Creating atom-nanoparticle quantum superpositions

A nanoscale object evidenced in a non-classical state of its centre of mass will hugely extend the boundaries of quantum mechanics. To obtain a practical scheme for the same, we exploit a hitherto unexplored coupled system: an atom and a nanoparticle coupled by an optical field. We show how to control the center-of-mass of a large $\sim500$nm nanoparticle using the internal state of the atom so as to create, as well as detect, nonclassical motional states of the nanoparticle. Specifically, we consider a setup based on a silica nanoparticle coupled to a Cesium atom and discuss a protocol for preparing and verifying a Schr\"{o}dinger-cat state of the nanoparticle that does no require cooling to the motional ground state. We show that the existence of the superposition can be revealed using the Earth's gravitational field using a method that is insensitive to the most common sources of decoherence and works for any initial state of the nanoparticle.

Introduction.-Quantum mechanics has been probed experimentally over a vast range of energies and scales. On the one side, down to subatomic distances using accelerators, while on the other side, spatial superpositions in the mesoscopic regime are being explored via quantum optomechanics. The former is ultimately expected to shed light on the basic building blocks of our universe, while the latter addresses the quantum-to-classical transition in the mesoscopic, a problem already highlighted by Schrödinger [1].
The field of optomechanics, and in particular levitated optomechanics [2], where the system is well isolated from deleterious effects of decoherence from the environment, has now reached the quantum regime [3,4] and is expected to soon test ideas from quantum foundations [5] and the nature of gravity [6][7][8]. Nonetheless, a challenge still remains how to prepare nonclassical motional states of the nanoparticle, such as the Schrödinger-cat state [9].
Possible approaches for nonclassical state preparation in levitated optomechanics are based on nonlinearities in the potential [10], as well as coupling to quantized fields along with possible usage of measurements [11][12][13][14][15][16]. Difficulties of these approaches include small single photon nonlinearities and/or detecting the effect of nonlinerities in the regime of small oscillations, where the motion is typically well described by a linear theory. Another promising strategy is to embed impurities in the nanoparticle and use that to control the nano-particle [17][18][19][20][21]. However, the placement, control and coherence of such impurities is experimentally very challenging. Hence any alternatives which are not susceptible to the above limitations are highly desirable.
Here we propose combining two hitherto disparate fields in an optimal way for the nonclassical state preparation of nano-objects: the long acquired ability to control the exceptionally coherent internal levels of trapped atoms (ions), and through them, their motional states [22] and the recently acquired expertise of controlling, to an exceptional level, the centre of mass of nano-objects [3,4]. We show how the addition of the highly controllable atom opens up feasible opportunities for the preparation of Schrödinger Cat states in the latter field. We consider the situation where the nanoparticle is trapped in a Paul trap and illuminated by a plane-wave optical field. The reflected light from the nanoparticle interferes with the incoming light and creates a series of dipole traps where atoms can be trapped. In particular, we consider one atom placed in a stiff trap such that displacing it also moves the center-of-mass of the atom-nanoparticle system. The induced effective coupling between the motional state of the nanoparticle and the internal state of the atom allows to directly apply the technical abilities from atomic physics to prepare nonclassical states of the nano-object. Moreover, the switchability of the coupling (simply by controlling the intensity of the optical field) enables release and recapture so as to exploit free-fall non-decoherent evolutions. This latter ability, for example, is absent in atom-micromechanical coupled systems [23][24][25][26]. We show that one can generate a small spatial superposition of the nanoparticle so that it is well protected from enviromental decoherence, and yet such a small superposition can be revealed using the Earth's gravitational field [19,27]. Moreover, we find that the protocol is insensitive to the initial state of the nanoparticle which will greatly facilitate the realization.
Atom-nanoparticle coupling.-The experimental setup consists of a nanoparticle trapped in Paul trap which is illuminated by a plane-wave optical field (see Fig. 1). We choose the light wavelength λ l to be comparable or smaller than the nanoparticle radius r, effectively making the nanoparticle a mirror-like object. The backscattered light from the nanoparticle interferes with the incoming light to form a standing wave in the rest frame of the nanoparticle (see Fig. 2) and the resulting intensity minima and maxima rigidly follow the motion of the nanoparticle. In one of the maxima we trap an atom exploiting an internal electronic transition in the red-detuned regime. Specifically, the potential is given by: where ω n (ω a ) is the frequency of the Paul (atomic) trap, m n (m a ) is the mass of the nanoparticle (atom),x n (x a ) is the nanoparticle (atom) position, and d is the distance between the two traps. The motional frequency of the atom is given by [28]: where I is the intensity of light at the trap center, w ∼ λ l /2 is the trap width, ω e is the electronic transition frequency, Γ is the decay rate from the excited state, ∆ = ω e − ω l is the detuning of the light field, ω l = 2πc λ l , and c is the speed of light. To obtain high trapping frequencies we can decrease the detuning ∆ at the cost of reducing the trapping time τ trap = mac 2 ω l 2 ∆ Γ . The trapped atom offers a new handle on motion of the nanoparticle. Particularity interesting is the situation when the atom is placed in a strong dipole trap, Figure 2. Simulated intensity using finite difference time domain methods [29,30]. We consider a nanoparticle of radius r = 500nm and an optical field with wavelength λ l = 1000nm propagating in the posive x-axis direction. The incoming field is polarized along the y-axis; other vertical planes shows a similar intensity profile. The colour bar is the enhancement in the square of the electric field. The large blue circle denotes the nanoparticle; the incoming field propagating from the bottom interferes with the backcattered field from the nanoparticle which creates dipole traps below the nanoparticle. The strongest dipole trap is located d ∼ 0.75µm below the center of the nanoparticle (first yellow patch below the blue circle).
resulting in a rigid atom-nanoparticle coupling. We then expect that any displacement of the atom will drag the whole atom-nanoparticle system, with only negligible excitation of the relative motion between the two. Mathematically, this translates to requiring that (i) the atom is placed in the motional ground state and (ii) the zeropoint motion of the atom, δ a , is small with respect to the one of the nanoparticle, δ n , such that when the nanoparticle is excited the atom remains in the ground state, i.e. we can writex a ≈x n − d.
Nanoparticle motion control.-In the considered regime we find the following interaction Hamiltonian between the motional state of the nanoparticle and the atomic hyperfine transition (in interaction picture) where we have introduced the nanoparticle modeâ, i.e.
x n = δ n (â † +â). Ω jk is the coupling of the stimulated Raman transition between the hyperfine states |j and |k , σ + = |k j|, η = kδ n is the Lamb-Dicke parameter, k = 2π λ = ω c with ω the frequency of the laser, δ = ω h − ω is the detuning that selects one of the sidebands or the carrier resonance, ω h is the hyperfine transition frequency, and φ is a phase that includes d λ . Here we limit the discussion to η 1, which puts a lower bound on the Paul trap frequency, i.e. 2mnλ 2 ω n . The coupling of the stimulated Raman transition is given by Ω jk ≡ g jk , where g jk = qE D jk , q is the electron charge, E is the amplitude of the electric field, and D jk is the transition dipole matrix element between the state j and k.
We are interested in two types of interactions, one that (a) controls the internal state without affecting the motional state, and one that (b) displaces the motional state without changing the internal one, both of which can be implemented in a Λ-type scheme using two lasers. In particular, using two-photon stimulated Raman transitions of type (a) and (b) we will consider three types of operations, where the coupling will be given by Ω jk ≡ g * jl g lk ∆ l , and ∆ l is the detuning from the intermediate state l [31].
To create a superposition of the hyperfine states we consider the carrier frequency, i.e. δ = 0, with a pulse of duration t = π/(2Ω ↑↓ ) using scheme (a), namely a π/2 pulse. This generates a beam splitter transformation, i.e. the hyperfine states evolve in the following way: Similarly, a π pulse using scheme (a) at the carrier corresponds to Ω ↑↓ t = π and δ = 0, which exchanges the hyperfine states, i.e. | ↑ → −| ↓ and | ↓ → | ↑ . On the other hand, to displace the motional state without modifying the hyperfine state we exploit scheme (b) at the first red sideband, i.e δ = ω n . This latter operation produces a displacement of the motional state by Ω ↓↓ ηt, where t is the duration of the pulse.
In summary, the discussed interactions have the same form as the ones exploited in atomic physics where in place of the motional state of the atom we have the motional state of the nanoparticle. We can thus adopt the experimentally well-established protocols from atomic physics to the nanoscale [22,31,32].
Schrödinger's cat.-Suppose the state of the system is |Ψ = |ψ h |ψ n , where |ψ h is the hyperfine state of the atom, and |ψ n is the motional state of the nanoparticle. Ideally, one would like to prepare a state of the form |ψ n ∼ | ↓ h |α top n + | ↑ h |α bottom n , where |α top n and |α bottom n denote states located at different heights in the Paul trap, i.e. a Schrödinger-cat state. Once such a state has been created we then want to ascertain its existence using as the readout the hyperfine state |ψ h .
A possible strategy is to cool the system to the ground state, i.e. |Ψ init = | ↓ h |0 n , and to apply the procedure described by Monroe et al [22], which consists of π/2, π, and displacement pulses. To make such a scheme work one would however need additional optical fields to control the motional state of the nanoparticle. In particular, cooling to the motional ground state can be achieved with a cavity-tweezer setup [3] and is expected to be soon available also in a tweezer setup [4,33].
However, a protocol that would not require cooling [26], but would rather work for a generic trapped state, such as the experimentally more readily available thermal state, is still desirable. A second attractive feature would be to have a reliable method to evidence that the nanoscale superposition has really been probed, for example, by relating the outcome of the experiment to one of its intrinsic properties such as the nanoparticle mass m n . A possible strategy to address both of these requirements has been outlined in [19], parts of which we now adapt to the hybrid atom-nanoparticle system. For simplicity of presentation we first consider the initial state |ψ init = |α ⊗ | ↓ , where the nanoparticle is prepared in the coherent state |α (but we show below that it applies for any initial state). The protocol consists of the following steps.
1. Trap a nanoparticle in the Paul trap at frequency ω 1 . Trap an atom in an intensity maxima below the nanoparticle using a plane wave and cool it to the ground state using resolved sideband cooling [31].

5.
Reduce the trapping laser power such that the radiation pressure force becomes small and the nanoparticle-atom system starts falling towards the Earth (matter-wave coherence is thus shielded from the deleterious effects of the laser photons and the system becomes a matter-wave sensor for the local Earth's gravitational acceleration ∼ g).
7. Increase the trapping laser power back to its initial value. Apply a displacement beam for a time δt to reverse the effect of step 4 and obtain a factorizable state |ψ ∼ |α ⊗ e −iφgrav | ↓ + | ↑ .
8. Apply a π/2 pulse to create the final state |ψ ∼ |α ⊗ |φ , where the hyperfine state is |φ = cos 9. Apply a laser field to drive a cycling transition and find the probability of being in the ground state P ↓ = cos 2 φgrav 2 .
10. After the measurement we recapture the nanoparticle by modulating the radiation pressure from the trapping laser and the Paul trap frequency.
The induced gravitational phase difference is given by where ∆x = δ n β = k 2mnω2 Ω gg δt is the superposition size of the nanoparticle and ∆t is the duration of the transient free fall motion. Since the nanoparticle mass m n is large we can have φ grav ∼ 1 already for small superposition sizes ∆x and for short free-fall times ∆t -a regime which is interesting on its own.
Let us now consider a generic initial state ρ init = ρ n ⊗ | ↓ ↓ |, where ρ n = d 2 αP n (α)|α α|, and P is Glauber's P quasi-probability distribution. Here we only require that the nanoparticle is initially trapped in the Paul trap, but the motional state can be otherwise completely generic. The steps 1-7 now result in the final state ρ final ∼ ρ n ⊗ |φ φ|, where ρ n is the final motional state of the nanoparticle, yet |φ is the same internal state obtained by considering an initial coherent motional state. Remarkably, the transient free fall dynamics entangles the motional and internal states in a simple way which can be readily disentangled at any time -this is a direct consequence of the uniform nature of the universal gravitational coupling, a feature which is absent already with a harmonic potential. Creating a superposition of an arbitrary motional state (such as of a thermal state) still fully retains its coherent properties, and once the gravitational phase is transferred to the internal state it can be then read out again using steps 8 an 9.
Discussion.-We can estimate the requirements to achieve φ grav ∼ 1 for a typical tabletop experiment using a nanoparticle of radius r = 500nm and mass m n ∼ 10 −15 kg in a Paul trap [34,35]. As discussed, we first trap an atom in a dipole trap near the nanoparticle, which induces a coupling between the two, while other interactions between the atom and the charged nanoparticle are negligible. For concreteness we consider a Cs atom and the D 2 transition 6 2 S 1 2 → 6 2 P 3 2 which has a transition dipole matrix element ∼ 4 × 10 −29 Cm and decay rate Γ ∼ 3 × 10 7 Hz.
We set the detuning of the trapping laser to ∆ ∼ 5 × 10 11 Hz to generate a far red-detuned dipole trap: we find a trap lifetime τ trap ∼ 1s ∆t and using Fig. 2 we estimate the atomic trap frequency to be ω a ∼ 5 × 10 6 Hz generated by an incoming (backscattered) intensity ∼ 5 × 10 12 Wm −2 (∼ 3 × 10 7 Wm −2 ). Such an intensity can be obtained using an unfocused laser beam at moderate power; at this intensity the radiation pressure force cancels the gravitational one (whilst not co-trapping the nanoparticle). We consider a short free fall-time ∆t ∼ ω −1 a ∼ 1µs in order to retain the atom's motional state which corresponds to a displacement of ∼ 5pm. The condition to excite the nanoparticle motion constrains the Paul trap frequency ω n from above, ω n 5 × 10 −4 Hz, and the Lamb-Dicke condition from below, ω n 5 × 10 −8 Hz. Specifically, we set the initial Paul trap frequency to ω 1 = 0.1kHz which is then softened to ω 2 = 5 × 10 −6 Hz. After the Paul trap is softened we create a spatial superposition of the nanoparticle by illuminating the atom with a short laser pulse of duration ∼ 100ps and detuning ∆ 3 ∼ 10 11 Hz. The requirement of unit phase, φ grav ∼ 1, fixes the intensity of the beam to I ∼ 1Wm −2 , resulting in a tiny nanoparticle superposition of size ∆x ∼ 10 −14 m. The control beam will illuminate also the nanoparticle (given its close proximity d ∼ 0.75µm), but such a tiny intensity will however not lead to any measurable dephasing. Larger as well as smaller superpositions can be created by varying the parameters of the setup, for example, by controlling the intensity and duration of the displacement beam one is expected to achieve superpositions of the size of the nanoparticle. Additionally, to further enlarge the size of the superposition -without extending the duration of the experiment -one could also introduce a boosting potential by adaptation of the coherent inflation method to the Paul trap [36].
The decoherence times for superposition sizes ∆x ∼ 10 −14 m exceed the duration of the experimental time ∆t ∼ 1µs at readily available pressures and temperatures -for concreteness we consider the vacuum chamber with pressure p ∼ 10 −2 mbar and temperature T ∼ 300K. Given the modest laser intensities, and the relatively high pressure, we can assume that both the center-of-mass and internal temperature of the nanoparticle remain below T ∼ 1000K [37] (for cooling the internal temperature see [38]). At such pressures/temperatures we find that gas collisions limit the coherence time to ∼ 6µs, while decoherence due to photon emission/absorption remains negligible -at T ∼ 300K the available coherence time is further extended [39][40][41].
For completeness we also estimate the emitted thermal radiation from the nanoparticle and its effect on the atom. Assuming black-body radiation from the nanoparticle with internal temperature T ∼ 1000K we find a radiated intensity ∼ 10 5 Wm −2 which is two orders below the intensity generating the atom's dipole trap (see above). Furthermore, the intensity of the thermal radiation in the narrow frequency range of the internal transition Cs D 2 (6 2 S 1 2 → 6 2 P 3 2 ) is ∼ 10 −6 Wm −2 which has to be compared with the intensity of the controlling lasers ∼ 1Wm −2 . We have to however re-scale the two intensities by the ratio of the duration of the experiment (∼ 1μs and of the controlling pulse and ∼ 100ps) which nonetheless still results in the coherent laser radiation dominating by 2 orders of magnitude over the thermal one. If instead one assumes an internal temperature T ∼ 300K the effect of thermal radiation becomes dwarfed by the controlling beams by about ∼ 20 orders of magnitude and can thus be again neglected.
Finally, we estimate the effect of voltage noise, S V , which gives rise to a force noise, S (vol) f ∼ qS V /D, where q is the net charge on the nanoparticle, and D is a characteristic distance to the electrodes. Specifically, assuming S V ∼ 10µV/Hz 1/2 , q ∼ 80e (we note that the charge on the nanoparticle can be controlled to a high degree [34]), and D ∼ 2.3mm we find S (vol) f ∼ 10 −23 N/Hz 1/2 [35]. By comparison the force noise due to gas collisions is S (gas) f ∼ √ 2k b T m n γ, where γ = 4πm g r 2 v t p/(3k b T m n )(1 + π/8) is the gas damping rate [42,43], m g is the molecular mass, and v t = 8k b T /(πm n ) is the thermal gas velocity -using T ∼ 300K and p ∼ 10 −2 mbar we find S (gas) f ∼ 10 −16 N/Hz 1/2 . As discussed above the thermal noise does not impede the witnessing of interference and hence voltage noise can be also safely neglected.
The insensitivity of the ten-step protocol to the environment can be explained by the fact that the characteristic wavelength of gas particles as well as the ones associated with laser and environmental photons, is much larger than ∆x, making the associated decoherence times long compared to the short free fall time.
In summary, we have shown that it is possible to create motional superposition of massive objects (a ∼ 500nm radius nano-object) by introducing a coupled atom-nanoparticle hybrid system and discussed how to detect them. It will extend the demostration of the superposition principle to unprecedented regimes of mass, 10 8 times the current record [44]. The method has several appealing features. It works for a generic initial state, the control and readout of the motional state is through well established versatile atomic protocols, and the created superposition is very well protected from deleterious decoherence effects.

Appendix A: Atom-Nanoparticle motion and internal transitions
We discuss the center-of-mass variables (Sec. A), which allows to reduce the problem to the effective interaction between the motional state of the nanoparticle (Sec. B) and the internal hyperfine state of the atom (Sec. C).

Center-of-mass motion
We introduce the center-of-mass (c.o.m.) variableŝ and using Eq. (1) we readily find the nanoparticle-atom Hamiltonian: We will be primarily interested in controlling the c.o.m. modeâ which to good approximation coincides with the motion of the nanoparticle. We consider the rigidcoupling regime discussed in the main text, i.e. we prepare the atom in the motional ground state and require δ n δ a . More specifically, we require that the displacement beam will not excite the atom's motional state, while sufficiently exciting the nanoparticle.
Some remarks about the approximations involved are in order. In Eq. (A3) we have neglected terms of order ∼ O(m a /m n ) which for typical atomic and nanoscale masses would correspond to a correction of 1 part in ∼ 10 8 . The analysis was also based on a semiclassical approximation, where the internal motion responsible for the atomic polarizability is assumed to reach a steady-state on a time-scale faster than the motional time-scale of the atom in the trap [45]. The full dynamics would require simultaneous integration of the optical Bloch equations together with the atom-nanoparticle motional dynamics as described by the quantum kinetic equations [46][47][48]. In the following we will also consider additional lasers for controlling the motional state of the atom; we will suppose that the atom remains stably trapped for the duration of the experiment [49,50].

Nanoparticle potential
The potential of the nanoparticle in the Paul trap is given byĤ where we have introduced the gravitational force m n g E as well as the radiation pressure force F generated by the trapping laser for the atom (see Fig. 1). We first trap the nanoparticle in a relatively stiff Paul trap ω n = ω 1 with the radiation pressure force F constrained by the requirement of stable trapping in the Paul trap. The latter is controlled by light intensity I which also sets the atomic trap frequncy ω a in Eq. (2). Given the large mass of the nanoparticle in comparison with the atom's mass we can have both a small radiation pressure force F ∼ m n g E as well as a high trapping frequency ω a for the atom -the latter is required to introduce a handle on the nanoparticle's motion.
We then release the nanoparticle by (i) softening the Paul trap frequency from ω n = ω 1 to ω n = ω 2 as well as (ii) reducing the radiation pressure such that F m n g E . The net result is a change of equilibrium position and for a transient period the nanoparticle is in free fall evolving according to the potential H nano ≈ m n g Exn . (A5) In a nutshell, the idea is to suddenly release the nanoparticle from the trap and use laser fields to create a spatial superposition exploiting the atom-nanopaticle coupling. We effectively create a Mach-Zehnder type interferometer for the nanoparticle: we exploit the Earth's gravitational acceleration ∼ g E to impart a phase difference on the spatial parts of the superposition, which is then transferred to the internal state and read out.

Two-photon stimulated Raman transitions
We consider two types of interactions, one that (a) controls the internal state without affecting the motional state, and one that (b) displaces the motional state of the nanoparticle without changing the internal one [31].
In the former case (a) one links the ground and excited hyperfine states, i.e. the states | ↑ and | ↓ , respectively, through a third hyperfine state |3 using lasers of frequencies ω 1 and ω 2 : on resonance we would have |ω 1 − ω 2 − ∆ 3 | = ω h with ∆ 3 a suitably chosen detuning from the state |3 . Furthermore, we assume that the corresponding wave-vectors, k 1 and k 2 , are such that their difference δk = k 1 − k 2 is parallel to the vertical x-axis with the projection denoted by δk. Formally the interaction Hamiltonian is again given by Eq. (3), where η = δkδ n , and the coupling is given by ∆3 . If we work at the carrier frequency, i.e. δt = 0, the dominant term in the Hamiltonian is insensitive to δk and the motional state remains unaffected, i.e. we only change the hyperfine state. In the latter case (b) one instead stimulates the transitions | ↓ → |3 and |3 → | ↓ , resulting in a coupling Ω ↓↓ ≡ g * ↓3 g 3↓ ∆3 .
Here we want to induce big displacements of the nanoparticle for which large values of δk are preferrable, e.g. δk ∼ |k 1 |,|k 2 |. The Hamiltonian is still the one in Eq. (3) with the formal replacement σ + → I, where I is the identity matrix: now the hyperfine state is unaffected and the motional state changes, i.e. a displacement beam.

Appendix B: Classical evolution
We consider the motion of a point particle of mass m in a harmonic trap with frequency ω in the Earth's gravitational field. In particular, the total Hamiltonian of the problem is given by where x 1 (p 1 ) denote the position and momentum observable, and g E is the gravitational acceleration. Here we will denote the Earth's gravitational acceleration by g E while reserving the symbol g for the corresponding coupling which depends on ω n (see Eq. B12). In Eq. (B1) the subscript 1 labels the reference frame. We also introduce a shifted reference, i.e. reference frame 2, where the positions and momenta are given by and the Hamiltonian is We are ultimately interested in the evolution described in reference frame 1, i.e. the evolution arising from Eq. (B1). However, as we will see when discussing the quantum case, it is instructive to compare it to description in the shifted reference frame 2, i.e. the evolution arising from Eq. (B3). Specifically, in reference 2 we find the solution to be a simple harmonic motion: Using Eq. (B2) we then immediately find the solution in reference frame 1: We now consider two different limits. We note that by taking the limit g E → 0 we recover simple harmonic motion, for example the whole experiment, including the trap, is in free fall, i.e. we recover Eqs. (B4) and (B5) with the formal replacement x 2 → x 1 , p 2 → p 1 . On the other hand, in the limit ω → 0, i.e. we switch off the trap, we find: as expected for free fall.
To relate the results to a quantum analysis we introduce the zero-point motions, δ x = 2mω and δ p = mω 2 , and the adimensional position and momentum, The gravitational potential becomes where the gravitational coupling is The transition from harmonic to free fall motion depends on the strength of the frequencies ω and g, which we now explore. We rewrite Eqs. (B6) and (B7) using Eqs. (B10): Taking the limit g → 0 amounts to vanishing third terms on the righthand side in Eqs. (B13) and (B14), which is the expected result as discussed above. On the other hand, naively taking the limit ω → 0 in Eqs. (B13) and (B14) does not give the free fall evolution: the reason is that these have been derived from Eqs. (B13) and (B14) by diving/mupltipliying with δ x and δ p which depend on the harmonic frequenciy ω. A similar problem is encountered also by using the modes Specifically, from Eqs. (B13) and (B14) we find: where we are again confronted on how to consider the limiting free-fall case. The problem of taking the limit ω → 0 can be avoided by considering small adimensional expansion parameters, gt and ωt -to study the free-fall case, we choose to expand to quadratic order. Following the latter procedure we find from Eq. (B16): If we move back to the position-momentum description we find: Eqs. (B18) and (B19) have extra ω-dependent terms which were absent in the ω → 0 limit (see Eqs. (B8) and (B9)). Unlike the former ω → 0 calculation, the approximation procedure is not state-independent, but depends on the value of x 1 (0) and p 1 (0). In order to recover exactly free-fall one is implicitly assuming that the initial position and momentum, x 1 (0) and p 1 (0), are small enough when taking the ω → 0 limit. However, as we will explicitly see in the next sections we can retain the additional ω-dependent terms as they do not change the induced gravitational phase -as long as ωt remains small. Furthermore, higher order harmonic terms -beyond the free fall approximation -are interesting on its own and could be used to ascertain the spatial superposition of large nanoparticles without resorting to a dynamical equilibrium change (see section E).

Appendix C: Quantum evolution
In this section we consider the quantum dynamics of a particle of mass m harmonically trapped and subject to the Earth's gravitational potential. We continue to use the notation of Sec. B where the observables, e.g. O, are promoted to operators, e.g. O →Ô. The classical analysis of the transition from harmonic to free fall motion -in particualr the approximations involved -carry over also to the quantum case. To simplify the notation we will omit the subscript 1 for quantities related to reference frame 1 most of the time.

Change of equilibrium
We consider the operator version of the Hamiltonian in Eqs. (B1) which we rewrite aŝ and an initial coherent state |α associated to theâ mode. We first recall the definition of the displacement operator:D and the multiplication rulê To find the time-evolution we restate the problem in a displaced frame: where δ ≡ g ω . In particular, we findĤ 2 = ωâ †â and using Eqs.(C2)-(C3) we find the time evolved state Figure 3. We consider the vertical motion of a particle in a Paul trap in an Earth-bound laboratory. (a) The nanoparticle is initially confined in a trap with frequency ω1 and kept close to the origin of the trap; the gravitational force mgE, where m is the mass of the nanoparticle and gE is the gravitational acceleration, is counter-balanced by a radiation pressure force. (b) We change the frequency to ω2 ω1 and create a small superposition of size ∆x = 2mω 2 ∆X. (c) We decrease the radiation pressure force making it negligible with respect to the gravitational one; this changes the equilibrium position to gE/ω 2 2 = 2mω 2 δh. (d) We let the system evolve for a short time t such that the motion of the particle is governed by the uniform gravitational field. This transient free fall regime can be understood graphically -we note that the small arc drawn at radius δh with subtended angle ω2t can be well approximated by the initial part of a parabolic curve.
We now go back to the original frame using the inverse transformation Using again Eqs. (C2) and (C3) we finally find the time evolution of the state in the original frame: We expand to order O(t 2 ) analogously as in the classical case: where we recognize in the first and second prefactors on the righthand side a boost and a translation, respectively. In particular, using Eq. (B12) the phase factors expressed become where x = δ x (α * + α) and p = iδ p (α * − α). Similary, the state of the system |α has now been been boosted by−gt as well as displaced by − ωgt 2 2 in accordance with the classical evolution in Eq. (B17).

Change of equilibrium and frequency
We consider the time-dependent Hamiltonian: wherex andp are the operators associated to the reference frame centered at the Paul-trap origin, i.e. reference frame 1. In particular, we have a sudden change of equilibrium position, d(t), and of the Paul trap frequency, ω(t), i.e., For ω 2 = ω 1 one finds the problem already discussed in the previous section C 1.
Here we consider the full dynamics with the Hamiltonian defined in Eqs. (C12)-(C14). We consider an initial coherent state |α associated to the modeâ = 2mω1 (x + ip) prepeared at time t = 0. The timeevolution for t > 0 can be explicitly computed [51]: where the operators are given bŷ and the time-dependent parameters are defined as follows We have two squeezing parameters: the customary one is given by r = 1 2 ln( ω2 ω1 ) and the dynamical one by z = |z|e iθ . The equilibirium position in adimensional units is given by δ = g2 ω2 which is contained in the timedependent parameter , where g 2 = g E m 2 ω2 is the coupling induced by the gravitational acceleration.
We want to expand Eq. (C15) to order O(t 2 ) during which the system is approximately in free fall as discussed in the previous sections. However Eq. (C15) is not yet in a suitable form as displacement and rotation operators preceed the squeezing one;Ŝ(z) applied on a displaced coherent state also changes its displacement. To avoid this problem we adapt the analysis from [51] to commute the operators:Ŝ We can thus rewrite Eq. (C15) using Eq. (C3) and Eq. (C22) as |α →e We first note that the dynamical squeezing parameter z in Eq. (C19) is only of order O(ω 1 t): where we have assumed ω 2 ω 1 . Wence we can neglect squeezing and setŜ(z) ∼ I by assuming ω 1 t 1 (and hence also ω 2 t 1) . Performing a series expansion, keeping only the relevant terms, we obtain from Eq. (C15) the following evolution: |α h − i ω 2 ω 1 g 2 t − ω 2 g 2 t 2 2 where the harmonic contribution to the eigenvalue is given by It is instructive to introduce the gravitational coupling g 1 = g E m 2 ω1 associated to the modesâ 1 , in particular, we note that g 2 = ω1 ω2 g 1 . From (C27) then readily obtain the final result: |α → e − i 2 (α * +α)g1t e 1 2 (α * −α)ω1 (C29) Relabelling ω1 and g 1 as ω and g, respectively, we recovered the result in Eq. (C9). In particular, we note that the phase evolution depends only on g E , but not on the freequencies ω1 or ω2 -see Eqs. (C10) and (C11). change the Paul trap frequency from ω n = ω 1 to ω n = ω 2 can make a big difference. This can be seen by recalling that ∆x = δ R β where δ R = 2mωn is the zero-point motion,β = Ω gg ηδt is the displacement generated by the controlling lasers, and η = kδ R is the Lamb-Dicke parameter (see main text). In particular, combing the formulae we readily find: where we explicitly see the ∼ 1 ωn dependency of the superposition size. In other words, applying the same displacement beam in a weaker Paul trap leads to larger displacements as both the zero-point motion δ R and the Lamb-Dicke parameter η contribute a factor 1 √ ω . The O(t 3 ) correction to gravitational phase in Eq. (D3) is given by If we require |φ (3) | |φ grav | we find the simple condition ω 2 t 1.
Let us expand the expression for small ∆x compared to x 2 (0) and to O(t), i.e. we are interested in the free-fall regime of tiny superpositions. We readily find ∆φ grav ≈ ∆xx 2 (0)mω 2 t (E5)