Hybrid optomechanical superconducting qubit system

We propose an integrated nonlinear superconducting device based on a nanoelectromechanical shuttle. The system can be described as a qubit coupled to a bosonic mode. The topology of the circuit gives rise to an adjustable qubit/mechanical coupling, allowing the experimenter to tune between linear and quadratic coupling in the mechanical degrees of freedom. Owing to its flexibility and potential scalability, the proposed setup represents an important step towards the implementation of bosonic error correction with mechanical elements in large-scale superconducting circuits. We give preliminary evidence of this possibility by discussing a simple state-swapping protocol that uses this device as a quantum memory element.


I. INTRODUCTION
In recent years, optomechanical systems, both in the optical and in the microwave regime, have become one of the most prominent platforms for the investigation of quantum mechanical phenomena.On the one hand, they have allowed scientists to explore foundational aspects of quantum theory [1,2]; on the other, they have provided the test bed for future technological applications of quantum mechanics [3].Prominent results in the field include sideband [4] and feedback [5] cooling to the ground state, squeezing [6,7], and entanglement [8,9] of mechanical resonators.In the context of coupling between qubits and mechanical resonators, the generation of quantum states of mechanical motion has been recently realized in high-overtone bulk acoustic-wave resonators (HBARs) where the generation of Fock [10] and cat [11] states were demonstrated.
In most of the examples mentioned above, the optomechanical system consists of a mechanical resonator (e.g. a nano-drum) whose position is parametrically coupled to a photon cavity.One of the outstanding goals in these systems has been the realization of the so-called singlephoton strong-coupling limit.In this regime, the parametric coupling energy between a single photon and the mechanical mode becomes comparable to the bare optical cavity linewidth and can therefore significantly alter the dynamics of the system [12][13][14][15][16][17][18][19][20].In microwave setups, several proposals suggest that the addition of nonlinear elements, in the form of Josephson junctions, can provide the resource needed to reach the strong-coupling regime; realizations along these lines include charge [12,21] and flux-mediated optomechanical circuits [22].
Another intriguing aspect of these systems is the possibility of realizing a quadratic coupling between the mechanical motion and the optical field.Arguably, its most prominent application is the detection of phonon * francesco.massel@usn.noFock states [23][24][25][26][27], even though two-photon cooling and squeezing both of the mechanical and the electromagnetic degrees of freedom have been predicted [27] as well.In the optical frequencies range, the quadratic coupling of an optical cavity with a mechanical mode has been realized, e.g. in membrane-in-the-middle [23] and ultracold gases setups [28].In the microwave regime, quadratic parametric coupling of a qubit to mechanical motion was recently realized with drumhead mechanical resonators coupled to superconducting circuits, exploiting the large mismatch of mechanical and qubit resonant frequency [29,30], where the generation of (non-Gaussian) numbersqueezed states was demonstrated.
In this work, we extend the nonlinear circuit approaches mentioned above, integrating a mechanical shuttling element into the design of the superconducting circuit of Fig. 1.The shuttling element consists here of a portion of superconducting material that is free to perform mechanical oscillations between two (superconducting) electrodes.Analogous shuttling devices were realized experimentally in normal (i.e.non superconducting) circuits [31][32][33], demonstrating the ability of such devices to "shuttle" electrons along with the oscillatory mechanical motion.In addition, an analogous shuttling mechanism for Cooper pairs was theoretically investigated for superconducting circuits [34].In our work, we explore how the dynamical properties of a superconducting shuttling element can be recast in terms of a (nonlinear) optomechanical coupling between a superconducting circuit and a mechanical mode, for which we believe this particular charge shuttling mechanism certainly offers a new degree of freedom [35].
More specifically, we show how, in a lumped-element description, we are able to define a system constituted by a superconducting qubit exhibiting an intrinsic quadratic coupling to the mechanical motion, in addition to a tunable linear one.The latter can be externally suppressed, leading to a dominant coupling that is quadratic in the mechanical degrees of freedom.At the same time, we will show that the tunability of the linear coupling term allows for a coherent state exchange between the qubit and the mechanical resonator.

II. THE DEVICE
Our device is constituted by a superconducting shuttle, which is free to oscillate between two terminals, as shown in Fig. 1.The terminals being gated to a voltage source V g through a gating capacitance C g .The addition of a shunting capacitance C b , meant to ensure protection from charge fluctuations, defines a transmon qubit-like device [36] (the X2MON) exhibiting nontrivial properties as a function of the mechanical shuttle dynamics.The displacement of the grounded island induces a shift in the Josephson and charging energies of the two Josephson junctions (JJs), which translates into a coupling between the superconducting circuit and the mechanical motion.As anticipated, in our device, the coupling between the mechanical degrees of freedom and a superconducting qubit can be externally tuned between a linear and a quadratic coupling, depending on the external magnetic flux through the loop defined by the two JJs.The setup we propose here differs from the design of Refs.[29,30] inasmuch our realization, for suitable values of the control parameter, is intrinsically quadratic in the mechanical displacement -i.e.not relying on the relative value of the mechanical and qubit frequencies-owing to the symmetry of the design.
From a quantum-computational perspective, the coupling between a qubit and a bosonic mode -represented here, as we will show, by a shuttling mechanical elementis an extremely promising candidate for quantum error correction [37,38].Most importantly, the tunability of the qubit/mechanical coupling of our setup allows for a great flexibility in the choice of different protocols for state-preparation and state-transfer between qubit and mechanics.Further advantages offered by mechanical resonators compared to microwave cavities as the bosonic mode are represented by their larger coherence times (mechanical linewidths ∼ 1 kHz [32] vs. cavity linewidths ∼ 100 kHz), the lack of "cross-talk" between the bosonic (mechanical) modes and, with specific reference to shuttling mechanical elements, the scalability of such platforms.
A. Lumped-element model Following Koch et al. [36], we describe the Josephson junction energy as the sum of a capacitive contribution E C and the Josephson energy E J .In our device, the lower portion of the circuit -the orange element in Fig. 1corresponds to the actual island.As a consequence, the charging and Josephson energies become dependent on the shuttle dynamics Owing to the symmetry of the device, the upper (lower) sign corresponds to junction 1 (junction 2).
The total charging energy for the circuit can be written as with C Σ = C J1 + C J2 + C b + C g where C J1 and C J2 are the capacitances of the two Josephson junctions, C g the gate capacitance and C b the shunting capacitance aimed at reducing the effects of charge noise, like in a conventional transmon setup.
In the following, we will assume that the geometric capacitances associated with the Josephson junctions C J1 and C J2 can be modeled by two (equal, parallel plate) capacitors C J 1,2 .= C J / (1 ± δ/x 0 ).Furthermore, we assume an exponential dependence of E J on the electrodes' separation.This assumption can be justified through the standard Ambegaokar-Baratoff formula [39,40] arguing that the normal-state resistance is given by R N = R N0 exp [x/ξ], (x thickness of the JJ), as a consequence of the exponential suppression of the tunneling probability through a potential barrier, combined with the Landauer formula [41], leading to with ξ = x 0 / log ∆RK 8EJRN0 and E J = E J1 (0) = E J2 (0) for symmetrical JJs.Here ∆ is the superconducting gap and R K = 2πℏ/e 2 the resistance quantum.For a typical NbN junction, we can assume ∆ = 4500 GHz, E J = 20 GHz, R N0 = 50 Ω, and x 0 = 1 nm, allowing us to estimate ξ ≃ 0.1 nm (ℏ = 1 throughout the manuscript).Following Ref. [36], we can write the system's Hamiltonian as where n and ϕ are the excess number of charge carriers and phase on the island, respectively, and For a symmetric setup, we have where E J is the Josephson energy associated with either junction.Furthermore, n g is related to the external bias voltage by n g = −C g V g /2e and the phase ϕ b is determined by the external flux bias Φ b = Φ 0 /2π ϕ b through the loop defined by the two JJs (Φ 0 = h/2e).Here, the JJs are assumed to be symmetrical, but the full calculation with general junctions is presented in the Supplemental Material.Finally, the term E (x m , p m ) in the energy associated with the dynamics of the center of mass of Cartoon picture of a (grounded) X2MON shuttle and a lumped-element description of a transmon-like setup, including the X2MON.Contacts between the shuttle (orange) and its terminals (blue) can be described by position-dependent Josephson junctions.
the shuttle, which, in our analysis, we model as a simple harmonic oscillator.
Expanding the Hamiltonian given in Eq. ( 3) in powers of φ up to the fourth order and in powers of the mechanical displacement up to the second order, focusing on the two lowest-charge states we have (see e.g.[42]) where ) are the lowering (raising) operators associated with the electrical and mechanical degrees of freedom, respectively.All coefficients in Eq. ( 5) (ω q , ω m , g 1 , g 2 ) depend on the external flux bias.The explicit dependence on Φ b is given in the Supplemental Material.As shown in Fig. 2, g 2 is the dominating interaction when the bias flux through the JJ loop is set to zero, since g 1 vanishes in this case.However, g 1 becomes the dominating term even for small deviations from the Φ b = 0 condition.As we will show below, this ability to control the type of interaction between the qubit and the mechanics makes this system very flexible in terms of possible applications, such as preparing the mechanical oscillator into a specific quantum state.For m ≃ 5 • 10 −19 kg, ω m = 1 GHz, and ω q ≃ 17 GHz (all other parameters defined above), we have that Also, note that the Hamiltonian given in Eq. ( 5) is valid in the case of equal JJs.If the JJs exhibit some degree of asymmetry, a linear coupling to σ z and a quadratic coupling to σ x appear, in addition to a small qubit rotation (zeroth-order term in σ x ).While the quadratic coupling to σ x is negligible for all parameters, the linear coupling between the displacement and σ z becomes comparable to the quadratic one when the asymmetry of the Josephson energies is ∼ xZPF 2ξ , which, for the parameters chosen here, corresponds to ∆E J = |E J1 − E J2 | ≃ 30 MHz.Furthermore, the zerothorder term in σ x is negligible whenever the asymmetry A discussion of the general case of different JJs is presented in the Supplemental Material.

III. STATE SWAPPING PROTOCOL
A promising application for the setup proposed here is bosonic error correction.On general grounds, bosonic error correction provides key advantages over quantum error correction schemes utilizing multiple physical qubits, inasmuch it eliminates the overheads and the potential issues arising from cross-talk of multiple physical qubits.More specifically, our mechanical implementation has further advantages over a microwave cavity: mechanical resonators offer better ringdown times than microwave cavities, shuttling resonators offer a more compact design and relatively straightforward scalability, and the qubit/mechanics (linear) coupling can be externally tuned.
As a first step in this direction, we consider a stateswap protocol that demonstrates the ability of this device to coherently transfer a qubit state to a quantum state of the mechanical resonator.Given the relatively large frequency mismatch between the qubit and the mechanics (ω q /ω m ≃ 20), direct transitions between mechanics and qubit are highly non-resonant.To induce such sideband transitions, we therefore modulate the flux through the JJs loop at a frequency ω, corresponding to a phase modulation given by ϕ b (t) = ϕ b,0 cos (ωt) (flux-driven sideband transitions).This technique is analogous to the one employed in Ref. [43] for the case of a transmon qubit coupled to a microwave cavity.In the context of optomechanical systems, a similar approach was also considered in [30], where the gate charge modulation was used to induce single-phonon transitions in a mechanical resonator coupled to a qubit.In general, these techniques go under the name of ac-dither techniques [44].
As a consequence of the phase modulation, the coefficients appearing in the definition of the Hamiltonian H in Eq. ( 5) become time dependent.
It is possible to show that, in a non-uniformly ro-tating frame for the qubit and the mechanics, if the driving frequency is chosen in such a way that ω = ωq − ω m , with ωq = ω q | ϕb=0 − δ q , δ q = ϕ 2 b0 ω p0 /16 and ω p0 = 16E C E J cos(ϕ b /2) for symmetrical JJs, the transformed Hamiltonian corresponds to a stateswapping Hamiltonian where g sw = ḡ1 J 0 (δ q /2ω) with J 0 (x) being the 0th-order Bessel function and ḡ1 the linear approximation of g 1 (t) with respect to ϕ b,0 .Given that ϕ b,0 is externally tunable, after the stateswapping transition described above, it is possible to externally suppress the linear coupling between the qubit and the mechanical resonator.Having this in mind, one can consider using the mechanical mode as a lowdecoherence memory element, owing to the combined effect of the on-demand suppression of lowest-order coupling between the qubit and the mechanical resonator, and the intrinsically long decoherence times of the mechanical element.
In Fig. 3, we depict a simple instance of the stateswap protocol.Starting from the state |σ z = 1, n m = 0⟩, we first perform a state swap between the qubit state and the mechanics (left), followed by the "free" evolution in the absence of qubit-mechanics coupling -ϕ b = 0, implying g 1 = 0-(center), transferring then the excitation back to the qubit (right).We have compared this stateswapping protocol with the free evolution of the |σ z = 1⟩ state in the presence of the same environment (qubit decay rate γ σ = 100 kHz) as for the state-swap protocol.The |σ z = 1, n m = 0⟩ initial state can be prepared resorting to a preliminary cooling step of the mechanical mode.
We would like to point out that, while the parameters chosen here push the boundaries of what is experimentally realizable, with values of the f • Q product of the order of 10 14 , record values of f • Q = O(10 18 ) have been reported for bulk acoustic wave resonators [45,46].

IV. EXPERIMENTAL OUTLOOK AND PERFORMANCE
From an experimental point of view, the realization of a shuttling island (orange box in Fig. 1) can be performed with some additional processing steps.However, fabrication of a superconducting island has turned out to be a challenge so far, since standard superconducting materials, such as Al, tend to oxidize through for small 50 3 nm 3 islands.Our latest work on that end shows that we can realize superconducting NbN-strips [47] with a T c = 9K, which avoids the aforementioned issue and is fully compatible with our processing techniques.Hence, the fabrication of superconducting islands in varying circuit combinations is possible now.One of the circuits to be implemented is shown in Fig. 1.
Additionally, the ability to perform precise ac-dither protocols with the magnetic flux is required to successfully operate the device as a platform for useful quantum operations.These techniques are, in principle, possible with tuning either the gate charge or the magnetic flux [44], and different kinds of modulation schemes have been experimentally demonstrated to be feasible, e.g.Ref. [30] for charge-based and Refs.[43,48] for flux-based procedures.
Note that, in this work, we do not focus on parameter optimization for our device, for example, to maximize the couplings or to minimize the noise in the circuit, but instead we use fairly typical values for superconducting circuits.As an example, one could introduce a larger shunting capacitance to avoid charge noise effects thanks to the reduced charging energy, placing the device firmly in the E J ≫ E C transmon regime.
Even though we are not aiming to optimize the performance of our device here, the accessible coupling strengths between the qubit and the mechanics are promising when considering the proposed device as a candidate for practical quantum computation protocols.We obtain a quadratic coupling g 2 that is about an order of magnitude larger than presently achievable linewidths for shuttling mechanical resonators using the parameters in Fig. 2 for a wide range of flux biasing, and g 2 /ω m ≈ 4 × 10 −5 which is comparable to different state-of-theart implementations of bosonic error correction schemes.For example, in a recent demonstration of quantum error correction beyond the break-even point using a bosonic (photonic) GKP code [38], the corresponding ratio was ∼ 1 × 10 −5 with a superconducting cavity as the bosonic system.However, we do not predict reaching the ultrastrong coupling limit as in Ref. [30], but in terms of the linear coupling g 1 , we still can attain significantly larger values than the resonator linewidth with g 1 /ω m > 0.01 already starting from a small flux bias.
We want to emphasize that the state swapping protocol explored in this work is not meant to be a comprehensive procedure for bosonic error correction, but instead a proof of concept that our device could be useful in such applications.Indeed, it is possible to create arbitrary phonon number states with detailed control of the state swapping Hamiltonian [49] and, additionally, other interesting and useful operations can be realized with this interaction alone, such as the preparation of a mechanical cat states [11].On top of this linear interaction between the qubit and the mechanics, we also have access to another resource, namely the quadratic coupling, that is largely unexplored in this work.

V. CONCLUSIONS
In our work, we introduce a novel device (the X2MON) which consists of a transmon coupled to a mechanical shuttle operating in the quantum regime.We discuss the nature of the coupling between the two.Furthermore, we demonstrate a state-swap protocol, which, owing to the properties of the shuttle, can be directly employed as a quantum memory.Our work paves the way for bosonic error correction with mechanical modes.
1. Schematic of the SUNMESH circuit.For the formal calculation, the voltage source is replaced by a capacitor C Vg allowing voltage V g to be induced on island 2.

DERIVATION OF THE X2MON HAMILTONIAN
We present here a full derivation of the Hamiltonian of the X2MON circuit (see Fig. 1) and of the couplings between the mechanical motion and the qubit that arise in the system.
The calculation here generalizes the discussion of the main text, we do not assume the Josephson junctions in the system to be symmetric.Note that we use the convention ℏ = 1.

Lumped-element description
We derive the Hamiltonian for the X2MON circuit with standard circuit quantum electrodynamics (circuit-QED) methods [1,2].The flux at the node i at time t is given by which defines the voltage of the node as The node flux and phase are connected with the following relation where Φ 0 = h 2e is the flux quantum.The indices of different nodes in our circuit are given in Fig. 1.
The energy related to the capacitive elements of the circuit is given by Here C J1 and C J2 are the capacitances of the Josephson junctions, C g is the gate capacitance, and C b the shunting capacitance that is large so that our qubit operates in the TRANS-MON regime.C Vg is an additional capacitance, replacing the voltage source for the formal calculation, whose only function in this formalism is to induce the voltage bias on the qubit island, see Ref. [1].Note that Φ3 is not present here, since Φ3 = 0 is due to that node being grounded.Only the Josephson junctions contribute to the inductive energy of the circuit, where ) and the Josephson energies are given by Anticipating our analysis of the device dynamics, in Eqs.(SI5,SI6), we have explicitly indicated the JJ energy dependence on the deviation (δ) from the equilibrium position(x 0 ) of the shuttle.

Josephson energy
We provide here a brief explanation of the forms of the Josephson energies of the two junctions given in Eq. (SI6).Let us assume that the normal state resistance of the junctions is stemming from the exponentially suppressed tunneling probability across the junction as a function of its thickness x.The Ambegaokar-Baratoff formula [3,4] provides the following expression for the JJ energy where R K = 2πℏ e 2 is the resistance quantum, ∆ the superconducting gap, and ϕ the superconducting phase difference across the junction.Focusing on the characteristic Josephson energy E J = |E J (ϕ = 0)|, we get Eq.(SI6) with the help of the Landauer formula [5] and Eq.(SI7), where the parameter ξ is given by

X2MON Hamiltonian
The Lagrangian of the system, including the Lagrangian of the shuttle L m (x m , p m ), can be written using the capacitance matrix The Hamiltonian can be expressed using this inverse capacitance matrix where E (x m , p m ) is the (elastic) energy of the shuttle, and the conjugate variable to flux Φ i are charges on each island . Note that the voltage on island 2, ∂H ∂Q 2 , can now be set to the gate voltage V g by choosing Now, defining the number of Cooper pairs on the qubit island, the charging energy of the qubit, and the gate charge the Hamiltonian in Eq. (SI13) can be written in the canonical form where the last term can be ignored due to it providing only a constant contribution to the total energy of the system and not affecting its dynamics.

Quantization of the Hamiltonian
Firstly, we expand the potential energy part of the Hamiltonian in Eq. (SI15) up to the fourth order in ϕ 1 We then promote the phase ϕ and n to quantum operators φ and n, respectively.These We can express them in terms of bosonic lowering (raising) operators a (a † ), which obey the standard bosonic commutation relation a, a † = 1, as Substituting these definitions into Eq.(SI17) implies for the zero-point fluctuations The quantized form of the Hamiltonian, Eq. (SI16), can then be written as Within the conventional harmonic description [1], recalling the identity Eq. (SI19), we can express the zero-point fluctuations as and the full expression of the Hamiltonian becomes where where Let us go through the expansion of the Hamiltonian, Eq. (SI22), term by term.The coefficient definitions are summarized below in Eq. (SI33) for the reader's convenience.
For the (a † a + 1 2 ) term, we obtain and the term related to (a † + a) 4 gives where the first term contributes to the renormalization of the qubit frequency.
The energy of the shuttle E (x 0 + δ, p m ) is expressed as the sum of a kinetic p m /2m and an elastic 1/2 mω 2 m (x 0 + δ) 2 contribution; it is quantized as ω m0 b † b where ω m0 is the bare mechanical frequency of the shuttle.The frequency ω m0 is renormalized partly by the contribution proportional to b † b from the following term arising from the coupling to the qubit.We now expand the coefficients appearing in Eq. (SI22).The coefficient corresponding to the (a † + a) term is and finally the term with (a † + a) 3 Gathering all the terms, the full Hamiltonian is In the following, we neglect mixing of states outside the computational basis and therefore consider normal ordering the qubit operators (a † +a) 3 → 3 (a † +a) and (a † +a) 4 → 12 a † a.A final mapping of bosonic operators onto Pauli matrices (a † a → 1/2 (σ z + 1) and a † + a → σ x ) allows us to write the final form of the Hamiltonian of the X2MON circuit as exhibiting both linear and quadratic coupling of the mechanical displacement to the σ x and σ z terms of the qubit.The relative strengths of the couplings can be tuned with the flux bias ϕ b and also already in the fabrication stage of the device by choosing a suitable asymmetry of the Josephson energies of the junctions.Notably, the quadratic coupling to the σ x component is weak compared to the other terms and can be disregarded in general.
Here, the renormalized qubit and mechanical frequencies are For completeness, the full list of the coupling coefficients is x ZPF ξ  x ZPF ξ Crucially, for symmetric Josephson junctions (E J1 = E J2 ≡ E J ), many of these couplings vanish, and we obtain g 01 = g 10 = g 12 = g 21 = g 30 = g 32 = 0, resulting in the Hamiltonian presented in the main text Eq. ( 5) with the notation g 1 ≡ g 11 + 3g 31 and g 2 ≡ g 22 + 12g 42 related to the couplings g 1 (b † + b)σ x and g 2 (b † + b) 2 σ z .For symmetric junctions, the coefficients to these couplings are expressed as (SI34)

APPROXIMATE STATE-SWAPPING HAMILTONIAN
We derive here the Hamiltonian H ′ given in Eq. ( 7) of the main text, by considering the effect of a flux modulation given by ϕ b = ϕ b0 cos (ωt).As a result, the numerical factors appearing in the Hamiltonian given in Eq. ( 5) of the main text will be time-dependent.Up to second order in ϕ b0 , and first order in x ZPF /ξ, is possible to write them as 1 + cos (2ωt) 2 = ωq − δ q cos (2ωt) where J k (z) is the n-th order Bessel function, we can write H ′ as H ′ = ḡ1 cos (ωt) b † e iωmt + be −iωmt e iωqt J 0 δ q 2ω + J 1 δ q 2ω e 2iωt . . .σ + + h.c. .

FIG. 3 .
FIG.3.Demonstration of the state swapping protocol.Left panel: the qubit state |σz = 1⟩ (red line) is transferred to the mechanical oscillator Fock state |n = 1⟩ (black line/black shade).Central panel: the linear coupling g1 is suppressed.The free evolution of the state |n = 1⟩ is subject to the (small) intrinsic dissipation of the mechanical resonator.Right panel: the state |n = 1⟩ is transferred back to the qubit.The fidelity of the swapping to the mechanical resonator and back, is compared with the free evolution of the |σz = 1⟩ state under subject to pure qubit decoherence (blue line).Parameters: EC = 1 GHz, EJ = 20 GHz, xZPF/ξ = 3 • 10 −3 , ϕ b,0 = 0.5, mechanical angular frequency ωm = 1 GHz and damping rate γ = 1 kHz, qubit damping rate γσ = 100 kHz.In the simulations, we have assumed a finite temperature of 10 mK, both for the qubit and for the mechanical bath, corresponding to a population of n σ,th ≃ 2 • 10 −7 and n m,th ≃ 0.87, respectively.The time axis is reset to zero for each step in the protocol to better illustrate the different time scales.

)
DERIVING THE COUPLING BETWEEN THE QUBIT AND THE MECHANICSWe now expand the Hamiltonian in Eq. (SI22) with respect to the mechanical displacement up to the second order and denote the quantized mechanical displacement (obeying canonical commutation relations) with δ = x ZPF (b † + b).The following shorthand notation is used to display the results in a more compact form:E C = E C (0), E J1 = E J1 (0), E J2 = E J2 (0), and d 0 = d(0).The Josephson capacitances are approximated with parallelplate capacitors, i.e. the charging energy has the form