Probing Lambda-Gravity with Bose-Einstein Condensate

TL;DR

This work proposes a tabletop test of Lambda-gravity using a trapped Bose-Einstein condensate (BEC) whose phonon dynamics are harnessed by a three-mode frequency interferometer with a tritter to sense both the Newtonian term $- \frac{GMm}{r^2}$ and the cosmological-constant term $\frac{\Lambda r m c^2}{3}$ in the general potential $F$. By embedding the gravitational signal in a Gaussian-state quantum metrology framework, the authors derive a quantum Fisher information that scales with two-mode squeezing and the tritter interaction, enabling sensitivity to accelerations as small as $\Delta a^{\mathrm{BEC}} \sim 10^{-18}$ m s$^{-2}$. For a realistic $^{87}$Rb BEC, they predict $\Delta G/G \sim 10^{-6}$ and an Earth-based bound on $\Lambda$ near $10^{-31}$ m$^{-2}$, along with characterization of the distance dependence of the two terms. If realized, this scheme would mark a significant advancement in laboratory tests of gravity, potentially constraining modified-gravity theories and offering new experimental access to fundamental constants.

Abstract

We propose a precise test of two fundamental gravitational constants using a detector concept that exploits the dynamics of quantum phononic excitations in a trapped Bose-Einstein condensate (BEC), operable at the scale of tabletop experiments. In this setup, the sensitivity is enhanced by approximately 2 orders of magnitude through the use of a tritter operation, which mixes phononic excitations with the BEC's ground state. The BEC exhibits unique sensitivity to the two key components of the gravitational potential in $Λ$-gravity: the Newtonian $GM/r$ term and the cosmological constant $Λr^2$, both entering the most general function following from a Gurzadyan's theorem. Using state-of-the-art experimental design, we predict that the gravitational constant $G$ could be measured with an accuracy up to $10^{-17}$ N m$^2$/kg$^2$, representing an improvement by 2 orders of magnitude over current measurements. Moreover, this experiment aims to establish the best Earth-based upper limit on $Λ$ at $<10^{-31}$ m$^{-2}$, marking the first laboratory-based probe of the cosmological constant. Additionally, the setup allows for the measurement of the distance-dependent behavior of each term in the gravitational potential, providing a means to test modified gravity theories.

Figures (4)

• Figure 1: Experimental setup: an oscillating sphere of mass $M$, frequency $\Omega$, and amplitude $\delta_R$ (right) at distance $R_0 \gg \delta_R$ from a BEC of length $L$ (left).
• Figure 2: Probe state preparation. Two phonon modes (green and blue), initially in the vacuum state, undergo squeezing $\hat{U}_{\textrm{sq}}(r)$ and are then mixed with the BEC ground state (red) via the tritter $\hat{U}_{\textrm{tr}}(\theta)$. The gravitational parameter is encoded by $\hat{U}_{g}(a^\mathrm{BEC})$. The inverse transformations are then applied, followed by phonon number measurement.
• Figure 3: Sensitivity $\Delta a^{\mathrm{BEC}}$ as a function of the number of initial squeezed phonons $N_p$. The plot shows $\Delta a^{\mathrm{BEC}}$ calculated using Eq. \ref{['eq:delta-a']} for varying BEC lengths, with $N_p$ ranging from 1 to 1100. Increasing $N_p$ corresponds to enhanced sensitivity to $a^{\mathrm{BEC}}$.
• Figure 4: Top panel: Simulated measurements of $a^{\mathrm{BEC}}$ (blue circles) as a function of $R_0$, with the fit (red curve) used to determine $G$ and constrain $\Lambda$ assuming sensitivity $\Delta a^{\mathrm{BEC}} = 4.8 \times 10^{-18}$ m/s$^{2}$. Bottom panel: Relative deviation $(a^{\mathrm{exp}} - a^{\mathrm{th}})/a^{\mathrm{th}}$ (blue circles) and expected relative accuracy $\Delta a^{\mathrm{BEC}} / a^{\mathrm{th}}$ (blue dashed line).