# Quantum electromechanics with levitated nanoparticles

## Abstract

Preparing and observing quantum states of nanoscale particles is a challenging task with great relevance for quantum technologies and tests of fundamental physics. In contrast to atomic systems with discrete transitions, nanoparticles exhibit a practically continuous absorption spectrum and thus their quantum dynamics cannot be easily manipulated. Here, we demonstrate that charged nanoscale dielectrics can be artificially endowed with a discrete level structure by coherently interfacing their rotational and translational motion with a superconducting qubit. We propose a pulsed scheme for the generation and read-out of motional quantum superpositions and entanglement between several levitated nanoparticles, providing an all-electric platform for networked hybrid quantum devices.

## Introduction

Opto- and electromechanical systems are at the cutting edge of modern quantum devices1,2,3, with great potential for technological application and fundamental tests4,5,6. Levitating nanoscale objects almost perfectly isolates them from their surroundings, enabling superior force sensitivity and coherence times7. Optically levitated nanoparticles have been successfully cooled into their motional quantum groundstate8, opening the door to free-fall center-of-mass9,10,11 and rotational12,13 quantum superposition tests.

Quantum experiments with trapped nanoparticles7 require schemes to control their rotational and translational quantum states. Continuous-wave optical techniques are limited by the detrimental impact of photon scattering decoherence14 and by internal heating due to photon absorption15,16. In addition, the fact that nanoscale particles lack the discrete internal spectrum of atoms or other microscopic quantum systems makes it difficult to address them coherently with laser pulses.

Here, we demonstrate that the states of a superconducting qubit can be used to manipulate and read out the quantum dynamics of a charged nanoparticle levitated in a Paul trap (see Fig. 1). While this has been achieved with clamped oscillators2,17,18,19, for levitated nanoparticles it requires decoupling their rotations from their center-of-mass motion. We argue that this task can be realistically achieved in the all-electrical setup proposed here. Electrically levitated nanorotors are thus ideally suited for trapped superposition experiments with a wide variety of particle geometries and charge distributions, and present a versatile alternative to other nanoparticle-qubit setups20,21,22,23.

Quadrupole ion traps provide exceptionally stable confinement for charged nanoparticles24. Moreover, the particle motion induces an electric current in the endcap electrodes25,26,27. We propose to use this current for first cooling the nanoparticle to milliKelvin temperatures and then interfacing its motion with a superconducting circuit. The resulting coupling between superconductor and particle scales as charge over square root mass28,29 and can thus be as strong as for a single atomic ion for realistic charge distributions.

We show that the proposed all-electrical platform enables cooling and interference experiments with levitated nanorotors, making it well suited for generating and reading-out entanglement between several particles and superconducting qubits, thus forming a building block of a larger quantum network. The motional quantum state can be prepared and observed by qubit manipulations with an ultra-fast pulsed scheme, operating on a timescale much shorter than the mechanical period and within the coherence time of the charge qubit. This pulse scheme allows to speed-up the observation of nanoscale quantum interference in a variety of opto- and electromechanical setups20,21,30,31.

## Results

### Ro-translational macromotion

A charged nanoparticle is suspended in a hyperbolic Paul trap of endcap distance 2z0 and radius $$\sqrt{2}{z}_{0}$$, where the ring electrode is put to the time-dependent potential $${U}_{{\rm{PT}}}(t)={U}_{{\rm{dc}}}+{U}_{{\rm{ac}}}\cos ({\Omega }_{{\rm{ac}}}t)$$ with respect to the floating endcaps (see Fig. 1). Due to the quadrupole symmetry of the electric field, the rotational and translational particle motion is fully determined by its total charge q, orientation-dependent dipole vector p(Ω), and quadrupole tensor $${\mathsf{Q}}\left(\Omega \right)$$. Here, Ω denotes the orientational degrees of freedom of the particle, e.g. parametrized by Euler angles; its center-of-mass position is r.

In general, the resulting time-dependent force and torque will lead to complicated and unstable dynamics of the nanoparticle. However, if the trap is driven sufficiently fast, its micromotion can be separated, and one obtains a time-independent effective trapping potential for the macromotion (see “Methods”)

$$\begin{array}{ll}{V}_{{\rm{eff}}}({\bf{r}},\Omega) &= \frac{{U}_{{\rm{dc}}}}{2{z}_{0}^{2}}\left(\frac{q}{2}{\bf{r}}\cdot {\mathsf{A}}{\bf{r}}+{\boldsymbol{p}}\cdot {\mathsf{A}}{\bf{r}}-\frac{1}{2}{{\bf{e}}}_{z}\cdot {\mathsf{Q}}{{\bf{e}}}_{z}\right)\\ & +\frac{{U}_{{\rm{ac}}}^{2}}{16{z}_{0}^{4}{\Omega }_{{\rm{ac}}}^{2}}\mathop{\sum }\limits_{i = 1}^{3}\frac{1}{{I}_{i}}{\left[{{\bf{n}}}_{i}\cdot \left({\boldsymbol{p}}\times {\mathsf{A}}{\bf{r}}+{{\bf{e}}}_{z}\times {\mathsf{Q}}{{\bf{e}}}_{z}\right)\right]}^{2}\\ & +\frac{{U}_{{\rm{ac}}}^{2}}{16M{z}_{0}^{4}{\Omega }_{{\rm{ac}}}^{2}}\left(q{\bf{r}}+{\boldsymbol{p}}\right)\cdot {{\mathsf{A}}}^{2}\left(q{\bf{r}}+{\boldsymbol{p}}\right).\end{array}$$
(1)

Here, M is the particle mass and the Paul trap symmetry axis is aligned with ez, such that $${\mathsf{A}}$$ = $${\mathbb{1}}$$ − 3ezez. The Ii denote the moments of inertia with ni the associated directions of the rotor principal axes. We dropped the orientation dependence of p, $${\mathsf{Q}}$$, and ni for compactness.

The effective potential (1) describes the coupled rotational and translational macromotion of an arbitrarily charged and shaped nanoparticle in a quadrupole ion trap, and is thus pertinent for ongoing nanoparticle experiments24,31,32. It shows that stable trapping can be achieved for sufficiently small bias voltages Udc (with frequencies $${\omega }_{z}=q{U}_{{\rm{ac}}}/\sqrt{2}M{\Omega }_{{\rm{ac}}}{z}_{0}^{2}$$ and ωx,y = ωz/2). In addition, the rotational and translational motion decouple for particles with vanishing dipole moment. Note that if the particle has a finite quadrupole moment its rotation dynamics can still be strongly affected by the trapping field. This may provide means to manipulate the rotation dynamics of nanoscale particles, a challenging task due to the nonlinearity of rotations32,33,34,35,36.

### Particle-circuit coupling

The rotational and translational motion of the particle induces mirror charges in the endcap electrodes. The latter can be quantified by extending the Shockley−Ramo theorem to arbitrary charge distributions (see “Methods”), yielding the capacitor charge Qind = −kez (qr + p)/z0 + CV, given the endcap capacitance C and voltage drop V. The geometry factor k, with values of 0 < k 1/2 for realistic electrode geometries, determines the approximately homogeneous field −kV/z0ez close to the trap center in the absence of the particle. (A perfect plate capacitor corresponds to k = 1/2).

The induced capacitor charge only depends on the total motional dipole moment qr + p along the Paul trap axis, and is thus independent of the particle quadrupole moment. A circuit connecting the electrodes picks up the ro-translational motion of the particle via the current I = dQind/dt. At the same time, the particle feels a voltage-dependent electrostatic force and torque depending on the circuit state. This can be used for resistive cooling and for coherently interfacing the particle with a superconducting qubit.

Nanoparticle resistive cooling can be achieved by joining the endcaps with a resistance R. The dissipation of the induced current in the resistor leads to thermalization of the particle motion at the circuit temperature. The timescale of this resistive cooling can be tuned by adding an inductance in series to the circuit29. In the adiabatic limit, it reacts almost instantaneously to the particle motion. Thus, the circuit degrees of freedom can be expanded to first order in the particle velocity and rotation speed, yielding the effective total cooling rate

$${\gamma }_{{\rm{ad}}}=\frac{R{k}^{2}}{{z}_{0}^{2}}\left(\frac{{q}^{2}}{M}+\mathop{\sum }\limits_{i = 1}^{3}\frac{1}{{I}_{i}}{\left[{{\bf{n}}}_{i}\cdot ({\boldsymbol{p}}\times {{\bf{e}}}_{z})\right]}^{2}\right).$$
(2)

This quantifies how fast an initially occupied phase-space volume contracts, predicting the timescale of rotational and translational thermalization with the circuit. The rate (2) is always positive and exhibits a q2/M-scaling, indicating that charged nanoparticles can be cooled as efficiently as atomic ions.

### Interfacing nanoparticle and charge qubit

The levitated nanoparticle can be coherently coupled to a superconducting Cooper-pair box37 by attaching the latter to the endcap electrodes (see Fig. 1), as also proposed for atomic ions28,38. The nanoparticle motion towards the endcaps modifies the voltage drop over the Cooper-pair box, whose charge state determines the force and torque acting on the particle. Preparing the Cooper-pair box in a superposition of charge states37 thus entangles the nanoparticle motion with the circuit. This can be used to generate and verify nanoscale motional superposition states.

The combined nanoparticle-Cooper-pair box Hamiltonian can be derived in a lengthy calculation from Kirchhoff’s circuit laws (see “Methods”). Operating the Cooper-pair box in the charge qubit regime37 of N and N + 1 Cooper pairs yields the nanorotor-qubit coupling

$${H}_{{\rm{int}}}=-\frac{2ek}{{C}_{\Sigma }{z}_{0}}(N+{\sigma }^{+}{\sigma }^{-})(q{\bf{r}}+{\boldsymbol{p}})\cdot {{\bf{e}}}_{z},$$
(3)

where CΣ is the effective capacitance of the circuit and the qubit raising and lowering operators are denoted by σ+, σ. This interaction couples the charge eigenstates of the box to the motional dipole moment of an arbitrarily shaped and charged nanorotor, implying that qubit charge states are conserved by the interaction and that rotations of the quadrupole or higher multipole moments of the nanoparticle are not coupled by the qubit.

In the experimentally realistic situation that the nanoparticle is almost homogeneously charged and inversion symmetric, its dipole moment is negligibly small (see “Methods”). The rotational and translational macromotion in the Paul trap (1) then decouple even for large quadrupole moments. The center-of-mass motion along the Paul trap axis further decouples from the transverse degrees of freedom, since only the motion towards the electrode is affected by the Cooper-pair box. The particle trapping potential in z-direction is slightly shifted and stiffened due to the charge qubit (with N ≠ 0), yielding the effective Hamiltonian

$${H}_{{\rm{1D}}}={E}_{{\rm{c}}}{\sigma }^{+}{\sigma }^{-}+\hslash \omega {a}^{\dagger }a-\hslash \kappa {\sigma }^{+}{\sigma }^{-}\left(a+{a}^{\dagger }\right),$$
(4)

with charge energy Ec and coupling strength $$\kappa =2ekq/{C}_{\Sigma }{z}_{0}\sqrt{2M\hslash \omega }$$, where the nanoparticle oscillation with frequency $${\omega }^{2}={\omega }_{z}^{2}+{q}^{2}{k}^{2}/{C}_{\Sigma }M{z}_{0}^{2}$$ is described by the ladder operators a, a (see “Methods”).

The Hamiltonian (4) demonstrates the important fact that the qubit can be used to generate quantum states as if the nanoparticle were non-rotating18,20,21,30. The absence of discrete internal transitions can thus be compensated by the nonlinearity provided by a superconducting circuit. The nanoparticle-qubit coupling strength is proportional to charge over square root mass, yielding appreciable coupling for highly charged nanoscale objects (see “Methods”).

Note that a finite bias voltage Udc applied to the ring electrode will not affect the Cooper-pair box, but produce an additional, approximately linear potential at the shifted trap center zs. It adds the term −Vext (a + a) to (4), with $${V}_{{\rm{ext}}}=q{U}_{{\rm{dc}}}{z}_{s}\sqrt{\hslash /2M\omega {z}_{0}^{4}}$$. This term will be used below to control the relative phase of the nanoparticle superposition state. The linear approximation is valid as long as the thermal width $$\sqrt{2{k}_{{\rm{B}}}T/M{\omega }^{2}}$$ is much smaller than zs.

### Generating and observing superpositions

Quantum interference of the nanoparticle motion on short timescales can now be performed by a rapid sequence of qubit rotations and measurements. At the beginning of the interference scheme, the charge qubit is prepared in its groundstate $$\left|N\right\rangle$$, while the nanoparticle is cooled to temperature T, $${\rho }_{0}=\left|N\right\rangle \left\langle N\right|\otimes \exp (-\hslash \omega {a}^{\dagger }a/{k}_{{\rm{B}}}T)/Z$$. After this initial state preparation Vext is switched to a constant value. The free dynamics governed by (4) with the external potential is then intersected by σx-rotations of the qubit at four different times:

1. (i)

a π/2-pulse at t = 0, which prepares the qubit in a superposition of charge states,

2. (ii)

a π-pulse at t = t1, which flips the qubit state,

3. (iii)

another π-pulse at t = t2, and

4. (iv)

a π/2-pulse at t = t3 with subsequent measurement of the qubit occupation σ+σ.

With a symmetric pulse scheme, i.e. t1 = t3 − t2 = τ, it is always possible to find a Δτ ≡ t2 − t1 such that the nanoparticle state evolves into a superposition and then recombines with maximal overlap (see “Methods”). The corresponding phase-space trajectories are illustrated in Fig. 2. In this case, the particle motion is first entangled with the qubit, generating a motional quantum superposition. This superposition is then reversed by steps (ii) and (iii), and finally recombined, undoing the entanglement. Through this sequence, the phase imprinted on the nanoparticle motion through the external voltage Vext is transferred onto the qubit state and read-out via its population,

$$\left\langle {{\mathsf{\sigma }}}^{+}{{\mathsf{\sigma }}}^{-}\right\rangle ={\cos }^{2}\left[\left(\frac{{\kappa }^{2}}{\omega }+\frac{2\kappa {V}_{{\rm{ext}}}}{\hslash \omega }-\frac{{E}_{{\rm{c}}}}{\hslash }\right)\left(\tau -\frac{{\Delta }_{\tau }}{2}\right)\right].$$
(5)

Varying Vext and observing the corresponding modulation of the qubit population can thus be used to verify that the nanoparticle existed in a spatial superposition state. The final population also oscillates as a function of the pulse time τ.

This pulse scheme enables the generation and observation of nanoscale quantum superpositions in harmonic potentials with pulse separations τ much shorter than the particle oscillation period. It is therefore applicable to various opto- and electromechanical systems20,30,31. The required accuracy of the pulse times, determined by the qubit frequency and the particle temperature T, must ensure that the phase κVext(2τ − Δτ)/$$\hbar$$ω is measurable.

To illustrate that nano- to microsecond motional superpositions can be realistically prepared and observed on the coherence timescale of a charge qubit39, we show in Fig. 3 the expected interference signal of a 106 amu particle at T = 1 mK. The nanoparticle is assumed to be cylindrically shaped, with a homogeneous surface charge of q = 200e and a realistic dipole moment (see “Methods”). It is stably levitated inside a sub-millimeter Paul trap, its motion well approximated by a harmonic oscillation with ω = 138 kHz. We find that the resulting strong coupling to the Cooper-pair box of κ = 16.8 MHz renders the nanoparticle particle sensitive to the presence or absence of a single Cooper-pair. A voltage of Udc = 25 V then suffices to imprint a relative phase on the motional superposition that shifts the interference pattern by a full fringe.

### Networking levitated nanoparticles

The proposed interference protocol can be extended to transfer qubit entanglement to nanoparticles. Levitated objects may thus be coherently integrated into superconducting quantum networks, e.g. for sensing and metrology applications. Here we illustrate how to entangle the nanoparticle with a second, separated charge qubit, or with another nanoparticle levitating in a distant Paul trap.

Entanglement of the nanoparticle with a second qubit is achieved by replacing the initial π/2-pulse with an operation that prepares a maximally entangled two-qubit state40,41. To verify the involvement of the particle in the nonlocal dynamics, one carries out the above interference protocol by performing all pulses on both qubits. During the pulse sequence one qubit is coupled to the nanoparticle, while the other qubit is isolated and can in principle be located at large distances. The occupation of the separated qubit, conditioned on having found the directly coupled one in the groundstate, is then given by

$$\left\langle {\sigma }^{+}{\sigma }^{-}\right\rangle ={\cos }^{2}\left[\chi \left(\tau -\frac{{\Delta }_{\tau }}{2}\right)\right],$$
(6)

with χ = κ2/ω + 2κVext/$$\hbar$$ω − (Ec1 − Ec2)/$$\hbar$$. This assumes that the qubits were initially prepared in the singlet state $$\left|{\Psi }^{-}\right\rangle$$. The external potential Vext, acting only on the nanoparticle, thus serves to fully control the measurement outcome of the distant qubit. Having established that the coherent dynamics extends from the particle to the distant qubit, entangled states of these two systems can be produced by carrying out step (iv) and the subsequent measurement of the directly coupled qubit at t3 < 2τ + Δτ, i.e. before the particle wave packets overlap.

An all-electrical protocol to entangle two distant levitated nanoparticles works along the same lines: We consider two distant nanoparticle-qubit setups of identical frequency ω, where the qubits are again initially in the state $$\left|{\Psi }^{-}\right\rangle$$. To verify the involvement of both particles in the nonlocal dynamics, one carries out the protocol on each nanoparticle-qubit setup until the time t3 = 2τ + Δτ of wave packet overlap. The occupation of the second qubit, conditioned on having found the first one in the groundstate, is then given by (6), with χ = χ1 − χ2, where $${\chi }_{i}={\kappa }_{i}^{2}/\omega +2{\kappa }_{i}{V}_{i}/\hslash \omega -{E}_{{\rm{c}}i}/\hslash$$. The interference pattern thus depends on the difference of the local nanoparticle phases. By measuring both qubits before the wave packets overlap, e.g. at τ + Δτ < t3 < 2τ + Δτ the two oscillators can be projected onto an entangled motional state (see “Methods”).

## Discussion

The coherent control of charged nanoparticles by superconducting qubits offers a new avenue for quantum superposition experiments with massive objects. The nanoparticle superposition state is generated and read-out by pulsed qubit rotations and measurements, enabling interference experiments on ultra-short time scales. All-electric trapping, cooling, and manipulation avoids photon scattering and absorption, the dominant decoherence sources in laser fields. In addition, the nanoparticle rotations decouple from the center-of-mass motion for realistic particle shapes and charge distributions, rendering this setup widely applicable. It holds the potential of bridging the mass gap in quantum superposition tests from current experiments with massive molecules42 to future guided interferometers with superconducting microscale particles43. Moreover, these hybrid quantum devices can serve as building blocks for larger networks connected by superconducting circuitry, distributing entanglement between multiple nanoparticles.

The presented qubit-nanoparticle coupling scheme is feasible with available technology for realistic particles. Beyond that, if it were possible to achieve qubit coherence times on the order of the oscillator period, the generation of more complex mechanical superposition states might become possible44. Similarly, combining qubit rotations with variable interaction times might enable implementing nonlinear phase gates acting on the nanoparticle state45. In addition, the degree of quantum control can be enhanced by fabricating particles with tailored dipole and quadrupole moments and by combining electric with optical techniques46,47,48,49. This may give rise to the observation of coherent effects between the rotational and translational nanoparticle degrees of freedom, and provide a platform for studying charge-induced decoherence in an unprecedented mass and complexity regime.

## Methods

### Ro-translational macromotion in a Paul trap

An arbitrarily charged nanoparticle moving and revolving at position R and orientation Ω in a hyperbolic Paul trap is subject to the time-dependent potential

$$V({\bf{R}},\Omega ,t)=\frac{{U}_{{\rm{PT}}}(t)}{2{z}_{0}^{2}}\left(\frac{q}{2}{\bf{R}}\cdot {\mathsf{A}}{\bf{R}}+{\boldsymbol{p}}\cdot {\mathsf{A}}{\bf{R}}-\frac{1}{2}{{\bf{e}}}_{z}\cdot {\mathsf{Q}}{{\bf{e}}}_{z}\right),$$
(7)

with $${\mathsf{A}}$$ = $${\mathbb{1}}$$ − 3ezez. Here, the dipole moment p and the quadrupole tensor $${\mathsf{Q}}$$ depend on the principal axes Ni of the nanoparticle with the associated moments of inertia Ii. The effective potential for the macromotion is obtained by setting R = r + ϵ, and Ni = ni + δ × ni, serving to separate the center-of-mass macromotion r from the much faster micromotion ϵr varying with zero mean. Similarly, the rotational micromotion δ 1 varies much faster than ni. The center-of-mass and angular momentum obey

$$m\ddot{{\bf{R}}}=-\frac{{U}_{{\rm{PT}}}(t)}{2{z}_{0}^{2}}{\mathsf{A}}\left(q{\bf{R}}+{\boldsymbol{p}}\right)$$
(8A)

and

$$\dot{{\bf{J}}}=-\frac{{U}_{{\rm{PT}}}(t)}{2{z}_{0}^{2}}\left({\boldsymbol{p}}\times {\mathsf{A}}{\bf{R}}+{{\bf{e}}}_{z}\times {\mathsf{Q}}{{\bf{e}}}_{z}\right).$$
(8B)

Taking macromotion to be approximately constant on the time scale of the micromotion and neglecting all small terms yields

$${\boldsymbol{\epsilon }}\approx \frac{{U}_{{\rm{ac}}}\cos ({\Omega }_{{\rm{ac}}}t)}{2M{z}_{0}^{2}{\Omega }_{{\rm{ac}}}^{2}}{\mathsf{A}}\left(q{\bf{r}}+{\boldsymbol{p}}\right)$$
(8C)

and

$${\boldsymbol{\delta }}\approx \frac{{U}_{{\rm{ac}}}\cos ({\Omega }_{{\rm{ac}}}t)}{2{z}_{0}^{2}{\Omega }_{{\rm{ac}}}^{2}}\mathop{\sum }\limits_{i = 1}^{3}\frac{1}{{I}_{i}}{{\bf{n}}}_{i}[{{\bf{n}}}_{i}\cdot \left({\boldsymbol{p}}\times {\mathsf{A}}{\bf{r}}+{{\bf{e}}}_{z}\times {\mathsf{Q}}{{\bf{e}}}_{z}\right)].$$
(8D)

The dipole and quadrupole moments here only include the macromotion, i.e. p = ∑pini and Q = ∑Qijninj, in contrast to (8A and 8B). The Mathieu parameter $${U}_{{\rm{ac}}}q/2M{\Omega }_{{\rm{ac}}}^{2}{z}_{0}^{2}$$ and its rotational analogs $${U}_{{\rm{ac}}}| {\boldsymbol{p}}| /2M{\Omega }_{{\rm{ac}}}^{2}{z}_{0}^{2}{l}_{{\rm{cm}}}$$, $${U}_{{\rm{ac}}}| {Q}_{\ell m}| /2{I}_{i}{\Omega }_{{\rm{ac}}}^{2}{z}_{0}^{2}$$ and $${U}_{{\rm{ac}}}| {\boldsymbol{p}}| {l}_{{\rm{cm}}}/2{I}_{i}{\Omega }_{{\rm{ac}}}^{2}{z}_{0}^{2}$$ in (8C and 8D) determine when the micromotion ϵ and δ is small. Here lcm is the length scale of the center-of-mass motion.

The effective force and torque of the macromotion can be obtained by inserting (8C and 8D) into (8A and 8B) and averaging over one micromotion cycle. A lengthy but straightforward calculation demonstrates that they can be expressed through the time-independent effective potential (1). We remark that the same potential (1) can also be derived quantum mechanically by adapting the method outlined in ref. 50 for the combined rotational and translational motion of the nanoparticle.

### Generalized Shockley−Ramo theorem

To calculate the current induced by an arbitrary, rigidly bound charge distribution moving and rotating between the endcap electrodes we use Green’s reciprocity theorem,

$${\int}_{{\mathcal{V}}}{d}V\,{\phi }_{{\rm{ref}}}\rho +{\int}_{\partial {\mathcal{V}}}{d}A\,{\phi }_{{\rm{ref}}}\sigma ={\int}_{{\mathcal{V}}}{d}V\,\phi {\rho }_{{\rm{ref}}}+{\int}_{\partial {\mathcal{V}}}{d}A\,\phi {\sigma }_{{\rm{ref}}}.$$
(9)

It relates the particle charge density ρ, the electrode surface charge density σ and the electrostatic potential ϕ to those of a reference system. Choosing the reference system to have no particle in the trap volume $${\mathcal{V}}$$, a vanishing potential on the ring electrode, and opposite potentials on the endcaps leads to an approximately linear potential ϕref near the trap center. This results in

$$\frac{k}{{z}_{0}}{{\bf{e}}}_{z}\cdot {\int}_{{\mathcal{V}}}{d}V\,{\bf{x}}\rho ({\bf{x}})+\frac{{Q}_{1}-{Q}_{2}}{2}=CV,$$
(10)

where Q1 and Q2 are the total charges on the top and bottom endcap and x originates from the trap center. The remaining integration yields the capacitance charge Qind= (Q1 − Q2)/2 = −kez (qr + p)/z0 + CV, and its time derivative the induced current.

### Cooper-pair box-nanoparticle Hamiltonian

Figure 4 shows how the Paul trap is connected with the circuit. The latter consists of a superconducting loop with two Josephson junctions modeled as a capacitance CJ and a tunneling junction in parallel. The loop of vanishing inductance encloses an external magnetic flux Φ.

The quantum state of the circuit is described by a macroscopic wave function whose phase jumps φ1, φ2 at the Josephson junctions satisfy φ2 − φ1 + 2πΦ/$$\hbar$$ = 2πm with m an integer. Josephson’s equations $${I}_{{\rm{J}}i}={I}_{{\rm{c}}}\sin {\varphi }_{i}$$ and $${U}_{{\rm{J}}i}=\frac{\hslash }{2e}\dot{{\varphi }_{i}}$$ relate them to the tunneling current and the voltage drop in each junction i = 1, 2. The loop is coupled capacitatively to the endcaps via Cc, and can be controlled with the external voltage U, applied via the gate capacitance Cg. The circuit equations of motion can be obtained starting from Kirchhoff’s laws,

$$V+{U}_{{\rm{J1}}}+{U}_{{\rm{c1}}}+{U}_{{\rm{c2}}}=0,$$
(11A)
$${U}_{{\rm{J1}}}+U+{U}_{{\rm{g}}}=0,$$
(11B)
$${I}_{{\rm{c1}}}+{I}_{{\rm{J1}}}+{I}_{{\rm{c2}}}+{I}_{{\rm{J2}}}=I+{I}_{{\rm{g}}}.$$
(11C)

Inserting the capacitance charge (10) into (11A), differentiating with respect to time and using that Uci = Qci/Cc and that $$I=\dot{Q}_{\rm ind}={\dot{Q}}_{{\rm{c}}i}$$ yields

$$\dot{Q}_{\rm ind}=-\frac{{k}}{{z}_{0}}\frac{{C}_{{\rm{eff}}}}{C}\left(q\dot{{\bf{r}}}+\dot{{\boldsymbol{p}}}\right)\cdot {{\bf{e}}}_{z}-\frac{{C}_{{\rm{eff}}}}{{C}_{{\rm{J}}}}{\dot{Q}}_{{\rm{J1}}},$$
(12)

with the effective capacitance Ceff = CCc/(Cc + 2C). In addition, (11B) and (11C) yield the relations

$${\dot{Q}}_{{\rm{g}}}=-\frac{{C}_{{\rm{g}}}}{{C}_{{\rm{J}}}}{\dot{Q}}_{{\rm{J1}}}-{C}_{{\rm{g}}}\dot{U},$$
(13)
$${\dot{Q}}_{\rm ind}+{\dot{Q}}_{{\rm{g}}}={\dot{Q}}_{{\rm{J1}}}+{\dot{Q}}_{{\rm{J2}}}+{I}_{{\rm{c}}}\sin {\varphi }_{1}+{I}_{{\rm{c}}}\sin {\varphi }_{2}.$$
(14)

Inserting (12) and (13) into (14), using flux quantization, and defining φ = φ1 − eΦ/$$\hbar$$ finally yields the circuit equation of motion,

$$\begin{array}{ll}\ddot{\varphi }=&-\frac{4e{I}_{{\rm{c}}}}{\hslash {C}_{\Sigma }}\cos \left(\frac{e\Phi }{\hslash }\right)\sin \varphi -\frac{2e{C}_{{\rm{eff}}}k}{\hslash C{C}_{\Sigma }{z}_{0}}\left(q\dot{{\bf{r}}}+\dot{{\boldsymbol{p}}}\right)\cdot {{\bf{e}}}_{z}\\ &-\frac{2e}{\hslash {C}_{\Sigma }}\left[{C}_{{\rm{g}}}\dot{U}+\left({C}_{{\rm{eff}}}+{C}_{{\rm{g}}}\right)\frac{\ddot{\Phi }}{2}\right],\end{array}$$
(15)

with CΣ = Ceff + Cg + 2CJ.

The ro-translational motion of the particle is driven by the endcap voltage V via the force F = −kVqez/z0 and the torque N = kVez × p/z0, in addition to the Paul trap force and torque. In the relevant limit of large Cc the Hamiltonian generating the coupled dynamics of circuit and particle takes the form

$$\begin{array}{ll}H&=\frac{2{e}^{2}}{{C}_{\Sigma }}{\left[\frac{\Pi }{\hslash }-\frac{k}{2e{z}_{0}}(q{\bf{r}}+{\boldsymbol{p}})\cdot {{\bf{e}}}_{z}-{n}_{{\rm{g}}}\right]}^{2}-{E}_{{\mathrm{J}}}\cos \varphi \\ &-\frac{k\dot{\Phi }}{2{z}_{0}}\left(q{\bf{r}}+{\boldsymbol{p}}\right)\cdot {{\bf{e}}}_{z}+{H}_{{\rm{rb}}}+{V}_{{\rm{eff}}}({\bf{r}},\Omega)\ ,\end{array}$$
(16)

where the canonical momentum Π conjugate to φ quantifies the number of Cooper pairs on the island and Hrb is the free rigid body Hamiltonian for the center-of-mass motion and rotation. Equation (16) involves the voltage-induced number of Cooper pairs $${n}_{{\rm{g}}}={C}_{{\rm{g}}}U/2e+\left(C+{C}_{{\rm{g}}}\right)\dot{\Phi }/4e$$, and the Josephson energy $${E}_{{\rm{J}}}=\hslash {I}_{{\rm{c}}}\cos \left(e\Phi /\hslash \right)/e$$.

We choose the flux Φ and applied voltage U so that the Cooper-pair box can be treated as an effective two-level system51 with N or N + 1 Cooper pairs on the island, Π = $$\hbar$$(N + σ+σ). Tuning ng, EJ, and $$\dot{\Phi }$$ to zero yields the Hamiltonian

$$\begin{array}{ll}H&=\frac{2{e}^{2}}{{C}_{\Sigma }}(2N+1){\sigma }^{+}{\sigma }^{-}-\frac{2ek}{{C}_{\Sigma }{z}_{0}}(N+{\sigma }^{+}{\sigma }^{-})(q{\bf{r}}+{\boldsymbol{p}})\cdot {{\bf{e}}}_{z}\\ &+\frac{{k}^{2}}{2{C}_{\Sigma }{z}_{0}^{2}}{[(q{\bf{r}}+{\boldsymbol{p}})\cdot {{\bf{e}}}_{z}]}^{2}+{H}_{{\rm{rb}}}+{V}_{{\rm{eff}}}({\bf{r}},\Omega),\end{array}$$
(17)

which leaves the charge eigenstates of the box unaffected. The charge-dependent potential shift given by the second term yields the nanorotor-qubit coupling (3) and will drive the particle into a ro-translational superposition if the charge states are superposed.

Neglecting the dipole moment and separating the nanoparticle transverse motion and rotations finally yields (4) with the potential minimum shifted to zs = 2ekNq/CΣz0Mω2 and the charge energy Ec = 2e2(1 + 2N − kqzs/ez0)/CΣ.

### Time evolution and measurement outcome

The time evolution generated by (4) with external potential Vext can be written, up to a global phase, as a combination of a qubit-dependent phase, qubit-dependent particle displacements and the free time evolution of the harmonic oscillator,

$$\begin{array}{ll}U(t)&=\exp \left[-it\left(\frac{{E}_{{\rm{c}}}}{\hslash }-\frac{{\kappa }^{2}}{\omega }-\frac{2\kappa {V}_{{\rm{ext}}}}{\hslash \omega }\right){\sigma }^{+}{\sigma }^{-}\right]\\ &\times \exp \left[\left(\frac{\kappa }{\omega }{\sigma }^{+}{\sigma }^{-}+\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right)\left({a}^{\dagger }-a\right)\right]\\ &\times \exp \left(-i\omega t{a}^{\dagger }a\right)\\ &\times \exp \left[-\left(\frac{\kappa }{\omega }{\sigma }^{+}{\sigma }^{-}+\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right)\left({a}^{\dagger }-a\right)\right].\end{array}$$
(18)

The qubit is initially prepared in its groundstate while the nanoparticle is in a thermal state of temperature T. A π/2-pulse rotates the qubit into the superposition $$\left(\left|N\right\rangle +i\left|N+1\right\rangle \right)/\sqrt{2}$$, so that the system after time t is given by $${\rho }_{t}=\mathop{\sum }\nolimits_{n = 0}^{\infty }\exp \left(-\hslash \omega n/{k}_{{\rm{B}}}T\right)\left|{\Psi }_{n}\right\rangle \left\langle {\Psi }_{n}\right|/Z$$ with $$\left|{\Psi }_{n}\right\rangle =\left(\left|N\right\rangle {U}_{{\rm{g}}}(t)+i\left|N+1\right\rangle {U}_{{\rm{e}}}(t)\right)\left|n\right\rangle /\sqrt{2}$$. This involves particle time-evolution operators associated with the ground and the excited state of the qubit,

$${{\mathsf{U}}}_{{\rm{g}}}(t)={\mathsf{D}}\left(\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right)\exp \left(-i\omega t{a}^{\dagger }a\right){\mathsf{D}}\left(-\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right)$$
(19)

and

$$\begin{array}{ll}{{\mathsf{U}}}_{{\rm{e}}}(t)&=\exp \left[-it\left(\frac{{E}_{{\rm{c}}}}{\hslash }-\frac{{\kappa }^{2}}{\omega }-\frac{2\kappa {V}_{{\rm{ext}}}}{\hslash \omega }\right)\right]{\mathsf{D}}\left(\frac{\kappa }{\omega }+\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right)\\ &\times \exp \left(-i\omega t{a}^{\dagger }a\right){\mathsf{D}}\left(-\frac{\kappa }{\omega }-\frac{{V}_{{\rm{ext}}}}{\hslash \omega }\right),\end{array}$$
(20)

where $${\mathsf{D}}(\alpha)=\exp \left(\alpha {{\mathsf{a}}}^{\dagger }-{\alpha }^{* }{\mathsf{a}}\right)$$ is the displacement operator. The scheme with π-pulses at times t1 and t2 then results in $$\left|{\Psi }_{n}\right\rangle =\left(\left|N\right\rangle {{\mathsf{U}}}_{+}+i\left|N+1\right\rangle {{\mathsf{U}}}_{-}\right)\left|n\right\rangle /\sqrt{2}$$ at time t3, where

$${{\mathsf{U}}}_{+}={{\mathsf{U}}}_{{\rm{g}}}({t}_{3}-{t}_{2}){{\mathsf{U}}}_{{\rm{e}}}({t}_{2}-{t}_{1}){{\mathsf{U}}}_{{\rm{g}}}({t}_{1}),$$
(21A)
$${{\mathsf{U}}}_{-}={{\mathsf{U}}}_{{\rm{e}}}({t}_{3}-{t}_{2}){{\mathsf{U}}}_{{\rm{g}}}({t}_{2}-{t}_{1}){{\mathsf{U}}}_{{\rm{e}}}({t}_{1}).$$
(21B)

The charge occupation of the box after a final π/2-pulse is thus given by

$$\left\langle {\sigma }^{+}{\sigma }^{-}\right\rangle =\frac{1}{2}+\frac{1}{2Z}\mathop{\sum }\limits_{n = 0}^{\infty }\Re \left[\left\langle n\right|{{\mathsf{U}}}_{+}^{\dagger }{{\mathsf{U}}}_{-}\left|n\right\rangle \right]\exp \left(-\frac{\hslash \omega n}{{k}_{{\rm{B}}}T}\right).$$
(22)

Noting that $${{\mathsf{U}}}_{+}^{\dagger }{{\mathsf{U}}}_{-}$$ displaces the particle state in phase space and using

$$\begin{array}{ll}&\frac{1}{Z}\mathop{\sum }\limits_{n = 0}^{\infty }\left\langle n\right|{\mathsf{D}}(\alpha)\left|n\right\rangle \exp \left(-\frac{\hslash \omega n}{{k}_{{\rm{B}}}T}\right)\\ &=\exp \left[-\coth \left(\frac{\hslash \omega }{2{k}_{{\rm{B}}}T}\right)\frac{| \alpha {| }^{2}}{2}\right],\end{array}$$
(23)

one obtains

$$\begin{array}{ll}\left\langle {{\mathsf{\sigma }}}^{+}{{\mathsf{\sigma }}}^{-}\right\rangle &=\frac{1}{2}+\frac{1}{2}\exp \left[-\frac{{\kappa }^{2}}{2{\omega }^{2}}\coth \left(\frac{\hslash \omega }{2{k}_{{\rm{B}}}T}\right)| d({t}_{1},{t}_{2},{t}_{3}){| }^{2}\right]\\ &\times \cos \left[\frac{\kappa }{\omega }\left(\frac{\kappa }{\omega }+\frac{2{V}_{{\rm{ext}}}}{\hslash \omega }\right){\rm{Im}}\ d({t}_{1},{t}_{2},{t}_{3})\right.\\ &\left.-\left(\frac{{\kappa }^{2}}{\omega }+\frac{2\kappa {V}_{{\rm{ext}}}}{\hslash \omega }-\frac{{E}_{{\rm{c}}}}{\hslash }\right)(2{t}_{1}-2{t}_{2}+{t}_{3})\right],\end{array}$$
(24)

where $$d({t}_{1},{t}_{2},{t}_{3})=2{e}^{i\omega {t}_{1}}-2{e}^{i\omega {t}_{2}}+{e}^{i\omega {t}_{3}}-1$$. Qubit dephasing is the dominant decoherence mechanism in Fig. 3 and is modeled by taking ng in (16) to be a random number with Lorentzian distribution of width γdCΣ$$\hbar$$/(2e)2. This effectively modifies the charge energy and thus the qubit oscillation frequency between the pulses, resulting in an exponential factor $$\exp (-{\gamma }_{{\rm{d}}}{t}_{3})$$ multiplied to the second term in (24).

Equation (24) shows that the envelope of the qubit occupation assumes its maximum at d(t1, t2, t3) = 0 and that the particle temperature determines the width of the peak. Operating the interference protocol at the point of maximal envelope (corresponding to a maximal overlap in the particle state of both superposition branches) can be achieved with the symmetric choice t1 = τ, t2 = τ + Δτ and t3 = 2τ + Δτ, where

$${\Delta }_{\tau }=\frac{1}{\omega }\arctan \left[\frac{2\sin (\omega \tau)[2-\cos (\omega \tau)]}{{[2-\cos (\omega \tau)]}^{2}-{\sin }^{2}(\omega \tau)}\right]$$
(25)

and τ < π/ω. Evaluating (24) for these times finally yields (5).

### Generation of nanoparticle entanglement

The time evolution involving two nanoparticles can be described by means of the respective particle operators

$${{\mathsf{U}}}_{+}^{(i)}\left|n\right\rangle ={e}^{i{\phi }_{+}^{(i)}}{{\mathsf{D}}}_{i}({\alpha }_{i})\left|n\right\rangle ,$$
(26A)
$${{\mathsf{U}}}_{-}^{(i)}\left|n\right\rangle ={e}^{i{\phi }_{-}^{(i)}}{{\mathsf{D}}}_{i}({\beta }_{i})\left|n\right\rangle .$$
(26B)

where the Di are phase-space displacement operators acting on nanoparticle i.

To entangle the particles one prepares the qubits in a Bell state and performs the trapped interference scheme on both subsystems. Measuring both qubits before the wave packets overlap, e.g. at τ + Δτ < t3 < 2τ + Δτ projects the two oscillators onto the outcome-conditioned state

$$\rho ^{\prime} \propto \mathop{\sum }\limits_{n,m = 0}^{\infty }\exp \left[-\frac{\hslash \omega }{{k}_{{\rm{B}}}T}(n+m)\right]\left|{\Psi }_{nm}\right\rangle \left\langle {\Psi }_{nm}\right|,$$
(27)

where

$$\left|{\Psi }_{nm}\right\rangle =\left[{{\mathsf{D}}}_{1}({\alpha }_{1}){{\mathsf{D}}}_{2}({\beta }_{2})\pm {e}^{i\phi }{{\mathsf{D}}}_{1}({\beta }_{1}){{\mathsf{D}}}_{2}({\alpha }_{2})\right]{\left|n\right\rangle }_{1}\otimes {\left|m\right\rangle }_{2}.$$
(28)

The amplitudes αi, βi and the phase $$\phi ={\phi }_{-}^{(1)}+{\phi }_{+}^{(2)}-{\phi }_{+}^{(1)}-{\phi }_{-}^{(2)}$$ depend on the pulse times, whereas the sign in (28) is fixed by the outcome of the qubit measurements. The values of αi and βi determine the amount of entanglement of the state (27), as quantified by a suitable entanglement measure.

### Experimental parameters

For calculating the interference pattern in Fig. 3 we consider a cylindrically shaped silicon nanoparticle (diameter of 4.7 nm, length of 42 nm, homogeneously and positively charged with q = 200 e 52. A 106 amu particle with this charge exhibits the same coupling strength as a doubly charged strontium ion. The proposed interference scheme works also for other particle shapes with appropriately adjusted trap and circuit parameters.

We assume a dipole moment of p = 200 eÅ, motivated by randomly distributing charges on the cylinder surface and accounting for the opposed internal polarization field induced by the charges. This dipole moment is well within the regime where rotations are negligible (see below), and exceeds reported values of a few 10 eÅ for neutral particles of the same size53,54,55,56.

The Paul trap, with an endcap distance of 2z0 = 0.5 mm and geometry factor k = 0.4 29,57, is driven by an AC voltage of Uac = 1 kV with frequency Ω = 2π × 250 MHz 58.

The empty Cooper-pair box has a capacitance of CΣ = 4.4 fF, yielding a charge energy of 2e2/CΣ ≈ 72 μeV. A relatively high occupation N = 10 shifts the potential minimum by the distance zs = 1.17 μm from the trap center. The fast box oscillations then require a measurement time resolution on the ps-scale51. The total duration of the experiment of 87 ns is on the expected coherence time scale of a charge qubit39. Specifically, qubit dephasing with timescale 1/γd = 100 ns decreases the visibility by 60% (see Fig. 3). Other decoherence sources such as gas collisions, black-body scattering, or thermal emission can be neglected on the short time scale of the experiment (see Supplementary Note II). For instance, on average only 0.016 gas collisions occur during the pulse sequence in a room-temperature nitrogen gas at 10−4 mbar.

An initial motional temperature of the particle of T = 1 mK can be achieved via resistive cooling29,59,60 (and potentially by electric feedback cooling29,46,47 or optical techniques8). Assuming a resistance of R = 100 MΩ, the adiabatic cooling rate is 163 Hz, corresponding to 1.54 × 105 quanta per second in thermal equilibrium. Surface noise61 for the present system with endcap electrodes at 77 K would yield a particle heating rate of 170 $$\hbar$$ω/s, which does not noticeably raise the temperature or lead to decoherence on the time scale of the experiment.

### Impact of dipole moments

In the effective potential (1) the initial thermal state shows no correlations between rotations and center-of-mass motion at mK temperatures, because even a dipole moment of 2000 eÅ suppresses all coupling terms. In addition, these rotational and translational coupling terms can also be neglected compared to the direct rotation-qubit interaction with rate κrot = ekp/$$\hbar$$z0CΣ; see Eq. (3). Approximating the cylinder as a linear rotor and ignoring the kinetic terms due to the large moment of inertia yields that the rotations affect the measurement outcome by (i) adding a phase to the cosine function in (24), which is negligible since q2/Mp2/I, and by (ii) reducing the contrast of the interference signal, which can be conservatively estimated as sinc[2κrot(2t1 − 2t2 + t3)]. This reduction is negligibly small for the proposed setup (see Supplementary Note I).

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## Acknowledgements

L.M., K.H., and B.A.S. acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—411042854. B.A.S. acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodovska-Curie grant agreement no. 841040. M.S.K. thanks the Royal Society, the UK EPSRC (EP/R044082/1) and KIST Open Research Program. J.M. is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 803277), and by EPSRC New Investigator Award EP/S004777/1.

## Funding

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Authors

### Contributions

All authors contributed conceptually to the proposal. L.M., K.H., and B.A.S. performed the analytic calculations and wrote the manuscript with input from J.M. and M.S.K.

### Corresponding author

Correspondence to Benjamin A. Stickler.

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Martinetz, L., Hornberger, K., Millen, J. et al. Quantum electromechanics with levitated nanoparticles. npj Quantum Inf 6, 101 (2020). https://doi.org/10.1038/s41534-020-00333-7

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