Levitated electromechanics: all-electrical cooling of charged nano- and micro-particles

We show how charged levitated nano- and micro-particles can be cooled by interfacing them with an $RLC$ circuit. All-electrical levitation and cooling is applicable to a wide range of particle sizes and materials, and will enable state-of-the-art force sensing within an electrically networked system. Exploring the cooling limits in the presence of realistic noise we find that the quantum regime of particle motion can be reached in cryogenic environments both for passive resistive cooling and for an active feedback scheme, paving the way to levitated quantum electromechanics.


Introduction
The control of nano-and micro-scale objects in vacuum is of great importance for a wide range of applications [1], from the study of individual proteins [2], viruses [3] and bacteria [4], to the simulation of astrophysical processes via the study of dusty plasmas [5], to the detection of tiny forces [6,7]. It is predicted that the motion of levitated nanoparticles can be controlled even at the quantum level [8][9][10][11], enabling studies of macroscopic quantum physics [12] and the contribution of levitated high quality-factor oscillators in quantum technologies as coherent storage or signal transduction devices.
Charged particles can instead be levitated in ion traps [23,24], with trapped atomic ions at the forefront of quantum information processing technology [25,26]. Their motion can be detected and cooled via their coupling to the trap electrodes [27,28], either resistively or using active feedback methods [29,30]. Extending these techniques to nano-and micro-particles will enable manipulation and control of massive charged particles by electronic circuitry. This opens the field of levitated electromechanics, holding the promise of scalability and network integration.
In this work, we present an all-electrical levitation, detection and cooling scheme for charged nano-and micro-particles, which will operate stably under vacuum conditions while avoiding optical scattering and absorption heating. Our simulations show that sub-Kelvin particle temperatures are achievable with room temperature circuitry via resistive and feedback cooling. The applicable range of sizes, from sub-nanometer to several micrometers, and materials, including metallic clusters, dielectric particles and biological objects, illustrates the universality of this technology. Under ambient conditions, levitated electromechanics will offer a simple and robust pre-cooling method for optical cavity cooling, and state-of-the-art force sensing. This scheme is suited to cryogenic environments, where millikelvin temperatures allow reaching the deep quantum regime.
We first derive the equations-of-motion of a charged particle coupled to an RLC circuit, and show how they can be quantized. Then we discuss the electronic detection of motion, followed by the potential to cool through passive-resistive and active feedback methods. Finally we discuss the application of levitated electromechanics to displacement and force sensing.

Equations of motion
We consider a spherical particle of mass M and charge q levitated in a potential V(z), with trapping frequency ω z , which is levitated between plates of separation d and capacitance C. As illustrated in figure 1 the two electrodes are joined via an inductance L to form an LC circuit. While in this work we envision that V(z) is provided by a Paul trap, with the capacitor formed by the endcap electrodes, we note that the levitating potential could also be optical. The equations of motion for a charged particle in a Paul trap are provided in appendix A. The interaction with electrostatic mirror charges formed in the endcap electrodes is negligible for the trap parameters considered below.
The dynamics of the charged particle can be derived using the Shockley-Ramo theorem [31,32]. It relates the current I flowing in the circuit with the particle momentum =ṗ Mz and the voltage drop U across the capacitor. For now we consider the conservative LC circuit neglecting resistance, yielding We use the circuit-charge Q on the capacitor as the generalized coordinate of the circuit, so that the magnetic flux through the inductor F =LQ acts as its associated canonical momentum, and exploit Kirchhoff's law, = -Ḟ U , to obtain the canonical equations-of-motion The first term in equation (2b) gives rise to a harmonic oscillation of frequency w = LC 1 LC , while the second term accounts for the linear coupling to the particle position z.
The back-action of the circuit dynamics onto the particle motion can be derived by noting that the voltage offset U across the capacitor induces a constant force -qQ Cd acting on the particle. Neglecting charging effects due to the formation of mirror charges, which are on the order of w  q MCd z 2 2 2 , one thus obtains the equations-of-motion for the particle: Equation (2d) describes the coupled classical dynamics of a charged particle trapped inside an LC circuit. The Hamiltonian function associated with these dynamics can be identified Circuit diagram for the electronic detection of charged particle motion for an object levitated in a quadrupole ion trap (gray diagram). The motion of the trapped particle is picked up by the endcap electrodes separated by d, and induces a current I in the circuit. A voltage U is generated across the resistance R of the circuit. Resonant detection is possible by forming either a series or parallel RLC circuit (with parallel illustrated here), where the capacitance C is due to the trap electrodes plus any parasitic capacitance.
For harmonic V(z) the particle-circuit system behaves as two linearly coupled harmonic oscillators.

Dissipation
We now consider the role of a resistor, of resistance R and temperature T R , connected in series with the LC circuit. Its action is described by adding the damping and diffusion term to equation (2b). Here, the damping rate of the circuit is Γ=R/L, and the second term with white noise describes circuit thermalization at temperature T. Due to the linear coupling between the circuit and the particle (equation (3)), the particle will thermalize with the circuit resistance at temperature T R . The corresponding timescale 1/γ is determined by the circuit parameters R, L and C as well as by the particle charge q and mass M. Dissipation in a parallel RLC circuit can be described in a similar fashion by adding a friction and diffusion term, with parallel friction rate Γ=1/RC, to the equation-of-motion for the circuit charge.

Adiabatic damping
If the circuit follows the particle motion adiabatically we can eliminate the former from the equations-of-motion in equation (2). Neglecting diffusion for the moment, one obtains from equations (2a) and (2b) the quasi-static expressions for a series RLC circuit These yield the non-conservative equations of motion for the particle which allows us to identify the adiabatic damping rate A similar calculation for the parallel RLC circuit yields γ ad =0 since the capacitor is effectively shorted out in this scenario. In the derivation of equation (5) we neglected the additional attractive force related to the appearance of electrostatic mirror charges. This contribution is negligible in the present case, and causes a fractional frequency shift of w~q MCd 10 z 2 2 2 6 .

Damping on resonance
If the circuit is on resonance with the particle motion, ω LC =ω z , the oscillatory dynamics of the particle can effectively cancel the inductance, leading to a boosted friction rate. This can be made evident by Fouriertransforming the particle-circuit equations-of-motion for the series RLC circuit and eliminating the circuit which increases with decreasing circuit dissipation. A similar calculation for the parallel RLC circuit yields the same result for the friction rate. This resonant enhancement of friction is in agreement with the damping rate observed for trapped atomic ions and electrons [33,34]. Note that the series adiabatic friction rate (6) is equal to the parallel on-resonance damping rate (8).

Quantum dynamics
We consider the situation where particle and circuit are cooled to their ground-state or close to it. The coupled quantum dynamics of the combined particle-series RLC circuit state ρ can be modeled by the Lindblad master equation where H is the quantized Hamiltonian (operators are denoted by sans-serif characters), and the Lindblad operator is The completely positive master equation (9) describes the Markovian thermalization dynamics of the particle-circuit state ρ. For large circuit temperatures, where the term proportional to 1/T R can be neglected, one obtains The second term on the right hand side describes damping of the circuit momentum Φ with friction rate Γ, while the third term describes charge diffusion. The corresponding diffusion constant GLk T R B is in accordance with the fluctuation-dissipation relation. If the trapping potential is harmonic the master equation describes thermalization towards the Gibbs equilibrium , and the expectation values of the canonical phase-space observables exhibit classical thermalization dynamics (equation (2)).

Detection of the particle motion
Experiments demonstrating trapping of single electrons [36,37], ions [38] and protons [30] as well as electron [27] and proton [39] clouds have noted that it is possible to detect the classical motion of the charged particle(s) via the image current I that they generate in the endcap electrodes of Paul or Penning traps [33]. We now show that this is also possible for spherical nano-and micro-scale particles. Under the assumption that Γ ? ω z the circuit adiabatically follows the particle motion, and equation (1) reduces in lowest order of the particle velocity to where we have introduced the geometrical factor η to account for the shape of the pick-up electrodes, which we assign the realistic value of η=0.8 for slightly parabolically shaped electrodes [28].

Resistive cooling
As demonstrated in section 2.1, connecting the endcap electrodes to an RLC circuit serves to dissipate the induced current and thus damp the particle motion. The ensuing friction rate (equation (8)) on resonance (ω LC =ω z ) can be written as Tuned circuits with Q f =25 000 have been used to cool + N 2 ions [40], and it is proposed to exploit high Q f quartz crystal oscillators connected in parallel with the endcaps to further boost resistive cooling [41]. For a modest Q f =100 and R=100 MΩ, an r S =1 μm, q=10 6 e silica sphere at room temperature would generate a signal of ∼100 mV for electrodes separated by 1 mm.
Resistive cooling is illustrated in figure 2, where the motion of a charged microsphere in a quadrupole ion trap is numerically simulated including all sources of noise, as outlined in appendix A. From equation (14), it is clear that to increase the damping rate, one can increase R eff or q, as verified in figure 2(a), where equation (14) is compared to the results of the numerical simulation, with the agreement illustrating that realistic experimental noise has little effect on the damping rate. The scaling of the damping rate g µ q M 2 is favorable for highlycharged massive particles.
The dissipation of energy across R is accompanied by heating due to Johnson-Nyquist noise [28,33,34], modeled as a white voltage-noise source  R of width k T R 4 R B e f f , where T R is the temperature of the circuit. For a description of how noise is added to the simulations of trapped particle motion, see appendix A. In the absence of other noise sources, the particle will equilibrate to the temperature of the circuit T R , as indicated by the thermal trajectory in figure 2(b). Hence, to reach low temperatures via resistive cooling, cryogenic circuitry is required [34]. Note that it is possible to design ion trap geometries to resistively cool all three degrees-offreedom, for example by using a ring electrode which is split into segments [28,42].
Even under ambient conditions, resistive cooling can be useful to stabilize trapped particles, for example against collisions with residual gas molecules [6,13,20], or to pre-cool massive particles, which in general possess initial motional energies far above room temperature due to the loading mechanism [15,43]. Furthermore, avoiding optical fields removes motional heating due to scattering of photons.
To consider the limits of resistive cooling, we consider operation in a state-of-the-art dilution refrigerator at 5 mK. By considering the quadrupole ion trap stability parameters, as defined by equation (23) in appendix A, an r S =1 μm silica sphere with q=10 6 e can have stable frequencies of ω z >2π×1 MHz, which would correspond to reaching a phonon occupancy of w = < ( ) n k T 100 z B CM .

Feedback cooling
Since we can detect the motion of a charged particle via the endcap voltage U, we can feed this signal back onto the endcaps to dynamically control the particle's motion. Figure 3 shows a schematic of this process, where an amplifier with gain G generates the voltage GU on the lower endcap electrode. With the appropriate phase shift between U and GU, this either amplifies or cools the motion [29,38,44].
In the scenario depicted in figure 3, the charged particle sees a voltage U fb which is generated by the voltage difference between the two endcap electrodes: At G=1 the circuit resistance that the particle sees = -( ) R G R 1 fb eff and the thermal noise is tuned out. Explicitly, in this simple model the temperature T CM of the particle is reduced to = - Simultaneously, the feedback damping rate g µ R fb fb goes to zero as G approaches unity, g g = - Charged nano-and micro-particles are stable in Paul traps on the timescale of months [15], somewhat negating the necessity for rapid damping in the absence of other noise sources.
For a more realistic description, it is necessary to include the voltage noise added by the feedback amplifier  fb , as illustrated in figure 3. Noise added due to collisions with gas molecules and electrode surface potentials is considered in the supplementary information is available online at stacks.iop.org/QST/4/024003/mmedia, and included in the simulations of the particle motion presented in figures 2 and 4, and can be neglected. This noise voltage has an associated noise temperature T fb n defined through  = k T R B 4 fb B fb n amp , where R amp is the resistance of the feedback amplifier, and B is its bandwidth. This noise adds in quadrature with the uncorrelated voltage noise  R [44], leading to a total  (14)). (b) Simulated particle trajectories for two different circuit temperatures T R , illustrating that the particle thermalizes with the circuit. The parameters in this figure unless stated otherwise are: T in =1000 K, r S =1 μm, q=10 5 e, d=1 mm, Q f =100, R=100 MΩ, T R =300 K.
2 fb 2 and a particle equilibrium temperature A simulation of the effect of feedback on the particle dynamics is shown in figure 4(a). A comparison of equation (17) and simulation is shown both for a variation in feedback gain G and in noise voltage  fb in figures 4(b) and (c) respectively. Equation (17) is minimized at » - The noise voltage  fb of commercial amplifiers can be sub-nanovolt, yielding sub-kelvin particle temperatures even with room temperature circuitry, as illustrated in figure 4(c).
Using state-of-the-art cryogenic SQUID amplifiers with noise temperatures below 20 μK [45] with T R =5 mK, it would be possible to use feedback to reduce the temperature to below 100 μK. This corresponds to the motional ground-state for few-MHz oscillation frequencies. Hence, with cryogenic operation, this system is in-principle suitable for cooling micron-sized charged particles to the quantum ground-state.   (17)). The parameters in this figure unless stated otherwise are: T in =1000 K, r S =1 μm, q=10 5 e, d=1 mm, Q f =100, R=100 MΩ,  = -10 fb 10 V, T R =300 K.

Position, size and force sensitivity
Opto-and electro-mechanical devices make excellent sensors, due to their low mass and ability to couple to a wide range of forces [46], and they have found application in genetics, proteomics, microbiology and studies of DNA [47]. Their sensitivity is limited by dissipation to the environment, a problem which gets worse with decreasing physical size [48]. By levitating the oscillator, many dissipation processes are removed, which has enabled force sensing with optically levitated microparticles on the zepto-Newton scale [49].
The smallest displacement of a charged particle that can be measured through an induced voltage in this system is limited by noise. The dominant source is the Johnson-Nyquist noise voltage  n where Δν is the detection bandwidth. Since one can make a phase sensitive detection [33], Δν can be very small. By considering the point at which the induced signal U is equal to  R , it is straightforward to show that the minimum detectable particle velocity on resonance is . The variation in z min with T R and q is shown in figure 5(a). Considering a r S =500 nm silica sphere, γ=1 kHz, and ω z =2π×1 MHz, even at room temperature (300 K) this corresponds to 10 −11 m Hz -1 2 resolution, and at a cryogenic temperature of 5 mK, < 10 −15 m Hz -1 2 resolution. The zero-point fluctuation of such a particle is When considering the force sensitivity of a harmonic oscillator, the minimum detectable force is , where γ CM is the damping rate on its motion [49]. In our case, the dominant damping without feedback is resistive γ, as defined in equation (14). This cannot be made arbitrarily low, since small γ means we detect less signal U, which must in turn be greater than the noise voltage  R . Hence, we have the requirement Using equations (13) and (14), and assuming that the particle is in thermal equilibrium with the circuit = T T R CM , we find the simple expression γ>4Δν, which is the value of the resistive damping required to measure above the thermal noise in a given bandwidth. In other words, the measurement time must be on the order of the ring-down time (1/γ). This yields a force sensitivity The variation in F min with particle size and temperature is shown in figure 5(b). Practically, choosing a low value of γ will lead to long thermalization times, requiring a stable experiment, although particles in ion traps are stable on month-timescales. Using equation (20), with a 1 Hz measurement bandwidth, for an r S =100 nm silica sphere, F min at 300 K is 8×10 −19 N, and at 5 mK is 3×10 −21 N, comparable to state-of-the-art optical systems [49]. It is worth noting that it is possible to levitate lower-mass particles in an ion trap than optically, due to an increased trap depth, making the electromechanical system attractive for sensing applications. Figure 5. Sensing: (a) minimum detectable particle displacement with a 1 Hz measurement bandwidth as a function of the circuit temperature T R , for two different values of particle charge q, with ω z =2π×1 MHz. (b) Minimum detectable force with a 1 Hz measurement bandwidth, as a function of particle size r S (silica sphere), for two different circuit temperatures T R , assuming that the particle thermalizes with the circuit. The parameters in this figure unless stated otherwise are: r S =1 μm, d=1 mm, Q f =100, R=100 MΩ.
Finally, using equations (13) and (19), it is possible to detect particles which satisfy the following relation As an example, this means that using realistic experimental parameters ( = R 100 MΩ, Q f =100, η=0.8, d=1 mm, Δν=1 Hz), it would be possible to detect singly charged particles with masses up to 5×10 6 amu.

Discussion
We have demonstrated that levitated electromechanics, where charged particles levitated in an ion trap are interfaced with an RLC circuit, can be used for electronic detection, cooling and precision sensing. Using feedback, sub-kelvin temperatures are achievable with room temperature circuitry, and we predict that working in a cryogenic environment will enable ground-state cooling for micron-sized particles.
Levitated electromechanics is compatible with optomechanical systems. A hybrid levitated opto-/electromechanical set-up is suitable for cooling deep into the quantum regime, and for the production of interesting mechanical quantum states, such as squeezed states [50]. Charged particles equilibrate with the trapping circuitry, which acts as a highly controllable thermal bath, making the system suitable for studies of lowdissipation thermodynamics [51].

A.1. Inclusion of noise
We consider the influence of several noise sources: gas collisions at temperature T gas through the damping rate γ gas (see supplementary information); the Johnson-Nyquist noise of the circuit at the temperature T R through the rate γ; and the noise of the feedback amplifier  fb with associated noise temperature T fb n and damping rate γ fb .
We include each of these in our model via the fluctuation-dissipation theorem t g dt for the gas, resistive, and feedback random force respectively. The electrode surface noise, which is discussed in detail in the supplementary information, is included by considering the fluctuating force , constructed noting that an electric field E generates a force qE on a particle of charge q [52].
Hence, the full equation-of-motion for the z-direction reads: