Mesoscopic Interference for Metric and Curvature (MIMAC)&Gravitational Wave Detection

A compact detector for space-time metric and curvature is highly desirable, especially if it could also detect gravitational waves. Here we show that quantum spatial superpositions of mesoscopic objects, of the type feasible with potential advancement of techniques, can be exploited to create such a detector. By using Stern-Gerlach (SG) interferometry with masses much larger than atoms, where the interferometric signal is extracted by measuring spins, we show that accelerations as low as $5\times10^{-16}\textrm{ms}^{-2}\textrm{Hz}^{-1/2}$, as well as the frame dragging effects caused by the Earth, can be sensed. The apparatus is constructed to be non-symmetric so as to enable the direct detection of curvature and gravitational waves (GWs). In the latter context, we find that it can be used as meter sized, orientable and vibrational (thermal/seismic) noise resilient detector of mid and low frequency GWs from massive binaries (same regimes as those targeted by atom interferometers and LISA).

Matter wave interferometry, very successful with atoms [1], and implemented already with macromolecules (10 4 amu mass) [2], is gradually progressing towards ever more macroscopic masses. Several viable ideas have been proposed to date to demonstrate quantum interferometry with larger masses, primarily with foundational motivations such as testing the limits of the superposition principle [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] or exploring the quantum nature of gravity [19,20]. It is thus worthwhile to question the extent to which a large object matter wave interferometer can detect the full classical gravitational effects in a location as quantified by the metric and curvature. This comes against a backdrop of proposals of smaller particle interferometers [21][22][23] or larger quantum optomechanical systems [24,25] to detect a g 00 metric component arising from a Newtonian potential, whose variations can be used to infer the associated component of curvature or to detect the Earth's rotation [26,27] or general relativistic effects [28]. The most challenging entities to detect are the GWs, the g ij metric components, whose detection has been a huge recent success using kilometre long optical interferometers [29,30], with future devices proposed in space [31]. On the other hand there are also proposals for usage of atomic interferometers [32][33][34][35] and various quantum resonators [36,37], but nothing yet on the potential of interferometers for propagating (untrapped) objects which are much more macroscopic than single atoms.
In this paper, we will employ mesoscopic-object interference for detecting metric and curvature (MIMAC), based on the Stern-Gerlach principle [13,38]. Here, although a spatial interferometry involving superpositions of separated motional states takes place, the output signal of the interferometer is encoded in a spin degree of freedom in a manner which is insensitive to noise in the motional state (which may have thermal or seismic origin). We demonstrate that it can be used to observe the metric and, by using a non-symmetric set-up, also "directly" observe the derivatives in the interferometric signal which determine the curvature of a perturbed Minkowski metric (as opposed to measuring the metric in proximal regions and inferring the curvature by taking appropriate derivatives). Additionally, these interferometers enable the measuring of the Earth's frame dragging and gravitational waves of certain strength and frequency range. In all these cases, it is remarkable, and indeed directly due to the high masses of the objects undergoing interferometry, that the interferometer is very compact (not needing to be any larger than a meter), as well as highly sensitive at a single object level, i.e., not requiring high fluxes of the interfered objects.
Interferometric setup: The interferometric system we consider is an asymmetric modification of that proposed by Wan et. al. [13] as shown in Fig. 1. We use a mesoscopic mass containing an embedded spin 1 degree of freedom (three spin states |+1 , |0 , |−1 ). An example is a diamond crystal of nanometer to micrometer dimensions with a NV-centre spin, which is being widely probed as a candidate for the type of experiment we propose [10,12,39,40]. Imagine it to be leaving a source in a motional wavepacket |ψ(0) centred at the velocity v = (0, v y , 0) and in a spin state |0 . At time t = 0 it is initialised (by the application of a sudden microwave pulse) in a superposition of spin eigenstates 1 √ 2 (|+1 + |0 ). The presence of a magnetic field gradient in the x direction induces an acceleration a = (a, 0, 0) on the |+1 spin state (i.e., couples the spin and motional states). The acceleration of the |+1 component is reversed at a time t = τ 1 by applying a microwave pulse which accomplishes |+1 ↔ |−1 and reversed again at t = τ 2 = 3τ 1 by another identical pulse so that at any generic time while a object traverses the interferometer, the combined state of its spin and motion is 1 √ 2 (|0 |ψ 0 (t) + |σ |ψ 1 (t) ), where σ = +1 for 0 < t < τ 1 and τ 2 < t ≤ τ 3 , and σ = −1 for τ 1 ≤ t ≤ τ 2 . Thus there are two interferometric spatial paths: our procedure will first lead to a maximum spatial superposition at time t = 2τ 1 so that at that time, the centres of the spatial states |ψ 0 (2τ 1 ) and |ψ σ (2τ 1 ) are separated by ∆x = aτ 2 1 and then automatically bring the two compo- nents back so that their motional states exactly overlap at time t = τ 3 = 4τ 1 , i.e., |ψ 0 (4τ 1 ) = |ψ σ (4τ 1 ) . This has two striking consequences [13]: (i) The relative phase ∆φ between the interferometric arms is mapped on to the spin state in the form 1 √ 2 |+1 + e i∆φ |0 , so that it can be measured by measuring the spin state alone (for example, from the probability of the state to be brought to the spin state |0 by the application of a third microwave pulse). (ii) The ∆φ depends solely on the difference between phases accumulated in the interferometric paths, and is quite independent of the initial motional state |ψ(0) of the object. Thus, as long as the condition of a uniform magnetic gradient can be fulfilled, the interferometric signal is unaffected by an initial mixed thermal state or other noise (e.g. seismic) in the initial motional state as these can always be modeled as probabilistic choices of |ψ(0) .
The non-relativistic action: The phase difference between the two interferometric paths ∆φ = ∆S/h, where ∆S is the difference in action between the two arms. Consider the space-time metric g µν = η µν + h µν where η µν is the standard Minkowski metric of signature (− + ++) and h µν is some small perturbation that may have space and time dependencies. Then the action for a particle of mass m along a trajectory in the nonrelativistic limit (where the laboratory time t will be used as a replacement for the proper time due to the fact we are in the non-relativistic limit) is (1) Already by inspection of the above formula for action it is evident that compared to the term with h 00 , the terms h 0j are harder to detect as c is replaced by a nonrelativistic velocity v j , while h ij is the most difficult to detect with c 2 replaced with v i v j . On the other hand, a high value of m helps in amplifying the action and hence the phase difference. We expand S to the second order in derivatives of h µν assuming a static and slowly varying metric. This gives the difference in the action between the two interferometric paths due to the different components h µν (µ, ν = 0, x, y, z) as where all truncated terms are not pertinent to the effects explored in this letter. Note that we detect the second derivatives of h µν in the phase as well, so that spacetime curvature characterised by the Riemann tensor can be extracted from these derivatives of the perturbation as . Newtonian potential: If we only consider the first non-Minkowski term we can make the substitution h 00 = 2M G/c 2 R with the x-axis being vertical, the experiment taken to be performed at ground level so that R will be the radius of the Earth, and M Earth's mass, a difference in action between the two arms up to the second order in Eq.5 is consistent with the expectation discussed in [41] that any curvature detection will be of the form U (L/R) 2 where U is the gravitational potential and L is the characteristic laboratory length (in the above case, L ∼ ∆x).
Despite this quadratic suppression of the curvatures effect, it is still detectable due to the inverse Plank's constant factor in the phase difference it leads to. As such, we can expect to observe even second order effects (curvature effects) as large phase shifts. Fig. 2 shows how these results scale with the mass of the object in the interferometer assuming certain spatial separation ∆x between the interferometric paths being possible due to the requirements to create and maintain the coherence of such a superposition for relevant time-scales, see the discussion below. From Fig.2 it can be seen that a mass of 10 −16 kg in a ∼ 1mm interferometer with interrogation time τ 1 ∼ 100ms gives a detection of acceleration with sensitivity down to 5 × 10 −16 ms −2 Hz −1/2 where, additionally, a flux of N = 200 objects at a time is used (in this case, inter-particle interactions give only a 5% error in the phase). Frame Dragging:To explore the detection of frame dragging effects the slowly rotating metric has to be considered, see [42] ds 2 = −H (r) c 2 dt 2 + J (r) dr 2 + r 2 dθ 2 + r 2 sin 2 (θ) (dφ − Ω dt) 2 (7) In descending order first order Newtonian (blue), second order Newtonian/curvature (orange), first order frame dragging (green) and second order Frame Dragging/curvature (red). As the mass m increases, the phase change increases as ∆x = aτ 2 1 can be kept to its highest value by allowing more time τ1. However, an optimal point is reached slightly after about m = 10 −16 kg after which the ∆x obtained with the maximum τ1 starts decreasing in inverse proportion to mass even for the fixed maximum feasible values of magnetic fields (10 6 Tm −1 ). where where the binomial expansion approximation has been used for being in the linearized limit, and Ω = 2M Gν/c 2 R is the scaled angular velocity of the central rotating mass, where once again M is the mass of the Earth, R is its radius and ν is its angular velocity. The relevant component of Eq. 7 is the cross term dφdt. The apparatus can be aligned along ∆r = ∆x and ∆θ = 0 which is to say, the x direction is 'up' and y direction is aligned parallel to earth's lines of latitude, specifically we will later assume to be located at the equator. We also have ∆y giving rise to dφ, we will make the small angle approximation given the trajectories length is relatively short, hence dφ/dτ ≈ v y /r. The phase difference, again to the second order in aτ 2 1 /R , is thus given by: This phase difference is plotted as a function of mass with a fixed interferometer size in Fig. 2. Substituting all known constants, assuming the interferometer is located on the surface of the Earth, gives ∆φ (h 0j ) = 4 × 10 21 − 6 × 10 3 mav y τ 3 1 + 6 × 10 −4 ma 2 v y τ 5 1 .
Once again we note that greater sensitivities can be achieved with larger mass particles used in the interferometer. We can also see that the frame dragging caused by Earth has a significantly more modest effect on phase. It also suggests high precision measurements would be needed to be able to measure the second order derivatives of the metric perturbations due to frame dragging due to Earth's rotation. In Fig. 2 we have plotted the phase ∆φ with respect to the mass of the object for our apparatus situated on Earth, where we have taken Earth's rotation and the radius. Gravitational waves (GWs): Our setup can also extract the phase from the transverse traceless perturbations around the Minkowski background: we assume that a GW is propagating along the x 3 = z direction perpendicularly to the interferometer with angular frequency ω, the two helicity states of the GWs are h + , h × 1. We will also ignore the kinetic energy component of the atoms in action, see Eq.(1), as it is not relevant for the purpose of detecting the phase. The GW induced phase difference for our apparatus is thus given by: where ψ 0 is the wave's phase at t = 2τ 1 and the expansion is around ωτ 3 ≈ 0, i.e., the GW is assumed to not vary appreciably over the length of the interferometer. Note that the h × component is not recorded in our interferometer, as it is proportional to v x v y which varies between positive and negative values, thus cancelling itself out. The h + component is a function of v 2 x , and therefore does not cancel in this way. Essentially to detect the h × component, one has to rotate our apparatus by 45 degrees.
At this point, it is worthwhile to compare our proposal with other interferometric schemes for GW detection, although we acknowledge that our scheme has much to develop as here we are only showing its "in principle" feasibility with certain achievable advances in technology. In the domain of atomic interferometry, one of the most advanced of these suggestions is the Atomic GW Interferometric Sensor (AGIS) as discussed in [43] which generates an approximate phase difference of ∼ 10 16 h + for the space based detector [35] with baseline size L ∼ 10 7 m compared to our value of ∼ 10 17 h + for a baseline size of ∆x = 1 m as shown in Fig. 3. Note however, that our proposal differs significantly from AGIS and so the phase difference they are referring to is between two different atom interferometers, while our value is the phase difference between the two arms of the one interferometer. As such this comparison, though worth making, is not intended to capture the entire effectiveness of these two proposals. Indeed single atom interferometers have also been suggested for GW detection [32][33][34]. With respect to those, our advantage stems purely from the much larger m of our massive particle interferometers as our Stern-Gerlach methodology opens up the scope to create a high enough ∆x, even as the mass is increased. As far as optical interferometric setups such as LISA are concerned, which is the frequency domain in which our interferometer is most effective, one can make a comparison by noting that in our case, the path length differences of ∼ h + L are essentially being measured in units of the matter wave de Broglie wavelength, which can be 10 −17 times smaller than typical optical wavelengths through our Stern-Gerlach scheme. Thus the lengths L required can be much smaller (a meter suffices).
With respect to the frequency spectrum observable using this technique, one can see from Eq.13 that the phase output will be independent of GW frequency provided ωτ 3 ∼ ωτ 1 1 as seen in Fig. 3. This and the higher frequency detectability scaling can be understood by noting it is susceptible to the average wave amplitude over the time-frame of the interferometer, which tends to zero for higher frequency waves. Note that here we define a detectable strain as one that gives ∆φ (h ij ) = 1. However, if there are several particles traversing the interferometer at once, as well as several interferometers in parallel, so that the phase signal is to be read from N atoms in one shot of the apparatus run, then smaller strain causing ∆φ (h ij ) = 1/ √ N is detectable. Further note that around 10 − 10000 Hz, at which LIGO is performing [44], our setup will not be able to compete. However it will serve as a complementary procedure in the range of eLISA [45] (10 −6 − 10 Hz).
Practical implementation: In the proposed system, a magnetic field gradient ∂ x B is used to create the spatial superposition of size ∆x = aτ 2 1 with a = g N V µ B ∂ x B/m where g N V is the Landé g factor, µ B is the Bohr magneton. For large mass interferometry to carry advantage over its atomic counterpart, ∆x must be kept significant even while m increases. Thus in proportion to the value of m we want to use, we have to increase (i) the magnetic field gradient, (ii) the coherence time of the spatial and the spin states. A magnetic gradient as high as 10 6 Tm −1 can be achieved at a distance of 1 µm from a 10µm sized superconducting magnet trapping a flux of 5 T (larger values have been shown to be feasible in experiments [46]). The difficult task of keeping the magnet consistently about 1µm from the interfering object can be achieved by shaping an appropriate elongated magnet or by moving the magnet in tandem with the motion of the object corresponding to the |+1 spin state. The spatial coherence offers a huge window under low values of pressures P = 10 −15 Pa and low internal temperatures 10 mK, as already used in previous proposals [19,47], being for a mass of ∼ 10 −17 kg (100 nm radius), using the results of [47], γ air ≈ 0.006Hz due to scattering of air molecules and γ rad ≈ 6 × 10 −4 Hz due to black-body photon emission. The electron spin coherence at 10 mK can also reach one second with dynamical decoupling [48,49] which is naturally present here due to the spin flipping pulses. This can be further extended by switching the accelerating/decelerating path from the |±1 state to the |0 state (for the times that it is in the state |0 , the mass essentially undergoes free motion with its acquired velocity used to increase/decrease separations). Considering the most difficult metric component to detect, namely the GWs, the greatest sensitivity of detecting h + ∼ 10 −17 / √ Hz will occur for the mass ∼ 10 −17 kg. We can further stretch this sensitivity to h + ∼ 10 −19 / √ Hz by considering a flux of N = 400 particles traversing the interferometer at once, which is consistent with the effect of their mutual gravitational and Casimir-Polder interactions on the phase to be negligible (∼ 10 −3 radians) over their apparatus traversal time. For detecting the frame dragging, we need a v y , which can be 10 ms −1 . These speeds can be achieved for polarizable particle such as nanodiamond using rapid acceleration in a pulsed optical field [50]. Note that the small mass flux was taken to be N = 10 6 taken from [28] for 87Rb atoms. For higher frequencies the relative phase between the paths undergoes several oscillations while the object traverses the interferometer, so cancelling itself out. The the final phase difference is then something that accumulates over a lower time, leading to to a lower sensitivity.
We have presented a protocol for a compact (meter scale) interferometer for objects of mass ∼ 10 −17 kg which can not only detect metric components of Newtonian potentials, but also the Earth's frame dragging and GWs of low frequency. The Stern-Gerlach principle implies that simply by changing the orientation of a magnet, the whole interferometer is re-oriented to identify the angular origin of sources. Moreover, the compactness also implies that a large number of interferometers can be built to identify localized noise sources such as gravity gradients and cancel them. On site, the sensi-tivity can be modulated by changing the magnetic field gradient (say, by moving the magnet) so as to identify terms of decreasing strength in succession starting from the Newtonian term and reaching up to the gravitational waves (re-orientation can also aid this). By construction, our interferometric signal only depends on the relative phase between the two arms and thereby is immune to thermal and seismic noise in the initial wavepacket of the mass. Moreover, here the two paths, separated at most by a meter, are unlikely to suffer independent motional noise, while, if a static (and appropriately shaped) magnet is rigidly connected to the source of the objects, then fluctuations of the combined system during the interferometry will not matter. We leave a quantitative anal-ysis of noises, following, for example, the procedures of Refs. [35,51] for the future. Though the proposed implementation seriously stretches the magnetic field gradients and coherence times, much lower values of both should suffice to detect the less demanding components such as h 00 or for functioning as an accelerometer (for example, B = 10 4 Tm −1 and τ 1 ∼ 70 ms can already detect both the Newtonian curvature, as well as the Earth's frame dragging, where the mass used is 10 −18 kgs and ∆x = 1 mm.). Furthermore, we may be able to test modifications of gravity at short distances [52,53], and aspects of selflocalization of the wavefunction in its own gravitational potential [54,55].