeezing. Its optimized Ramsey sensitivity is

$n+1$5

and the paper states directly that, for gravimetry, one wants to transduce gravity into an effective spring-constant modulation rather than a static linear force on $n+1$6 (Qiao et al., 17 Jul 2025).

The general metrological analysis of squeezed probes makes the same caution from another direction. Canonical phase-space squeezing can fail to achieve a quantum Fisher information surpassing the shot-noise limit regardless of the interaction time, whereas position-momentum correlated input states can overcome this limit, with optimal sensitivity attained through projective momentum measurements combined with a time-dependent adjustment of the squeezing phase (Araujo et al., 15 Oct 2025).

4. Sensitivity, Fisher information, and scaling laws

For displacement sensing, the central metrological quantity is the variance of the generator conjugate to the displacement. In the dissipative optomechanical setting, a small displacement along $n+1$7 generated by $n+1$8 has

$n+1$9

for a pure probe. For squeezed number states,

$$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$0

so the sensitivity to a static acceleration is enhanced exponentially with $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$1 because squeezing amplifies the conjugate momentum fluctuations that determine the quantum Fisher information for $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$2-displacement estimation (Brunelli et al., 2018).

The squeezed-Kerr proposal casts this enhancement in qubit language. In the weak-force regime, the quantum Fisher information is

$$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$3

with $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$4, and the first optimum interrogation time is $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$5. The time-normalized quantum Cramér–Rao bound at that point is

$$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$6

The explicit scalings are $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$7, $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$8, and $$|\phi_n\rangle=\mathcal N_n D(\xi_n/\sqrt2)\sum_{k=0}^n {n\choose k} c_n^{-k}|k\rangle,$$9 (Yousefjani et al., 27 May 2026).

For direct mechanical qubits without auxiliary systems, the baseline Fock-qubit sensitivity is

$D(-\sqrt{n+1/2})|n\rangle$0

and the mechanical cat-qubit sensitivity is

$D(-\sqrt{n+1/2})|n\rangle$1

The corresponding QFI expressions are $D(-\sqrt{n+1/2})|n\rangle$2 and $D(-\sqrt{n+1/2})|n\rangle$3 at the optimal times (Huo et al., 16 Apr 2026). These results make explicit that a direct mechanical sensor can reach what that work calls the mass-limited and resource-limited standard quantum limits simultaneously.

For general squeezed probes in a uniform gravitational field, the exact QFI depends on both the squeezing amplitude and the squeezing phase. The reported asymptotics are especially important: for $D(-\sqrt{n+1/2})|n\rangle$4, $D(-\sqrt{n+1/2})|n\rangle$5 at short time and $D(-\sqrt{n+1/2})|n\rangle$6 at long time; for $D(-\sqrt{n+1/2})|n\rangle$7, the behavior is reversed; for $D(-\sqrt{n+1/2})|n\rangle$8, $D(-\sqrt{n+1/2})|n\rangle$9 at short time and $H_g=-mgx,$0 at long time (Araujo et al., 15 Oct 2025). A common misconception is therefore that “more squeezing” is automatically synonymous with better gravimetry. The published analyses show instead that the gain is coupling-dependent, phase-dependent, and measurement-dependent.

The same resource scaling appears in state-engineering experiments not yet configured as gravimeters. For displacement estimation, the HBAR squeezed-Kerr platform reports

$H_g=-mgx,$1

for squeezed-Fock states $H_g=-mgx,$2 (Marti et al., 2023). This suggests that, once a suitable gravity-to-displacement transduction or up-conversion protocol is supplied, squeezed-Fock encoding can combine the $H_g=-mgx,$3 number-state factor with the $H_g=-mgx,$4 squeezing factor.

5. Platforms, demonstrations, and implementation pathways

Current work spans proposals, partial demonstrations, and platform studies rather than a single finished device. The main experimental and near-experimental directions can be organized as follows.

Platform or route	Demonstrated or proposed resource	Gravimetric relevance
Three-tone cavity optomechanics	Dissipative stabilization of displaced finite Fock superpositions approaching displaced Fock states	Non-Gaussian steady-state probe; direct force-to-displacement sensing
Superconducting circuit optomechanics	Ground-state cooling to $H_g=-mgx,$5, mechanical squeezing of $H_g=-mgx,$6, millisecond-scale nonclassical evolution	Long-coherence continuous-variable platform with explicit gravimetry estimates
Squeezed Kerr mechanical oscillator	Two-phonon squeezing, tunable Kerr, Wigner negativity, high QFI	Hardware route to squeezed-Fock qubits and non-Gaussian sensing states
Quadratic CPB–mechanics coupling	Energy-squeezed state with $H_g=-mgx,$7 and $H_g=-mgx,$8	Number-squeezed, phase-insensitive sensing and per-phonon qubit spectroscopy
Levitated Duffing oscillator	Proposed squeezed-Fock qubit gravimeter with direct CM coupling	Preserves $H_g=-mgx,$9 gravimetric scaling and anti-squeezed signal amplification
SQUID–transmon mechanical gravimeter	Proposed geometric-phase gravimeter; no squeezed or Fock states realized in that work	SI-traceable readout architecture that can be extended with squeezing and sideband state preparation

The superconducting circuit optomechanical platform with ultra-low decoherence is especially relevant as an enabling system. It reports $mgx_{\mathrm{zpf}}\propto g\sqrt m$0, $mgx_{\mathrm{zpf}}\propto g\sqrt m$1, $mgx_{\mathrm{zpf}}\propto g\sqrt m$2, $mgx_{\mathrm{zpf}}\propto g\sqrt m$3, ground-state mean phonon number $mgx_{\mathrm{zpf}}\propto g\sqrt m$4, and squeezing of $mgx_{\mathrm{zpf}}\propto g\sqrt m$5 below zero-point fluctuation. Using the reported parameters, the same work gives $mgx_{\mathrm{zpf}}\propto g\sqrt m$6 for a $mgx_{\mathrm{zpf}}\propto g\sqrt m$7 shot and $mgx_{\mathrm{zpf}}\propto g\sqrt m$8 at $mgx_{\mathrm{zpf}}\propto g\sqrt m$9 integration when squeezing and finite efficiency are included (Youssefi et al., 2022). That study did not prepare Fock states, but it explicitly argues that the measured ultra-low decoherence and kHz-scale linearized coupling enable sideband-resolved protocols for $1/\sqrt m$0, $1/\sqrt m$1, and squeezed-Fock superpositions.

The quadratic-coupling CPB experiment demonstrates a different sensing resource: nonclassical energy squeezing. It stabilizes a state with $1/\sqrt m$2 and Fano factor $1/\sqrt m$3, giving sub-Poissonian number fluctuations of approximately $1/\sqrt m$4 phonons, and it resolves vibronic sidebands arising from qubit-state-dependent mechanical recoil (Ma et al., 2020). Because the readout measures energy rather than position, this route is naturally phase-insensitive and well matched to acceleration signals that appear as changes in oscillation radius rather than phase.

The chip-scale SQUID–transmon gravimeter provides a high-bandwidth, SI-traceable readout chain but does not itself realize squeezed or Fock mechanical states. Its longitudinal Hamiltonian,

$1/\sqrt m$5

leads to revival-time sensitivity

$1/\sqrt m$6

and the paper projects sensitivities down to the $1/\sqrt m$7 range with kilohertz-rate sampling (Wani et al., 1 Jan 2026). It also states explicitly that the same architecture could be extended to mechanical squeezing via $1/\sqrt m$8 flux modulation and to mechanical Fock states via sideband couplings.

6. Constraints, misconceptions, and outlook

The first recurring misconception is that squeezing always improves gravimetry. The published record shows a more conditional picture. In direct displacement sensing, $1/\sqrt m$9-squeezed number states enhance the QFI because $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$00 and $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$01 (Brunelli et al., 2018). In the levitated squeezed-Fock qubit gravimeter, aligning gravity with the anti-squeezed quadrature gives $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$02 and improves sensitivity (Yousefjani et al., 27 May 2026). In the direct squeezed-Fock extension of the mechanical qubit, however, $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$03-squeezing gives $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$04 and worsens Rabi-based sensitivity, while anti-squeezing improves it (Huo et al., 16 Apr 2026). In the spring-constant-sensing proposal, squeezing improves $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$05 but degrades the direct linear-force matrix element (Qiao et al., 17 Jul 2025). In general free-fall metrology, canonical squeezing can fail to beat the vacuum shot-noise reference for all times, whereas position-momentum correlated squeezing with momentum readout can (Araujo et al., 15 Oct 2025).

The second misconception is that large squeezing can be increased without penalty. In fact, each architecture carries a stability or decoherence threshold. The dissipative optomechanical protocol requires $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$06, and the exact displaced Fock-state limit $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$07 is explicitly unstable (Brunelli et al., 2018). Parametric Duffing gravimeters require $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$08, $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$09, and rotating-wave constraints on the discarded quartic terms (Yousefjani et al., 27 May 2026). The same levitated analysis shows that squeezing converts ordinary damping into anisotropic qubit noise,

$$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$10

so practical operation is limited by the competition ratio $$\hat S(r,\theta)=\exp[(r/2)(e^{-i2\theta}a^2-e^{i2\theta}a^{\dagger2})]$$11 (Yousefjani et al., 27 May 2026).

The third misconception is that a full mechanical squeezed-Fock qubit gravimeter has already been experimentally realized. What has been realized are key ingredients: dissipative stabilization of Fock-like steady states in theory, milli-second mechanical coherence and squeezing in superconducting optomechanics, squeezed-basis control in trapped ions, Wigner negativity and Kerr-engineered non-Gaussianity in superconducting phononics, and number-squeezed phase-insensitive states under quadratic coupling (Brunelli et al., 2018, Youssefi et al., 2022, Kienzler et al., 2016, Marti et al., 2023, Ma et al., 2020). The fully integrated gravimeter remains an active synthesis problem.

A plausible implication is that the most direct implementation path is not unique. One route is to combine a long-lived superconducting mechanical platform with sideband preparation and qubit-assisted readout. Another is to use a levitated Duffing oscillator, where gravity acts directly on the center of mass and the mass resource is preserved. A third is to graft squeezed or Fock-state preparation onto the already SI-traceable SQUID–transmon gravimeter architecture. Across these routes, the literature identifies the same bottlenecks: thermal decoherence, pump phase and amplitude noise, residual cavity or qubit heating, finite readout efficiency, and the need to calibrate gravity-to-mechanical transduction with high stability (Youssefi et al., 2022, Wani et al., 1 Jan 2026, Yousefjani et al., 27 May 2026).

The topic therefore sits at the intersection of non-Gaussian state engineering, mechanical-qubit control, and quantum-limited force metrology. What distinguishes it from generic quantum gravimetry is not merely the use of squeezing, but the attempt to place squeezing, Fock-state structure, and qubit selectivity into the same mechanical degree of freedom.