Testing the weak equivalence principle for nonclassical matter with torsion balances

TL;DR

This work extends the weak equivalence principle (WEP) into the quantum regime by promoting inertial and gravitational masses to operators and analyzing their impact on free-fall and torsion-balance dynamics. By deriving the approximate free-fall acceleration operator $\hat{a}$ and expressing its mean and variance in terms of internal-state coherence, the authors identify the variance as a robust quantum signature of WEP violations and show how torque measurements in a quantum torsion balance can reveal coherence effects. They develop two experimental schemas: an Eötvös-type torsion-balance test with quantum test masses and a dynamical Cavendish setup with time-dependent gravitational fields, both linking observable moments (mean and variance) to the mass-operator parameters $r_1,r_2,r$, and internal coherence described by Bloch-vector components. Feasibility analyses indicate current tabletop sensitivity could bound $|r|$ to $\mathcal{O}(10^{-3})$–$\mathcal{O}(10^{-4})$, with significant gains possible via cryogenic operation and advanced quantum-control techniques, offering a complementary approach to atom-interferometry in probing quantum aspects of gravity and the WEP.

Abstract

We propose tests of the weak equivalence principle (WEP) using a torsion balance, in which superposition of energy eigenstates are created in a controllable way for the test masses. After general considerations on the significance of tests of the WEP using quantum states and the need for considering inertial and gravitational masses as operators, we develop a model to derive the matrix elements of the free-fall operator, showing that the variance of the acceleration operator, in addition to its mean, enables estimation of violations of the WEP due to quantum coherence in a way that is robust with respect to shot-to-shot fluctuations. Building on this analysis, we demonstrate how the validity of the WEP may be tested in a torsion balance setup, by accessing the mean and variance of a torque operator we introduce and quantize. Due to the long acquisition times of the signal as compared to the timescale on which coherent superposition states may survive, we further propose a dynamical setting, where the torsion balance is subject to a time-dependent gravitational field, and measurements of angular acceleration encode possible violations of the WEP.

Figures (6)

• Figure 1: (a) Torsion balance in which each arm comprises a collection of two-level quantum systems (e.g., a solid-state high-density NV ensemble) whose state, $\hat{\rho}_A$ and $\hat{\rho}_B$, can be coherently controlled and placed into a superposition of internal energy states. ${\bf F}_A$ and ${\bf F}_B$ are the net forces acting on each arm at distances ${\bf r}_A$ and ${\bf r}_B$ from the center of the balance and ${\bf T}$ is the tension in the fiber. (b) Depicted are the forces acting on one arm of the balance, labeled by $j \in \{A,B\}$, in the lab-centered, noninertial frame. ${\bf F}_{C,j}$ is the centrifugal force, which is proportional to the inertial mass $M_{i,j}$, while ${\bf F}_{\mathSun,j}$ and ${\bf F}_{\mathTerra,j}$ are the gravitational force of the Sun and Earth, respectively, which are proportional to the gravitational mass $M_{g,j}$; the $z$ axis is chosen to align with the local gravitational field of the Earth. Not taken into account are the gravitational force of the Moon, the centrifugal force due to the orbital motion of the Earth around the Sun, and perturbations induced by other Solar System objects.
• Figure 2: Relative geometry between the Earth and Sun when the magnitude of ${\bf d}(t)$ is smallest and Earth is at the winter solstice in the northern hemisphere. An Earth-centered coordinate system is used in which the $z$ axis is orthogonal to the ecliptic plane and the Sun lies along the positive $x$ axis at the winter solstice in the northern hemisphere. The latitude of the torsion balance is $\lambda$ and $\varepsilon \approx 23.4^\circ$ is the axis tilt of the Earth relative to the ecliptic plane.
• Figure 3: Schematics for the proposed dynamical Cavendish experiment. As in the static case, the internal states of the balance's arms are described by density operators $\hat{\rho}_A$ and $\hat{\rho}_B$; in addition, two identical sources with mass $m_s$ generate time-dependent gravitational fields ${\bf g}_A(t)$ and ${\bf g}_B(t)$ [Eqs. \ref{['gA']}--\ref{['gB']}] by rotating with a constant angular frequency $\Omega$. Here, $\Theta$ represents the average angular displacement of the balance referred to the $x$ axis, and by construction we have ${\bf R}_t = (R_t \cos \Theta, R_t \sin \Theta, 0)$, ${\bf R}_s= (R_s \cos \Omega t, R_s \sin \Omega t, 0)$. The red wavy lines represent the laser pulses required to prolong the coherence time of the two test masses, and possibly dynamically freeze them in their target initial state.
• Figure 4: Bounds on the off-diagonal matrix element $r$ in the regime where thermal fluctuations are small compared to the quantum variance of the acceleration operator, evaluated for $N = 10^5$ and $\sqrt{\Delta \alpha_{\mathrm{cl}}^{2}} /\alpha_\mathrm{cl}(t) = 10^{-5}$, close to the constraint set by the current relative precision in the measurement of $G_N$, with the dashed lines representing contours of constant $\mathrm{qSNR}$ as defined in Eq. \ref{['SNRQ']}. (Left) Dependence of $|r|$ upon the magnitude of the Bloch vector $n$ for different values of the $\mathrm{qSNR}$, in the optimized case of $\theta = \pi/2$ and $\varphi_r + \phi = 0$. Notice that the $\mathrm{qSNR}$ is a monotonically increasing function of both $n$ and $|r|$. For a given $n$, if a signal is not detected for a given $\mathrm{qSNR}$, all values larger than the corresponding $|r|$ in the vertical axis are ruled out. (Right) Contour plot of the $\mathrm{qSNR}$ in the $\varphi_r-|r|$ plane for the choice of $n=0.9$, $\theta = \pi/2$, and $\phi=0$.
• Figure 5: (Left) Distribution of values of $\cos \gamma$ with $10^5$ realization of uniformly distributed real numbers for $\gamma\in [-\pi, +\pi]$. (Right) Average of $\cos \gamma$ and its standard deviation versus the number of realizations, showing that its average value is compatible with zero within two standard deviations already for $10^3$ realization, and quickly converges to zero, reaching $\langle \cos \gamma \rangle =(2.3 \pm 22.4) \times 10^{-4}$ for $10^5$ realizations.