Macroscopic superpositions and gravimetry with quantum magnetomechanics

Precision measurements of gravity can provide tests of fundamental physics and are of broad practical interest for metrology. We propose a scheme for absolute gravimetry using a quantum magnetomechanical system consisting of a magnetically trapped superconducting resonator whose motion is controlled and measured by a nearby RF-SQUID or flux qubit. By driving the mechanical massive resonator to be in a macroscopic superposition of two different heights our we predict that our interferometry protocol could, subject to systematic errors, achieve a gravimetric sensitivity of Δg/g ~ 2.2 × 10−10 Hz−1/2, with a spatial resolution of a few nanometres. This sensitivity and spatial resolution exceeds the precision of current state of the art atom-interferometric and corner-cube gravimeters by more than an order of magnitude, and unlike classical superconducting interferometers produces an absolute rather than relative measurement of gravity. In addition, our scheme takes measurements at ~10 kHz, a region where the ambient vibrational noise spectrum is heavily suppressed compared the ~10 Hz region relevant for current cold atom gravimeters.

for the magnetized sphere described in the main paper is where V is the sphere volume. Since B = ∇ × A in cylindrical coordinates we have (A2) To calculate the vertical trapping frequency ω, we require the force exerted on the resonator ring as a function of displacement from the equilibrium point. This force arises due to the fact that when the ring moves, the magnetic flux threading through it will change. As flux lines can't pass through the superconducting ring, however, a current arises in the ring to restore the flux, and this current gives rise to a Lorentz force.
Since the resonator ring is horizontal, the flux through it is given by the line integral Changes in current in the resonator as it moves vertically are related to changes in flux via where L r is the self inductance of the ring and is given by L r = µ 0 R r (ln[8R r /a] − 2).
Taking the equilibrium vertical position point z = z eq , then for small displacements the current in the resonator is I r = (z − z eq ) · dI r /dz| z=z eq where, using (A4), dI r dz z=z eq = 3µ 0 MVR 2 r z eq 2L(R 2 r + z 2 eq ) 5/2 .
The Lorentz force from the current, magnitude I r , flowing through a small element dl of the wire is given by dF = (I r dl) × B. Assuming the resonator is circular, sitting horizontally, and is co-axial with theẑ−axis, the line element dl will always be perpendicular to B = B radialρ + B axialẑ . Hence the vertical force on the resonator for small vertical displacements from equilibrium z − z eq , is Finally, the equation of motion in the z direction is for small displacements, providing an harmonic restoring force. Comparing (A6) and (A7) we find the vertical oscillation frequency We also need to consider transverse trapping and oscillations firstly to establish that the resonator is indeed trapped in all three directions, and secondly to determine if there is any coupling between the vertical and horizontal motions. If this coupling exists then by cooling the vertical motion one cools the entire motion of the resonator, but such couplings can also lead to unwanted energy leakage from the coherent vertical dynamics to the transverse modes, leading to decoherence of our vertical superposition states.
When considering the horizontal movement of the resonator we break the cylindrical symmetry, meaning it is easier to work in Cartesian coordinates. In this coordinate system the magnetic vector potential is given by (A9) Due to the coordinate system, rather than a circular resonator, we consider a square resonator of width 2w, and wire radius a, and assume it is displaced sideways along the x−axis a small amount δx.
We can calculate the flux through the resonator at a displaced position δx via Eq. (A3), and expand the result in a Taylor series in δx. To third order we get The zeroth-order term is a constant for motion along the x−direction and can be ignored. Using (A4) modified for x−directional motion we obtain the dependence of the induced current on δx, where we have used the fact that self-inductance of a square loop is L = 2µ 0 w(log[w/a]−0.774)/π.
Using the Lorentz force law as in the previous section, we can integrate the loop current in the presence of the magnetic field and obtain the resulting force. Renaming the small displacements δx → x and similarly for y, z from the equilibrium point (0, 0, z eq ), we find that to lowest order the x−component of this force is F x = −β x 3 with β > 0, and at equilibrium the resonator is transversely trapped in a pure anharmonic potential. As these forces come from a conservative potential we can integrate along paths to obtain the leading terms for the potential of the system which describes a type of cross-mode coupling. For parameters described in Table I  In order to put our resonator into a cat state and use it as a gravimeter, it is necessary to ensure that we can begin with it in the motional ground state. This in turn requires that we have mechanism to cool it from its initial non-equilibrium state to the ground state by removing energy.
Details of the cooling scheme we use can be found in Refs. [1,2], which we briefly summarize here. We cool by coupling a two level system (the qubit) to the resonator, with the qubit coupled to a bosonic thermal bath. The Hamiltonian for the coupled system iŝ where Ω is the Rabi frequency with which we drive the qubit, δ is the detuning of the driving field from resonance with the qubit frequency splitting ω q ,â is the annihilation operator for the resonator oscillation modes and theσ x,z are the standard spin-1/2 Pauli operators.
The open systems dynamics of the qubit-resonator system is described in Sec. E and is characterised by Γ and Γ ⊥ , the amplitude damping rates of the resonator and qubit respectively. The initial state of the resonator is modelled as a coherent state with amplitude α = √ N th where the initial occupation number is N th = (e ω/k B T r − 1) −1 with T r an effective bath temperature for the environment of the resonator. In the limit where λ Γ ⊥ , ω, the final phonon occupation number for the resonator, n f , is given by Here ζ = Γ/Γ c (0) and the renormalized cooling rate is . The qubit polarization Fourier components, S z 1 and S z −1 , are given by the solutions to the Bloch equations for the qubit. In the Lamb-Dicke regime (λ √ N th + 1/2 Γ ⊥ , ω) one can obtain an effective master equation for the resonator after tracing out the qubit. This gives a new effective resonator damping rate where S (ν) denotes the qubit fluctuation spectrum and is given by where · 0 denotes the steady state expectation. The resulting steady state phonon occupation of the resonator in the Lamb-Dicke regime is [2] where In Figure 1 we plot the performance of this cooling scheme for our system, showing both the full cooling solution and a simplified cooling solution that makes the assumption that we are always in the Lamb-Dicke regime, i.e. Eq. (B4) holds for all initial resonator temperatures. The plot shows that even with initial phonon occupation numbers as high as ∼ 10 9 we can cool the resonator to the ground state, with an average final occupation number of 0.16.
The timescale governing the cooling is given by the effective resonator cooling rate Γ cool . For our system, using parameters given in the caption of Figure 1, we obtain Γ cool = 27 kHz. We note this cooling rate scales as λ 2 , and we have chosen a very conservative coupling rate of λ = 10 kHz. We have assumed the standard resonator parameters given in Table I, and take Ω = ω/2, δ = − √ ω 2 − Ω 2 , λ/2π = 10 4 Hz, decoherence times T 1 = T 2 = 70 µs, ω q /2π = 6 GHz, and Γ = 2.70 × 10 −8 Hz. See Table I for the definitions of system parameters.
As our system is capable of coupling strengths of up to λ ∼ 1 GHz, the cooling can be made much faster if required.
Appendix C: Ancillary parameter determination While our measurement protocol and phase estimation scheme gives us a phase, this phase must still be converted to a value for g via Clearly, in order to obtain a precise estimate of g, we must know the parameters m, ω, ω q and λ to the same level of precision. These quantities can be measured offline with any additional resources, and will not affect the time taken for the phase estimation protocol.
One way to obtain information on these parameters is to observe the effect on the evolution of the qubit. If the qubit is not driven, i.e. Ω = 0, then the equations of motion associated with the full coupled HamiltonianĤ If we assume the resonator starts in the ground state then we have â(0) +â † (0) = 0. Denoting where These solutions have intricate time-dependent structure, meaning an arbitrary number of independent datapoints can be obtained by measuring, say,σ x on the qubit. Provided the qubit preparation and measurement process has only statistical errors and not systematic ones, arbitrarily precise values of ω q , ω and λ can be obtained by fitting a suitably large number of measurement results against the theoretically expected profile.
In order to measure the mass of the resonator, techniques such as those described by Schilling are likely to perform well [3,4]. These schemes utilize electro-optical measurement of oscillation period of a levitated superconducting oscillator, exactly the same situation as described by our scheme.
Of course, if other simpler or more precise methods are available that can provide values for any of these parameters, they can be used in the calibration process and reduce the number of parameters that need to be fitted.
Appendix D: Decoherence

Quality Factor
In the subsequent discussions we make use of the quality factor Q of the mechanical oscillations of our resonator. One usual definition of Q is given by where P is the power loss, ω is the oscillation frequency and ω is the energy of the system. In our protocol, however, the resonator is not in the motional ground state -the resonator is oscillating back and forth with a large amplitude (several nanometers). As we begin the interferometry protocol (or slosh), with the resonator high up on a potential hill, we have V(l) = 1 2 mω 2 l 2 where l = λz 0 /ω is the displacement from equilibrium, and z 0 = √ /2mω is the harmonic oscillator ground state extent. This means for our system we have V = λ 2 /4ω. Associating this potential energy with the energy in (D1) we obtain To compute an associated decoherence rate we use We also note that for all the calculations in this section we use the system parameters described in Table I.

Qubit dephasing
The effect of qubit decoherence on the evolution of the joint qubit-resonator system is solved for in Supp Material E, The main result is that the off diagonal elements of the qubit density matrix, which carry the gravitationally induced phase accumulation, will experience exponential decay by a factor e −τ/T 2 where T 2 is the qubit dephasing time. Dephasing rates vary greatly with the superconducting circuit architecture. Recent experiments with superconducting flux qubits in 3D microwave cavities have reported decoherence times of T echo 2 > 19 µs [5], while decoherence times of T echo 2 > 100 µs have been reported for transmon qubits in 3D cavities [6].

Decoherence due to eddy currents in the magnet
As the resonator oscillates, carries currents, and is in close proximity to the magnetised sphere, it will inductively induce eddy currents in the sphere, which will result in power loss as the magnetic material has electrical resistance. In order to estimate this effect we consider infinitesimal where r 0 is the minimum distance from the bottom of the sphere to the centre of the resonator, and ρ is the resistivity of the magnetic material. Our sphere is composed of YIG, which has ρ = 10 12 Ωm; we take the I r to be the largest current reached in the resonator (occurring at full displacement), i.e. I rmax ∼ 48 µA. This gives the power loss due to eddy currents in the YIG sphere as P = 6.2 × 10 −38 W, which via (D2) corresponds to a quality factor of Q = ω 2 /P = 3.1 × 10 22 and a decoherence rate of Γ eddy = 8.1 × 10 −19 s −1 .

Dipole radiation
An oscillating loop carrying current will emit electromagnetic radiation, dissipating energy from our system. We treat our resonator loop as a dipole, with a current given by I = I rmax e iωt .
The power loss of an oscillating dipole due to radiation is given by P = R rad I 2 /2, where R rad = π 6 R r ω c 4 Z, where Z = 377 Ω is the impedance of the vacuum. Using the parameters in Table I we obtain a power loss of P = 2.5 × 10 −41 W, corresponding to Q = 7.5 × 10 25 , and an associated decoherence rate of Γ rad = 3.3 × 10 −22 s −1 .

Background gas collisions
In the limit where the mean free path of the gas molecules is sufficiently large, the damping rate is given by [7] where ρ gas is the density of the gas, A = 2πR r 2a is the cross-sectional area of the resonator interacting with the gas, m g is the mass of a gas molecule, and u av = 2k B T/m g is the average velocity of a gas molecule. In order to be in this limit, the system must have a Knudsen number Kn > 10 [8]. Using the parameters in Table I

Coupling to torsional modes
Coupling of the vertical (z) centre of mass (z−COM) oscillation mode to other bending/twisting/torsional modes of the resonator also allows energy to leak from the z−COM phonon mode. Of these alternative motional modes one can consider, the lowest frequency mode is the torsional mode which has frequencies where n > 0 is the integer valued mode number, A is the cross sectional area of the wire, µ = ρπa 2 is the mass per unit circumference, and E is the Young's modulus of the wire. We take E = 16 × 10 9 Pa, ρ = 11340 kg/m 3 giving µ = 3.56 × 10 −8 kg/m. This gives the lowest frequency mode as ν = 1.89 × 10 8 rad/s, which is ∼ 1200 times larger than ω, indicating cross-coupling to other modes is negligible.

Appendix E: Open System Dynamics
The open systems dynamics of the joint qubit-resonator system is given by the master equatioṅ

Free evolution is governed by the Hamiltonian
and amplitude damping of the resonator and amplitude and phase damping of the qubit are de- with the mapD defined asD The equilibrium phonon occupation of the qubit environment is N q = (e − ω q /k B T q − 1) −1 where T q is the qubit phonon bath temperature. The decay rates are related to the usual decoherence times according to T −1 1 ≡ Γ ⊥ (2N q + 1) and T −1 2 ≡ T −1 1 /2 + Γ . We treat the environment of the resonator as zero temperature meaning the resonator only loses energy to the environment. This is justified as it is not clamped to any material and we assume the surrounding cavity is in the electromagnetic vacuum state. Any temperature dependence of damping due to background gas collisions can be encorpeated into the value of damping rate Γ gas as described in Sec. D 5.
At each measurement run, the joint state of the qubit and resonator is prepared in the initial where |0 r is the motional ground state of the resonator. At this point we can make some simplifications. We are interested in obtaining a worst case scaling for the decoherence of our protocol which would occur when the size of the initial Schrödinger cat state is largest, i.e. l = l max . The time evolution is only over one period of oscillation τ = 2π/ω of the resonator and we assume that Γ ⊥ , Γ < ω and Γ ω. It is convenient to divide the Louivillian into two parts:L =L 1 +L 2 : During evolution generated byL 1 , the operatorσ z is a conserved quantity and we can solve for the joint evolution of the qubit and resonator exactly. Evolution generated byL 2 describes amplitude damping of the qubit. We approximate the evolution of the system over one resonator oscillation period τ as the composition of maps: We first consider evolution byL 1  To derive the evolution during decay we use the characteristic function where the trace is taken over the resonator's motional degree of freedom such thaṫ Using the relations To solve for the dynamics, we make the ansatz: From the reflection symmetry of the state dependent traps, the magnitudes of the coherent states correlated with the qubit states are equal at all time so we can write β M I (t) = α M I (t). Evaluating the time derivative of X(t) and setting this equal to Eq. (E7) we can solve for the dynamics. The diagonal terms evolve as The off-diagonal terms evolve as Transforming back to the Schrödinger picture, the state written explictly in the qubit basis is: where |α M (t) = |(1 + e −Γt/2 e iωt )λM/2ω , the coherently evolved phase is We seek a form for the joint state after one oscillation period τ = 2π/ω. Since 2πΓ/ω = Q −1 1, we can approximate α M (2π/ω) ≈ α M (0) and 1 − e −Γ2π/ω ≈ Γτ, so that where the coherent phase is and the decoherence is governed by the factor As expected, the dephasing grows with the square of the cat state separation.
Evolution according toL 2 is a map that acts only on the qubit and can be solved for explicitly.
The full evolution over one oscillation period returns the joint system to a product state of the resonator in the vacuum motional state and the qubit in a mixed state: This is the expression used in the overall measurement fidelity in Eq. (33) in the main text where the gravitationally induced phase φ is obtainable by measuring qubit coherences.