Charged oscillator quantum state generation with Rydberg atoms

We explore the possibility of engineering quantum states of a charged mechanical oscillator by coupling it to a stream of atoms in superpositions of high-lying Rydberg states. Our scheme relies on the driving of a two-phonon resonance within the oscillator by coupling it to an atomic two-photon transition. This approach effectuates a controllable open system dynamics on the oscillator that permits the dissipative creation of squeezed and other non-classical states which are central to applications such as sensing and metrology or for studies of fundamental questions concerning the boundary between classical and quantum mechanical descriptions of macroscopic objects. We show that these features are robust to thermal noise arising from a coupling of the oscillator with the environment. Finally, we assess the feasibility of the scheme finding that the required coupling strengths are challenging to achieve with current state-of-the-art technology.


I. INTRODUCTION
The interface between different types of quantum systems has been the subject of much attention in the quest for complex quantum technologies [1][2][3][4]. In order to combine advantages of various platforms, such as long coherence time, strong interactions or low-loss transport [5], one has to be able to transfer quantum state between different systems. Alternatively, the interactions between two different quantum systems can be exploited to produce and probe quantum states [6,7].
Mechanical systems in particular have seen rapid experimental progress. Nowadays, micro-and nanomechanical oscillators can be cooled down to the quantum regime, where the quantised dynamics of the oscillator motion and controlled interaction with other quantum systems have become possible [5,[8][9][10]]. An alternative to the typically used optomechanical interaction is to exploit electric forces to couple an atom to a charged oscillator [11][12][13][14][15]. The strong dipole moment of atoms excited to high principal number Rydberg states [16], allows strong free-space interaction between single atoms and a charged oscillator, without the need for a mediating cavity. Atomic dipole -oscillator dipole coupling allows single atom cooling and the construction of complex superpostions of phononic Fock states [17]. Moreover, efficient coupling between Rydberg atoms and microwave cavities [18], acceleration of flying atoms [19] and creation of superpositions between different Rydberg states [20] all constitute well established technologies. At the same time, results in the fabrication of micromechanical oscillators with resonance frequencies matching Rydberg transitions in atomic systems, and with high quality factor are promising, particularly using single-crystal diamonds [21,22]. Additionally, these oscillators can be superconducting, and thus become chargeable on demand [23].
In this paper we exploit the coupling between flying Rydberg atoms and a charged mechanical oscillator. We show that when the oscillator is driven at two-phonon resonance and if the coupling between the atoms and the oscillator is sufficiently strong, the system dynamics results in a non-classical state of the oscillator, whose nature can be tuned by a suitable choice of the initial atomic state. The desired oscillator states are obtained after the passage of only tens of atoms corresponding to the initial transient period of an effective dissipative dynamics. Specifically we show that under the strong coupling condition one can create a squeezed or Schrödinger cat states of the oscillator which are robust with respect to realistic thermal noise. These states are particularly useful for fundamental tests of quantum physics and decoherence processes [24,25], quantum information and quantum simulation [26], metrology and sensing of small forces [27] or even for dark matter detection [28] or to probe quantum gravity inspired models [29]. While squeezed states of micromechanical oscillators have been produced [30][31][32], the creation of large and robust Schrödinger cat states of macroscopic mechanical oscillators is yet to be achieved. We perform a feasibility study and find that with current state-of-the-art technology it is challenging to access the strong coupling regime. To ultimately reach the desired coupling strengths may necessitate further developments, such as the use of collectively enhanced coupling through oscillator arrays or atomic ensembles.
The article is structured as follows. We introduce the system and describe its dynamics in Sec. II. We study the effect of thermal fluctuations and experimental contraints in Sec. III. Finally we comment on the implications of those constraints and discuss possible future directions in Sec. IV.

II. THE SYSTEM
The system under consideration is shown in Fig. 1(a). It consists of a stream of single Rydberg atoms coupled to a charge Q at the tip of a micromechanical oscillator, which oscillates in the z direction around the origin. We denote byẑ = z osc (â +â † ) the displacement operator of the oscillator, whereâ † andâ are the bosonic phonon creation and annihilation operators, z osc = /(2m eff ω osc ) the characteristic oscillator length, m eff the effective mass of the oscillator and ω osc is the mechanical oscillation frequency. The atoms move along a path R(t) = (X(t), Y (t) = 0, Z(t)) such that only one atom is interacting with the oscillator at a time.

A. Single atom dynamics
In this article we consider two distinct situations: a single-phonon and two-phonon resonance (see Fig. 1). In the first case the atomic ground state |s = |S 1/2 , 1/2 , the excited state |p = |P 1/2 , 1/2 , ω a is the |s − |p transition frequency and the interaction is described by the interaction HamiltonianV = −μ ·Ê[R(t)], see Fig.  1(b). Here,μ is the atomic dipole of the |s − |p transition andÊ[R(t)] is the electric field at the position R(t) created by the oscillator charge. In the latter case, the two-phonon oscillator transition couples to a two-photon transition between Rydberg levels |s = |S 1/2 , 1/2 and |s = |S 1/2 , 1/2 , which are S states with different principal quantum number, via an off-resonant manifold of P states. We denote by ω a the P − S transition frequency and by ∆ = ω a −ω osc the atom-oscillator detuning, which is assumed to be much larger than the energy separation of states within the P manifold. The interaction Hamiltonian in this case readsV = −(μ 2 +μ 2 ) ·Ê[R(t)], wherê µ 2 (μ 2 ) is the dipole moment of the S − P (P − S ) transition (see Appendix A).

B. Single-phonon resonance
The first scenario we are studying is that of a singlephonon resonance, where ω osc = ω a . Under the assumption of small oscillator displacement as compared to the distance between the oscillator and the flying Rydberg atom, z R, where R = |R(t)|, one can expand the electric field in powers ofẑ. Using the rotating wave approximation, the interaction picture Hamiltonian reads (see Appendix A) where γ(t) = Qµ0zosc is the time dependent coupling strength [33]. For this resonant case, the time evolution can be solved exactly with the propagator Here n is the oscillator phonon occupation number, Θ n = √ n + 1 G, G =  (2) is written in the {|p, n , |s, n + 1 } basis. This is a situation corresponding to the micromaser physics as described for example in Ref. [34].
The atoms are prepared identically and interact one at a time with the oscillator (see Fig. 1(a)) such that the evolution of the oscillator can be evaluated according tô U (t f , t i ) after the passage of each single atom. The initial state of each atom is assumed to be a superposition of the form with the amplitude β = 1 − |α| 2 e iθ . The state of the oscillator can be determined at an arbitrary time iteratively as follows: the state of the oscillator ρ (k) osc after k atoms have passed can be obtained by time-evolving the initial product state ρ a ⊗ ρ (k−1) osc (where ρ a = |ψ a ψ| a is the initial state (3) of the atom) withÛ and subsequently tracing out the atomic degrees of freedom The propagatorÛ gives the exact evolution of the system as an atom travels past. However, it is useful to describe the dynamics of the oscillator in terms of an approximate master equation. We derive the master equation in the limit where the change in the oscillator state due to the interaction with a single atom is small such thatρ osc ≈ r∆ρ osc and r is the rate by which the atoms fly by the oscillator. The master equation approach has the advantage that it provides useful insights in the dynamics of the system without explicit exact solution. It also allows for adding directly the coupling to a thermal bath [34], as we shall discuss in detail in the case of the two-phonon resonance.
Next, assuming Θ n 1, the propagator (4) can be expanded to second order in Θ n which yields the effective open system dynamicṡ where

C. Two-phonon resonance
In order to move beyond a displaced thermal state and achieve quantum states that are more complex we consider coherent two-phonon transitions of the oscillator that generate explicitly quantum effects. On the atomic side we consider two possible coupling mechanisms: direct single-photon and intermediate states mediated twophoton transition between a pair of atomic levels. As both situations lead to equivalent form of the effective interaction Hamiltonian, we first focus only on the latter which we analyze in detail. We then invoke the former in Sec. III for the sake of quantitative comparison.
The situation for atomic two-photon transition via an intermediate manifold of states coupled to a two-phonon transition of the oscillator is depicted in Fig.1 (c). When the intermediate manifold of states is detuned far enough from resonance with a single phonon it remains unpopulated and can be eliminated from the dynamics leaving an effective two-level system.
In the following we consider the case of a two-phonon resonance with the initial atomic state |ψ a = α |s +β |s as described in Fig.1 (c). On two-phonon resonance (ω a + ω a = 2ω osc ) the intermediate P levels are adiabatically eliminated and the interaction between the atom and the oscillator is described by the interaction picture Hamiltonian As in the single-phonon resonance case, the time evolution of the system can be solved exactly using which is now written in the basis {|s , n , |s, n + 2 } , Θ n,2 = (n + 1)(n + 2) G 2 and G 2 = t f ti dtγ 2 (t). Note that the evolution in the odd/even n subspaces of the oscillator are independent of each other.
The two-phonon coupling between the atom and the oscillator is reminiscent of two-photon micromasers [35][36][37], and we show here that it allows the creation of squeezed states, as suggested by the form of the Hamiltonian (6) [38]. For the quantification of squeezing we introduce the standard quadrature observable The quadrature angles φ = 0, π/2 correspond to the X and P quadratures, and the state is squeezed along φ if ∆χ 2 φ < 1/2. The squeezing of mechanical motion was in fact achieved in recent experiments [30][31][32]. The manipulation of the oscillator state using Rydberg atoms at two-phonon resonance however goes beyond the squeezed state preparation and allows for creation of various other kinds of non-classical states. In order to quantify the nonclassicality of the created states we use the negativity of the Wigner quasi-probability distribution is the spatial wavefunction of the oscillator [39]. The negative volume of the Wigner function then reads [40] The exact evolution of the system can be solved by iteratively applying (4) whereÛ is replaced byÛ 2 and we take ρ (0) osc = |0 0|. The resulting state depends on the number k of atoms that pass by the oscillator. The exact value of k is not particularly important, as long as the number of atoms is sufficient to reach the desired non-classical state. For the following calculations, we fix k = 30, which fulfills this conditions for all considered states.
We now turn to numerical simulation of the exact evolution as described by eqs. (3),(4) and (7). The results of the simulation are summarized in Fig. 2. Fig. 2(a) shows the minimum variance ∆χ 2 φmin of the state of the oscillator as a function of the integrated coupling strength G 2 and the atomic excited state population |β| 2 . The angle φ min minimizing ∆χ 2 φ depends only on the relative phase θ between the atomic states (see Appendix C). For θ = 0 used in the simulation, φ min = π/4. The negative volume of the Wigner function V neg (9) is plotted in Fig. 2 Finally, Fig. 2(c-f) show the Wigner function for specific values of G 2 and |β| 2 denoted by × in Fig. 2 Fig. 2(d) has the qualitative features of a cat state [41], which is of particular interest as it is used in metrology for small force sensing [27] and in fundamental test of quantum mechanics [25].

III. THERMAL FLUCTUATIONS AND EXPERIMENTAL CONSIDERATIONS
We now investigate how robust the production of these quantum states is in the presence of thermal fluctua- tions. Combining the master equation for the interaction with the passing atoms, derived analogously to the single phonon case (see Appendix B), with the thermal processes givesρ where the atomic part is and the thermal part is Here Γ m is the coupling of the oscillator to the thermal bath andn th = 1 e ωosc /k B T −1 is the mean phonon number of the bath at temperature T .
To demonstrate how the coupling to the thermal bath deteriorates the oscillator quantum states, we solve the master equation (10) numerically for a total time corresponding to the passage of 30 atoms and initial thermal state withn th [42] . In Fig. 3(a) we plot the negative volume of the Wigner function V neg as a function of thermal coupling Γ m and mean thermal occupation number n th . The used parameters G 2 = 0.2, |β| 2 = 0.2 correspond to the cat state in Fig. 2(d). Fig. 3(b) shows the minimum variance ∆χ 2 φmin for parameters corresponding to the squeezed state in Fig. 2(c). It follows from Fig. 2(a,b) and Fig. 3(a,b) that in order to create a non-classical state one requires G 2 ∼ 0.1 and the atom passage rate r should be maximized while minimizing Γ m andn th . With the help of specific examples, we demonstrate the performance of the scheme below.
Considering Γ m = 2π × 500 Hz and the state-of-the-art temperature T = 10 mK corresponding ton th = 0.1, we find for the cat state of Fig 3(a) that V neg = 0.25V neg,0 . Here V neg,0 denotes the value of V neg for the system not coupled to a thermal bath (V neg,0 = 0.24 for the parameters used in Fig. 3(a)). Similarly, using (10) with the parameters from Fig. 3(b), we find that for squeezing to be achieved one needsn th 6 corresponding to T 150 mK.
Next, in order to assess what couplings can be achieved in a realistic experiment, we consider the following parameters: 133 Cs Rydberg atoms with a transition between n = 100 and n = 101 which are separated by ω a + ω a ≈ 2π × 6 GHz [43] corresponding to an oscillator resonant frequency ω osc = 2π × 3 GHz, which are achievable e.g. with clamped mechanical beams [44] (although with smaller quality factors than assumed in this work) or single-crystal diamond nanobars [22]. The detuning between the oscillator frequency and the P − S transition frequency is ∆ = ω a − ω osc ≈ 2π × 300 MHz, while the splitting P 3/2 − P 1/2 < 20 MHz [43]. We take the oscillator characteristic length z osc = 10 −13 m, the charge on the tip of the oscillator Q = 200e (compatible e.g. with ∼ aF capacitances of micron size electromechanical resonators operated with ∼ V voltages [45,46]) and thermal bath coupling strength Γ m = 2π × 500 Hz (corresponding to a quality factor Q = 6 × 10 6 of the oscillator [22]). For n ≈ 100 Rydberg states the atomic size is ≈ 10 4 a 0 ≈ 1 µm, and the corresponding dipole moments are µ 0 ≈ µ 0 ≈ 10 4 ea 0 (a 0 is the Bohr radius). For the atomic motion, we consider a simple linear trajectory R(t) = (vt, 0, Z 0 ) with t going from −∞ to ∞, where we neglect any deflection of the atom's path due to the interaction with the oscillator (the static monopole part of the field resulting from the charge Q can always be compensated by additional static charges; see Appendix A for details of the interaction). Assuming the atom-cantilever distance to be Z 0 = 5 µm we choose the atomic speed v = 10 m/s and the rate of atoms r = 10 5 atoms per second, giving the separation between successive atoms of 100 µm and the interaction time of couple of µs. This guarantees, to a good approximation, that only one atom is interacting with the oscillator at a time and that one can neglect the decay of the Rydberg states which have lifetime of 100 µs [47,48]. We then obtain for the integrated coupling We now turn our attention to the direct single-photon two-phonon resonance provided by the atomic dipoleoscillator quadrupole coupling as we show in Appendix A. Here, an analogous derivation leads to the integrated coupling strength G 2,quad = Qµ0z 2 osc 4π 0 2 3vZ 3 0 ≈ 10 −9 which is smaller by orders of magnitude compared to G 2 in the two-photon two-phonon resonance scheme.

IV. DISCUSSION AND OUTLOOK
We have explored a method of creating squeezed and non-classical states of a charged macroscopic mechanical oscillator. Such on-demand quantum state preparation constitutes a basic element of the mechanical oscillators state manipulation toolbox using atoms. Specifically, the squeezed and Schrödinger cat states that can be in principle generated might find applications as probes of decoherence processes of macroscopic bodies, in quantum information processing or in sensing and metrology. The values of the estimated couplings that are achievable with current state-of-the-art technology and typical parameter regimes turn out to be too small to be of a practical use. Further improvement might be sought e.g. by increasing the charge of the oscillator or by more suitable choice of the employed Rydberg states which would increase the dipole moment and decrease the two-photon detuning ∆. Another possibility is to exploit the enhancement of the coupling when considering an ensemble of atoms coupled to an array of oscillators which we leave for future investigations.
Remark: After finishing our manuscript, we became aware of a related work [49].
with R = |R| and the last line introduces notation for the coupling strengths γ j that are used in the following. The first term in eq. (A3) corresponds to a Coulomb interaction, which can be cancelled by additional static charges with opposite sign (see also Fig. 1(a)) and thus we omit it in the following. The matricesM x,y,z depend on the specific atomic transitions that couple to the electric field of the oscillator. We compute the matrix elements using the standard angular momentum theory as [50,51] (M α ) where α = x, y, z, L is the electron angular momentum, J the total angular momentum and m J the projection of the total angular momentum on the z axis. The operatorsχ are given by the relationsχ ±1 = ∓ 1 √ 2 (χ x ± iχ y ) andχ 0 = χ z . When expressed in the coordinate basis, they are simply rescaled spherical harmonics χ ± = 4π 3 Y 1,±1 (θ, φ), 0 (θ, φ). The dipole matrix elements are then obtained with the help of the relation where q = −1, 0, 1, S = J − L is the total spin and · · · · · · , · · · · · · are the Wigner 3j and 6j symbols respectively.
Note that we have absorbed the radial part of the dipole transition elements into the dipole moment amplitude µ 0 .

Single-phonon transition
For a single-phonon transition we consider resonant transitions the S and P manifolds of an atom within the same principal quantum number, as shown in figure 4(a). The transition matrices in the |L J , m J = {|S 1/2 , −1/2 , |P 1/2 , −1/2 , |S 1/2 , 1/2 , |P 1/2 , 1/2 } basis read We calculate the coupling strengths for a position R(A) = {AZ 0 , 0, Z 0 }. We find that γ y (R(A)) = 0 and that γ z (R(A))/γ x (R(A)) = 2−A 2 A . Note that the last ratio is independent of Z 0 . Fig. 4(b) shows the dipole coupling strengths γ j (R(A)) as a function of the scaled coordinate A, where the coupling strengths have been normalized to the maximum value of γ z . Since γ z γ x we neglect γ x . This simplifies the description so that one can use only two of the four levels and we choose |s ≡ |S 1/2 , 1/2 and |p ≡ |P 1/2 , 1/2 .
With this two-level system the atom-oscillator Hamiltonian can be written asĤ =Ĥ 0 +V , whereĤ 0 = ω oscâ †â + ω aσ z andV whereσ z = |p p| − |s s|,σ x = |p s| + |s p| and we have usedẑ = z osc (â +â † ). When ω osc = ω a the |s -|p transition of the atom is resonant with the one-phonon transition of the oscillator and the interaction picture Hamiltonian H I = exp(−iĤ 0 t/ )Ĥexp(iĤ 0 t/ ) reads where F and F contain only terms oscillating at ω osc or higher frequency, which can be neglected through the rotating wave approximation and we have introduced the single-phonon coupling strength γ(t) = Qµ0zosc
We are now in a position to apply the methods of degenerate perturbation theory [52] to find an effective Hamiltonian in the space spanned by {|s , |s }. Defining the projectorP = |s s| + |s s | and its complementQ = 1 −P , the Hamiltonian is partitioned into the block diagonal part H D =PĤ IP +QĤ IQ and the off-diagonal perturbation V x =PĤ IQ +QĤ IP . We find a unitary transformationÛ = eĜ, withĜ = −Ĝ † , such thatĤ eff =ÛĤ IÛ † is block diagonal, i.e.Ĥ eff =PĤ effP +QĤ effQ andĜ = ∞ j=0 1 ∆ j G (j) . The first non-zero contribution to the effective Hamiltonian is first order in 1 ∆ (second order in the expansion): The resulting Hamiltonian in the space {|s , |s } iŝ The diagonal terms are the dispersive frequency shifts. A quasi-perfect two-photon-two-phonon resonance is achieved if 3 n ∆ (γ − γ + + γ z 2 ) − (γ − γ + + γ z 2 ) , with n the phonon number, is negligibly small as compared to the off-diagonal terms in (A14). For sufficiently small n which is the situation of this article, and under the realistic assumption of µ 0 ≈ µ 0 the quasi-perfect resonance can be achieved and we thus consider only the off-diagonal terms of (A12). The effective two-phonon coupling rate γ 2 is given by the off-diagonal terms and the resulting interaction picture Hamiltonian readŝ H I,2 (t) ≈ γ 2 (t)|s s |(â † ) 2 + γ * 2 (t)|s s|â 2 (A16) The integrated coupling strength, for an atom taking a path R(t) = {vt, 0, Z 0 } then becomes (A17)

Atomic dipole -oscillator quadrupole coupling
In principle, the two-phonon resonance condition with interaction Hamiltonian similar to (6) can be achieved by exploiting the coupling between the atomic dipole and the oscillator quadrupole as we now show. The oscillator quadrupole corresponds to theẑ 2 term in the expansion ofÊ(r). Specifically, the O(ẑ 2 ) term in (A1) reads Under the two-phonon resonance condition ω osc = ω a /2, (A18) dominates the atom-oscillator interaction, and the resulting interaction picture Hamiltonian readŝ H I, quad = 1 2 Qµ 0 z 2 osc 4πε 0 R 7 Z(5Z 2 − 3R 2 ) |s p|(â † ) 2 + |p s|â 2 + F ≈ γ 2,quad (t)|s p|(â † ) 2 + H. c., where F contain only terms oscillating at ω osc or higher frequency, which can be neglected through the rotating wave approximation, and γ 2,quad = We start the derivation of the master equation for the single-phonon resonance by using (2) and the atomic initial state (3) and find the state of the oscillator after the passage of a single atom. For brevity we will write the state before the kth atom has passed ρ = ρ (k−1) osc ⊗ ρ a . Expanding (4) yields where we have used |α| 2 + |β| 2 = 1, and for the last line the index k has been suppressed, as none of the dynamics depend on it. The derivation of the master equation for the two-phonon resonance follows the same lines, withâ(â † ) replaced byâ 2 ((â † ) 2 ).