Investigating the environmental dependence of ultralight scalar dark matter with atom interferometers

Abstract

We study the environmental dependence of ultralight scalar dark matter (DM) with linear interactions to the standard model particles. The solution to the DM field turns out to be a sum of the cosmic harmonic oscillation term and the local exponential fluctuation term. The amplitude of the first term depends on the local DM density and the mass of the DM field. The second term is induced by the local distribution of matter, such as the Earth. And it depends not only on the mass of the Earth, but also the density of the Earth. Then, we compute the phase shift induced by the DM field in atom interferometers (AIs), through solving the trajectories of atoms. Especially, the AI signal for the violation of weak equivalence principle (WEP) caused by the DM field is calculated. Depending on the values of the DM coupling parameters, contributions to the WEP violation from the first and second terms of the DM field can be either comparable or one larger than the other. Finally, we give some constraints to DM coupling parameters using results from the terrestrial atomic WEP tests.

Appendices

Calculation of velocity and position of atoms

Solving Eq. (36) is as follows. We first integrate on both sides to get the velocity,

$$\begin{aligned} {\dot{z}}(t')= & {} {\dot{z}}(t_{i})\!-\!g_{0}(t'\!-\!t_{i})\!-\!\alpha _E I\left( \frac{R_{E}}{\lambda _{\text {eff}}}\right) \frac{GM_{E}}{c^2} \int ^{t'}_{t_i} \Bigg [\alpha _{A}\Bigg (c^2\!+\!\frac{({\dot{z}}^{(0)}(t))^2}{2}\!+\!g_0z^{(0)}(t)\Bigg )\nonumber \\&\cdot \,\Bigg (\frac{1}{(R_E\!+\!z^{(0)}(t))^2}\!+\!\frac{1}{(R_E+z^{(0)}(t))\lambda _{\text {eff}}}\Bigg ) +g_0\alpha _E\Bigg (\frac{z^{(0)}(t)}{(R_E+z^{(0)}(t))^2}\nonumber \\&+\,\frac{1}{R_E+z^{(0)}(t)}\Bigg (1+\frac{z^{(0)}(t)}{\lambda _{\text {eff}}}\Bigg )\Bigg )e^{-\frac{R_E+z^{(0)}(t)}{\lambda _{\text {eff}}}}\Bigg ]dt \nonumber \\&+\,\varphi _{0}\int ^{t'}_{t_i}\Bigg [k\Bigg (\Bigg (c^2-\frac{({\dot{z}}^{(0)}(t))^2}{2}\Bigg )\alpha _{A}+g_0z^{(0)}(t)(\alpha _{A}+\alpha _E)\Bigg )\sin \Bigg (k(R_{E}\nonumber \\&+\,z^{(0)}(t))-\omega t+\delta \Bigg ) -g_0(\alpha _{A}+\alpha _E)\cos \Bigg (k(R_{E}+z^{(0)}(t))-\omega t+\delta \Bigg )\Bigg ]dt \nonumber \\&-\,\alpha _{A}\varphi _{0}\Bigg (\cos \Big (k(R_E+z^{(0)}(t'))-\omega t'+\delta \Big ){\dot{z}}^{(0)}(t') -\cos (k(R_E+z^{(0)}_i)\nonumber \\&-\,\omega t_i+\delta ){\dot{z}}^{(0)}(t_i)\Bigg )\, . \end{aligned}$$

(51)

Here, $z^{(0)}(t)$ denotes the unperturbed atomic trajectory, which is nothing but the freefall trajectory

$$\begin{aligned} z^{(0)}(t)=z_i^{(0)}+v_i^{(0)}(t-t_i)-\frac{1}{2}g_0(t-t_i)^2 \, \end{aligned}$$

(52)

where $t_i$ is the initial time, $z_i^{(0)}$ and $v_i^{(0)}$ are respectively the initial position and velocity for each segment of the freefall trajectory.

To finish the integration in Eq. (51), we do the following approximation,

$$\begin{aligned} e^{-\frac{R_E+z^{(0)}(t)}{\lambda _{\text {eff}}}}\simeq & {} \left( 1-\frac{z^{(0)}(t)}{R_E}\right) \, e^{-\frac{R_E}{\lambda _{\text {eff}}}} \nonumber \\ \frac{1}{R_E+z^{(0)}(t)}\simeq & {} \frac{1}{R_E}\, \left( 1-\frac{z^{(0)}(t)}{R_E}\right) \nonumber \\ \frac{1}{(R_E+z^{(0)}(t))^2}\simeq & {} \frac{1}{R_E^2} \, \left( 1-\frac{2z^{(0)}(t)}{R_E}\right) \end{aligned}$$

(53)

Then we can get the result

$$\begin{aligned} {\dot{z}}(t)={\dot{z}}(t_i)-g_0(t-t_i)+{\dot{z}}_{\mathrm{exp}}+{\dot{z}}_{\mathrm{bg}}\, , \end{aligned}$$

(54)

where

$$\begin{aligned} {\dot{z}}_{\mathrm{exp}}=&-\alpha _Eg_0I\left( \frac{R_E}{\lambda _\mathrm{eff}}\right) e^{-\frac{R_E}{\lambda _{\mathrm{eff}}}}\frac{1}{R_E\lambda ^2_\mathrm{eff}}\Bigg \{\alpha _A\,(t\!-\!t_i)\Bigg [R_{E}\,{\lambda _{\mathrm{eff}}}^{2}+{R_{E}}^{2}\lambda _{\mathrm{eff}} \nonumber \\&-\,z_{i}^{(0)}\, \Big (\! 2\,{\lambda _{\mathrm{eff}}}^{2}\!+\! \big ( 2\,R_{E}\!-\!2\,z_{i}^{(0)} \big ) \lambda _{\mathrm{eff}}\!+\!R_{E}\, \big (\! R_{E}\!-\!z_{i}^{(0)} \big ) \!\Big ) \!+\!\frac{1}{10}\left( \lambda _{\mathrm{eff}} \!+\!\frac{R_{E}}{2} \right) {g_{0}}^{2}{t_{i}}^{4} \nonumber \\&-\,\frac{2}{5}\left( \! tg_{0}\!-\!\frac{5}{4}v_{i}^{(0)} \right) \left( \!\lambda _{\mathrm{eff}}\!+\! \frac{R_{E}}{2} \!\right) g_{0}{t_{i}}^{3}\!+\!\Bigg ( \!\frac{3}{5} \left( \! \lambda _{\mathrm{eff}}\!+\!\frac{R_{E}}{2} \!\right) {g_{0}}^{2}{t }^{2} \!-\!\frac{3}{2}v_{i}^{(0)} \left( \lambda _{\mathrm{eff}}\!+\!\frac{R_{E}}{2} \right) g_{0}t \nonumber \\&+\,\frac{1}{3}g_{0}{\lambda _{\mathrm{eff}}}^{2}\!+\! \Big (\!\left( \! \frac{R_{E}}{3}\!-\!\frac{2}{3}z_ {i}^{(0)} \!\right) g_{0} \!+\!\frac{2}{3}\,{v_{i}^{(0)}}^{2} \!\Big ) \lambda _{\mathrm{eff}} \!+\!\frac{1}{6} \Big (\!\big ( R_{E}\!-\!2z_{i}^{(0)} \big ) g_{0}\!+\!2{v_{i}^{(0)}}^{2} \!\Big ) R_ {E} \!\Bigg ) {t_{i}}^{2} \nonumber \\&+\,\Bigg (\!-\!\frac{2}{5} \left( \lambda _{\mathrm{eff}}\!+\!\frac{R_{E}}{2} \right) {g_{0}}^{2}{ t}^{3} \!+\!\frac{3}{2}v_{i}^{(0)} \left( \lambda _{\mathrm{eff}}\!+\!\frac{R_{E}}{2} \right) g_{0} {t}^{2}\!+\! \bigg ( \!-\frac{2}{3}\,g_{0}{\lambda _{\mathrm{eff}}}^{2}+ \Big ( \Big ( \frac{4}{3}z_{i}^{(0)} \nonumber \\&-\,\frac{2}{3}R_{E} \Big ) g_{0}-\frac{4}{3}{v_{i}^{(0)}}^{2} \Big ) \lambda _{\mathrm{eff}} -\frac{1}{3} \Big (\!\big ( R_{E}\!-\!2z_{i}^{(0)} \big ) g_{0}\!+\! 2{v_{i}^{(0)}}^{2} \Big ) R_{E} \!\bigg ) t\!+\!v_{i}^{(0)}\Big ( {\lambda _{\mathrm{eff}}}^{2} \nonumber \\&+\, \big ( R_{E}-2z_{i}^{(0)} \big ) \lambda _{\mathrm{eff}} +\frac{1}{2}R_{ E} \big ( R_{E} -2\,z_{i}^{(0)} \big ) \Big ) \Bigg ) t_{i}+\frac{1}{10}\, \Big ( \lambda _{\mathrm{eff}}+\frac{R_{E}}{2} \Big ) {g_{0}}^{2}{t}^{4} \nonumber \\&-\,\frac{1}{2}\,v_{i}^{(0)}\, \Big ( \lambda _{\mathrm{eff}}\!+\!\frac{R_{E}}{2} \Big ) g_{0}\,{t}^{3}\!+\! \bigg ( \frac{1}{3}\,g_{0}\,{\lambda _{\mathrm{eff}}}^{2} \!+\! \left( \left( \frac{R_{E}}{3}- \frac{2}{3}\,z_{i}^{(0)} \right) g_{0}\!+\!\frac{2}{3}\,{v_{i}^{(0)}}^{2} \right) \lambda _{\mathrm{eff}} \nonumber \\&+\,\frac{1}{6}\, \Big ( \big ( R_{E}-2\,z_{i}^{(0)} \big ) g_{0}\!+\!2\,{v_{i}^{(0)}}^{2} \Big ) R_{E} \bigg ) {t}^{2} -v_{i}^{(0)}\, \bigg ( {\lambda _{\mathrm{eff}}}^{2 }+ \big ( R_{E}-2\,z_{i}^{(0)} \big ) \lambda _{\mathrm{eff}} \nonumber \\&+\,\frac{1}{2}\,R_{E}\, \big ( R_{E}-2\,z_{i}^{(0)} \big ) \bigg ) t \Bigg ] \end{aligned}$$

$$\begin{aligned}&+\,\frac{\alpha _A\big (t-t_i\big )}{12c^2}\,\big (2\,g_{0}\,z_{i}^{(0)}+{v_{i}^{(0)}}^{2}\big )\Bigg [ 6\,{R_{E}}^{2}\lambda _{\mathrm{eff}}+6\,R_{E}\,{\lambda _{\mathrm{eff}}}^{2}-6\,z_{i}^{(0)}\, \Big ( 2\,{\lambda _{\mathrm{eff}}}^{2} \nonumber \\&+\, \big ( 2\,R_{E}-2\,z_ {i} ^{(0)}\big ) \lambda _{\mathrm{eff}}+R_{E}\, \big ( R_{E}-z_{i}^{(0)} \big ) \Big ) +\frac{3}{10}\, \left( t-t_{i} \right) ^{4} \big ( 2\,\lambda _{\mathrm{eff}} +R_{E} \big ) {g_{0}}^{2} \nonumber \\&+\, \left( t\!-\!t_{i} \right) ^{2} \bigg ( \frac{3}{2}\,v_{i}^{(0)}\, \left( 2\,\lambda _{\mathrm{eff}}\!+\!R_{E} \right) t_{i}\!-\!\frac{3}{2}\,v_{i}^{(0)}\, \left( 2\,\lambda _{\mathrm{eff}}\!+\!R_{E} \right) t\!-\! \big ( 2\,R_{E} \!+\!4\, \lambda _{\mathrm{eff}} \big ) z_{i}^{(0)} \nonumber \\&+\,{R_{E}}^{2}\!+\!2\,R_{E}\,\lambda _{\mathrm{eff}}\!+\!2\,{\lambda _{\mathrm{eff}}}^{2} \!\bigg ) g_{0}\!+\!2\,{v_{i}^{(0)}}^{2} \left( 2\,\lambda _{\mathrm{eff}}\!+\!R_{E} \!\right) {t_{i}}^{2} \!+\!3\,v_{i}^{(0)}\, \bigg (\!\! \!-\!\frac{4}{3}\,v_{i}^{(0)}\, \big (\! 2\,\lambda _{\mathrm{eff}} \end{aligned}$$

$$\begin{aligned}&+\,R_{E} \big ) t\!+\! \left( \! -\!2\,R_{E}-4\,\lambda _{\mathrm{eff}} \right) z_{i}^{(0)}\!+\!{R_{ E}}^{2}\!+\!2\,R_{E}\,\lambda _{\mathrm{eff}}+2\,{\lambda _{\mathrm{eff}}}^{2} \bigg ) t_{i}+2\,{v_{i}^{(0)}}^{2} \big ( 2\,\lambda _{\mathrm{eff}}\nonumber \\&+\,R_{E} \big ) {t}^{2}-3\,v_{i}^{(0)}\, \bigg ( \left( -2\,R_{E}-4\,\lambda _{\mathrm{eff}} \right) z_{i}^{(0)}+{R_{E}}^{2}+2\,R_{E}\,\lambda _{\mathrm{eff}}+2\,{ \lambda _{\mathrm{eff}}}^{2} \bigg ) t \Bigg ] \nonumber \\&-\,\frac{\alpha _Eg_0\,\big (t-t_i\big )}{20c^2}\Bigg [20\,{R_{E}}^{2}{\lambda _{{ \mathrm eff}}}^{2}\!-\!40R_{E}z_{i}^{(0)}{\lambda _{\mathrm{eff}}}^{2}\!-\!40\,{R_{E}}^{2}z_{i}^{(0)}\lambda _{\mathrm{eff}}+20{R_{E}}^{2}{z_{i}^{(0)}}^{2} \nonumber \\&-\, 20R_{E}{z_{i}^{(0)}}^{3}\!+\!60R_{E}{z_{i}^{(0)}}^{ 2}\lambda _{\mathrm{eff}}\!-\!40{z_{i}^{(0)}}^{3}\lambda _{\mathrm{eff}}\!+\!40{z_{i}^{(0)}}^{ 2}{\lambda _{\mathrm{eff}}}^{2} \!+\!\frac{5}{14}\left( t-t_{i} \right) ^{6} \big ( 2\lambda _{\mathrm{eff}}\nonumber \\&+\,R_{E} \big ) {g_{0}}^{3} \!+\!\left( t\!-\!t_{i} \right) ^{4} \left( \!-\!\frac{5}{2}v_{i}^{(0)}t\!+\!\frac{5}{2}v_{i}^{(0)}\,t_{i}+R _{E}\!-\!3\,z_{i}^{(0)}\!+\!\lambda _{\mathrm{eff}} \right) \left( 2\,\lambda _{\mathrm{eff }}\!+\!R_{E} \right) {g_{0}}^{2} \nonumber \\&-\,5\,g_{0}\left( t\!-\!t_{i} \right) ^{2} \left( \!-\!\frac{6}{5}{v_{i}^{(0)}}^{2} \left( 2\,\lambda _{\mathrm{eff}}+R_{E} \right) {t_{i}}^{2}\!-\!v_{i}^{(0)}\big ( 2\,\lambda _{\mathrm{eff}}\!+\!R_{E} \big ) \left( \! -\!{\frac{12\,v_{i}^{(0)}t}{5}}\!+\!R_{E}\right. \right. \nonumber \\&\left. -\,3z_{i}^{(0)}\!+\!\lambda _{{ \mathrm eff}} \!\right) t_{i}\!-\!\frac{6}{5}\,{v_{i}^{(0)}}^{2} \left( 2\lambda _{\mathrm{eff}} +R_{E} \!\right) {t}^{2}\!+\!v_{i}^{(0)}\left( 2\lambda _{\mathrm{eff}}\!+\!R_{E} \! \right) \!\left( \! R_{E}\!-\!3z_{i}^{(0)}\!+\!\lambda _{\mathrm{eff}} \!\right) t \nonumber \\&+\, \left( \!-\!\frac{4}{3}\,R_{E}\!+\!\frac{8}{3}\,z_{i}^{(0)} \right) {\lambda _{\mathrm{eff}}}^{2}+ \left( -\frac{4}{3} \,{R_{E}}^{2}+4\,R_{E}\,z_{i}^{(0)}-4\,{z_{i}^{(0)}}^{2} \right) \lambda _{\mathrm{eff }}+\frac{4}{3}\,{R_{E}}^{2}z_{i}^{(0)} \nonumber \\&-\,2\,R_{E}\,{z_{i}^{(0)}}^{2} \bigg )+5\,{v_{i}^{(0)}}^{3} \left( 2\,\lambda _{\mathrm{eff}}+R_{E} \right) {t_{i}}^{3}+\frac{20}{3}{v_{i}^{(0)}}^{2} \left( -\frac{9}{4}\,v_{i}^{(0)}\,t+R_{E}-3\,z_{i}^{(0)}\right. \nonumber \\&\left. +\,\lambda _{\mathrm{eff}} \right) \big ( 2\,\lambda _{\mathrm{eff}}+R_{E} \big ) {t_{i}}^{2}-\frac{40}{3}v_{i}^{(0)}\, \left( -\frac{9}{8}{v_{i}^{(0)}}^{2} \big ( 2\,\lambda _{\mathrm{eff}}+R _{E} \big ) {t}^{2}+v_{i}^{(0)}\, \big ( 2\,\lambda _{\mathrm{eff}}\right. \nonumber \\&\left. +\,R_{E} \right) \big ( R_{E}\!-\!3\,z_{i}^{(0)}\!+\!\lambda _{\mathrm{eff}} \big ) t+ \left( \!-\!\frac{3}{2}\,R_{E}\!+\!3\,z_{i}^{(0)} \right) {\lambda _{\mathrm{eff}}}^{2}\!+\! \left( -\frac{3}{2}\,{ R_{E}}^{2}\!+\!\frac{9}{2}\,R_{E}\,z_{i}^{(0)}\right. \nonumber \\&\left. -\,\frac{9}{2}\,{z_{i}^{(0)}}^{2} \right) \lambda _{\mathrm{eff}} +\frac{3}{2}\,R_{E}\,z_{i}^{(0)}\, \left( R_{E}-\frac{3}{2}\,z_{i}^{(0)} \right) \bigg ) t_{ i}-5\,{v_{i}^{(0)}}^{3} \left( 2\,\lambda _{\mathrm{eff}}+R_{E} \right) {t}^{3} \nonumber \\&+\,\frac{20}{3}{v_{i}^{(0)}}^{2} \big ( 2\,\lambda _{\mathrm{eff}}\!+\!R_{E} \big ) \big ( R_{E} -3\,z_{i}^{(0)}\!+\!\lambda _{\mathrm{eff}} \big ) {t}^{2}\!+\!20\,v_{i}^{(0)}\, \bigg ( \left( \!-\!R_{E}\!+\!2\,z_{i}^{(0)} \right) {\lambda _{\mathrm{eff} }}^{2} \nonumber \\&+\, \left( -{R_{E}}^{2}+3\,R_{E}\,z_{i}^{(0)}-3\,{z_{i}^{(0)}}^{2} \right) \lambda _{\mathrm{eff}} +R_{E}\,z_{i}^{(0)}\, \left( R_{E}-\frac{3}{2}\,z_{i}^{(0)} \right) \bigg ) t \Bigg ]\Bigg \} \end{aligned}$$

(55)

and

$$\begin{aligned} {\dot{z}}_{\mathrm{bg}}=&\frac{k\varphi _{0}\,\alpha _{A}\,{c}^{2}}{{\omega }^{3}}\Bigg [\frac{1}{2}\, \bigg ( \Big ( g_{0}\, \left( t-t_{i} \right) ^{2}+ \left( -2\,t +2\,t_{i} \right) v_{i}^{(0)}-2\,z_{i}^{(0)} \Big ) {\omega }^{2}-2\,g_{0} \bigg ) k \nonumber \\&\cdot \, \sin \left( R_{E}\,k-\omega \,t+\delta \right) +\bigg ( \omega -k \Big ( g_{0}\, \left( t-t_{i} \right) -v_{i}^{(0)} \Big ) \bigg ) \omega \,\cos \left( R_{E}\,k-\omega \,t+\delta \right) \nonumber \\&-\, \left( k\omega v_{i}^{(0)}\!+\!{\omega }^{2} \right) \cos \left( R_{E}k\!-\! \omega t_{i}+\delta \right) \!+\!k \left( {\omega }^{2 }z_{i}^{(0)}+g_{0} \right) \sin \left( R_{E}k-\omega \,t_{i}+\delta \right) \Bigg ] \nonumber \\&-\,\frac{3\alpha _A\varphi _0}{2\omega ^5}\Bigg [\Bigg (\bigg ( \frac{1}{3}\, \Big ( \left( t-t_{i} \right) ^{2}{g_{0}}^{2}+ \big ( \left( -2\,t+2\,t_{i} \right) v_{i}^{(0)}-z_{i}^{(0)} \big ) g_{0}+\frac{1}{2}\,{v_{i}^{(0)}}^ {2} \Big ) \nonumber \\&\cdot \left( g_{0}\left( t-t_{i} \right) ^{2}+ \left( -2t+ 2t_{i} \right) v_{i}^{(0)}-2z_{i}^{(0)} \right) {k}^{2}-\frac{2}{3}g_{0} \bigg ) { \omega }^{4}+2g_{0} \Big ( g_{0} \big ( t-t_{i} \big ) \nonumber \\&-\,v_{i}^{(0)} \Big ) k{ \omega }^{3}\!-\!4g_{0}\,{k}^{2} \bigg ( \left( t\!-\!t_{i} \right) ^{2}{g_{0 }}^{2}\!+\! \Big ( \left( \!-\!2t\!+\!2\,t_{i} \right) v_{i}^{(0)}\!-\!\frac{z_{i}^{(0)}}{2} \Big ) g_ {0}+\frac{3}{4}{v_{i}^{(0)}}^{2} \bigg ) {\omega }^{2} \nonumber \\&+\,8{g_{0}}^{3}{k}^{2} \Bigg )\sin \left( R_Ek\!-\!\omega \,t\!+\!\delta \right) \!+\!\Bigg (\bigg ( \left( t\!-\!t_{i} \right) ^{2}{g_{0}}^{2}\!+\! \Big ( \left( \!-\!2t\!+\! 2t_{i} \right) v_{i}^{(0)}\!-\!\frac{4}{3}z_{i}^{(0)} \Big ) g_{0} \nonumber \\&+\,\frac{1}{3}{v_{i}^{(0)}}^{2} \bigg ) {\omega }^{3}\!-\!\frac{4}{3}\Big ( \!g_{0} \left( t\!-\!t_{i} \right) \!-\!v_{i}^{(0)} \Big )\bigg (\! \left( \! t\!-\!t_{i} \!\right) ^{2}{g_{0}}^{2}\!+\! \Big (\! \left( -2t\!+\!2t_{i} \right) v_{i}^{(0)}\!-\!\frac{3}{2}z_{i}^{(0)} \Big ) g_{0} \nonumber \\&+\,\frac{1}{4}{v_{i}^{(0)}}^{2} \bigg ) k{ \omega }^{2}\!-\!2{g_{0}}^{2}\omega +8{g_{0}}^{2} \Big ( g_{0} \left( t\!-\!t_{i} \right) \! -\!v_{i}^{(0)} \Big ) k \Bigg )k\omega \cos \left( R_{E}k-\omega \,t+\delta \right) \nonumber \\&+\,\Bigg ( \bigg ( \frac{1}{3}z_{i}^{(0)} \left( -2g_{0}z_{i}^{(0)}+{v_{i}^{(0)}}^{2} \right) {k}^{2}+\frac{2}{3}g_{0} \bigg ) {\omega }^{4}+2v_{i}^{(0)}g_{0}\,k{ \omega }^{3}+ \big ( -2{g_{0}}^{2}z_{i}^{(0)} \nonumber \\&+\,3g_{0}{v_{i}^{(0)}}^{2} \big ) {k}^{2}{\omega }^{2}\!-\!8{g_{0}}^{3}{k}^{2} \Bigg ) \sin \left( R_{E}k\!-\!\omega \,t_{i}\!+\!\delta \right) \!-\!\frac{1}{3} \Bigg ( \left( \!-\!4g_{0}z_{i}^{(0)}+{v_{i}^{(0)}}^{2} \right) {\omega }^{3 } \nonumber \\&+\,kv_{i}^{(0)}\left( \!-\!6g_{0}z_{i}^{(0)}\!+\!{v_{i}^{(0)}}^{2} \right) {\omega }^{2}\!-\!6 {g_{0}}^{2}\omega \!-\!24v_{i}^{(0)}{g_{0}}^{2}k \Bigg ) k\omega \cos \left( R_{E}k\!-\!\omega \,t_{i}\!+\!\delta \right) \Bigg ] \end{aligned}$$

$$\begin{aligned}&+\,\frac{\alpha _E\varphi _0g_0}{\omega ^5}\Bigg [\Bigg ( -\frac{1}{4}\bigg ( 2+ \Big ( g_{0}\left( t-t_{i} \right) ^{2}-2 tv_{i}^{(0)}+2v_{i}^{(0)}t_{i}-2z_{i}^{(0)} \Big ) k \bigg ) \bigg ( -2 \nonumber \\&+\,\left( g_{0} \left( t\!-\!t_{i} \right) ^{2}\!-\!2tv_{i}^{(0)}+2v_{i}^{(0)}t_{i}\!-\! 2z_{i}^{(0)}\right) k \bigg ) {\omega }^{4}-2 \Big ( g_{0} \left( t-t_ {i} \right) -v_{i}^{(0)} \Big ) k{\omega }^{3} \nonumber \\&+\,3\left( \left( t\!-\!t_{i} \right) ^{2}{g_{0}}^{2}\!+\! \left( \!-\!2tv_{i}^{(0)}\!+\!2v_{i}^{(0)}t_{i}\!-\!\frac{2}{3}z_{i}^{(0)} \right) g_{0}\!+\!\frac{2}{3}{v_{i}^{(0)}}^{2} \right) {k}^{2}{\omega }^{2}\!-\!6{g_{0}} ^{2}{k}^{2} \Bigg ) \nonumber \\&\cdot \,\sin \left( R_{E}\,k-\omega \,t+\delta \right) + \Bigg ( \Big ( -g_{0}\, \left( t-t_{i} \right) ^{2}+2\,tv_{i}^{(0)} -2\,v_{i}^{(0)}\,t_{i}+2\,z_{i}^{(0)} \Big ) {\omega }^{3} \nonumber \\&+\, \Big ( g_{0}\, \left( t-t_{i} \right) -v_{i}^{(0)} \Big ) \left( g_{0}\, \left( t-t_{i} \right) ^{2}-2\,tv_{i}^{(0)}+2\,v_{i}^{(0)}\,t_{i}-2\,z_{i}^{(0)} \right) k{\omega }^{2}+2\,g_{0} \,\omega \nonumber \\&-\,6g_{0} \Big ( g_{0} \left( t\!-\!t_{i} \right) \! -\!v_{i}^{(0)} \Big ) k \!\Bigg ) \!k\omega \cos \left( R_{E}k\!-\!\omega \,t+\delta \right) \!+\!\Bigg (\!\!\! \left( {k}^{2}{z_{i}^{(0)}}^{2}\!-\!1 \right) {\omega }^{4}\!-\!2v_{i}^{(0)}k{ \omega }^{3} \nonumber \\&-\,2{k}^{2} \left( -g_{0}z_{i}^{(0)}+{v_{i}^{(0)}}^{2} \right) { \omega }^{2}+6{g_{0}}^{2}{k}^{2} \Bigg ) \sin \left( R_{E}\,k-\omega t_{i}+\delta \right) - \Bigg ( 2v_{i}^{(0)}k{\omega }^{2}z_{i}^{(0)} \nonumber \\&+\,2\,{\omega }^ {3}z_{i}^{(0)} +6v_{i}^{(0)}\,g_{0}\,k+2\,g_{0}\,\omega \Bigg ) k\omega \cos \left( R_{E}\,k-\omega \,t_{i}+\delta \right) \Bigg ] \nonumber \\&-\,\alpha _{A}\,\varphi _{0}\, \Bigg ( \cos \left( R_{E}\,k-\omega \,t+ \delta \right) -\sin \left( R_{E}\,k-\omega \,t+\delta \right) k \Big ( z_{i}^{(0)}+v_{i}^{(0)}\, \left( t-t_{i} \right) \nonumber \\&-\,\frac{1}{2}\,g_{0}\, \left( t-t_ {i} \right) ^{2} \Big ) \Bigg ) \Big ( v_{i}^{(0)}-g_{0}\, \left( t-t_{i} \right) \Big ) +\alpha _{A}\,\varphi _{0}\, \Bigg ( \cos \left( R_{E} \,k-\omega \,t_{i}+\delta \right) \nonumber \\&-\,\sin \left( R_{E}\,k-\omega \,t_{i}+ \delta \right) kz_{i}^{(0)} \Bigg ) v_{i}^{(0)}\, . \end{aligned}$$

(56)

It is clear that ${\dot{z}}_{\mathrm{exp}}$ and ${\dot{z}}_{\mathrm{bg}}$ denote effects from the exponential term and the oscillation background term of the DM field, respectively.

For later use, we give the following velocities. At the time of applying the $\pi$-pulse, the velocity for atoms in the lower arm is ${\dot{z}}_{1l}\equiv {\dot{z}}(t)\vert _{t=T}$ with $t_i=0$ and ${\dot{z}}(t_{i})=v_L$, while the velocity for atoms in the upper arm is ${\dot{z}}_{1u}\equiv {\dot{z}}(t)\vert _{t=T}$ with $t_i=0$ and ${\dot{z}}(t_{i})=v_L+v_R(\varphi )$. At the time of applying the second $\frac{\pi }{2}$-pulse, ${\dot{z}}_{2l}\equiv {\dot{z}}(t)\vert _{t=2T}$ with $t_i=T$ and ${\dot{z}}(t_{i})={\dot{z}}_{1l}+v_R(\varphi )$, while ${\dot{z}}_{2u}\equiv {\dot{z}}(t)\vert _{t=2T}$ with $t_i=T$ and ${\dot{z}}(t_{i})={\dot{z}}_{1u}-v_R(\varphi )$.

Next, we do the time integration on Eq. (54) to get the solution for the trajectory

$$\begin{aligned} z(t)= z(t_{i})+\int ^{t}_{t_i}{\dot{z}}(t') dt' \, . \end{aligned}$$

(57)

For later use, we give the following positions. At the time of applying the $\pi$-pulse, the position for atoms in the lower arm is $z_{1l}\equiv z(t)\vert _{t=T}$ with $t_{i}=0$, $z(t_{i})=0$ and ${\dot{z}}(t_{i})=v_L$, while the position for atoms in the upper arm is $z_{1u}\equiv z(t)\vert _{t=T}$ with $t_{i}=0$, $z(t_{i})=0$ and ${\dot{z}}(t_{i})=v_L+v_R(\varphi )$. At the time of applying the second $\frac{\pi }{2}$-pulse, $z_{2l}\equiv z(t)\vert _{t=2T}$ with $t_{i}=T$, $z(t_{i})=z_{1l}$ and ${\dot{z}}(t_{i})={\dot{z}}_{1l}+v_R(\varphi )$, while $z_{2u}\equiv z(t)\vert _{t=2T}$ with $t_{i}=T$, $z(t_{i})=z_{1u}$ and ${\dot{z}}(t_{i})={\dot{z}}_{1u}-v_R(\varphi )$.

Calculation of the DM-induced phase shift in AI experiments

The total phase shift can be written as a sum of three components [33], the propagation phase shift, the laser phase shift, and the separation phase shift,

$$\begin{aligned} \Delta \phi =\Delta \phi _{prop}+\Delta \phi _{laser}+\Delta \phi _{sep} \, . \end{aligned}$$

(58)

For each segment of the atomic trajectory, the atom accumulates a propagation phase

$$\begin{aligned} \phi _{prop}=\int ^{t_f}_{t_i}L\, dt \, , \end{aligned}$$

(59)

where $t_f$ is the final time for each segment, and L is the Lagrangian (31). The propagation phase shift $\Delta \phi _{prop}$ is the difference in the propagation phase between the two arms,

$$\begin{aligned} \Delta \phi _{prop}=\sum _{\mathrm{upper}}\phi _{prop}-\sum _{\mathrm{lower}}\phi _{prop} \, . \end{aligned}$$

(60)

The laser phase shift comes from the interaction of laser pulses with atoms. At each interaction point, the laser field transfers its phase to the atom. Then, $\Delta \phi _{laser}$ is the difference in the accumulated laser phase between the upper and lower arms

$$\begin{aligned} \Delta \phi _{laser}&=\sum _{\mathrm{upper}}\phi _{laser} - \sum _{\mathrm{lower}}\phi _{laser} \nonumber \\&=c\int _{0}^{\frac{z_{i}}{c}}k_{eff}(t)dt-c\int _{T}^{T+\frac{z_{1u}}{c}}k_{eff}(t)dt \nonumber \\&\quad -c\int _{T}^{T+\frac{z_{1l}}{c}}k_{eff}(t)dt+c\int _{2T}^{2T+\frac{z_{2l}}{c}}k_{eff}(t)dt\, , \end{aligned}$$

(61)

where $z_i$ is the initial position of atoms at the time of applying the first $\frac{\pi }{2}$-pulse.

Since the two arms do not exactly intersect at the final laser pulse, then the separation phase shift $\Delta \phi _{sep}$ appears.

$$\begin{aligned} \Delta \phi _{sep}=\frac{m_{A}}{2\hbar }({\dot{z}}_{2u}-v_R+{\dot{z}}_{2l})(z_{2l}-z_{2u}) \, . \end{aligned}$$

(62)

With Eqs. (54) and (57), we can calculate $\phi _{prop}$ along each segment of the lower and upper arms, and thus compute $\Delta \phi _{prop}$. Similarly, $\Delta \phi _{laser}$ and $\Delta \phi _{sep}$ can also be computed. Summing them together, one can get the final result for the DM-induced phase shift. We find that $\Delta \phi$ consists of a static component $\Delta \phi _{\delta \varphi }$, an oscillatory component $\Delta \phi _{bg}$, and the well-known term $-g_{0}T^2k_{\text {eff}}$,

$$\begin{aligned} \Delta \phi =-g_{0}T^2k_{\text {eff}}+\Delta \phi _{\delta \varphi }+\Delta \phi _{bg} \, . \end{aligned}$$

(63)

The first term is the known phase shift for atoms in freefall, where ${\mathbf{k }}_{\mathrm{eff}}$ has been taken to be parallel to ${\mathbf{g }}_0$. The $\delta \varphi$-contribution to $\Delta \phi$ is given by

$$\begin{aligned} \Delta \phi _{\delta \varphi }&=\!-\!g_{0}T^2k_{\text {eff}}\Bigg [\Bigg (\frac{\frac{7}{6}g_{0}T^2-(2v_{L}+v_{R})T}{\lambda _{\text {eff}}}\!+\!\left( 1+\frac{R_{E}}{\lambda _{\text {eff}}}\right) \!+\!\frac{ v_{L}(v_{R}+v_{L})}{2c^2}\Bigg )\alpha _{A} \nonumber \\&\quad +\,\Bigg (\frac{g_{0}(2\,v_{L}\!+\!v_{R})T\!-\!g_{0}R_{E}}{c^2}\!-\!\frac{7\,g^2_{0}\,{T}^{2}}{6c^2}\Bigg )\alpha _{E} \!+\!\frac{1}{\lambda _{\text {eff}}c^2}\Bigg (\Bigg (\Bigg (\frac{7}{12}v_{L}(v_{L}\!+\!v_{R})\nonumber \\&\quad -\,\frac{1}{12}v_{R}^2\Bigg )g_{0}T^2\!-\!v_{L}(v_{L}\!+\!v_{R})\Bigg ((v_{L}\!+\!\frac{1}{2}v_{R})T -\frac{1}{2}R_{E}\Bigg )\Bigg )\alpha _{A} +\Big (-{\frac{31}{20}}{T}^{3}{g_{0}}^{2}\nonumber \\&\quad +\frac{9}{2}\, \Bigg ( v_{L}+\frac{v_{R}}{2} \Bigg ) g_{0}\,{T}^{2}-\Bigg (\, \frac{7}{6}\,g_{0}\,R_{E}+\frac{7}{2}\,{v_{L}}(v_{L}+\,v_{R})+{v_{R}}^{2} \Bigg ) T+R_{E}\big (2v_{L}\nonumber \\&\quad +\,v_{R}\big ) \Bigg ) g_{0}T\alpha _{E}\Bigg )\Bigg ]I\left( \frac{R_{E}}{\lambda _{\text {eff}}}\right) \alpha _{E}e^{-\frac{R_{E}}{\lambda _{\text {eff}}}} \nonumber \\&\quad +\,g_{0}k_{\text {eff}}T^2\Bigg [\frac{1}{c^2}\Bigg (\frac{17}{2}\,{g_{0}}^{2}{T}^{2}+\frac{1}{2}\, \left( -26\,v_{L}\,T-15\,Tv_{R}+2\,R_{E} \right) g_{0}+ \big (4 v_{L}\nonumber \\&\quad +\,v_{R} \big ) \left( v_{R}+v_{L} \right) \Bigg )+\frac{1}{\lambda _{\text {eff}}c^2} \Bigg ({\frac{161}{4}\,{g_{0}}^{3}{T}^{4}} -93\, \left( v_{L}+{\frac{37\,}{62}v_{ R}} \right) {g_{0}}^{2}{T}^{3}\nonumber \\&\quad +\,\frac{1}{2}\,g_{0}\, \left( 17\,g_{0}\,R_{E}+138\,{v_{L}}^{2}+168\,v_{R}\,v_{L}+49 \,{v_{R}}^{2} \right) {T}^{2}-\frac{1}{2}\, \big ( 26\,g_{0}\,R_{E}v_{L}\nonumber \\&\quad +15\,g_{0}\,R_{E}v_{R}+32\,{v_{L}}^{3}+60\, v_{R}\,{v_{L}}^{2}+34\,{v_{R}}^{2}v_{L}+6\,{v_{R}}^{3} \big ) T +\big ( 4\,v_{L}\nonumber \\&\quad +\,v_{R} \big ) R_{E} \left( v_{R}+v_{L} \right) \Bigg ) \Bigg ]I\left( \frac{R_{E}}{\lambda _{\text {eff}}}\right) {\tilde{d}}\alpha _{E}e^{-\frac{R_{E}}{\lambda _{\text {eff}}}} \end{aligned}$$

(64)

The $\varphi _{bg}$-contribution to $\Delta \phi$ is given by

$$\begin{aligned} \Delta \phi _{bg}&=-k_{\text {eff}}\frac{c^2k\alpha _{A}\varphi _{0}}{\omega ^2}\Bigg (\sin (kR_{E}-2\omega T+\delta )-2\sin (kR_{E}-\omega T+\delta ) \nonumber \\&\quad +\,\sin (kR_{E}+\delta )\Bigg ) +\alpha _{A}\frac{2g_{0}k_{\text {eff}}T}{\omega }\varphi _{0}\Bigg (\sin (kR_{E}-\omega T+\delta )-\sin \big (kR_{E} \nonumber \\&\quad -\,2\omega T+\delta \big )\Bigg ) +\left( \alpha _{E}+2\alpha _{A}\right) \frac{g_{0}k_{\text {eff}}}{\omega ^{2}}\varphi _{0}\Bigg (\cos (kR_{E}+\delta )-2\cos \big (kR_{E} \nonumber \\&\quad -\,\omega T+\delta \big )\!+\!\cos (kR_{E}-2\omega T+\delta )\Bigg ) \!-\!\alpha _{A}\left( \frac{k_{\text {eff}}(v_{L}+\frac{v_{R}}{2})}{\omega }\right) \varphi _{0}\Bigg (\sin \big (kR_{E} \nonumber \\&\quad +\,\delta \big )+2\sin (kR_{E}-\omega T+\delta )-\sin (kR_{E}-2\omega T+\delta )\Bigg ) \nonumber \\&\quad -k_{\text {eff}}\frac{k\varphi _{0}}{\omega ^3}\Bigg [ \big ( 4g_{0}T-2v_{L}-v_{R} \big ) \bigg ( \big ( g_{0}T^2-v_{L}T-\frac{v_{R}}{2}T \big ) {\omega }^{2}\alpha _{A}-\frac{9}{2}\,g_{0} \alpha _{A} \nonumber \\&\quad -2\,g_{0}\,\alpha _{E} \bigg )\cos (kR_{E}-2\omega T+\delta ) -\big ( g_{0}T-v_{L}-\frac{1}{2}v_{R} \big ) \bigg ( \big ( g_{0}T^2-2v_{L}T \nonumber \\&\quad -\,v_{R}T \big ) {\omega }^{2}\alpha _{A}-18\,g_{0} \alpha _{A}-8\,g_{0}\,\alpha _{E} \bigg )\cos (kR_{E}-\omega T+\delta ) +g_{0}\big (2\alpha _{E} \nonumber \\&\quad +\,\frac{9}{2}\alpha _{A}\big ) (2v_{L}+v_{R})\cos (kR_{E}+\delta )\Bigg ] +k_{\text {eff}}\frac{k\varphi _{0}}{\omega ^4}\Bigg [\Bigg (\bigg ( 2g_{0}\,T \big ( 2\,g_{0}T-2\, v_{L} \nonumber \\&\quad -\,v_{R} \big )\left( \alpha _{E}+3\,\alpha _{A} \right) + \frac{3}{2}\,v_{L}\, \left( v_{L}+v_{R} \right) \alpha _{A} +\dfrac{1}{2}\,{v_{R}}^{2} \alpha _{A} \bigg ) {\omega }^{2}-12\,{g_ {0}}^{2} \Bigg ( \alpha _{A} \nonumber \\&\quad \!+\!\frac{\alpha _{E}}{2} \Bigg ) \Bigg ) \sin (kR_{E}-2\omega T+\delta )\!+\!\Bigg (\bigg ( 2g_{0}\,T \left( 2\, v_{L}\!+\!v_{R}\!-\!g_{0}T \right) \left( \alpha _{E}+3\,\alpha _{A} \right) \nonumber \\&\quad -\, 3\,v_{L}\, \left( v_{L}\!+\!v_{R} \right) \alpha _{A}\! -\!{v_{R}}^{2} \alpha _{A} \bigg ) {\omega }^{2}+24\,{g_ {0}}^{2} \Bigg ( \alpha _{A}+\frac{\alpha _{E}}{2} \Bigg ) \Bigg ) \sin \big (kR_{E}-\omega T \nonumber \\&\quad +\,\delta \big )\!+\!\Bigg (\big (\frac{3}{2}v_{L}(v_{L}\!+\!v_{R})\!+\!\frac{1}{2}v_R^2\big )\omega ^2\alpha _{A} -12g_{0}^2(\alpha _{A}\!+\!\frac{\alpha _{E}}{2})\Bigg )\sin (kR_{E}+\delta )\Bigg ] \nonumber \\&\quad -\,\frac{k_{\text {eff}}T^2k{\tilde{d}}\varphi _{0}}{ c^2}\Bigg [ \left( 4\,{T}^{2}{g_{0}}^{2}-\left( 8c+4v_{L}+2v_{R} \right) g_{ 0}\,T+4c \left( c+v_{L}+v_{R} \right) \right) \nonumber \\&\quad \cdot \left( Tg_{0} \!-\!v_{L}\!-\!\frac{1}{2}v_{R}\right) ^{2}\sin \left( kR_{E}\!-\!2\,T\omega +\delta \right) +\Bigg ( -\frac{1}{8}\,{g_{0}}^{3}{T}^{3}+\frac{1}{4}\, \big ( c+3\,v_{L}\nonumber \\ \end{aligned}$$

$$\begin{aligned}&\quad +\frac{3}{2}\,v_{R} \big ) {g_{0}}^{2}{T}^{2}-g_{0}\,T \bigg ( \frac{3}{2}\,{v_{L}}^{2}+ \Bigg ( c +\frac{3}{2}\,v_{R} \Bigg ) \Bigg ( v_{L}+\frac{v_{R}}{2} \Bigg ) \bigg ) + \bigg ( {v _{L}}^{2}+v_{L}\,v_{R} \nonumber \\&\quad +\,\frac{1}{2}\,{v_{R}}^{2} \bigg ) c+\frac{1}{2}\,{v_{L}}^{3}+\frac{1}{2} \, \left( v_{L}+v_{R} \right) ^{3}\Bigg ) \left( g_{0}\,T-2\,c \right) \sin (kR_{E}-\omega T+\delta ) \Bigg ] \nonumber \\&\quad -\frac{k_{\text {eff}}T{\tilde{d}}\varphi _{0}}{ \omega c}\Bigg [ \left( k{g_{0}}^{2}{T}^{2}- \left( v_{L}+\frac{v_{R}}{2} \right) g_{0}\,kT+c\, \omega \right) \left( 2\,Tg_{0}-2\,v_{L}-v_{R} \right) \nonumber \\&\quad \cdot \cos \left( kR_{E} -2\omega \,T+\delta \right) +\Bigg ( -\frac{1}{4}\,k{g_{0}}^{3}{T}^{3}+\left( v_{L}+\frac{v_{R}}{2} \right) { g_{0}}^{2}k{T}^{2}- \bigg ( \Bigg ( {v_{L}}^{2}\nonumber \\&\quad +v_{L}\,v_{R}+\frac{1}{2}\,{v_{R }}^{2} \Bigg ) k+c\,\omega \bigg ) g_{0}\,T- \left( 2 v_{L}+v_{R} \right) c\,\omega \Bigg ) \cos \left( kR_{E}-T\omega +\delta \right) \Bigg ] \end{aligned}$$

(65)