Enhanced force sensitivity and entanglement in periodically driven optomechanics

F. Cosco, J. S. Pedernales, and M. B. Plenio1

1Institut fur Theoretische Physik und IQST, Albert-Einstein-Allee 11, Universitat Ulm, D-89081 Ulm, Germany

Squeezing is a resource that enables precision enhancements in quantum metrology and can beused as a basis for the generation of entanglement by linear optics. While strong squeezing ischallenging to generate in optical fields, here we present simple periodic modulation protocols inoptomechanical systems that can generate large squeezing of their mechanical degrees of freedomfor realistic system parameters. We then proceed to show how such protocols can serve to improvethe measurement precision of weak forces and enhance the generation of entanglement between testmasses that are subject to any kind of weak interaction. Moreover, these protocols can be revertedto reduce the amount of injected energy, while preserving the generated entanglement and makingit more resilient to noise. We present the principle at work, discuss its application in a varietyof physical settings, including levitated and tethered mechanical harmonic oscillators, and presentexample applications to Casimir and gravitational forces.

Introduction.—The quantum control of the mechani-cal degrees of freedom of ever more massive objects isone of the permanent ambitions of quantum physics. Onthe lower end of the mass spectrum, trapped ion tech-nologies [1, 2] and atom interferometers [3] represent, ar-guably, the state of the art on the control of the me-chanical properties of a particle. Above that level, therealm of physical systems for which exquisite control hasbeen achieved extends across many orders of magnitudein mass, from the atomic to the mesoscopic scale. Thisincludes macromolecules containing up to 104 amu [4],Bose-Einstein condensates of ∼ 109 atoms [5], as wellas solid state systems that can be either clamped res-onators in a wide variety of implementations [6], e.g.pendula of remarkable stability [7], or levitated nanopar-ticles [8, 9] that promise extended coherence times dueto their extraordinary isolation from the environment.Despite the many challenges that come with each massregime and setup, the intense experimental activity ofthe last decade in the field of optomechanics has lead tothe conquest of encouraging milestones. These includethe ground state cooling of clamped [10, 11] as well aslevitated resonators [12], or the remote entanglement be-tween two tethered massive oscillators mediated by op-tical fields [13, 14], suggesting that extensive quantumcontrol of mechanical properties well beyond the atomicscale might be within reach in the near future. Thiswould have two main implications: on the one hand, itwould open the door to experimentally testing proper-ties of quantum mechanics such as its linearity at an un-precedented scale of mass and size, and with it allow toverify or falsify extensions of the Schrodinger equationlike collapse models [15]. On the other hand, attainingquantum control of the mechanical degrees of freedom ofa massive resonator would entail applications in preci-sion sensing of minute forces [16–19], and in particular offorces that are proportional to the size of the detector,like Casimir-Polder or gravitational forces [20, 21].

In the light of the high force sensitivities that such de-vices augur, a natural question arises: would two such

0B(x d0/2) B(x + d0/2)

d0

x

a)

b)

t

12 2

cycle1

21

forward protocol inverse protocol

N ln

r!2

!1

ln

r!2

!1

N

!1 !2 !1 !2 !1 !1 !2 !1 !2 !1!1

12 21

FIG. 1. (a) Pictorial representation of a possible setup. Twodiamagnetic nanoparticles potentially hosting a spin degreeof freedom, are trapped in two linear magnetic field gradi-ents. The particles oscillate around equilibrium positions de-termined by the zeros of the magnetic fields, which are sep-arated by a distance d0, and interact via a weak force. (b)Squeezing protocol. Each particle is subject to a sequenceof jumps between two frequencies ω1 and ω2 properly spacedin time by intervals τ1 and τ2, respectively. After N cycles,this leads to squeezing of the mechanical degrees of freedomof each oscillator, parametrized by a squeezing parameter ξNthat grows linearly in time. The squeezing can be reverted byapplying the inverse protocol.

resonators be sensitive to the mutually induced weakforces that emerge when placed sufficiently close to eachother? If the answer is positive, measuring the generatedcorrelations would provide information on the nature ofthese forces [22–24] in possible experiments based on op-tomechanical and other technologies [18, 19, 25–32]. Notonly that, provided that entanglement could be mediatedby such weak forces, a complete toolbox of QED oper-ations would become immediately available for massiveoptomechanical setups, with a plethora of applications inquantum information and metrology. However, the lowenergy of these interactions inevitably challenges this am-bition.

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In this Letter, we show that two weakly couplednanomechanical resonators can develop sizeable entangle-ment between their mechanical degrees of freedom evenin the presence of dissipation, provided that these aresubject to a continuously applied protocol that squeezesthem in suitable quadratures, and that this squeezing isremoved sufficiently fast after the desired entanglementhas been generated. We describe analytically and sim-ulate numerically such a protocol, consisting of period-ically applied local shifts of the resonator frequencies.We show how our protocol can generate squeezing in thesystem to accelerate the generation of entanglement, andsubsequently reduce the squeezing again, while retainingthe generated quantum correlations, in order to avoida deleterious enhancement of the sensitivity to environ-mental noise. This extraction of local squeezing also fa-cilitates the verification of entanglement, which can becarried out quantitatively by the detection of correlationsin local measurements [33–35].

Setup.—The central findings of our analysis are ap-plicable to any setup consisting of two weakly interact-ing massive resonators operating at low temperatures.This includes nanoparticles levitated, either by opticalmeans [12], in Paul traps [36], or diamagnetically [37], aswell as tethered oscillators such as massive pendula [38].The origin of the weak interaction can be, among others,a surface force like the Casimir-Polder force or a gravita-tional force. For the sake of simplicity, we consider twoidentical oscillators, that is, with the same mass m andtrap frequency ω, and consider the motion in only onespatial direction, e.g. the x direction. Assuming that thecenters of the harmonic traps confining the two oscilla-tors are separated by some distance d0, see Fig. 1 (a),the Hamiltonian of the system is given by

HD = UW(x1, x2) +

2∑

i=1

p2i

2m+

1

2mω2(xi + di)

2, (1)

where UW represents the weak coupling between the res-onators and d1/2 = ±d0/2. For the sake of generality,we consider that the interaction energy is a function ofthe separation distance between the centers of mass ofthe two resonators, d = |x1 − x2|, and follows an inversepower law with exponent n, UW = C/|d|n, such that wecan expand it around the point d = d0, provided that|d− d0| d0, as

UW =C

dn0[(1− n

d0(d−d0)+

n(n+ 1)

2d20

(d−d0)2 + . . . ]. (2)

A coupling between the mechanical degrees of freedomof the two nanoparticles is established by the third termin the expansion. In this quadratic form the interactionHamiltonian is very general and allows to replicate inthis platform well established protocols from quantuminformation processing platforms like trapped ions [39,40]. Our purpose is to estimate the time scale required

to generate detectable entanglement mediated by such aweak interaction and check if it is compatible with thestate of the art. This would allow us to entangle twomassive oscillators without the use of charges or magneticimpurities, which are potential sources of noise, and couldhelp us determine experimentally the ‘quantumness’ ofthese forces.

For the specific parameters in Eq. (1), when run-ning numeric simulations, we will consider values cor-responding to magnetically levitated diamagnetic parti-cles. These have promising prospects in terms of coher-ence when compared to tethered oscillators, as the de-tachment from the substrate dramatically reduces theirinteraction with the environment, and also when com-pared to optically levitated nanoparticles, as the passivetrapping fields remove the dissipation mechanisms associ-ated with the recoil and absorption of photons from theoptical trapping field. Stable levitation of diamagneticnanoparticles has been demonstrated employing mag-netic field gradients from permanent magnets in high vac-uum, with center of mass motion temperatures below 1mK [37, 41]. Such setups represent a promising platformto implement matter-wave interferometry at macroscopicscales, for high-sensitivity quantum metrology, and forinvestigating the fundamental limits of quantum mechan-ics [15, 42, 43].

For a diamagnetic particle in a magnetic field B(x),

the potential energy is given by UB = −mχB(x)2

2µ0ρ, where

m is the mass of the particle, ρ its density, µ0 the vac-uum permeability, and χ is the material magnetic sus-ceptibility. In case of a linear magnetic gradient such asB(x) = B′x, and negative magnetic susceptibility (dia-magnetism), the magnetic energy acts as an effective har-

monic potential with frequency ω =√− χµ0ρ

B′, propor-

tional to the magnetic field gradient B′. For diamondsthe magnetic susceptibility is χ = −2.1 × 10−5 and thedensity ρ = 3500 Kg/m

3, allowing trapping frequencies

of the order of ∼ 2π 100 Hz, for magnetic field gradientsof the order of ∼ 104 T/m. We stress that consideringdifferent materials leaves the principal conclusions of thisanalysis unchanged.

As a particular instance of the weak interaction inEq. (2), we will consider the Casimir-Polder force [44, 45],which arises between any two surfaces in proximity, as aconsequence of the quantization of the electromagneticfield. Originally derived for two parallel metallic plates,it can also be computed for compact objects of arbitraryshape and material [46]. For two identical spheres, theleading contribution to the interaction energy takes theform [46] UC(x1, x2) = αR6

0/|x1 − x2|7, where α is a func-tion of the electric and magnetic permeabilities of thespheres [47].

Entanglement.—In order to highlight the challengesinvolved, let us consider two oscillators of frequency ωprepared in their ground states. Under a coupling term

3

of the form of Eq. (2), with C = αR60 and n = 7,

corresponding to a Casimir interaction, the system willdevelop entanglement between its mechanical degreesof freedom, which quantified by the logarithmic nega-tivity [48, 49], oscillates in time reaching a maximumvalue of Emax

N ' 56αR60/(mω

2d90 ln 2). For two diamonds

with a radius of 250 nm, which are held at a distanced0 = 5µm and trapped with a frequency ω/2π = 100 Hz,we estimate a maximum logarithmic negativity Emax

N '10−6. However, this macroscopic manifestation of quan-tum effects is very fragile and is dramatically reducedwhen a realistic dissipative dynamics of the oscillators isconsidered. This sets daunting perspectives in the feasi-bility of detecting such an entanglement unless suitablecountermeasures are taken.

Thereby, here we resort to strategies that can increasethe effective interaction between the two subsystems. Anestablished way to enhance the sensitivity of a system toexternal interactions consists in squeezing its mechani-cal degrees of freedom. For each oscillator, one methodto achieve this is to suddenly relax the frequency ofthe trap from ω1 to a lower value ω2, wait for a quar-ter of a period of the new trap frequency, τ2 = π

2ω2,

then switch the frequency back to the initial value ω1

and wait for another quarter of a period of the currenttrap frequency, τ1 = π

2ω1. This results in the squeez-

ing of the initial state of the oscillator ρin by an amount

ξ = ln(√

ω2/ω1

), where the new state is S(ξ)ρinS(ξ)†,

with S(ξ) the squeezing operator [50]. However, in ex-periments, the range of available frequencies is, typically,limited. In particular, for magnetic traps, this limita-tion is imposed by the attainable magnetic field gradi-ents, which have maximum values, usually, on the orderof ∼ 104 T/m. Nonetheless, it is possible to enhance thegeneration of squeezing beyond ξ by periodically repeat-ing such a frequency jump protocol. As a matter of fact,by repeatedly switching between two frequencies, ω1 andω2, the amount of squeezing can be made to increase lin-early in time, provided that frequency jumps are prop-erly timed [51–54], see Fig. 1(b). After N cycles, eachone lasting τ1 + τ2, the resulting squeezing parameter is

ξN = N ln(√

ω2/ω1

). With this proposed scheme, a

restricted choice of experimentally accessible frequenciescan still allow for the generation of a large amount ofsqueezing.

In the presence of interaction, local squeezing enhancesthe effective coupling as S(ξ)†x1x2S(ξ) = e2rx1x2, withξ = −r and r real [55, 56]. A variety of protocols rely-ing on squeezing have been developed and proposed asa standard scheme to enhance the generation of entan-glement in different platforms, from charged particles astrapped ions [39, 40] to neutral objects as massive bodiesinteracting via gravitational interaction [31]. In Fig. 2(a), we show the evolution of entanglement for two nan-odiamonds interacting via Casimir-Polder forces, which

0 20 40 60 80 100 1200.00.10.20.30.40.5

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0.50 0.55 0.60 0.65 0.7010-4

0.0010.0100.100

0.1101000105

nph(c)

(a)

0 1000 2000 3000 4000 5000 600005

101520

t

LN

0.05 0.06 0.07 0.08 0.09 0.10012345

t

LN

50 100024

75

(b) nph

EN

EN

EN

t(s)

t(ms)

t(ms)

FIG. 2. Time evolution of the logarithmic negativity betweentwo trapped nanodiamonds of radius R0 = 250 nm, held at adistance d0 = 5µm and coupled through Casimir interaction.The system is initially in the ground state and undergoes asequence of periodic frequency shifts between ω1 = 2π 100 Hzand ω2 = 0.5ω1. (a) Long-time dynamics of the logarith-mic negativity for different durations of the squeezing pro-tocol, without dissipation. In dot-dashed blue, dotted red,and dashed green, we show the entanglement generated af-ter squeezing trough 8, 9, and 10 cycles, respectively. In theinset, the short-time dynamics, displaying in black (uppersolid line) the evolution during the squeezing protocol andin orange (lower solid line) the case in which dissipation isconsidered for a protocol with 10 cycles. (b) Dynamics ofentanglement for a protocol consisting of N = 10 cycles for-ward and another 10 cycles backward (solid blue line) andfor the case with no backward cycles (dashed blue line). Inorange the number of phonons in the oscillators during theprotocols, dot-dashed orange for the complete protocol, anddotted orange for the evolution without the reversal. (c) Log-arithmic negativity at the end of a protocol as in (b), as afunction of the ratio between the two alternating frequencies(blue circle markers). In orange (square markers), the num-ber of phonons before the reversal (dashed) and at the endof the protocol (solid). In (b) and (c), we have included thedecoherence effects induced by the interaction with a thermalbath with n = 100 phonons and coupled to the system witha rate Γ = ω1/Q, for a quality factor of Q = 108 [68].

are initially subject, for different periods of time, to thefrequency jump protocol described above. We observe

4

that entanglement oscillates with a period determined bythe strength of the interaction and an amplitude deter-mined by the amount of injected squeezing [57, 68], whichin turn is proportional to the duration of the protocol.In the inset, the orange line shows the evolution of theentanglement for the same system, but now in the pres-ence of dissipative noise of the oscillators. We observethe strong deleterious effect that even an optimistic dis-sipation rate has on the entanglement. While, during theaction of the protocol, entanglement is still generated inthe presence of sufficiently weak dissipation, albeit at amuch slower rate, it quickly decays as soon as the proto-col is stopped.

As the decay rate of the entanglement is expected tobe proportional to the generated squeezing, we extendour protocol to contain an inverse sequence of frequencyjumps, which is able to reduce the local squeezing thathas built up during the protocol, and with it reduce theenvironmental sensitivity of the system. Despite the con-tinued presence of the mutual interaction, this protocolis found not to affect the entanglement that has builtup. This is one of the main findings of our analysis, andshows that the complete sequence of pulses constitutes avalid sensing protocol able to the selectively enhance thesignal over the noise. Besides, the reduction of the sys-tem energy has the additional beneficial side effect thatit tends to facilitate the measurement of observables atthe end of the protocol, as the number of energy levelsthat requires control is reduced. More specifically, thereversal is achieved by running the inverse sequence offrequency jumps after a waiting time of half of a periodof the trap frequency, see Fig. 1 (b). In the absence ofnoise, control imperfections, and of interaction betweenthe particles, after running such an inverse sequence, allthe injected squeezing would be extracted, and the sys-tem would return to its initial state. For a single par-ticle, this would constitute a witness of coherence overthe spatial length scale of the maximally squeezed statereached at the middle of the protocol, which could beused to set bounds on collapse models [15, 68]. In thepresence of interaction, however, the system will retainthe acquired entanglement, and will, consequently, notreturn to its initial state. Nevertheless, the amount ofexcitations gained over the complete protocol can be re-garded as the smoking gun for the presence of the weakinteraction.

In Fig. 2 (b), we plot the time evolution of the logarith-mic negativity for the same setup discussed in (a), butnow applying the reverse protocol. The continuous blueline shows how the inversion of the protocol is able todramatically reduce the decay rate of the entanglement,as compared to the case where no inversion protocol isapplied, dashed blue line. The orange dot-dashed anddotted lines show the evolution of the excitations in theresonators for the protocol with and without the inver-sion part, respectively. It is verified that the inversion of

the protocol strongly reduces the energy of the system,albeit a number of excitations is retained in the systemas a consequence of the generated entanglement. Theprecise energy reduction achieved by the the reverse pro-tocol is provided in Fig. (2c) as a function of the ratioof frequencies employed in the protocol. In blue circles,the entanglement achieved at the end of the protocol foreach case is displayed [60].

On the other hand, recent developments in tetheredpendula [7, 38] with remarkable stabilities, encourage usto consider similar protocols in such setups, now, withgravitationally mediated interactions. This is motivatedby the idea that the ‘quantumness’ of a mediating forcecan be assessed by its ability to act as a quantum chan-nel [22–24]. Consider the pendulum in Ref. [38], withan eigenfrequency of ω1 =

√g/l = (2π) 2.2 Hz. A fre-

quency shift can, for example, be induced by suddenlypulling the base of the pendulum upwards with an accel-eration aup. Ignoring finite material stiffness, this would

result in a new oscillation frequency ω2 =√

(g + aup)/l.For example, an acceleration of aup = g would result in afrequency shift ω2/ω1 =

√2, while the base of the pendu-

lum would be displaced by an amount gτ22 /2 = (π/4)2l,

after a quarter of the new period. We estimate that twosuch pendula, which have a mass of m = 7 mg, placedat a distance of 2 mm from each other, and interactingonly gravitationally, would develop in a time of t = 10s an entanglement of EN ' 0.5, in the presence of a de-coherence process characterized by n/Q = 10−10, wheren is the average phonon number of the thermal bath,and Q the quality factor of the pendula. This confirmsthe extremely challenging isolation conditions required toobserve gravitationally mediated entanglement betweenmassive resonators, which, as expected, remains far moredemanding than the earlier discussed Casimir mediatedcase.

Once a significant amount of entanglement has devel-oped in the system, the question arises on how to detectit. As the full state may not be Gaussian due to smallexperimental imperfections as well as higher order con-tributions in Eq. 2, one needs to adopt the approach de-veloped in [33] and determine the least entangled state,quantified by the logarithmic negativity, that is compat-ible with, for example, the measured covariance matrix.An alternative approach is available when the oscillatorscan be coupled to spin degrees of freedom [62, 67]. Thiscan be the case, for example, if one considers the oscilla-tors to be nanoparticles hosting a color center, e. g. NVcenters in diamond [58, 59, 61]. In such a setup, the spinscan be coupled to the motion of the oscillators by placinga spatially inhomogeneous magnetic field. This interac-tion can then be exploited, either to map the alreadygenerated entanglement onto the spins, or to make thespins interact through the weak force between the oscilla-tors. In the second case, an enhancement of the interac-tion can again be achieved by squeezing the mechanical

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resonators with the protocol introduced here and com-bined with other techniques designed to increase noiseresilience [63–66]. The specific details of such a protocolare outside of the scope of this work and will be describedin coming publications.

Conclusions.— Optomechanics is ushering in a new eraof quantum resonators with unprecedented quality fac-tors that is opening the door to quantum optical experi-ments in new mass regimes. We believe that it is possibleto extend these platforms to accommodate more than onemassive oscillator that interact with each other via weakforces. We have presented a protocol that can both, in-ject and extract large amounts of local squeezing in theresonators mechanical degrees of freedom with modestresources, by periodically modulating their frequencies.This combination of squeezing injection and extractionallows to amplify the sensitivity of the system to weakinteractions for a short time window and then attenu-ate it again, in order to recover resilience to noise with-out loosing the acquired entanglement. This establishesan interaction channel between massive nanomechanicalresonators that is not destroyed by its weakness or noisesensitivity, and which, if quantum in nature, it can beused to generate quantum correlations between the spa-tially separated particles, thus providing means to exam-ine the very nature of these interactions [18, 19, 22–32].Furthermore, this offers a potential interface between op-tomechanics and existing quantum platforms which canbe used to import many quantum control techniques thathave already proven successful. We believe that our find-ings set the ground for a multi-particle quantum platformoperating in a new mass regime.

Acknowledgments.— We acknowledge support by theERC Synergy grant HyperQ (Grant No. 856432), the EUprojects HYPERDIAMOND (Grant No. 667192) andAsteriQs (Grant No. 820394), the QuantERA projectNanoSpin, the BMBF project DiaPol, the state of Baden-Wurttemberg through bwHPC, the German ResearchFoundation (DFG) through Grant No. INST 40/467-1FUGG, and the Alexander von Humboldt Foundationthrough a postdoctoral fellowship.

Note added. While preparing this work for journalsubmission we became aware of [69] which pursues closelyrelated ideas.

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tum state transmission and non-locality tests, New J.Phys. 11, 023007 (2009).

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[47] The explicit expression is

α = − ~cπ

234

[(ε−1ε+2

)2

+

(µ−1µ+2

)2]− 7

2

(ε−1ε+2

)(µ−1µ+2

).

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7

ter spin in optically levitated nanodiamond, J. Opt. Soc.Am. B 34, C31–C35 (2017).

[60] For every simulation, we check that√〈(x1 − x2)2〉

d0 at all times, such that the harmonic approximationemployed for the Casimir interaction is always valid. Fora thermal initial state with nph phonons, this translates

to the condition (ω1ω2

)N√

~2mω1

(2nph + 1) d0, where

we assume that ω1 > ω2.[61] M. Geiselmann, M. Juan, J. Renger, J. M. Say, L. J.

Brown, F. J. Garcıa de Abajo, F. Koppens, and R.Quidant, 3D optical manipulation of a single electronspin, Nat. Nanotechnol. 8, 175 (2013).

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[68] See the Supplemental Material for a more detailed de-scription of the interaction Hamiltonian, numerical sim-ulations, noise analysis, and bounds on collapse models.It includes Refs. [7,36,48,55,61-64].

[69] T. Weiss, M. Roda-Llordes, E. Torrontegui, M. As-pelmeyer, and O. Romero-Isart, Large Quantum Delocal-ization of a Levitated Nanoparticle using Optimal Con-trol: Applications for Force Sensing and Entangling viaWeak Forces, e-print arXiv:2012.12260 (2020)

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8

Entanglement under xx interaction

An interaction of the form Hint = λx1x2, as consideredin the main text, can be divided into two terms: the beamsplitter term, Hbs = λx2

0(a†b + ab†), and the two-modesqueezing term, Htms = λx2

0(ab+a†b†), where a, a† andb, b† are ladder operators for each of the two modes,such that x1 = x0(a + a†) and x2 = x0(b + b†). Whileboth forms of interaction have the ability to entangle aninitially separable state, if considered separately, the dy-namics of the generated entanglement is rather different.For a time evolution under the beam splitter type of inter-action, the amount of created entanglement will dependon the initial state, for example, if applied to vacuumsuch an interaction will generate no entanglement, while,when applied on two squeezed states, these will effectivelyevolve, at short times, as under a two-mode squeezing op-eration acting on the vacuum [57]. Nevertheless, for anyinitial state, the maximum amount of entanglement isbounded in time, and will oscillate with a period deter-mined by the coupling strength, T = 2π/(λx2

0). On theother hand, an interaction of the type Htms will entanglethe two modes irrespective of the presence of squeezingin the initial state and, furthermore, this entanglementis not bounded in time, that is, it will grow linearly in-stead of oscillate. Moreover, numerical simulations sug-gest that for the same amount of entanglement a stategenerated by applying the two-mode squeezing interac-tion to the vacuum contains less energy and is more re-silient to dissipation than a state generated by inputtingtwo squeezed states to the beam splitter type of interac-tion. This is not all that surprising, as the latter containssingle-mode squeezing, which makes it more sensitive todissipation.

For the xx-type of interaction, in the presence of trap-ping potentials of similar frequency for each of the twomodes, the beam splitter component of the interactionwill be dominant, while the two-mode squeezing compo-nent will be off resonant. This explains the oscillationsobserved in the long-time dynamics of the logarithmicnegativity, in Fig. (2a) of the main text. However, itwould be desirable to have the opposite situation, thatis, the two-mode squeezing part of the interaction to beresonant, while the beam splitter part is off resonant,as this form of interaction leads to a constant growth ofthe entanglement, reaching higher values and being moreresilient to dissipation. One natural way to reach this sit-uation is to invert the frequency of one of the resonatorsω → −ω, however, such an operation may not be avail-able in many setups. An alternative approach is to mimicdynamical decoupling techniques for spins [63] where, byflipping the sign of spin operators in resonance with theiroscillation frequency, one is able to bring specific Hamil-tonian terms in and out of resonance, either for noiseresilience or to amplify specific interactions [64–66]. In

spin systems, the way to change the sign of a given spinoperator is to apply a fast π-pulse in a direction thatis orthogonal to the operator. For bosonic modes, theequivalent effect can be achieved by suddenly changingthe frequency of the oscillator to a value that is muchlarger, such that ω → ω with ω ω, and after waitingfor half of a period of the new frequency, τ = π

ω , switchingback to the original frequency. This protocol would havethe effect of mapping b, b† → −b, b†, in a time scalethat can be regarded as instantaneous for the frequencyof the modes, τ 1/ω. Using this technique to flip thesign of one of the two resonators with the periodicity of ωone would achieve a resonant two-mode squeezing type ofinteraction. We relegate the analysis of the performanceof such a protocol under realistic experimental conditionsto future work.

Numerical simulations

To compute the time evolution of the interacting res-onators, we use the covariance matrix formalism, as wealways consider initial Gaussian states, and the Hamil-tonian is quadratic. We consider a system of interactingresonators evolving under a Hamiltonian of the form

H =1

2~rT

mω2 λ 0 0λ mω2 0 00 0 1

m 00 0 0 1

m

~r, (3)

with ~r(t) =(x1 x2 p1 p2

)T, and define the covariance

matrix of a quantum state ρ as

Vij(t) =1

2〈ri(t)rj(t) + rj(t)ri(t)〉 − 〈ri(t)〉 〈rj(t)〉 . (4)

It can be shown that the covariance matrix obeys aMarkovian master equation of the form [39]

d

dtV = KV + V KT − ΓV + ΓV∞, (5)

where Γ is a Markovian decay rate, used to define thequality factor as Q = ω/Γ, and V∞ is the covariancematrix of the system in thermal equilibrium with the en-vironment, that is, the tensor product of two thermalstates of resonators of frequencies ω1 and ω2 at a giventemperature T . The matrix K describes the unitary evo-lution induced by (4) and has the form

K =

0 0 1m 0

0 0 0 1m

−mω2 −λ 0 0−λ −mω2 0 0

. (6)

The formal solution of (5) is found to be

V (t) = eK(t−t0)V (t0)eKT (t−t0)e−Γ(t−t0)

+Γ

∫ t

t0

dt′e−Γ(t−t′)e−K(t−t′)V∞e−KT (t−t′).

(7)

9

For a fixed set of Hamiltonian parameters, the integralin Eq. (7) can be analytically solved. For the data re-ported in the manuscript, we numerically solve the equa-tions of motion including the periodic switch betweentwo values of the oscillator frequencies. Once obtainedthe covariance matrix, the entanglement at a given timet is computed from it in the form of the logarithmic neg-ativity. In general, for a system of m modes in the Gaus-sian quantum state ρ, equivalently characterised by thecovariance matrix V , the logarithmic negativity quanti-fying the entanglement between two complementary sub-systems of the modes is given by

EN (ρ) = −2m∑

k=1

log2[min(1, 2|νk|)], (8)

where νk are the symplectic eigenvalues of the covariancematrix V Γ of the partially transposed states ρΓ, that is,the eigenvalues of the matrix iΩV Γ, where Ω is the sym-plectic matrix

Ω =

(0 Im−Im 0

), (9)

with Im the identity matrix of dimension m, and thesuperscript Γ represents the partial transposition withrespect to one of the two subsystems [49]. The covariancematrix of the partially transposed density operator ρΓ

with respect to the the first q modes is given by V Γ =PV P where P = Im⊕Iq⊕(−Im−q). For the case of onlytwo modes, this expression can be reduced to

EN (ρ) = −2 log2[min(1, 2|νmin|)], (10)

where νmin is now the minimum symplectic eigenvalueof the covariance matrix after partial transposition w.r.t.one of the two modes.

Sensitivity due to control errors

In the lab, a frequency-jump protocol as the one dis-cussed in the main text will be subject to a finite pre-cision in the ability to set a specific frequency in eachstep of the protocol. In this section, we analyse thesensitivity of our protocol to control errors on the per-formed frequency jumps, and bound the precision re-quired to observe entanglement. For that, we assumethat after each frequency jump in the protocol the at-tained frequency deviates from the ideal one by a randomamount with a finite variance σω, such that the imple-mented frequency is randomly picked from a Gaussianprobability distribution centered around the ideal value,

p(ω) = 1√2πσ2

ω

exp(− (ω−ω1/2)2

2σ2ω

). We then calculate nu-

merically the logarithmic negativity obtained from thecovariance matrix after averaging it over this source of

![mHz]

EN

(!)/

EN

(0)

N=9N=10N=12 N=110.1 0.2 0.5 1 2

0.00.20.40.60.81.0

σω [mHz]

LN/L

N(0)

9 10 11 120.250.5

12

N

σ ω*[m

Hz]

N

![m

Hz]

(a)

(b)

FIG. 3. A system of two particles of radius R0 = 250 nm, withequilibrium positions separated by d0 = 5µm and coupledthrough Casimir interaction, is prepared in the ground stateof its uncoupled Hamiltonian and periodically undergoes a se-quence of imperfect frequency shifts between the ideal valuesω1 = (2π) 100 Hz and ω2 = 0.5ω1. (a) Normalized logarith-mic negativity at the end of a protocol consisting of N cyclesforward and another N cycles backward as a function of thevariance in frequency. The normalization is with respect tothe ideal case (zero variance) for each protocol. (b) Values forthe variance above which no entanglement is observed, σ∗

ω, asa function of the number of cycles N in the protocol. In bluewe display σ∗

ω for the forward protocol only, and in green wedisplay σ∗

ω at the end of the protocol.

noise. Here, we make the simplifying assumption thatthe averaged state is still Gaussian and, therefore, fullycharacterized by the averaged covariance matrix. Theresults of our simulations are displayed in panel (a) ofFig. 3, where we show the logarithmic negativity after acomplete protocol (consisting of the forward and back-ward sequence of pulses) as a function of the variancein the frequencies. As expected, we observe that an un-certainty on the frequencies employed in the sequencereduces the amount of entanglement retained at the endof the protocol. Specifically, in the regime of parametersconsidered here (see the caption of Fig. 3 for details), wenotice that a precision below mHz is required in orderto observe entanglement. Furthermore, we see that thetolerable noise depends on the length of the protocol. Inpanel (b) of Fig. 3 we display σ∗ω, defined as the thresh-old value of the variance for which the entanglement iscompletely destroyed. We see how this threshold val-ues decrease with the number of cycles. This behaviourcould be qualitatively understood as follows: a frequencyshifted by ∆ω, acting on the system for a time t, rotates

10

the state of the system in phase space by an extra angle∆ωt. Due to this rotation, after a quarter of a period ofthe nominal frequency, the position variance will changeby an amount proportional to the momentum variance,

that is, by an amount ∼ ∆p2

m2ω21e2r sin

(π∆ωπ4ω1

). In order to

observe entanglement, this fluctuation must be smallerthan the uncertainty of the vacuum state. This impliesthat, once the average is taken into account, the vari-ance on the frequency should satisfy (qualitatively) thecondition ∆ω 4ω1

π e−2r, where r = N log(ω1/ω2). Wenumerically verify that this condition predicts the correctorder of magnitude.

The requirements for the case of gravitational inter-action between pendula as reported in Ref. [7, 38] withω1 = (2π)2.2 Hz and a mass of m = 7 mg as discussed inthe main text are even more stringent.

Bounds on collapse models

When considering a single particle, and in the absenceof noise and control imperfections, after running bothforward and backward sequences of our protocol, all thegenerated squeezing is extracted, and the system returnsto its initial state. The recovery of the initial state atthe end of our protocol can be regarded as a witness ofcoherence over the spatial length scale of the maximallysqueezed state, reached at the middle of the protocol. Acertification of such a coherence could then be used to setbounds on collapse models or other decoherence mecha-nisms in general. As a figure of merit for this coherence,we resort to the purity of the quantum state at the endof the protocol, which we can easily compute from thecovariance matrix as

P =1√

detV. (11)

For a massive particle in a spatial superposition, col-lapse models, such as the mechanism of continuous spon-taneous localisation (CSL), predict a coherence timeτCSL = 1

ΛCSLd2, which depends on the mass of the particle

and the size of the coherent superposition [71]. Here, d isthe size of the spreading and ΛCSL a coefficient depend-ing on the number of nucleons and the free parametersof the CSL model

ΛCSL =m2

m20

γ0CSL

4aCSLf(R/aCSL), (12)

where m is the mass of the body, m0 the mass of a singlenucleon, in the following example taken to be the mass of

a carbon atom, and f(x) = 6x2

[1− 2

x2 +(1 + 2

x2

)e−x

2]

[70]. γ0CSL and aCSL are the free parameters of the model

that need to be experimentally determined, and whichare conventionally predicted to take values on the orderof γ0

CSL = 10−16 Hz and aCSL = 100 nm [15, 71].

In our scheme, the system reaches its maximum spa-tial spread, σmax, at the middle of the protocol. How-ever, at this point the system is rotating in phase spaceat frequency ω1 and, therefore, the system spends thereonly an infinitesimal amount of time. To set bounds onthe free parameters of collapse models, both the spatialspread of the superposition and its duration in time arerelevant. To that end, we determine the amount of timethat the system spends in a state with a position vari-ance that is σ ≥ σmax/2, τ = 2π/(3ω1) = 4/3τ1, andwe require that the collapse model allows for a coherencetime that is at least one order of magnitude larger thanthat, τCSL > 10τ . This condition upper bounds the CSLparameter γ0

CSL as

γ0CSL < γCSL =

4m20aCSL

10τm2σ2maxf(R/aCSL)

. (13)

The experimental certification of coherence at the endof our protocol for growing values of σmax would settighter and tighter bounds on γ0

CSL. We display inFig. (4) the purity retained after the squeezing and un-squeezing sequences in the presence of dissipation dueto a thermal environment. In panel (a), we show thevalue of the purity as a function of the total duration ofthe protocol and the maximum spreading σmax reachedat the middle of the protocol. As expected, we observehow the purity decreases with increasing protocol lengthand squeezing. We indicate the approximated boundγCSL that reaching such a delocalization would set. Inpanel (b), in order to explore the role of environmen-tal decoherence, we display the purity as a function ofthe quality factor for a protocol of maximum spreadingσmax ' 80 nm for a particle with a mass of m ' 10−16 kg(corresponding roughly to 1010 nucleons), which wouldset a bound of γCSL ' 2 · 10−17 Hz.

It is worth to mention, that the impact of backgroundgas molecules with the levitated particle will destroy thesuperposition, thus hindering the test of collapse mod-els. Specifically, the collision rate with air molecules ona spherical particle of radius R can be approximated as[8]

Rair ≈πvPR2

kBT, (14)

where P is the chamber pressure, v =√

3KBTma

the av-

erage velocity of the air molecules with ma ∼ 28.97amu their mass. For the particle sizes considered here,R = 250 nm, and at temperatures of T ∼ 100 K, we finda collision rate Rair ≈ 1010P [Hz]. This implies that werequire ultra-high vacuum below 10−10 Pa in order to geta collision rate ∼ 1 Hz.

11

105 106 107 108 109 10100.0010.0050.0100.0500.1000.500

1

Q

0.1 1 10 100 10000.0050.010

0.0500.100

0.5001

15 30 45 60 75 90 105 120

[ ]

t [ms](a)

(b)

PP

Q

max[nm]

C

SL'

1·1

0

15H

z

C

SL'

3·1

0

16H

z

C

SL'

8·1

0

17H

z

C

SL'

2·1

0

17H

z

T[ms]

FIG. 4. A single particle of radius R0 = 250 nm is prepared inthe ground state and periodically undergoes a sequence of fre-quency shifts between the ideal values ω1 = (2π) 100 Hz andω2 = 2ω1. (a) Purity at the end of a protocol of duration Tand corresponding maximum spreading σmax, in the presenceof a thermal bath with n = 100 phonons and coupled to thesystem at a rate Γ = ω1/Q, with quality factor of Q = 108.On the vertical lines we display the parameter γCSL whichwould predict, for the corresponding spreading, a coherencetime one order of magnitude larger than 4/3τ1. (b) Purityas a function of the quality factor for a protocol of durationτ ' 100 ms with maximum spreading σmax ' 80 nm.