The presence of χ particles can be detected by the momentum they impart to a test particle via elastic collisions with the constituent nucleons; this recoil is measurable in either a classical detection scheme or via the reduction in fringe visibility in a matter-wave interferometer. Our candidate DM particles are sufficiently light and numerous that we do not expect to resolve individual scattering events; rather, we expect an overall drift in the direction of the DM and a very small Brownian-like diffusion.

Scaling with target particle size

A consequence of the low χ mass is that the de Broglie wavelength is large compared to the internuclear separation in normal matter, . The χ hence scatters coherently from the constituent nuclei. For small particles under the Born approximation, all nuclei are subject to the same field from the incident χ and we find an effective cross-section σ eff = σN2. Conversely, for large particles, the flux is attenuated and the cross-section is the projected surface area σ eff ∝ N2/3. In the intermediate regime, details of the interaction depend strongly on particle shape and on whether the underlying interaction is attractive or repulsive. For illustration, we consider a spherical particle with an attractive potential and we calculate the interaction via partial waves, described in further in the supplementary material; the expected acceleration a = σ eff P/M, where M is the particle mass, as shown in FIG. 2 a, reduces to the Born approximation and to the geometrical approximation in the respective limits. Details of size-dependent acceleration in the intermediate regime, if observed, will allow for an independent measurement of the χ DM pressure P and collisional cross-section σ.

Figure 2 (a), Acceleration of a silicon test particle (nucleon number density 1.4 · 1030 m−3) across the size regimes for χ de Broglie wavelength . For small particles ( ), the Born approximation holds and acceleration is proportional to nucleon number; for large particles ( ), the force is proportional to projected area and thus increases slower than the inertia. In the intermediate regime ( ), acceleration depends strongly upon the particle shape: for illustration we have chosen a spherical particle with an attractive interaction; the repulsive case is similar. Resonances, which distract from the main argument, have been smoothed by a few times their width. Similar plots are obtained for other de Broglie wavelengths and the limiting cases are unaffected. (b), Reduction in sinusoidal fringe visibility due to elastic collisions for a range of m χ . Experiments with a similar geometry and path separation are indicated: state-of-the-art experiments have demonstrated 104 nucleons21; an experiment with 106 is proposed22; and space-based ‘MAQRO’23 will span the necessary range. For , the Born approximation for scattering χ particles is not well satisfied and further theoretical work is needed to fully describe the decoherence. Full size image

Dark Matter optics

For macroscopic objects, χ particles experience an average potential and, in close analogy with neutron optics24, the interaction may be described using a refractive index , where we identify the ‘critical wavelength’ , with n being the number-density of nucleons in the material and the scattering length a s = ±2 fm is found via the low-energy limit in which . The uncertainty in sign (and thus whether λ c is real or imaginary) arises because the cross-section is insensitive to whether the underlying interaction is attractive (−) or repulsive (+). For typical materials, and we expect χ particles to be strongly reflected.

Acceleration of a mesoscopic particle

Given the possibility of a measurable effect upon nanometre-sized particles and the uncertainty about whether χ particles will penetrate the Earth's atmosphere, we propose a space-based experiment, as illustrated in FIG. 3. Particle radii in the range 10 nm ≤ r ≤ 1 µm are expected to show accelerations , with possibly much higher values and a rich size-dependent structure. Recently, 140 nm particles have been held in vacuum in a 120 kHz harmonic trap provided by a tight laser focus and feedback ‘cooled’ to reduce the uncertainty in both their position (<1 nm) and velocity (500 µm/s)25. For a thermal state, the velocity uncertainty is the product of trap frequency and position uncertainty and, in ultra-high vacuum where gas collisions are negligible, one may decrease the trap frequency considerably; for a 10 kHz trap frequency, we expect a velocity uncertainty below 50 µm/s. After several minutes of free-flight under these conditions, the positional uncertainty will be sub-millimetre while acceleration from collisions with χ particles will give a millimetre-sized displacement. The effect is also observable without any such improvements; the displacement will be revealed in the statistics of position measurements.

Figure 3 Illustration of the suggested experiment, the hardware for which can be provided by the proposed ‘MAQRO’ space-craft23. (a), Location of the space-craft at Lagrange point 2 in the context of our solar system (not to scale). (b), Close-up of the optical arrangement: a compound objective lens provides high numerical aperture focusing for laser light to create a gradient-force dipole trap for a micron-scale particle. Light, which diverges strongly after the particle, is collected by a lens. Interference between the laser light and the light scattered coherently by the particle gives rise to a difference in intensity across the cross-section which, when measured by balanced photodiodes (PDs), provides sub-wavelength position information in three dimensions25. (c), A further close-up, showing s-wave scattering of a χ DM particle, with an approximately plane-wave incident wavefunction and an example scattering outgoing direction with the associated recoil of the test particle. Full size image

Matter-wave decoherence

The prediction of an acceleration is based upon the assumption that the Earth moves through the local DM distribution at some appreciable speed. However this local distribution is uncertain particularly for this yet-to-be-simulated DM candidate, so here we propose a detection scheme which does not rely on some overall drift.

Elastic scattering events can be interpreted as revealing partial which-way information or, in a more complete treatment including recoil, diffusing momentum in a quantum Brownian Motion26. While individual collisions may not affect the visibility significantly, many such events will have a measurable effect. A proposed space-based matter-wave nanoparticle interferometer23 will be sensitive to this decoherence mechanism and the effect can be controllably extinguished by shielding the nanoparticle from the DM flux. We analyse the decoherence for a similar interferometer22, where a nanoparticle, prepared in a thermal state of a harmonic oscillator via feedback cooling, provides a point-like source for a near-field (Fresnel region) Talbot interferometer using a phase grating of period Λ provided by a standing light-wave.