Sympathetic cooling and squeezing of two co-levitated nanoparticles

Levitated particles are an ideal tool for measuring weak forces and investigating quantum mechanics in macroscopic objects. Arrays of two or more of these particles have been suggested for improving force sensitivity and entangling macropscopic objects. In this article, two charged, silica nanoparticles, that are coupled through their mutual Coulomb repulsion, are trapped in a Paul trap, and the individual masses and charges of both particles are characterised. We demonstrate sympathetic cooling of one nanoparticle coupled via the Coulomb interaction to the second nanoparticle to which feedback cooling is directly applied. We also implement sympathetic squeezing through a similar process showing non-thermal motional states can be transferred by the Coulomb interaction. This work establishes protocols to cool and manipulate arrays of nanoparticles for sensing and minimising the effect of optical heating in future experiments.


I. INTRODUCTION
The ability to cool and control the centre-of-mass (CoM) motion of levitated nanoparticles and microparticles, coupled with their extreme isolation from the environment, makes them ideal candidates for measuring weak forces.They have been proposed as detectors in the search for dark matter candidates [1][2][3], for investigating the macroscopic limits of quantum mechanics [4][5][6][7][8] and for measuring short-range forces [7][8][9][10][11].To date, only a few investigations have focused on cooling and controlling more than a single particle levitated in vacuum [12,13].Arrays of levitated nanoparticles are of interest as they can be used to enhance the detection of dark matter candidates [2,3], measure vacuum friction [14] and for evidencing the quantumness of gravity via entanglement [15,16].Even arrays as small as two particles can be useful for increasing the isolation from external noise sources [17].A first step towards utilising arrays of levitated particles is the development of tools to control the motion of co-levitated particles.
Sympathetic cooling has been used extensively in ion trapping experiments to cool atomic and molecular species where no favourable internal transitions for laser cooling are available [18][19][20][21][22]. It is made possible by the coupling from the Coulomb interaction between cotrapped ions.Coupling between two levitated nanoparticles in vacuum has been demonstrated with optical binding [12] and via the Coulomb interaction [13].The ability to cool and control all particles via a single particle using light, while not illuminating the other co-trapped particles, can be used to minimise heating [23,24], particularly when they contain internal atomic like systems such as nitrogen vacancy centers, whose internal state manipulation is highly temperature dependent.
The coupling between the particles in the array allows, in principle, to transfer other more complex motional states as well.Among these, squeezed states are considered the simplest, more easily accessible non-classical states [25,26].In the quantum regime and in presence of a multimode system, as is considered here, squeezed states are an important resource which can allow the generation and observation of entanglement between different mechanical degrees of freedom [27,28].Even far from the quantum domain, squeezed states have important applications for enhanced force sensing [29,30].
In this paper, we co-trap a pair of silica nanoparticles in a linear Paul trap that are coupled through their mutual Coulomb repulsion.By implementing a velocity damping scheme [31][32][33][34][35][36][37][38][39] on just one particle we sympathetically cool the motion of the second particle to achieve sub-kelvin normal mode temperatures.This differs from previous work [13] were both particles were cooled simultaneously.Importantly, we also show that the Coulomb interaction can transfer other states between co-trapped particles by squeezing the normal modes of the system with a parametric drive [27,28,[40][41][42][43][44][45] on just one particle.

II. THEORY
The equations of motion (EoM) for two harmonic oscillators coupled via the electrostatic force are given by where z i are the positions of the particles (i = {1, 2} denote the particle), γ i are the damping constants, ω i are the natural frequencies, m i are the masses, F f luct,i are the thermal force noises defined by arXiv:2111.03123v3[quant-ph] 8 Nov 2022 where k B is the Boltzmann constant and T 0 is the temperature of the surrounding thermal bath, Q i are the charges of the particles and 0 is the permittivity of free space.Several techniques for levitating multiple nanoparticles exist but here we will focus on the case of particles co-trapped in a single linear Paul trap.Paul traps confine charged particles using a combination of static and oscillating electric fields.
Considering the axial direction of a linear Paul trap where the trap is formed by only static fields, the uncoupled secular frequencies of the two particles are given by ω i = 2QiκU0 miz 2 0 .Including the Coulomb interaction, the total potential for the trapped particles is given by [46] where . The equilibrium positions of each particle, z eq i , can be calculated by setting ∂V ∂zi = 0 and solving for z i .From these, an equilibrium separation of can be calculated which is mass independent.Moving to a coordinate system given by the particles' deviations about their equilibrium positions, s i = z i − z eq i , and assuming s i z eq sep , the interaction term in Eq. 3 can be expanded to second order about the equilibrium positions.By ignoring damping and external forces, the Euler-Lagrange EoM are found to be where Assuming the oscillator motion takes the from s i = s i,0 e −iωt then the problem is reduced to finding the eigenvalues and eigenvectors of matrix V which describe the normal modes of the system.The eigenvalues are given by where and Provided the eigenvalues are positive (ω 2 ± > 0) then the motion is stable and the normal mode frequencies are given by ω ± .The normalised eigenvectors are given by where The product r + r − = −m 1 /m 2 therefore these eigenvectors are only orthogonal when m 1 = m 2 .In this case, the values of r ± becomes mass independent.The eigenvectors define the normal modes in terms of the displacement of the individual particles such that where z + and z − are the amplitudes of the normal modes.
For two particles with the same charge and mass (like atomic ions) the eigenvalues and eigenvectors reduce to: where In this case, we can consider e + the in-phase CoM motion and e − the out-of-phase stretching motion of the two particle system.
By assuming the same mass and size for both particles then the uncoupled EoM for the two normal modes amplitudes can be written as where k, m = {+, −}.Each mode will thermalise to the energy of the surrounding thermal bath i.e. mω 2 ± z 2 ± = k B T 0 .The motion of each particle will contain a fraction of the energy from each mode that is determined by the charge of that particle.By considering that energy is proportional to the variance of the displacement and using Eq. 13 we find the relations where E i represents the total energy of particle i and E i,k represents the energy in particle i coming from mode k.From this it can be seen that each particle will contain a total energy equal to the thermal bath.As the charge difference increases, |r ± | → ∞ and r ∓ → 0 and the particles no longer display normal modes.For Q i Q j we find ω − = ω i and ω + = √ 3ω j so the particle with large charge oscillates at its trap frequency and the particle with small charge is strongly affected by the electrostatic repulsion.
The radial motion of trapped nanoparticles will also couple to form normal modes, however, the coupling scales much more strongly with charge difference than the axial modes.For large charge differences both particles are almost completely unaffected by the other and oscillate close to their trap frequencies.The radial normal modes can be calculated in a similar manner to the axial normal modes [47].

III. EXPERIMENTAL METHOD
In this experiment we use a linear Paul trap with four parallel rod electrodes for radial trapping and two "endcap" electrodes for axial trapping.The four parallel rods are held in place by two gold coated printed circuit boards which contain the electrical connections for the rods and have an endcap electrode etched into each [48].The parabolic coefficients were calculated using finite element modelling and found to be r 0 = 1.1 mm, z 0 = 3.5 mm, κ = 0.071 and η = 0.82.Typical trap parameters are Silica nanoparticles, with charges of up to 6000e, were loaded into the trap at ∼ 10 −1 mbar using the electrospray technique [48,49].Two particles were either trapped simultaneously or, after trapping one particle, more nanoparticles were sprayed into the trapping region until a second was caught.Particles were monitored on a CMOS camera using scattered light from a 637 nm diode laser.Since both particles were illuminated by the 637 nm laser (fig.1), timetraces of the particle motion could also be recorded on the CMOS camera at 1000 frames per second [48,50].The timetraces were calibrated by moving the camera a fixed distance with a translation stage and recording the resulting displacement of the image.Both particles were recorded simultaneously in the same camera image so that the phase difference between the displacement of the particles was known.The camera acted as an out-of-loop detector for measuring the temperature when feedback cooling the particles.
Real-time detection of the particle motion was done using a quadrant photodiode.Individual arms of the 637 nm beam illuminated each particle such that just the motion of one particle was measured on the quadrant photodiode.The signal from the quadrant photodiode was fed to a Red Pitaya FPGA to generate a feedback signal to either cool or squeeze the particle motion.The PyRPL software package was used to filter the position signal of the particle around the appropriate mode then either delay the signal (to cool the motion) or mix the signal with a sinusoidal wave at twice the central frequency of the mode (to squeeze the motion) followed by amplification.The feedback signal was then used to modulate the power of the 1030 nm diode laser and create a force on the particle.Despite relatively high intensities of 2 × 10 8 Wm −2 for the 1030 nm laser, the trapping frequencies of particles were shifted by less than 2 % due to the additional laser.
IV. PARTICLE CHARACTERISATION Fig. 1b shows two particles trapped in the Paul trap.Using the CMOS camera, the equilibrium separation of the particles was measured to be z eq sep = 198 ± 1 µm.This is much larger than the expected amplitude of a single particle in thermal equilibrium with a frequency of ω 0 = 2π × 200 Hz ( q 2 = 6.8 µm) so the Taylor expansion used in Eq. 5 is valid.The power spectral density (PSD) of two coupled particles can be seen in fig.2a) taken at a pressure of 1.3 × 10 −2 mbar.Unlike a single uncoupled particle which would display only one mode [48,50], both particles display a mix of the normal modes of system showing they are axially coupled and it can be seen that they do not contain equal mode energy suggesting each particle has a different charge.The normal modes, z ± , are constructed from the measured timetraces using the linear transform described in Eq. 11.To find the normal modes of the system the values of r ± are varied until the PSD of the z ± mode shows a minimal amount of the z ∓ mode.Fig. 2b) shows the PSDs of the normal modes for r + = 0.6 and r − = −1.6.Each normal mode is clearly seen with no component of the other mode suggesting the normal modes are orthogonal and the masses of each particle are approximately equal.The total energy in each particle was calculated by integrating the area under the PSDs of the individual particles and used to calculate their masses by assuming each particle is in thermal equilibrium with the surrounding gas.The particle radii are measured to be r 1 = 195 ± 3 nm and r 2 = 192 ± 3 nm assuming a density of 1850 kg/m 3 .These values both agree with one another and agree with the nominal radius of 193.5 nm.Together, this is clear evidence that two single particles of approximately equal mass were trapped.Other pairs of trapped particle were measured to also have the mass of single particles with separations ranging from 150 µm to 200 µm.
The individual particle charges are different enough such that no coupling between the radial modes of each particle can be seen.This means the radial frequencies can be used to measure the individual charge-to-mass ratios of the two particles in the same manner as for a single trapped particle.By varying the frequency and voltage of the AC signal supplied to the rod electrodes whilst measuring the frequencies of the radial modes, charges of Q 1 = 2135 ± 58 e and Q 2 = 906 ± 15 e were calculated using the mass values determined earlier.We can verify the charges by using them to calculate the theoretical values of r ± and E i,+ /E i for the axial modes and comparing them to the measured values.We find r − = −1.60 ± 0.03 and r + = 0.61 ± 0.

V. SYMPATHETIC COOLING
By modulating the power of a 1030 nm laser focused onto just one of the particles (particle 1) the normal modes can be cooled.Through the Coulomb interaction, the same modes of the other particle are also cooled.If a force proportional to the velocity of the z + is applied to particle 1, the resulting EoM for the two normal modes are where γ f b is the feedback gain and δ ż+ is imprecision noise in the detection.Provided ω − − ω + γ 0 + γ f b , the γ f b ż+ term in Eq.21 will have a negligible effect on the z − mode and can be ignored.Additionally, in practice the feedback signal is bandpass filtered such that the γ f b δ ż+ term does not affect the z − mode.Thus, the z + mode is cooled whilst the z − mode remains unaffected.Similar equations can be found for the two modes (by interchaging all ± subscripts) when cooling the z − mode.Since the particles are the same mass, the temperature compression ratio for a mode should be equal in each particle [47].The cooled mode will have a temperature given by [32,35,37,39] where T 0,+ is the initial temperature of the CoM mode and S nn,+ = ∞ −∞ δz + (t)δz + (0)e iωt dt is the spectral density of the imprecision noise and is assumed to be white and Gaussian over the linewidth of the oscillator mode.A similar expression is found for the z − mode.By increasing the feedback gain a minimum temperature will be found that is dependent on the frequency of the mode.
In the experiment, the feedback was implemented by applying a bandpass filter to the position signal of the particle to remove noise and other modes from the feedback signal then amplifying and adding a π/2 phase shift to estimate the current velocity of the particle [35,39].Fig. 3a) shows the spectra of the two modes cooled to minimum temperatures of T + = 200 ± 10 mK and T − = 190 ± 40 mK (calculated from the area under the PSDs) at a pressure of P = 3.2 × 10 −7 mbar and spectra of the modes with no cooling (T + = T − = 293 K) at P = 1.3 × 10 −2 mbar.As expected, the temperature compression ratio of a mode was found to be the same for each particle.The final temperatures reached are approximately twice as high as the expected temperatures from measurements of detection noise and pressure.By comparing the measured mode temperature as a function of feedback gain to the theoretical prediction (fig.3b) we see that the temperatures are higher at low gain than expected.This suggests additional heating of the particle motion which calculations show could be partly due to voltage noise in the electronics.The voltage noise is increased compared to a single particle since the particles are pushed off-centre by their Coulomb repulsion.Alternative trap geometries could be used to reduce this effect.At a pressure of 10 −1 mbar, where the particle mass was measured, this level of voltage noise would have a negligible effect on the motion of the particle.
To decrease the temperature further, the detection noise could be improved or the pressure could be reduced further.The additional force noise would have to be removed before reducing the pressure since white, Gaussian noise sources increase the temperature of the oscillator scaling with 1/γ 0 whereas velocity damping only scales with √ γ 0 .Here, we measure a detection noise of S nn = 3 × 10 −15 m 2 Hz −1 which limits the final temperature at a given pressure.Recent experiments with single particles in Paul traps have demonstrated detection noise as low as 2.9 × 10 −24 m 2 Hz −1 [51].Using a similar detection technique and a numerical aperture (NA) limited by the trap geometry (NA = 0.5) would allow us to reach a minimum occupancy of n ∼ 0.5 [52] with similar additional optical losses to other experiments [53].In order to remain in the underdamped regime where the theory is valid, the oscillator frequency would have to be increased and the background gas damping would have to be re- duced.An oscillator at a frequency of ∼ 500 Hz and a background pressure of ∼ 10 −11 mbar would be sufficient provided other noise sources are kept negligible.

VI. SYMPATHETIC SQUEEZING
Squeezing the motion of a mechanical oscillator has been achieved with several methods including parametric modulation [27,28,[40][41][42][43][44][45], non-adiabatic shifts [54][55][56] and back-action evading measurements [57][58][59][60].Parametric modulation is usually implemented by modulating the oscillator spring constant at twice the mode resonance frequency.In the Paul trap this would drive both particles.In order to show a sympathetic operation, we use a measurement based scheme to parametrically drive only one particle.This scheme also avoids acting on either particle with a linear drive that would appear from modulating the trap potential [61,62].A measurement of the particle position was filtered around the appropriate mode using a Lorentzian bandpass filter with a bandwidth of 152 Hz then mixed with a sinusoidal signal at twice the resonance of that mode to produce a signal to modulate the 1030 nm laser.An additional delay was added to the signal to minimise errors to the phase response of the filter.The laser imparted a force F ∝ z ± sin(2ω ± t) onto particle 1.If we were, for example, squeezing the z + mode this results in the following EoM for the normal modes z+ + γ 0 ż+ + ω where G is a gain applied to the signal.Similar to the case for sympathetic cooling, the z − mode remains unaffected by the applied force provided ω − − ω + γ 0 .In this instance, δz + z + and therefore has a negligible impact on the dynamics of the system.Thus, the z + mode experiences a parametric drive whilst the z − mode is unaffected.In the case of squeezing with a parametric drive, the variance of the X and Y quadratures is given by [41] where g = G × ω 0 /2γ 0 .It can be seen that this limits the maximum achievable squeezing to -3 dB before the onset of parametric instability in the X quadrature [63,64].Fig. 4 shows phase space plots of the quadratures of the z + mode of both particles with and without a parametric drive applied to particle 1 at a pressure of P = 1.2 × 10 −2 mbar.Squeezing was performed at a relatively high pressure to reduce the measurement time required to accurately sample the squeezed thermal state.The quadratures were extracted from the displacement data by applying a demodulation followed by a 50 Hz lowpass filter to the z + mode and sweeping the demodulation frequency until maximal squeezing was observed.A clear signature of squeezing can be seen in the elongations of the phase space distributions in both particles.Anomalous heating of the mode was also seen as the gain of the squeezing operation was increased.Simulations based on the experimental parameters, implemented with the leapfrog algorithm, suggest that suggest this is due to using a tightly focused beam to generate the force for squeezing the particle.Since the beam waist is comparable to the amplitude of particle motion the particle feels an additional position dependent component to the feedback force.Taking the heating into account, we were able to measure −1.70 ± 0.05 dB and −1.69 ± 0.05 dB of squeezing in the z + mode of particle 1 and 2 respectively.The squeezing gain was limited by the non-linear effect of the tight beam generating a spiralling in the phase space plots [56].Both the z + and z − mode could be squeezed using this method.
Transferal of the thermal squeezed state between the particles shows that non-thermal states can also be transmitted via the Coulomb force.For an array of coupled nanoparticles this property could be used to control and readout the state of the array from only a single nanoparticle.For large arrays, this could be technically easier than reading out the state of every single nanoparticle.

VII. CONCLUSIONS
We have demonstrated and characterised coupling between two charged nanoparticles that produces two orthogonal normal modes of motion.Both sympathetic cooling and squeezing of the motion of one particle were shown through interaction with another coupled particle.This demonstrates thermal and non-thermal states can be transferred via the Coulomb interaction.Such techniques could be extended to a mixed species system with particles of similar masses or to an array of many nanoparticles.As a result, this work represents an important tool in implementing the creation of macropscopic quantum superpositions [15,[65][66][67][68] via coupling of the normal modes of a co-trapped silica-nanodiamond pair to the internal quantum state of a nitrogen vacancy cen-tre in the nanodiamond.Such a technique can be applied to experiments which have previously been limited by internal heating of samples [24].
Our results are a first step towards schemes that utilize the coupling between levitated particles.Large arrays of coupled particles would have a range of motional frequencies that scale with the number of particles.This would be ideal for sensing over a wide range of frequencies simultaneously such as in ultra-light dark matter searches [3].Using sympathetic techniques would allow control and read-out of the entire array from a single particle simplifying the experimental procedure compared to an uncoupled array.

FIG. 1 .
FIG. 1. (a)The experimental set-up for sympathetic cooling and squeezing.Two 387 nm diameter, silica particles are trap simultaneously in a Paul trap.One arm of a 637 nm laser is focused onto each particle individually.The power in each arm is balanced such that the scattering from each particle is approximately equal when measured on the CMOS camera.One of the arms is also focused onto a quadrant photodiode for real-time detection of the motion of one particle.A force is applied to the same particle to either cool or squeeze the normal modes by modulating the power of a 1030 nm laser.The feedback signal is generated from the real-time measurement of the particle position.BS: Beam splitter, QP: Quadrant photodiode, SPF: Short-pass filter.(b) Two co-trapped particles in the Paul trap.The separation between these particles is 198 ± 1 µm.

FIG. 3
FIG.3.a) Spectra for the cooled z+ (left) and z− (right) modes at a pressure of P = 3.2 × 10 −7 mbar (dark lines) shown alongside the spectra of the modes with no cooling at P = 1.3 × 10 −2 mbar (light lines) where the modes are expected to be in thermal equilibrium with the surrounding gas.b) Temperature of the normal modes in the experiment (markers) and theory (lines) against feedback gain.The disagreement between theory and experiment suggests there is an additional source of heating for the normal modes.

FIG. 4 .
FIG.4.Phase space diagrams showing a thermal and squeezed state for the z+ mode of both particles at a pressure of P = 1.2 × 10 −2 mbar.Both particles display squeezed states despite interaction with only particle 1.The particles used here have approximately equal charges therefore the mode energies should be equal as they appear here.