Quantum Metrology of Newton's Constant with Levitated Mechanical Systems

Abstract

Newton's constant is the least well-measured among the fundamental constants of Nature, and, indeed, its accurate measurement has long served an experimental challenge. Levitated mechanical systems are attracting growing attention for their promising applications in sensing and as an experimental platform for exploring the intersection between quantum physics and gravitation. Here we propose a mechanical interferometric scheme of interacting levitated oscillators for the accurate estimation of Newton's constant. Our scheme promises to beat the current standard by several orders of magnitude.

Figures (5)

• Figure 1: Sketch of the set-up. Two objects of mass $m$ separated by a distance $d$ and interacting through gravity levitate in harmonic traps of frequency $\omega_0$, generated e.g. by trapping small ferromagnets in superconducting boxes. Here $x_{1,2}$ are the displacements from the respective equilibrium positions.
• Figure 2: QFI and CFI for different measurement schemes. (a) Projective measurements: the black curve is the QFI, calculated using the CM; the red curve represents the CFI for $s\to 0$ (projective position measurement) while the blue curve the CFI for $s \to \infty$ (projective momentum measurement). Finally, the green curve is the CFI for $s \to 1$ (projective heterodyne measurement). We also report in yellow the CFI for intensity measurements. The temperature is $T=0$ ($\bar{n}=0$). (b) Continuous measurements at finite temperature ($T=1$mK, i.e., $\bar{n}=1.31\times10^6$) and quality factor ($\mathcal{Q}=10^7$). The measurement rate is chosen as $\Gamma_m=1\times10^{-2}$ with perfect detector efficiency. Here we report the CFI for the different measurement strategies compared to the corresponding effective QFI [cf. Appendix for details]. The black dashed line represent the effective and conditional QFI of the conditional state of the system. The CFI of the position and momentum measurements are plotted overlapping in blue and red. In panels (a) and (b), the input state is chosen as the tensor product of two thermal coherent states with parameters $\omega_0=100\ \text{rad}\ \text{s}^{-1}$, $m=1\times10^{-4}$kg, $d=5\times10^{-3}$m, $\alpha_1\approx3.374\times10^{12}$, $\alpha_2=0$. (c) QFI as a function of the squeezing $s_1$ and thermal occupation number $\bar{n}$ for a displaced squeezed thermal state at $t=100$s. Other parameters as in panels (a), where the input amplitude is defined through the initial displacement as $\langle x_i(0)\rangle=\sqrt{2}\alpha_i$.
• Figure 3: QFI against time for different values of the ${\cal Q}$ factor (see the plot legend). All curves shown are for the input state $|\alpha_1,0\rangle_{a_1(\omega_0)}\otimes|0,0\rangle_{a_2(\omega_0)}$ with parameters $\omega_0=100\,\text{rad}\ \text{s}^{-1}$, $m=1\times10^{-4}$ kg, $d=5\times10^{-3}$ m, $\alpha_1\approx3.374\times10^{12}$, $\alpha_2=0$, $\bar{n}=0$.
• Figure 4: Comparison of the exact QFI (Black) and the approximation Eq.(10) in the main text, which utilizes the corotating frame for input state $|\alpha_1,0\rangle_{a_1(\omega_0)}\otimes|0,0\rangle_{a_2(\omega_0)}$ with parameters $\omega_0=100\,\text{rad}\ \text{s}^{-1}$, $m=1\times10^{-4}$ kg, $d=5\times10^{-3}$ m, $\alpha_1\approx3.374\times10^{12}$, $\alpha_2=0$, $\bar{n}=0$.
• Figure 5: Here we compare the left-hand side of eq. \ref{['effQFI']}, representing the effective QFI (black curves) with the second term on the right-hand side of eq. \ref{['effQFI']} --- representing the conditional QFI (dashed black lines) --- and the first term on the right-hand side of the same equation that is the CFI (colored curves). The three panels show these quantities for the three measurement strategies considered. The parameters are as in the main text with $\bar{n}=1.31\times 10^{6}$ ($T=1\,\text{mK}$) and $\Gamma_m=1\times10^{-2}$.