Relative acceleration noise mitigation for nanocrystal matter-wave interferometry: Applications to entangling masses via quantum gravity

Matter-wave interferometers with large momentum transfers, irrespective of specific implementations, will face a universal dephasing due to relative accelerations between the interferometric mass and the associated apparatus. Here we propose a solution that works even without actively tracking the relative accelerations: putting both the interfering mass and its associated apparatus in a freely falling capsule, so that the strongest inertial noise components vanish due to the equivalence principle. In this setting, we investigate two of the most important remaining noise sources: (a) the noninertial jitter of the experimental setup and (b) the gravity-gradient noise. We show that the former can be reduced below desired values by appropriate pressures and temperatures, while the latter can be fully mitigated in a controlled environment. We finally apply the analysis to a recent proposal for testing the quantum nature of gravity [S. Bose et al., Phys. Rev. Lett. 119, 240401 (2017)] through the entanglement of two masses undergoing interferometry. We show that the relevant entanglement witnessing is feasible with achievable levels of relative acceleration noise.

Figure 1

Conceptual scheme of the experiment as seen by a distant observer. Here we focus on the horizontal $x$ motion where the objects of mass $m$ are placed in a spatial superposition. Both the system (here depicted as two adjacent interferometers) and the experimental apparatus (here illustrated as a box of mass $M$) follow approximately geodetic motion. The deviation from ideal geodetic motion is due to gas collisions and photon scattering (here we have illustrated only dust particles outside the experimental container). To describe the experiment we consider an ideal free-falling observer and an observer attached to the experimental container. For the ideal free-falling observer also the experimental box becomes a dynamical degree of freedom (to account for its motion about the geodesic), while the observer attached to the experimental container will describe it using an accelerated reference frame with a time-dependent acceleration $a$. In addition, any external mass $m_{\text{ext}}$ will generate a small gravity gradient over the finite extension of the experiment. On the other hand, the uniform potential generated by the same mass $m_{\text{ext}}$ vanishes due to the equivalence principle: for the experimental observer both the system and the apparatus fall at the same rate toward any external mass.

Figure 2

Paths for a single-particle interferometry experiment as described from the viewpoint of an observer stationary with the experimental box; we have indicated by $a$ the residual acceleration which arises due to the collisions of the experimental box with gas particles (see Fig. 1). The paths are predominantly determined by the magnetic field gradient forces (and hence still symmetric), while the phases can have also unknown random contributions from noninertial and gravity-gradient terms. Such phases can induce a dephasing channel when there is momentum transfer between the system and experimental apparatus (i.e., when there is relative motion between the system and experimental apparatus and the two are coupled). The interferometric loop has three parts: (i) creation of superposition, (i) central part when the system is completely decoupled from the experimental apparatus, and (iii) recombination of the superposition. Interference and dephasing can be discussed only when the full interferometric loop is taken into account (see text).

Figure 3

Plot of the condition to witness entanglement given by ${\mathrm{\Phi }}_{\text{eff}}>{\mathrm{\Gamma }}_{\text{jitter}}(p,T,l,M)$ where ${\mathrm{\Phi }}_{\text{eff}}$ is the effective entanglement phase, and ${\mathrm{\Gamma }}_{\text{jitter}}$ is the damping of coherences due to noninertial jitter; the horizontal (vertical) axis denotes the box mass $M$ (box size $l$, i.e., edge length). Using the experimental values (see Sec. 4) we find the effective entanglement phase ${\mathrm{\Phi }}_{\text{eff}}\sim 0.01$ which gives the constraint ${\mathrm{\Gamma }}_{\text{gg}}\ll 0.01$. We consider the outside of the experimental box to be at room temperature, $T=300\text{K}$, and set the pressure, $p$, to the following values, ${10}^{-3}\text{Pa}$, ${10}^{-6}\text{Pa}$, ${10}^{-9}\text{Pa}$, and ${10}^{-12}\text{Pa}$; the corresponding excluded parameter space is depicted in shades of gray. The pressure inside the box is set to $\sim {10}^{-16}\text{Pa}$ and hence its effect on center-of-mass motion of the experimental box can be neglected. In addition, we have colored in light blue the region which would require capsule densities larger than of lead (Pb); the allowed parameter regime is thus restricted to the upper part of the plot. We find that noninertial jitter is successfully suppressed as long as the pressure, $p$, is low enough for a given box mass and size. Ideally we would like to have the lightest capsule mass, $M$, which would allow for simple experimental manipulation; we have indicated ideal small and heavy capsules by green points. The limiting cases are given by a $\sim 1.5\text{mm}$ ($\sim 25\text{cm}$) size capsule which would require ${10}^{-12}\mathrm{Pa}$ (${10}^{-3}\mathrm{Pa}$) depicted by the green dot in lower left (upper right) corner. We have also indicated in the plot a reference point corresponding to the ZARM short capsule, i.e., $l\sim 0.9\text{m}$ and $M\sim 200\text{kg}$ [44], well inside the allowed parameter space at $p\sim {10}^{-6}\text{Pa}$. Lowering the outside pressure and temperature would even further relax the constraints on the mass and size of the experimental box.

Figure 4

Modified QGEM protocol. A perfectly conducting plate is placed at the origin which cancels the Casimir-Polder interaction between the two masses, allowing for smaller initial separation, $d$. In particular, one can generate a higher entanglement phase for a given particle mass. However, we have to take into account the deviation of the particle trajectories due to the attractive force with the plate; we denote the displacement of the inner trajectories toward the plate by $s$ (states associated with spins ${|\downarrow \rangle }_1$ and ${|\uparrow \rangle }_2$, where the subscript denotes the particle).

Figure 5

Plot of GGN phase fluctuations, ${\mathrm{\Gamma }}_{\text{gg}}$, as a function of the minimum allowed distance of the respective GGN source from the experiment, $r_{\text{min}}$, that would still allow the detection of entanglement. Using the experimental values (see Sec. 4) we find the effective entanglement phase ${\mathrm{\Phi }}_{\text{eff}}\sim 0.01$ which gives the constraint ${\mathrm{\Gamma }}_{\text{gg}}\ll 0.01$; the allowed region is the one colored in white in the lower part of the figure. We find that seismic and atmospheric activity does not pose a significant limitation to the experiment (blue dashed line; $r_{\text{min}}\sim 0.01\text{m}$). Similarly, a human is only restricted from walking in the immediate vicinity of the experiment (light green and dark green dot-dashed lines for a jerking motion and continuous motion, respectively; $r_{\text{min}}\sim 2\text{m}$). Finally, cars and planes have to be distant by more than $\sim 10$ m and $\sim 60$ m, respectively (dark orange and light orange dotted lines, respectively).