Testing weakest force with coldest spot

The GPE could be derived from the variation of the action PhysRevLett.77.5320 ; PhysRevA.56.1424 ; Gupta:2015cta

with respect to $\delta\psi$ and $\delta\psi^{*}$. For a normalized wave function ansatz,

the width parameter $\sigma(t)$ and phase parameters $\gamma(t)$ and $B(t)$ could be solved from the corresponding Euler-Lagrange equations of Lagrangian

where

will be analytically evaluated later with specific form of the gravitational potential.

The total time-derivative term $\hbar\dot{\gamma}(t)$ in (II.1) does not admit a dynamical equation, and the rest of Euler-Lagrange equations for $\sigma(t)$ and $B(t)$ read

respectively, which could be combined into a single equation of motion (EOM)

After adopting following dimensionless quantities,

the EOM (16) becomes

which effectively describes a particle moving in a dimensionless effective potential

To evaluate $L_{G}$ in (19), we first make replacements of variables as

so that $r_{1}^{2}+r_{2}^{2}=r^{2}+R^{2}$, and then $L_{G}$ becomes

After employing the following dimensionless quantities

with the Planck mass $m_{\mathrm{Pl}}\equiv\sqrt{\hbar c/G}$, one arrives at

where the integration $I(\nu)$ reads

Therefore, the EOM could be solved for given effective potential of form

If $U(\nu)$ admits a local minimum $\nu_{\mathrm{min}}$, the width of BEC $\sigma(t)=\nu(t)a_{0}$ would experience the shape oscillation with frequency $\omega=\sqrt{U^{\prime\prime}(\nu_{\mathrm{min}})}\omega_{0}$ around $\nu_{\mathrm{min}}$. An illustration for the effective potential is shown in Fig. 1 with magnetically manipulated scattering length $a=0$ (as we will assume later), and the cases without gravity (black solid line), with the Newtonian gravity only (blue solid line), and with extra Yukawa interaction (red dashed line), which will be presented in detail in the following sections in addition to the cases with the extra power-law potential and the fat-graviton potential.

To estimate the contributions of each term in the effective potential (25), one could adopt the typical value of $a_{0}\sim 10^{-4}\,\mathrm{cm}$ and $N\sim 10^{6}-10^{8}$, which is experimentally achievable currently, as well as $a_{0}^{G}$ taken to be

where the atomic mass number $A$ is usually around order of hundred, and the trapping frequency $\omega_{0}$ could be adjusted in the range of $50-10,000$ Hz. Therefore, for typical choice of scattering length $a\approx a_{0}\gg a_{0}^{G}$ due to the $m/m_{\mathrm{Pl}}$ suppression in the ratio $a_{0}^{G}/a_{0}\sim 10^{-24}-10^{-22}$, the second term would dominate over the third term in (25), namely, the $s$-wave scattering would totally erase the trace of gravitational potential. To manifest the gravitational effect, one could tune down the scattering length $a$ magnetically to zero Cornish:2000zz (Hereafter we will assume this idealistic condition for our preliminary studies) so that we can get rid of the second term in (25), and the effect of additional gravitational potential could be extracted from the frequency deviation of shape oscillation PhysRevA.47.4114 ; Inouye1998 ; PhysRevLett.81.69 ; PhysRevLett.81.5109 in the absence of gravity.

For an additional power-law potential (5) to the Newtonian gravity, the corresponding effective potential is obtained as

for $k<3$. For $k\geq 3$, the integral (24) is not well-defined. Similar to (31), the oscillation frequency could be found to be deviated from $2\omega_{0}$ by

where the local minimum $\nu_{\mathrm{min}}$ is determined by $U^{\prime}(\nu)=0$ as

For an additional Yukawa interaction (6) to the Newtonian gravity, the corresponding effective potential is obtained as

where $\beta\equiv a_{0}/d$ and $\mathrm{Erfc}(z)\equiv 1-\mathrm{Erf}(z)$ is the complementary error function. The oscillation frequency could be found to be deviated from $2\omega_{0}$ by

where the local minimum $\nu_{\mathrm{min}}$ is determined by $U^{\prime}(\nu)=0$ as

For an additional fat-graviton potential (8) to the Newtonian gravity, the corresponding effective potential is obtained as

where $\beta\equiv a_{0}/\lambda$ and $\,{}_{p}F_{q}(a_{1},\cdots,a_{p};b_{1},\cdots,b_{q};z)$ is the generalized hypergeometric function. The oscillation frequency could be found to be deviated from $2\omega_{0}$ by

where the local minimum $\nu_{\mathrm{min}}$ is determined by $U^{\prime}(\nu)=0$ as

For the power-law potential (5), we will take $k=2$ for illustration. The deviation of the oscillation frequency (43) with respect to the gravity-free oscillation frequency $2\omega_{0}$ is shown in the left panel of Fig. 2 for the particle numbers $N=10^{9}$ (red line) and $N=10^{12}$ (blue line), where the shaded regions $\beta_{2}>1.3\times 10^{-3}$ Hoskins:1985tn , $\beta_{2}>4.5\times 10^{-4}$ Adelberger:2006dh , and $\beta_{2}>3.7\times 10^{-4}$ Tan:2020vpf are excluded by the existing experiments. Unfortunately, despite of the smallness of frequency deviation, it is also insensitive to the size of coefficient $\beta_{k}$. Therefore, the ultra-cold BEC experiment in space cannot constrain the modified gravity with power-law gravitational potential.

The experimental perspective for the gravitational potential with Yukawa interaction (6) are presented in Fig. 3. The deviations of the oscillation frequency (III.2) with respect to the gravity-free oscillation frequency $2\omega_{0}$ are shown for the particle numbers $N=10^{9}$ (the second panel) and $N=10^{12}$ (the fourth panel) with respect to the current exclusion limits Hoyle:2004cw ; Kapner:2006si ; Hoskins:1985tn ; Decca:2005qz ; Chen:2014oda ; Chiaverini:2002cb ; Geraci:2008hb ; Long:2002wn ; Tu:2007zz ; Yang:2012zzb ; Tan:2016vwu ; Tan:2020vpf ; Lee:2020zjt . The frequency deviations are insensitive to both $d$ and $\alpha$ when $\alpha\lesssim\mathcal{O}(10)$, which are as small as $10^{-14}$ (the second panel) and $10^{-11}$ (the fourth panel), respectively. For $d\gtrsim 10^{-6}\,\mathrm{m}$, the frequency deviation grows with power-law in the range of $\alpha\gtrsim\mathcal{O}(10)$. For $d<10^{-6}\,\mathrm{m}$, the frequency deviation flips a sign around $\alpha\sim\mathcal{O}(10)$, after which the absolute value of frequency deviation still grows with power-law in $\alpha$. This could also be seen in the third panel for some illustrative values of $N$ and $d$, where the dashed lines denote for the negative value of the frequency deviation $\omega/\omega_{0}-2$. Note that the regime of flipped sign appears when $d$ is smaller than the characteristic length scale $a_{0}\simeq 10^{-6}\,\mathrm{m}$ of the BEC state. Except for the region of flipped sign, the frequency deviation would be insensitive to the interaction range $d$ for given $\alpha$, which could greatly push current exclusion limits.

To detect such a small frequency deviation, for example in the regime of $\alpha\lesssim\mathcal{O}(10)$ with $\omega/\omega_{0}-2\simeq 10^{-14}$ for $N=10^{9}$ and $\omega/\omega_{0}-2\simeq 10^{-11}$ for $N=10^{12}$ , we need the resolution of frequency in the measurement as

where the trapping frequency is usually in the range of $50-10,000$ Hz. The above requirement for the resolution of frequency could be either relaxed in the regime where the absolute value of frequency deviation $|\omega/\omega_{0}-2|$ grows with the power-law in $\alpha$, or enhanced in the regime where the frequency deviation $\omega/\omega_{0}-2$ flips a sign.

However, the actual requirement for the resolution of frequency measurement could be further relaxed universally in the whole parameter space of $d$ and $\alpha$ by virtue of the long lifetime of ultra-cold BEC in space. In fact, the difference between the measured time-evolution $\nu(\tau)$ of oscillating BEC and the gravity-free solution $\nu_{0}(\tau)$ grows with power-law in time $\tau$ as seen in the first panel of Fig. 3. Furthermore, for trapping frequency in the range of $50-10,000$ Hz, the measurement of BEC oscillations up to $\tau\sim 10^{4}$ corresponds to a condensate lifetime of $1-100$ seconds that is exactly comparable to the free-fall time achieved with the help of microgravity environment. Therefore, even if the current resolution of frequency-measurement is not high enough to determine the oscillation frequency within a single oscillation period, the long-time measurement of oscillating BEC in space allows us to calibrate the measured frequency from previous oscillation period to higher precision over the next numerous oscillation periods. In our specific case, the measurement of oscillations up to $\tau\sim 10^{4}$ corresponds to $n\sim 10^{3}$ periods, which could further relax the resolution of frequency measurement down to

As matter of a preliminary investigation, we also study the Yukawa interaction in a largely extended parameter space supported by the interaction range $d$ down to $10^{-12}$ m and strength $\alpha$ up to $10^{30}$. For the cases of $d>a_{0}$ and $d<a_{0}$, the potential minimum $\nu_{\mathrm{min}}$ experiences a cross-over and first-order phase transitions with increasing $\alpha$ as shown with blue and red curves in the fifth panel of Fig. 3, respectively. The absolute deviation of oscillation frequency in this extended parameter space is depicted in the last panel of Fig. 3, which is relatively small for the blue and purple regions so that the impact on the relative error of the oscillation frequency from the Newtonian gravitational potential is negligible compared to the measured deviation of oscillation frequency according to (41). Therefore, this parameter regions could be promising to be ruled out by future ultra-cold atom experiments. The rest of the parameter regions, although largely affected by the Newtonian potential, were already ruled out mostly by the current constraints reviewed in ANTONIADIS2011755 ; Mostepanenko:2020lqe and references therein. Furthermore, the idealistic condition we assume for neglecting the second term in (25) could be relaxed for large $\alpha$ in the extended parameter space, and the further investigation for the full competitions of all three terms in (25) will be reserved for future work.

For the fat-graviton potential (8), the deviation of oscillation frequency (III.3) with respect to the gravity-free oscillation frequency $2\omega_{0}$ is shown in the right panel of Fig. 2 for the particle numbers $N=10^{9}$ (red line) and $N=10^{12}$ (blue line), where the dashed lines denote for the negative value of the frequency deviation $\omega/\omega_{0}-2$, and the shaded region $\lambda>98\,\mu\mathrm{m}/0.914$ is excluded by the experiments Kapner:2006si ; Adelberger:2006dh . The other shaded region $\lambda<20\,\mu\mathrm{m}/0.914$ is argued by naturalness in Sundrum:2003jq . Similar to case with power-law potential, the frequency deviation is also insensitive to $\lambda$ regardless of the flipped sign around $\lambda\simeq 2\times 10^{-6}\,\mathrm{m}$, which is also roughly the characteristic length scale $a_{0}\simeq 10^{-6}\,\mathrm{m}$ of BEC state. Therefore, the ultra-cold BEC experiment in space also cannot constrain the modified gravity with fat-graviton potential in the regime of interest $20\,\mu\mathrm{m}/0.914<\lambda<98\,\mu\mathrm{m}/0.914$.