Decoherence as a way to measure extremely soft collisions with dark matter

A new frontier in the search for dark matter (DM) is based on the idea of detecting the decoherence caused by DM scattering against a mesoscopic superposition of normal matter. Such superpositions are uniquely sensitive to very small momentum transfers from new particles and forces, especially DM with a mass below 100 MeV. Here we investigate what sorts of dark sectors are inaccessible with existing methods but would induce noticeable decoherence in the next generation of matter interferometers. We show that very soft but medium range ($0.1\mathrm{nm}–1\mu \mathrm{m}$) elastic interactions between nuclei and DM are particularly suitable. We construct toy models for such interactions, discuss existing constraints, and delineate the expected sensitivity of forthcoming experiments. The first hints of DM in these devices would appear as small variations in the anomalous decoherence rate with a period of one sidereal day. This is a generic signature of interstellar sources of decoherence, clearly distinguishing it from terrestrial backgrounds. The OTIMA experiment under development in Vienna will begin to probe Earth-thermalizing DM once sidereal variations in the background decoherence rate are pushed below one part in a hundred for superposed 5-nm gold nanoparticles. The proposals by Bateman et al. and Geraci et al. could be similarly sensitive although they would require at least a month of data taking. DM that is absorbed or elastically reflected by the Earth, and so avoids a greenhouse density enhancement, would not be detectable by those three experiments. On the other hand, the aggressive proposals of the MAQRO collaboration and Pino et al. would immediately open up many orders of magnitude in DM mass, interaction range, and coupling strength, regardless of how DM behaves in bulk matter.

Figure 1

Existing experimental constraints on possible new Yukawa forces. The region shaded solid gray has been excluded directly by neutron-lead scattering [19] (light green), measurements of the neutron electric form factor [20] (orange), neutron optics experiments [19] (purple), Casimir forces measurements [21] (red), and torsion balances [21] (dark green). The region hashed gray is excluded by the strong but somewhat model-dependent constraint arising from the rate of stellar cooling [22, 23] (blue). The dashed box encloses our region of primary interest, investigated in Figs. 4 and 5.

Figure 2

Interferometers vs. resonators. Matter interferometers, (a), superpose objects over a distance comparable to or larger than their physical size (nanometers, generally). In contrast, quantum nanomechanical resonators, (b), are typically much larger (microns) but are superposed over much shorter distances (femtometers). Scattered environmental particles can carry off which-path ($L$ or $R$) information, decohering the superposition. However, a particle scattered from a region jointly shared by both parts of the superposition intuitively does not acquire which-path information and hence does not contribute to the decoherence rate. This suppression of decoherence can be confirmed analytically and applies even for momentum transfers too small to be localized within the superposed object.

Figure 3

Sidereal daily variation. Consider the OTIMA experiment at lattitude $48°\mathrm{N}$ (Vienna, Austria) with the spatial coherence vector $\mathbf{\Delta }\mathbf{x}$ (i.e., the direction connecting different slits) pointed horizontal and oriented at 70° with respect to geographic north. Let the DM be characterized by $M=1\mathrm{keV}$ and $m=200\mathrm{eV}$ (inset: $M=1\mathrm{MeV}$ and $m=20\mathrm{eV}$). In accordance with the Solar System’s velocity through the Milky Way, the apparent wind blows at $v_{\odot }\approx {\overline{v}}_{\mathrm{DM}}$ from a $+38°$ declination. The total DM flux (black, solid) varies over the course of the day due to shielding by the Earth. The flux is largest when the DM wind blows down from overhead, and smallest 12 hours later when the experiment is inside the partial shadow cast by the Earth. The variation in the DM-induced decoherence rate (colored lines) tracks the variation in the overall flux, both when the DM isotropizes (blue, dotted) or does not isotropize (red, dashed) in the atmosphere before reaching ground level. Variations in the typical DM momentum with respect to $\mathbf{\Delta }\mathbf{x}$ give only modest corrections. There is a small relative phase shift between the DM flux and the unisotropized decoherence rate in the main plot because the dominant momentum transfer is comparable to $\mathrm{\Delta }x$, so the rate of decoherence is sensitive to the orientation of the DM distribution even with the energy spectrum fixed. This is not true for the heavier DM shown in the inset, which consequently does not exhibit the asymmetry. These order-unity daily variations of the decoherence rate are a generic feature of all DM scenarios coupled strongly enough to be shielded by the Earth. The absolute phase with respect to the solar day drifts together smoothly over the course of the year because the DM has interstellar origins and tracks the sidereal day, allowing it to be distinguished from terrestrial sources of decoherence.

Figure 4

Interferometer sensitivity to MeV dark matter. Each point in the $m,{\alpha }_{\mathrm{M}}$ plane corresponds to a DM scenario with fixed $M=1\mathrm{MeV}$, ${\alpha }_{\mathrm{DM}}=1$. The characteristic length scale ${ƛ}_m=m^{-1}$ of the Yukawa potential is given by the inverse mass of the associated mediator. The solid black line bounds the gray and hashed-gray regions that have already been excluded, as detailed in Fig. 1. Three gray dotted curves depict the progressively larger minimum couplings ${\alpha }_{\mathrm{M}}$ for which DM incident on the atmosphere scatters at least once, is isotropized, and thermalizes before it reaches the ground. The first two coincide when the interaction is short range, ${ƛ}_m\ll {ƛ}_{\mathrm{DM}}\sim 0.26\mathrm{nm}$. Under the conservative assumption that all DM incident on the Earth’s crust is absorbed, the solid colored lines denote the limit of sensitivity for five proposed interferometer experiments: OTIMA [29, 31, 32], Bateman et al. [37], Geraci et al. [28], the MAQRO satellite [34, 35, 36], and Pino et al. [38]. (Because of atmospheric shielding, the experiments could only detect DM where the sensitivity line dips below the isotropization curve, as discussed in Appendix pp2.) Nonexcluded regions for which DM would induce detectable decoherence in the experiments are shaded in the corresponding color, and account for atmospheric effects. If the surface of the Earth isotropically reflects incident DM, then the sensitivity lines would shift slightly downward (more sensitive) by roughly a factor of ${\log }_{10}(2)\approx 0.3$ and would not need to pass below the isotropization curve. The sensitivity lines end when the Born approximation breaks down; as described in Appendix pp6, this indicates that the cross section has become of the order of the geometric cross section, and the sensitivity would generally saturate for arbitrarily large ${\alpha }_{\mathrm{M}}$ although there can be sensitivity enhancements for resonant scattering with attractive potentials. The dashed colored lines denote the increased sensitivity if there is a DM greenhouse effect, i.e., if rather than being absorbed or reflected, DM incident on the Earth’s surface thermalizes to the temperature of the upper crust and is re-emitted (see Appendix pp3). We do not include greenhouse lines for the MAQRO satellite or the Pino et al. proposal; the former operates away from the Earth, and the latter would need to account for scattering lengths that could exceed the dimensions of the atmosphere or the Earth.

Figure 5

Interferometer sensitivity for other DM masses. The same quantities displayed in Fig. 4 with $M=1\mathrm{keV}$, 10 keV, 100 keV, and 10 MeV. The dot-dashed lines for MAQRO and Pino et al. when $M=1\mathrm{keV}$ bound the regions with a detectable phase shift from the DM wind [14] that exceeds the decoherence rate. (See Appendix pp4.) Due to scattering in the atmosphere at this mass, the DM distribution seen by the other four experiments would always be isotropic, implying no net DM wind and hence no phase shift. The greenhouse scenario (dashed colored lines) only increases the sensitivity when the Earth is cooler than the temperature of incident DM, translating to $M\gtrsim 120\mathrm{keV}$. For smaller masses, the dominant source of decoherence is just the unthermalized component of the DM distribution, so greenhouse effects have little effect on sensitivity.