Optomechanical Tests

Optomechanical experiments couple a mechanical degree of freedom to an optical or microwave field. They can test quantum control of motion, search for unexpected force noise, and constrain models of spontaneous localization. Ground-state cooling, a spatial superposition, and an exclusion bound are distinct achievements. The device dynamics are developed in Optomechanics; this page concerns the inference from measured motion to a foundational claim.

Zero-point motion is a scale, not a cat state

For mechanical frequency $\Omega_m$ and effective mode mass $m_{\rm eff}$,

$$x_{\rm zpf}=\sqrt{\frac{\hbar}{2m_{\rm eff}\Omega_m}}, \qquad x=x_{\rm zpf}(b+b^\dagger).$$

Radiation pressure produces the interaction $H_{\rm int}=-\hbar g_0a^\dagger a(b+b^\dagger)$ in the elementary single-mode model. Cooling toward the oscillator ground state reduces thermal occupation. The ground state still has a positive Gaussian Wigner function and is not a superposition of two well-separated localized wave packets in the usual cat-state sense.

For $m_{\rm eff}=10^{-15}$ kg and $\Omega_m=2\pi\times10^5\ {\rm s}^{-1}$, $x_{\rm zpf}\simeq2.9\times10^{-13}$ m. This illustrates why a device with a visibly large physical size can have extremely small coherent displacement scales. The effective mass of one mode is also not automatically the total apparatus mass.

From a spectrum to an added-noise limit

A calibrated linear displacement measurement probes the mechanical susceptibility

$$\chi_m(\omega)= \frac{1}{m_{\rm eff}(\Omega_m^2-\omega^2-i\gamma_m\omega)}.$$

In a simplified model with uncorrelated noises,

$$S_x(\omega)=|\chi_m(\omega)|^2 [S_F^{\rm th}(\omega)+S_F^{\rm add}(\omega)] +S_x^{\rm imp}(\omega).$$

Here the spectra must share a declared one-sided or two-sided convention. In the high-temperature classical limit, the one-sided thermal force spectrum is $S_F^{\rm th}=4m_{\rm eff}\gamma_m k_BT$. Quantum backaction and its correlations with imprecision may invalidate the uncorrelated-noise simplification.

An unexplained excess can bound a candidate force-noise model only after calibration, thermalization, vibrational pickup, laser intensity noise, and feedback-induced correlations have been treated. Fitting an excess noise parameter is not itself evidence for collapse.

Relating diffusion to localization

For an illustrative momentum-diffusion term

$$\dot\rho=-\frac{D_{pp}}{\hbar^2}[x,[x,\rho]],$$

a position coherence separated by $d$ decays at rate $D_{pp}d^2/\hbar^2$. The same term increases $\langle p^2\rangle$ at rate $2D_{pp}$ and the oscillator energy at rate $D_{pp}/m_{\rm eff}$. This links a heating measurement to a model of coherence loss without requiring a prepared spatial cat.

A realistic collapse calculation integrates the resonator’s mass distribution and mode shape. It cannot in general replace that distribution by one point mass. Dated best bounds and changing feasibility assessments should be read as research evidence, rather than inferred from this stable response model.

Exercise

If an experiment bounds additional phonon heating by $\dot n<r$, what bound follows for $D_{pp}$ in the diffusion model?

Solution

Since $E=\hbar\Omega_m(n+1/2)$, one has $\dot n=D_{pp}/(m_{\rm eff}\hbar\Omega_m)$. Thus $D_{pp}<r\,m_{\rm eff}\hbar\Omega_m$. Translating this into collapse parameters requires the spatial coupling and geometry of the selected theory.

References

• Aspelmeyer, M., Kippenberg, T. J., and Marquardt, F., “Cavity optomechanics,” Reviews of Modern Physics 86, 1391–1452 (2014), doi:10.1103/RevModPhys.86.1391 — response, cooling, and readout.
• Bassi, A., et al., “Models of wave-function collapse, underlying theories, and experimental tests,” Reviews of Modern Physics 85, 471–527 (2013), doi:10.1103/RevModPhys.85.471 — diffusion and test strategies.