Multiloop atom interferometer measurements of chameleon dark energy in microgravity

Chameleon field is one of the promising candidates of dark energy scalar fields. As in all viable candidate field theories, a screening mechanism is implemented to be consistent with all existing tests of general relativity. The screening effect in the chameleon theory manifests its influence limited only to the thin outer layer of a bulk object, thus producing extra forces orders of magnitude weaker than that of the gravitational force of the bulk. For pointlike particles such as atoms, the depth of screening is larger than the size of the particle, such that the screening mechanism is ineffective and the chameleon force is fully expressed on the atomic test particles. Extra force measurements using atom interferometry are thus much more sensitive than bulk mass based measurements, and indeed have placed the most stringent constraints on the parameters characterizing chameleon field. In this paper, we present a conceptual measurement approach for chameleon force detection using atom interferometry in microgravity, in which multiloop atom interferometers exploit specially designed periodic modulation of chameleon fields. We show that major systematics of the dark energy force measurements, i.e., effects of gravitational forces and their gradients, can be suppressed below all hypothetical chameleon signals in the parameter space of interest.

Figure 1

Exclusion plots of chameleon parameters. (a) $(\beta ,\mathrm{\Lambda })$ exclusion plot for $n=1$. Region near the upper left corner (yellow) is excluded by the torsion balance experiment [10]. Region bounded by the blue curve is excluded by the atom interferometer experiment in Ref. [8]. The blue diamond marks the parameter set of $(n,\mathrm{\Lambda },\beta )=(1,0.1\mathrm{meV},{10}^3)$, and the blue dashed line across it labels parameters that would produce the same chameleon force. Failure of detecting accelerations calculated in this work based on this parameter set will exclude the shaded region above the dashed line. $\mathrm{\Lambda }={\mathrm{\Lambda }}_0$ is of cosmological relevance, and is shown as the horizontal line. (b) $(n,\beta )$ exclusion plot for $\mathrm{\Lambda }={\mathrm{\Lambda }}_0$. Regions excluded by the corresponding experiments are labelled. The red diamond marks the embracive parameter set of $(n,\mathrm{\Lambda },\beta )=(10,{\mathrm{\Lambda }}_0,1)$, and the red dashed curve across it labels parameters that would produce the same chameleon force. Failure of detecting accelerations calculated in this work based on this parameter set will exclude the shaded region above the dashed curve.

Figure 2

Chameleon fields inside spherical shells of different radii $R$ as a function of distance to the center of the shell. Upper figure: $(n,\mathrm{\Lambda },\beta )=(1,0.1\mathrm{meV},{10}^3)$. Lower figure: The embracive set $(n,\mathrm{\Lambda },\beta )=(10,2.4\mathrm{meV},1)$.

Figure 3

Depiction of dual AIs in a spherical shell. Left: Two AIs moving from the center towards the sides driven by the same retroreflected laser pulses. Right: Space-time diagram of two AIs. In the Ramsey-Bordé configuration (bold) four $\pi /2$ pulses are applied with separation time $T,T_1$, and $T$, while in the Mach-Zehnder configuration (dashed) a $\pi /2\text{−}\pi \text{−}\pi /2$ pulse sequence is applied with $T_1=0$.

Figure 4

A periodic structure and the corresponding chameleon field. (a) Depiction of a tube with six dividers and a center bore. (b) The profile of the chameleon field inside a structure with the following dimensions: tube length of 20 cm, tube inner radius of 10 cm, center bore diameter of 1 cm, and divider thickness of 1 mm. The chameleon field is for parameters $(n,\mathrm{\Lambda },\beta )=(1,0.1\mathrm{meV},{10}^3)$. Note that additional end caps at the top and the bottom of the tube are assumed in the calculation. (c) Chameleon acceleration on the atoms along the symmetry axis.

Figure 5

Axial acceleration for the embracive chameleon parameters $(n,\mathrm{\Lambda },\beta )=(10,2.4\mathrm{meV},1)$. Blue: axial acceleration on the symmetry axis. Red: axial acceleration 2.5 mm away from the symmetry axis.

Figure 6

Space-time diagram of dual 10-loop AIs using two-photon beamsplitters $(n_b=1)$. Arrows on the plot indicate AI pulses at various times, with longer and shorter arrows representing $\pi$-pulses and $\pi /2$-pulses, respectively. Lower traces (blue and red) form a 10-loop AI, while upper traces (green and yellow) form another 10-loop AI. Two AIs are derived from a single source by a $\pi /2$-pulse applied in advance of the AI start time ($t=0$).

Figure 7

AI phases of 10-loop AIs. (a) Phase $\psi$ of a 10-loop AI as a function of $T$ and $d$. (b) Differential phase $\delta \psi$ between dual 10-loop AIs as a function of $T$ and $\delta d$, assuming one starts at $d=0$.

Figure 8

A periodic structure with 12 dividers and trim masses, and the corresponding gravity and chameleon accelerations. (a) Sketch of the structure. The dimensions are: Tube length 20 cm, radius 1 cm, thickness 1 mm, center bore diameter 1 cm, divider thickness 0.1 mm, trim ring thickness 0.962 mm ($1/16$ of the divider spacing) radially extended by 8.8 mm from the tube body. The material density is ${\rho }_{\mathrm{wall}}=2.7\mathrm{g}/{\mathrm{cm}}^3$ (aluminium) rather than $7\mathrm{g}/{\mathrm{cm}}^3$ (steel) in previous calculations. (b) The axial chameleon accelerations for the embracive parameters. Shown in red are accelerations 2.5 mm away from the symmetry axis. The calculation is performed with end caps. (c) The axial gravitational acceleration. Additional trim blocks are added on both ends to null the gravity gradient at the middle of the tube, for better visualization of the acceleration modulation. Shown in red are accelerations 2.5 mm away from the symmetry axis. Note that the gravitational acceleration is smaller in amplitude and has different periodicity from the chameleon acceleration.

Figure 9

Differential phase between dual 10-loop AIs as a function of $T$ and $\delta d$, with one AI starting at $d=0$. Both plots assume $a_p=10\mathrm{pm}/{\mathrm{s}}^2$, $v_0=0$, and $n_b=2$. (a) $1/K=0.2/13\mathrm{m}$, corresponding to the periodicity of the chameleon acceleration shown in Fig. 8. (b) $1/K=0.2/26\mathrm{m}$, corresponding to the periodicity of the gravitational acceleration shown in Fig. 8.