Coherent Scattering-mediated correlations between levitated nanospheres
CCoherent Scattering-mediated correlations between levitated nanospheres
I. Brand˜ao, ∗ D. Tandeitnik, † and T. Guerreiro ‡ Departamento de F´ısica, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro, 22451-900 Rio de Janeiro, RJ, Brazil (Dated: February 19, 2021)We explore entanglement generation between multiple optically levitated nanospheres interactingwith a common optical cavity via the Coherent Scattering optomechanical interaction. We derivethe many-particle Hamiltonian governing the unitary evolution of the system and show that it givesrise to quantum correlations among the various partitions of the setup, following a non-Markoviandynamics of entanglement birth, death and revivals. We also consider the effects of coupling thesystem to external environments and show that under reasonable experimental conditions entangle-ment between the mechanical modes can survive even at room temperature. Its dependence uponthe number of nanoparticles, their initial temperature and coupling strength is studied. A numeri-cal toolbox to simulate the closed and open dynamics of Gaussian optomechanical states and theirinformational measures is developed.
I. INTRODUCTION
Testing quantum mechanics in novel regimes, such asobserving quantum effects in systems with many con-stituents or a large number of degrees-of-freedom, is oneof the cornerstones of fundamental science and a promis-ing achievement towards new technologies. A numberof experiments have contributed along that directionby studying the quantum mechanics of nano- and mi-croscale objects. For instance, entanglement of hundredsof ions has been observed and controlled [1], interferomet-ric systems have achieved micron-spaced superposition ofatomic wavefunctions [2], coherence in Bose-Einstein con-densates has been observed [3] and ground state coolingof micron-sized cantilevers and their coupling to super-conducting quantum electronics demonstrated [4].Optically levitated nanoparticles allow exceptionalcontrol over translational [5–7] and rotational [8–10]degrees-of-freedom and achieve excellent environmentalisolation [11, 12], thus providing a promising setup forpushing the boundaries of quantum theory towards un-explored regimes. Proposals for generating spatial super-postion of levitated nanoparticles have been put forward[13–15], as well as for testing collapse models [16] andwitnessing nonclassicality through recurrence of opticalsqueezing [17] and optical entanglement [18]. Moreover,levitated systems can give rise to steady-state entangle-ment [19–23] and help in the search for new physics [24].On the experimental front, the possibility of detectingnonclassical correlations in levitated particles has beendemonstrated [25]. Effective 3D cooling [26, 27], groundstate cooling [6, 7] and strong light-matter coupling havebeen realized [28]. All of these are essential requirementstowards entering and controlling the mesoscopic quantumregime.Ground state cooling of levitated nanoparticles along asingle axis was first enabled through the so-called coher- ∗ [email protected] † [email protected] ‡ [email protected] ent scattering interaction [6, 29], and it has been theoret-ically shown that simultaneous 2 D ground state coolingis possible with the same technique [30]. In this cool-ing scheme, motion of the particle coherently scattersphotons from the trapping beam into an optical cavitytuned to enhance scattering of photons that carry awayenergy from the trapped object [6, 31]. Following the re-cent interest on entanglement dynamics in optomechani-cal systems [18, 32–35], coherent scattering has also beenconsidered as a platform for generating mechanical en-tanglement [23, 36]. The present work builds along thatdirection and investigates how the coherent scatteringinteraction between a single cavity mode and an arbi-trary number of levitated nanoparticles can give rise toquantum correlations among the various partitions of thesystem even at room temperature.We begin by deriving the many-particle coherent scat-tering Hamiltonian in close analogy to [31, 37], wherean arbitrary number of nanoparticles share a commonoptical cavity. By appropriate positioning of the parti-cles with respect to the cavity nodes one can minimizethe dispersive optomechanical interaction and favor thecoherent scattering terms. The unitary dynamics gener-ated by the Hamiltonian is responsible for creating en-tanglement in the system. We discuss the closed systemevolution through numerical simulation of the so-calledLyapunov equation in the absence of photon loss anddecoherence. In the unitary regime, we show the occur-rence of periodic entanglement birth, death and revivalsevidencing the non-Markovian nature of the evolution.Real-life quantum systems, however, are open. For thisreason, we model the environmental interactions througha set of quantum Langevin equations and the associatedLyapunov equation; both closed and open dynamics aresimulated with a custom numerical toolbox [38]. Weproceed to study entanglement generation in the pres-ence of collisional decoherence and photon recoil heating.We demonstrate the persistence of entanglement oscilla-tions and non-Markovianity even in the presence of noiseand contact to bosonic and heat baths, and discuss theprospects of experimentally verifying these quantum cor-relations with current technology. a r X i v : . [ qu a n t - ph ] F e b ... ω c ≈ ω t x x x ω t x N κ ω , γ ω N , γ N Laser ... xzy x e t, y x θ j j FIG. 1. Schematics of N optically levitated nanoparticles,and its corresponding tweezers inside a common optical cavity.Every tweezer is considered to have the same frequency, whichis tuned close to the cavity resonance frequency, such that thescattered photons can survive inside the cavity. It is assumedthat the tweezers are sufficiently spaced apart such that beamoverlap can be neglected. Information about the system canbe retrieved through the leaking field from the rightmost endmirror of the cavity. II. HAMILTONIAN
The system we are interested in is comprised of N optically trapped dielectric nanoparticles (NP) of mass m j , each with radius R j on the order of magnitude of100 nm, refractive index n R ,j , homogeneous and isotropicpermitivitty (cid:15) j ≈ n ,j , and polarizability α j ≡ (cid:15) (cid:15) j − (cid:15) j +2 .Each NP is optically trapped by an independent opticaltweezer (OT) and placed on the axis of a Fabry-P´erotcavity of length L and resonance frequency ω c , as de-picted in Figure 1. The tweezers are assumed to be suf-ficiently apart such that any overlap and cross-talk be-tween the traps can be neglected. All OTs propagateperpendicularly to the cavity axis, have the same fre-quency ω t = 2 πc/λ t and their polarization vectors e t ,j can be decomposed as e t ,j = cos( θ j ) e x + sin( θ j ) e y alongthe cavity axis.Following [31], the total Hamiltonian governing thesystem dynamics can be written asˆ H = ˆ H NP + ˆ H field + ˆ H int , (1)where the first term is the energy of the free NPs, thesecond term is the total energy stored in the electromag-netic (EM) field and the third term represents interac-tions between the NPs and the EM field. To place thisHamiltonian in a suitable form, we consider the differ-ent contributions from the electric field present in thesystem and work on approximating each of these termsindividually.The total electric field is considered to be approxi-mately given by a sum of contributions from the fieldsof each OT, E t ,j , and the intracavity field ˆ E c , ˆ E ( r ) (cid:39) ˆ E c ( r ) + N (cid:88) j E t ,j ( r ) . (2) The OTs are considered to be in strong coherent states[31] and thus well described by a classical field. Themean value of the j -th OT’s electric field operator in anappropriate rotating frame is given by E t ,j ( r , t ) = (3)12 (cid:15) t w ,j w j ( z ) e − ( x − x ,j )2+ y w j e ik t z e iφ G,j ( z ) e iω t t e t + c . c ., where P t ,j is the power, k t = 2 π/λ t the wave-number, e t ,j the polarization vector, w ,j the waistand (cid:15) t ,j , w j ( z ) , φ G ( z ) , z R ,j are the field amplitude, beamwidth, Gouy phase and Rayleigh range, respectively.These quantities are given by (cid:15) t ,j = (cid:115) P t ,j w ,j π(cid:15) c , (4) w j ( z ) = w ,j (cid:113) z /z ,j ,φ G ( z ) = − arctan( z/z R ,j ) ,z R ,j = k t w ,j / . The intracavity electric field is a standing wave describedquantum mechanically by the operator ˆ E c ( r ) = (cid:15) c (cid:0) ˆ a † + ˆ a (cid:1) cos( k c x ) e y , (5)where k c = 2 π/λ c is the wave vector, (cid:15) c = (cid:113) (cid:126) ω c (cid:15) V c thesingle photon electric field for a cavity of mode volume V c , ˆ a the time-dependent annihilation operator and e y the cavity field polarization. We note that the externalfree EM field also plays a role in the dynamics of the NPs.However, as shown in the Appendix B, the effect of inter-action with this field is negligible if the NPs are properlypositioned within the cavity and sufficiently cooled down.We will therefore drop any term involving the free EMfield in what follows.In the long wavelength approximation, given by R j (cid:28) λ c , λ t ,j , the interaction Hamiltonian can be expressedas [31, 37, 39],ˆ H int = − (cid:90) P ( r ) E ( r ) d r (cid:39) − N (cid:88) j =1 α j | ˆ E (ˆ r j ) | (6) (cid:39) − N (cid:88) j =1 α j (cid:12)(cid:12)(cid:12)(cid:12) ˆ E c (ˆ r j ) + E t ,j (ˆ r j ) (cid:12)(cid:12)(cid:12)(cid:12) , where P j ( r ) = α j E ( r ) is the j -th NP polarization vec-tor and α = αV = 4 π(cid:15) R (cid:15) j − (cid:15) j +2 . Here ˆ r j = R ,j + ˆ R j denotes the center-of-mass (COM) position operator ofthe j -th particle, with R ,j = ( x ,j , ,
0) being themean position of the j -th OT along the cavity axis and ˆ R j = ( ˆ X j , ˆ Y j , ˆ Z j ) the fluctuations of the particle around R ,j . The interaction Hamiltonian is simplified by as-suming that the overlap term proportional to E t ,i (ˆ r j ) isnegligible for i (cid:54) = j , i.e., E t ,i (ˆ r j ) ≈ δ ij E t ,i (ˆ r j ).Terms proportional to | E t ,j (ˆ r j ) | give rise to a 3 D har-monic potential on the NPs, effectively levitating themwith trapping frequencies ω α,j , α = x, y, z , given by ω x,j ω y,j ω z,j = (cid:113) α j P t ,j m j w ,j π(cid:15) c (cid:113) α j P t ,j m j w ,j π(cid:15) c (cid:113) α j P t ,j m j w ,j z ,j π(cid:15) c . (7)Moreover, terms proportional to | E c (ˆ r j ) | result in threecontributions: a shift in the natural frequency of the cav-ity, a radiation pressure coupling and a driving term onthe j -th NP. Finally, the last term proportional toRe (cid:8) E t ,j (ˆ r j ) E c (ˆ r j ) (cid:9) generates the coherent scatter-ing (CS) interaction [31], effectively 2D coupling theNPs with the cavity field, and a drive term in the cavityfield. The nature of this interaction can be intuitivelyunderstood as due to the OT’s photons being coherentlyscattered by the trapped NPs into the cavity mode, thuspopulating it [27, 29, 31].Gathering all the terms, disregarding constant energyshifts and moving into the appropriate rotating frameat the frequency of the OTs (see Appendix A for moredetails), we find the full Hamiltonian for the CS systemwith N particles:ˆ H = (cid:126) ∆ˆ a † ˆ a + N (cid:88) j =1 (cid:40) ˆ P j m j + ˆ R Tj Ω j ˆ R j − (cid:126) G j sin( θ j ) cos( k c x ,j ) (cid:0) ˆ a † + ˆ a (cid:1) − (cid:126) g ,j k c x ,j ) ˆ X j − (cid:126) g ,j sin(2 k c x ,j )ˆ a † ˆ a ˆ X j + (cid:126) G x,j sin( k c x ,j ) sin( θ j ) (cid:0) ˆ a † + ˆ a (cid:1) ˆ X j + i (cid:126) G z,j cos( k c x ,j ) sin( θ j ) (cid:0) ˆ a † − ˆ a (cid:1) ˆ Z j (cid:41) , (8)where Ω j ≡ diag( ω x,j , ω y,j , ω z,j ) describe the trapfrequencies for the j-th NP, ∆ ≡ ω c − ω t − (cid:80) Nj =1 δ j cos ( k c x ,j ) is the shifted cavity frequency, g ,j = k c δ j are the dispersive couplings with δ j = α j ω c (cid:15) V c , G x,j = k c G j and G z,j = k t z R ,j − z R ,j G j are the bare CS cou-plings in the x and z directions for the j -th particle with G j = α j (cid:15) t (cid:15) c (cid:126) , and ˆ P j = ( ˆ P x,j , ˆ P y,j , ˆ P z,j ) is the momentumoperator for the j -th particle.In order to favour the CS couplings over the dispersiveones, we place the NPs’ mean position at the cavity nodesby acting on their respective OT. Consequently the totalHamiltonian simplifies toˆ H/ (cid:126) ≈ ∆ˆ a † ˆ a + N (cid:88) j =1 ω j ˆ b † j ˆ b j + N (cid:88) j =1 g j (ˆ a † + ˆ a )(ˆ b † j + ˆ b j ) , (9)where ˆ b j (ˆ b j ) is the annihilation (creation) operator forthe j -th NP. Note that the shifted cavity frequency sim-plifies to ∆ = ω c − ω t and g j ≡ x ZPF ,j G x,j sin( θ j ) is theCS coupling, with x ZPF ,j = (cid:112) (cid:126) / (2 m j ω j ) the zero pointfluctuation for the j -th NP.Note that for the case of a single NP, the Hamiltonianis symmetric between optical and mechanical modes. Wenote that it describes also the situation within the lin-earized dispersive optomechanical approximation [40], aswell as the linearized membrane-membrane coupling [41]. III. EQUATIONS OF MOTION
The Hamiltonian (9) dictates closed unitary dynamics.We now proceed to discuss the open quantum dynamicsof the system through a set of quantum Langevin equa-tions [20, 42–45]. We consider that one of the cavity mir-rors is not perfect, resulting in a finite cavity linewidth κ and allowing photon exchange between the cavity fieldand the external free field [46]. Moreover, each NP isconsidered to be in contact with its own thermal bath,at temperature T env ,j , resulting in a quantum Brown-ian motion for the NP [47]. We define the dimension-less position and momentum quadratures for each par-ticle ˆ x j = ˆ b † j + ˆ b j , ˆ p j = i (ˆ b † j − ˆ b j ), and for the cavityfield ˆ Q = ˆ a † + ˆ a , ˆ P = i (ˆ a † − ˆ a ), such that the quantumLangevin equations read˙ˆ Q = +∆ ˆ P − κ Q + √ κ ˆ x in , (10)˙ˆ P = − ∆ ˆ Q − κ P + √ κ ˆ p in − N (cid:88) j =1 g j ˆ x j , (11)˙ˆ x j = + ω j ˆ p j , (12)˙ˆ p j = − ω j ˆ x j − γ j ˆ p j + ˆ f j − g j ˆ Q , (13)where ˆ x in = ˆ a † in + ˆ a in , ˆ p in = i (ˆ a † in − ˆ a in ) are the zero-averaged delta-correlated optical input noise terms sat-isfying (cid:104) ˆ a in ( t )ˆ a † in ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) [46]; γ j is the dampingrate for the j -th NP, which is under the influence of zero-averaged stochastic thermal noise ˆ f j [47] with correlationfunctions given by (cid:104) ˆ f j ( t ) ˆ f k ( t (cid:48) ) (cid:105) = 2 γ j ω j (cid:90) ω j e − iω ( t − t (cid:48) ) ω (cid:20) coth (cid:18) (cid:126) ω k B T env ,j (cid:19) + 1 (cid:21) dωπ δ j,k Here, the reservoir cut-off frequency is ω j , and k B de-notes the Boltzmann constant. From the form of thesecorrelators, the stochastic noise automatically satisfiesthe fluctuation-dissipation theorem and in the high-temperature regime, k B T env ,j (cid:29) (cid:126) ω j , becomes delta-correlated (cid:104) ˆ f j ( t ) ˆ f k ( t (cid:48) ) (cid:105) ≈ γ j (2 n th ,j + 1) δ ( t − t (cid:48) ) δ j,k [43, 44, 47], where n th ,j = (cid:2) − exp[ (cid:126) ω j / ( k B T env ,j )] (cid:3) − ≈ k B T env ,j / ( (cid:126) ω j ) is the thermal occupation number for the j -th heat bath.We are interested in the case where the j -th NP isinitially in a thermal state at temperature T j with oc-cupation number n ,j and the cavity field starts in thevacuum state. The linear nature of the Langevin equa-tions preserves the Gaussianity of the initial states, allow-ing the use of the Gaussian quantum information tool-box [48]. In particular, since Gaussian quantum statesare completely characterized by their first and secondmoments, we can focus directly on the dynamics of thecovariance matrix. See Appendix C for a brief introduc-tion to the structure of Gaussian states and definitionsof useful quantities used in remaining of this work suchas the logarithmic negativity, von Neumann entropy andmutual information.Note that first moments can be easily calculated oncewe recast the Langevin equations in a more compactform, ˙ ˆ X ( t ) = A ˆ X ( t ) + ˆ N ( t ) , (14)where ˆ X = ( ˆ Q, ˆ P , ˆ x , ˆ p , ... ) T is the quadrature vector, ˆ N = ( √ κ ˆ x in , √ κ ˆ p in , , ˆ f , ... ) T is the input noise vectorand A is the diffusion matrix.Consider the formal expression for the quadratures ˆ X ( t ) = e At ˆ X ( t ) + (cid:90) tt e A ( t − s ) ˆ N ( s ) ds . (15)Equipped with the fact that the initial states and in-put noise have zero average, direct calculation shows that (cid:104) ˆ X ( t ) (cid:105) = for all times.The second moments of the system can be representedby the covariance matrix (CM), with components definedas V i,j = (cid:104) ˆ X i ˆ X j + ˆ X j ˆ X i (cid:105) . Using Equation (15), we seethat the CM satisfies the Lyapunov equation˙ V = AV + V A T + D , (16)where D l,k δ ( t − t (cid:48) ) ≡ (cid:104) ˆ N l ( t ) ˆ N k ( t (cid:48) ) + ˆ N k ( t (cid:48) ) ˆ N l ( t ) (cid:105) , suchthat D = diag (cid:0) κ, κ, , γ (2 n th , + 1) , , γ (2 n th , +1) , . . . (cid:1) is the drift matrix. We can use the above dynam-ical equations to study entanglement within our systemboth in the closed and open regimes. IV. UNITARY ENTANGLEMENT DYNAMICS
It is expected that the unitary dynamics generated by(9) exhibits entanglement between the various mechani-cal and optical modes in the system. To gain some ana-lytical insight into this entanglement generation considerthe formal limit ω j (cid:28) ∆. For N = 2 NPs the Hamilto-nian reduces to,ˆ H / (cid:126) ≈ ∆ˆ a † ˆ a + g (ˆ a † + ˆ a )(ˆ x + ˆ x ) , (17)where we assume for simplicity that both NPs coupleequally to the optical mode, g = g ≡ g . This sim-plified Hamiltonian can be exponentiated exactly usingthe same techniques employed in calculating the unitaryevolution operator for a dispersive optomechanical sys-tem [18]. The unitary time evolution operator for thiscase is U ( t ) = exp (cid:16) − i ˆ H t/ (cid:126) (cid:17) , which readsˆ U ( t ) = e − i ˆ a † ˆ a ∆ t e (ˆ aη ( t ) − ˆ a † η ( t ) ∗ )( g/ ∆)(ˆ x +ˆ x ) ×× e − i ( g/ ∆) (ˆ x +ˆ x ) sin(∆ t ) e ig (ˆ x − ˆ x ) ∆ t (18)where η ( t ) = 1 − e − i ∆ t . Note that the two exponen-tial terms in the second line contain effective interac-tions among the NPs, given by products of the ˆ x , ˆ x terms. The presence of these optically mediated inter-actions lead to generation of entanglement between theNPs within this simplified approximation. Moreover, thethe fact that the unitary operator is written in terms ofperiodic functions hints at entanglement death and re-vivals.In order to verify the entanglement generation in thisregime, we numerically solve the Lyapunov equation inthe absence of noise and losses. Figure 2 shows numer-ical plots of the Logarithmic negativity (LN) and vonNeumann entropies for the various partitions of a systemcomprised of two NPs and one optical mode, all initiallyin the ground state. The parameters used for this sim-ulations are shown in Table II, except for ∆ = 10 × ω j , κ = 0 and γ j = 0 ∀ j . From now on, unless explicitlystated otherwise, all NPs are taken to be identical andTable II dictates all the parameters considered in the sim-ulations throughout the rest of this work; see Section VI.We observe that in the unitary CS scenario cyclic en-tanglement birth and death are present, analogous to thedispersive optomechanical case [18]. Moreover, while theentropy of each bipartition is synchronized, the LN in themechanical bipartition only achieves local maxima whenthe optomechanical LN are at their local minima. Thispoints towards the idea that under certain circumstancesentanglement can flow through different partitions of thesystem, in this case back and forth between the opto-mechanical and mechanical modes. It is also instructiveto consider the optical field as an environment for the twoNPs. Under this point of view, we can understand en-tanglement and entropy oscillations as a consequence ofthe non-Markovian nature of the subsystems’ evolution. S c a v , S c a v , S , FIG. 2. Simulation with N = 2 identical levitated NPs ini-tially in the vacuum state following a closed unitary dynamics.Time evolution of logarithmic negativity (blue) and von Neu-mann entropy (red) for each bipartition of the system. In thissimulation, we considered ∆ = 10 × ω j , ensuring the formallimit ω j (cid:28) ∆, while κ = 0 and γ j = 0 ∀ j making the systemclosed. Due to instabilities in the numerical simulation of thevon Neumann entropy in the mechanical bipartition, we showboth the calculated values (dark red points) and smoothedtrace (light red line). V. ENTANGLEMENT IN A NOISYENVIRONMENT
In any realistic experimental scenario the system un-der study is always interacting with its environment. Forthis reason it is important to study how the unitary en-tanglement dynamics is modified when the optomechan-ical system is placed in contact with uncontrolled exter-nal degrees of freedom. For high-vacuum environments( p < − mbar), it has been shown that CS mediatedmechanical entanglement can resist photon scattering de-coherence [36] and in moderate vacuum ( p ∼ − mbar)steady state entanglement is only possible at low envi-ronment temperatures around T env ,j ≈
15 K [23]. In thelatter regime, the crucial question of whether mechani-cal entanglement could exist for realistic environmentaltemperatures before the system achieves its steady stateremains open.For a start, consider the more experimentally challeng-ing scenario where each NP begins in the ground stateand in contact with a cryogenic environment at temper-ature 130 K [6]. This setting allows for entanglement ofa large number of NPs. Figure 3 shows the time evolv-ing LN as a function of the number of identical NPs inthe cavity. We note that in this case the LN is sym-metric over all possible mechanical bipartitions providedthe particle parameters are identical, e.g. mass, couplingstrength, bath temperature. As the number of NPs inthe cavity is increased we observe the maximum LN de-creases: the additional NPs act as an environment forthe bipartite subsystem. This can also be understood asa consequence of monogamy of entanglement.In contrast to the result of Section IV, we observe thatquantum correlations between mechanical modes are onlynon-zero for a brief interval of time, while their oscillatingnature is washed out by interactions with the environ-
FIG. 3. Time evolving LN between a pair of particles for anincreasing number of particles sharing the same cavity. EveryNP is considered to be in the ground state and in contactwith a cryogenic environment at temperature T env ,j = 130 K,resulting in γ j = 0 .
957 mHz. ment. Since the environment is traced out and treatedeffectively as damping and stochastic forces in a Marko-vian approximation, quantum correlations between thesystem’s degrees of freedom are never recovered [49].Moreover, as we consider a higher number of interactingNPs, both the maximum of the LN trace and the timeinterval during which it is non-zero decreases, i.e., theentanglement dilutes over the system due to monogamyconstraints.In the following we restrict our attention to a systemcomprised of N = 2 identical levitated NPs. This allowsto study the system in a setting where entanglement gen-eration is maximized. The particles are considered to becooled to a thermal state at temperature 4 . µ K, close totemperatures achieved in current optomechanical exper-iments [6]. We consider the thermal environment to becryogenic, at 130 K. In Figure 4(a), we illustrate tracesof the LN and entropy evolving in time for each possiblebipartition. Once again, we observe loss of entanglementoscillations for the mechanical modes due to interactionswith the external environment. Note, however, that theopto-mechanical entanglement persists over some oscilla-tions before dying out in the steady state at long times(not shown). Note also that the non-Markovian featuresof the evolution such as oscillations of entropy, albeitpresent, are strongly attenuated due to the Markoviannature of environmental interactions.We can speak more broadly about the correlationswithin the system employing the mutual information, ameasure of the total classical and quantum correlations[50]. See Appendix C for details on the definition. In Fig-ure 4(b), we plot the time evolving mutual informationfor the total system, I tot , and the reduced system com-posed solely of the NPs, I particles . The parties are initiallyuncorrelated as expected from the form of the separablestates at t = 0. It later becomes correlated during thesystem’s coherence time. In the reduced mechanical bi-partition, CS-mediated correlations are generated beforequantum entanglement comes into play and persists afterentanglement death. As a final remark, we have solvedthe Lyapunov equation for the steady state, given by S S S FIG. 4. Simulation with N = 2 identical levitated NPs with initial temperature T j = 4 . µ K in contact with a cryogenicenvironment at T env ,j = 130 K, γ j = 0 .
957 mHz. (a) Time evolution of logarithmic negativity (blue) and von Neumann entropy(red) for each bipartition of the system. (b) Mutual information for the total system (yellow) and subsystem with only thelevitated NPs (blue) evolving in time. AV + V A T + D = 0, and find that E j,kN = 0 ∀ j, k . This isconsistent with previous results in the literature [23]. Thesteady state, however, displays non-zero total mutual in-formation I tot (cid:39) .
314 and I particles (cid:39) .
223 meaningthat although the LN cannot detect steady state entan-glement, general correlations among subsystems due tothe CS interaction are present.So far, our focus has been centered on how coherentscattering-mediated entanglement appears and evolves intime. We now shift our attention to the LN’s dependencyon experimentally controlled parameters in the search fora configuration allowing room temperature mechanicalentanglement. An in depth discussion of the experimen-tal feasibility of the parameters used in the simulationsis presented in Section VI.
FIG. 5. Maximum of the logarithmic negativity within thesystem’s coherence time for two particles as a function of theirinitial temperatures. Notice that we need highly cooled par-ticles to generate mechanical entanglement.
Figure 5 shows the maximum of the LN between thetwo NPs within the coherence time of the system, τ (cid:39) . µ s as quantified in Appendix D, for different val-ues of initial temperature and occupation number in thepresence of an environment at room temperature (300 K).We can immediately see that in a realistic scenario me- FIG. 6. Time evolving logarithmic negativity for differentcoupling strengths, g j = g for all j . Here we considered everyNP starting in a thermal state at T j ∼ . µ K (occupationnumber n ,j = 0 . g ∼ ω , entanglement birth, death and revivals arepresent. chanical entanglement would only occur for extremelycooled particles. For T j > µ K we see the LN vanishes,and we note that such temperature corresponds to onetenth of the occupation number achieved by [6], imply-ing that experimental verification of coherent scatteringgenerated entanglement at room temperature is a chal-lenging achievement.This difficulty could be partially circumvented by in-creasing the CS coupling. In Figure 6, we show thetime-dependent LN as a function of the optomechani-cal coupling strength for an initial NP temperature of T j ∼ . µ K (occupation number n ,j = 0 . g j /ω j ∼ . π ×
110 kHz could be achieved by increasing the NP ra- t = 1.488 -5 0 5 x -505 p -5 0 5 x -505 p t = 1.116 -5 0 5 x -505 p t = 0.744 -5 0 5 x -505 p t = 0.372 -5 0 5 x -505 p t=0 t FIG. 7. Time evolution of Wigner function for N = 2 identical levitated NPs. As the particles are identical, they posses identicalfunctions. Black dashed lines indicate the Wigner function semi-axes. Solid black lines denote contours at half of its maximuminitial value. As the particles are initially in a thermal state the squeezing degree starts at η (0) = 1 and becomes increasinglysmaller as time progresses. The squeezing degree for each time stamp is respectively η = (1 , . , . , . , . dius and using higher power tweezers; in such regime,where g j /ω j ∼ .
36, entanglement starts to appear inthe system. For higher values of optomechanical cou-pling, we enter the strong coupling regime g j ≥ ω j wherebirth, death and revivals of entanglement occur.As a concluding remark, we note that another inter-esting feature of the CS interaction is the generation ofmechanical squeezing during the system’s time evolution.Generation of squeezing using the CS Hamiltonian for asingle particle has been shown in [51]. We are able toshow that squeezing is also generated if more than oneparticle is present in the cavity. As an example, considerthe case of two NPs in the cavity initially in a thermalstate. Figure 7 displays the Wigner function for a NP atdifferent instants of time. As we consider identical NPs,their Wigner functions are also identical and only a sin-gle one is shown. We observe the generation of squeezingthrough the appearance of elliptical shapes of the Wignerdistribution and the calculated lower than one squeezingdegree η . We define the squeezing degree η as the ra-tio of the squeezed and the antisqueezed quadratures,see Appendix D for details. Squeezing is then presentif η <
1. The squeezed Wigner function also rotates asa result of time evolution. The emergence of squeezingin multi-particle CS could find interesting applications inquantum metrology of tiny forces [15].
VI. EXPERIMENTAL PARAMETERS
The parameters used in our simulations were adaptedfrom Ref. [6]. There, a silica nanosphere (density ofSiO : 2200 kg / m [31]) was trapped by an OT withina high-finesse Fabry-P´erot cavity mounted in a vacuumchamber and cavity cooling through CS was performedto cool the CM of the NP into its motional ground state.The parameters values of this experimental realizationare presented in Table I.In order to achieve a higher coupling between the NPsand the cavity field than previously reported in [27], wepropose slight adaptations of the experimental parame- TABLE I. Experimental parameters reported in Ref. [6]Parameter Unit ValueTweezer power P t mW 400Tweezer wavelength λ t nm 1064Tweezer waist (x axis) µ m 0 . µ m 0 . ω x kHz 2 π × F — 73 , κ kHz 2 π × L mm 10 . w µ m 41 . π × m fg 2 . R nm 71 . x ZPF pm 3 . p gas mbar 10 − Environmental temperature T gas K 300 ters. First, we increase the radius of each NP to 100 nm;second we raise the power of each trapping tweezer to 1 Wand consider a Gaussian beam waist of w ,j = 0 . µ m,such that the trapping frequency remains approximatelythe same ω j ≈ π ×
305 kHz. In Appendix D, we showthat this change does not greatly affects the coherencetime of the system, before a NP becomes populated bya single phonon. Table II shows the resulting proposedparameters.
TABLE II. Parameter values used in this workParameter Unit ValueMechanical natural frequency ω kHz 2 π × . π × . g kHz 2 π × . κ kHz 2 π × . γ mHz 2 π × . T µ K 12 . T gas K 300
Unless explicitly stated otherwise, these are the valuesconsidered in the simulations throughout this work withall NPs identical to each other.
VII. CONCLUSION
We have extended the CS Hamiltonian to an arbitrarynumber of particles, describing N mechanical oscillatorsinteracting with a single cavity mode. The resulting uni-tary dynamics has been shown to generate quantum cor-relations in every bipartition of the system following cy-cles of entanglement birth and death. Moreover, withineach revival, the Hamiltonian appears to steam a flowof quantum correlations between the bipartitions, withmechanical entanglement maximized exactly when opto-mechanical entanglement is minimized.In realistic experimental conditions, one has to con-sider the environmental effects on the system. Followingan open quantum dynamics, we have show that entangle-ment generation can still persist even in room tempera-ture environments for some time within experimentallyreasonable parameters. Although in the steady statethese quantum correlation die out [23], we have shownthat general correlations are still present in the system.We studied the dependence of mechanical entangle-ment on experimentally controlled parameters. We dis-covered that increasing the number of NPs interactingwith the optical field dilutes the entanglement over thecomplete system and is detrimental to the creation of bi- partite quantum correlations. Such correlations are ex-pected to only appear when the particles are extremelycool, below the minimum occupancy achieved with cur-rent state-of-the-art technology [6]. The CS couplingstrength also plays a significant role in entanglement gen-eration, and revivals of entanglement only come into playin the high-coupling regime, where g j > ω j . Squeezingis also generated by the many-particle CS Hamiltonian.In summary, the coherent scattering interaction propelsoptomechanics to the domain of complex quantum sys-tems within realistic scenarios. It will be interesting tosee what new experiments are enabled by this many-bodymesoscopic quantum toolbox. ACKNOWLEDGEMENTS
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The free energy for the NPs and the the EM field [39] are, respectively, given byˆ H NP = N (cid:88) j =1 ˆ P j m j , (A1)ˆ H field = (cid:15) (cid:90) ˆ E ( r ) + c ˆ B ( r ) d r (cid:39) (cid:126) ω c ˆ a † ˆ a , (A2)where, in the second equation, we made use of the approximation of the EM field given in Eq. (2) and neglectedconstant energies contributions, as will be done hereforth. The interaction Hamiltonian, Equation (6), can be brokendown into three pieces: ˆ H int = N (cid:88) j =1 − α j (cid:110) | (cid:126)E c (ˆ r j ) | + 2Re (cid:16) (cid:126)E c (ˆ r j ) (cid:126)E ∗ t ,j (ˆ r j ) (cid:17) + | (cid:126)E t ,i (ˆ r j ) | (cid:111) . (A3)In order to evaluate the final form of the interaction energy, we use the definitions given to the cavity’s and OTs’fields given by Eqs. (3) and (5), respectively. The gaussian tweezers will effectively trap the NPs close to their focus,confining them close to R ,j , thus we can approximate the interaction Hamiltonian through a series expansion around( ˆ X j , ˆ Y j , ˆ Z j ) = in each term. Moreover, as the optical modes’ frequencies ( ω t ,j ≈ ω c ) are much higher than thecoupling rates present in the system , we also take a rotating-wave approximation (RWA) at these frequencies ineach interaction term [31]. Finally, we disregard the constant energy shifts as they do not affect the dynamics of thesystem.The tweezer-tweezer interaction terms yields a 3 D trapping potential − α | ˆ E t ,j ( ˆ R ) | ≈ (cid:88) i = x,y,z m Ω j,i ˆ R j,i (A4)The cavity-cavity interaction results in three terms: − α j | E cav (ˆ r j ) | ≈ − (cid:126) δ j cos ( k c x ,j ) ˆ a † ˆ a + (cid:126) k c δ j sin(2 k c x ,j ) ˆ X j (cid:20) ˆ a † ˆ a + 12 (cid:21) , (A5)one proportional only to the cavity’s number operator, responsible for the cavity frequency shift due to the presenceof the j -th particle, other proportional to the j -th NP position quadrature, acting as a constant drive in the NP’smomentum, and an interacting term resulting in the radiation pressure effect on the j -th NP by the cavity field.Finally, the cavity-tweezers interaction gives rise to the coherent scattering interaction [31], effectively 2D couplingthe NPs with the cavity field and a drive in the cavity field. The nature of this interaction is due to the j -th tweezer’sphotons being coherently scattered by the particle inside the cavity, populating it [27, 29, 31]: − α j Re (cid:16) ˆ E c (ˆ r j ) E ∗ t ,j (ˆ r j ) (cid:17) ≈ − α j (cid:15) t (cid:15) c (cid:0) ˆ a † + ˆ a (cid:1) (cid:40) cos( ω t t ) (cid:18) cos( k c x ,j ) − k c ˆ X j sin( k c x ,j ) (cid:19) − ˆ Z j k t z R j − z R j sin( ω t t ) cos( k c x ,j ) (cid:41) sin( θ j ) . (A6)Note that this last term has a time dependency, which we can get rid off by moving into a rotating reference frameat the frequency of the tweezers according to ˆ H → ˆ U ˆ H ˆ U † − i (cid:126) ˆ U ∂ ˆ U † ∂t , where ˆ U ( t ) ≡ exp (cid:0) iω t ˆ a † ˆ at (cid:1) . We apply another RWA by neglecting all rapidly oscillating terms at frequency 2 ω t .Gathering all these terms results in the general Hamiltonian for this system, presented in Equation (8).1 Appendix B: Interaction with free field
The free EM modes can be described by ˆ E free ( r ) = (cid:88) k , e (cid:15) k ( e k e i k · r ˆ c k , e + H.c. ) = (cid:88) l (cid:15) l ( e l e i k · r ˆ c l + H.c. ) , (B1)where (cid:15) l = (cid:113) (cid:126) ω l (cid:15) V f , V f stand for the quantization volume and ˆ c k , e is the annihilation operator of a free EM modewith wave-vector k and polarization (cid:15) k . To simplify the notation, the index l is used to denote the set { k , e k } .If we were to consider this additional EM field, the field Hamiltonian, in Equation (A2), and the interactionHamiltonian, in Equation (6), would transform intoˆ H (cid:48) field = ˆ H field + (cid:126) (cid:88) l ω l ˆ c † l ˆ c l , (B2)ˆ H (cid:48) int ,j = ˆ H int ,j − α j (cid:32) | E f (ˆ r j ) | + 2 Re (cid:8) E f (ˆ r j ) E ∗ t ,j (ˆ r j ) (cid:9) + 2 Re { E f (ˆ r j ) E ∗ c (ˆ r j ) } (cid:33) . (B3)Consequently, the extra interaction terms that would appear on the complete Hamiltonian areˆ H intf − f ,j = − α j | E free (ˆ r j ) | = − α j (cid:126) (cid:15) V f (cid:88) l (cid:88) l (cid:48) √ ω l ω l (cid:48) e l · e l (cid:48) (e i k · ˆ r j ˆ c l + H.c. )(e i k (cid:48) · ˆ r j ˆ c l (cid:48) + H.c. ) , (B4)It was shown that in the long-wavelength approximation ˆ H intf − f , j can be safely neglected [52]. Moreover, the cavity-freefields interaction is ˆ H intc − f ,j ≈ − (cid:126) (cid:88) l G cf ( l ) (cid:0) ˆ a † + ˆ a (cid:1) (e i k · R ,j ˆ c l + e − i k · R ,j ˆ c † l ) cos( k c x ,j ) , (B5)where G cf ( l ) = α j (cid:15) l (cid:15) c e l · e y / (cid:126) . Notice that this terms dies out when we place the mean position of NP at the cavitynodes. Lastly, the tweezers-free fields interaction is the source of recoil heating [28] of the NP as they incoherentlyscatters light off the tweezers into free space. These interactions are given byˆ H intt − f ,j ≈ − (cid:126) (cid:88) l G tf ( l ) (cid:34) (e i k · R ,j ˆ c l + e − i k · R ,j ˆ c † l ) (cid:18) cos( ω t t ) − ˆ z j k t z R ,j − z R ,j sin( ω t t ) (cid:19) + i k · ˆ R j (e i k · R ,j ˆ c l − e − i k · R ,j ˆ c † l ) cos( ω t t ) (cid:35) , (B6)where G tf ( l ) = α j (cid:15) l (cid:15) t e l · e y / (cid:126) . By moving this term into the rotating reference frame at the tweezers’ frequency ω t ,through the operator ˆ U f ( t ) ≡ exp (cid:0) iω t t (cid:80) l ˆ c † l ˆ c l (cid:1) , we would also introduce a shifted free-field frequency ∆ l = ω l − ω t and the field Hamiltonian and effective interaction with the free field would becomeˆ H (cid:48) field → ˆ H (cid:48) field + (cid:126) (cid:88) l ∆ l ˆ c † l ˆ c l , (B7)ˆ H intt − f ,j → − (cid:126) (cid:88) l G tf ( l ) (cid:34)(cid:18) ˆ c l e i k · R ,j +ˆ c † l e − i k · R ,j (cid:19) − i (cid:18) ˆ c † l e − i k · R ,j − ˆ c l e i k · R ,j (cid:19)(cid:18) k · ˆ R j + ˆ z j k t z R ,j − z R ,j (cid:19)(cid:35) . We observe a 3D coupling between the nanoparticles with the free electric field. For typical experimental values[6, 53], one usually has w ,j ≈ . µ m such that k t z R ,j − z R ,j ≈ k t (cid:39) . · − nm − , ω j ≈ π ·
305 kHz, and the mass of asingle nanoparticle m j ≈ .
83 fg such that x ZPF ,j ≈ . k t · x ZPF ,j ∼ k · x ZPF ,j (cid:28) T = 10 K, through the equipartition theorem we could assert that initially (cid:113) (cid:104) x j (cid:105) = (cid:113) k B T / ( m j ω j ) (cid:39) .
85 nm, suchthat we would still have k t · (cid:113) (cid:104) x j (cid:105) ∼ k · (cid:113) (cid:104) x j (cid:105) (cid:28) Appendix C: Gaussian Quantum Information Toolbox
A quantum system whose states live in a infinite-dimensional Hilbert space described by observables with continuousspectra is called a continuous variable systems [48]. The physical setup under study falls into such category: it iscomprised of M = N +1 modes, one optical field and N mechanical modes, each with their corresponding annihilationand creation operator. These can be conventionally arranged in a vectorial operator ˆ c = (ˆ a, ˆ a † , ˆ b , ˆ b † , ˆ b , ˆ b † , . . . ) T whichsatisfy the bosonic commutation relations expressed as [ˆ c j , ˆ c k ] = Ω jk and it immediately follows that the vectorialoperator ˆ X , defined in Section III, satisfy the commutation relations (cid:104) ˆ X j , ˆ X k (cid:105) = 2 i Ω jk , where j, k = 1 , . . . , M andΩ is the symplectic form 2 M × M matrix given byΩ = M (cid:79) k =1 (cid:20) − (cid:21) . (C1) Gaussian states
We are interested in a special class of states for such system called gaussian states , whose Wigner function isGaussian; here we refer the more interested reader once again to [48] for a more complete description of this topic.The crucial property of these states for us is the fact that a gaussian state with density matrix ρ can be completelycharacterized by its first, (cid:104) ˆ X (cid:105) = tr (cid:16) ˆ X ρ (cid:17) , and second moments represented by its covariance matrix, V i,j = (cid:104) ˆ X i ˆ X j +ˆ X j ˆ X i (cid:105) . This greatly simplifies our numerical treatment of these system, because instead of dealing with an infinite-dimensional density matrix, we need only to worry about a 2 M × M matrix. In the following, we present thetheoretical tools for gaussian states used in our simulations. Partial trace
Consider a density matrix ρ AB describing a multipartite gaussian state, which we choose to subdivide into thesubsystems A and B , respectively, with m and n modes. Let r = ( r A , r B ) be its first moments and V = (cid:20) V A V AB V TAB V B (cid:21) (C2)be its 2( m + n ) × m + n ) covariance matrix. Then, the reduced density matrix ρ A = tr B ( ρ AB ) describing solely thesubsystem A is also a gaussian state with first moments r A and covariance matrix V A . Logarithmic Negativity
The choice for employing the logarithmic negativity is due to the fact that it is an entanglement monotone andeasily computable for bipartite gaussian states, given their covariance matrix. We denote the LN between the j -thand k -th modes as E j,kN , j, k = cav , , , . . . , N and follow the prescription of [19] to evaluate it. First, we extract the4 × V jk from the total system covariance matrix V by taking its block matrices relative only to the modes j and k , which can be written in block form as V j,k = (cid:20) A CC T B (cid:21) , (C3)where A , B , C are 2 × V j,k = ( ⊗ σ z ) V jk ( ⊗ σ z ) associatedwith the partially transposed density matrix for the bipartition with the j -th and k -th mode [19]. For a bipartitesystem, this symplectic eigenvalue can be written directly by the form of V j,k given above through [54]˜ ν min = (cid:113) σ/ − (cid:112) σ − V ) / , (C4)where σ = det( A ) + det( B ) − C ), is the 2 × σ z = diag(1 , −
1) is a Pauli matrix.Finally, the LN is given by: E j,kN = max [0 , − log(˜ ν min )] . (C5) von Neumann entropy The von Neumann entropy S for a gaussian state with associated covariance matrix V is a function of its symplecticeigenvalues ν k : S = M (cid:88) k =1 g ( ν k ) , g ( x ) = x + 12 log (cid:18) x + 12 (cid:19) − x −
12 log (cid:18) x − (cid:19) (C6)where ν k , k = 1 . . . M , can be computed from modulus of the 2 M eigenvalues of i Ω V [48]. Mutual Information
The mutual information of the total system defined as I = M (cid:88) j =1 S j − S tot (C7)where S tot and S j respectively denote the von Neumann entropy of the total system and of the j -th single mode [55]. Squeezing degree
In order to quantify the mechanical squeezing generated by the CS interaction, we follow the procedure outlinedin [51]. First we perform a partial trace over the full system in order to arrive at the 2 × V j describing solely the j -th NP; secondly, we quantify the amount of squeezing by finding the variances of the squeezedand antisqueezed quadratures, respectively V sq and V asq : V sq = min(eig( V j )) V asq = max(eig( V j )) , (C8)where eig( V j ) denotes the eigenvalues of V j . The squeezing degree is, then, η ≡ V sq /V asq ≤ Numerical simulations
The numerical simulations for the time evolution and calculations of the theoretical tools for Gaussian statespresented above were performed with the developed numerical toolbox in MATLAB [38].
Appendix D: Decoherence mechanisms
In this article, we consider two major forms of decoherence/heating for the nanoparticles: thermal decoherence fromthe collisions with the environmental gas surrounding each NP and recoil heating as each NP incoherently scatterslight from its tweezer into free space.4Current experiments with CS interactions operate in moderate vacuum [6, 26, 53] achieving pressures ranging from10 − mbar [6] to 1 . j -th NP is linear in the gaspressure, p gas ,j , following [28, 56], γ j ≈ . R j p gas ,j m j v gas ,j (D1)where v gas ,j = (cid:113) k B T env ,j m gas is the root-mean-square velocity of a gas molecule with mass m gas . The associated thermaldecoherence rate from the collisions with the residual gas then becomes Γ gas ,j = γ j n th ,j [56].In regards to the recoil heating, we will only consider contributions arising from the scattering of photons from thetrapping tweezer of each particle. The reasoning for this simplification is twofold. First, as the cavity is not activelydriven, its field intensity is much smaller than each tweezer intensity [53], diminishing its contribution. Secondly, aswe are considering non-overlapping tweezers, the intensity of the j -th OT on the n -th NP, n (cid:54) = j , should be negligibleand, thus, disregarded. Therefore, the recoil heating rate on j -th NP reads [56]Γ recoil ,j = 15 P scatt ,j m j c ω t ω j (D2)where c is the speed of light and P scatt ,j = I ,j σ scatt ,j is the scattered power of the j -th NP from its tweezer; I ,j = 2 P t ,j / ( πw ,j ) is its intensity at xits NP’s mean position and σ scatt ,j = | α j | k / (6 π(cid:15) ) the scattering crosssection of with the NP.For simplicity, we may consider identical environments surrounding each NP, p gas ,j = p gas and T env ,j = T gas ∀ j , asthe residual gas and temperature should a priori be the same everywhere inside the vacuum chamber, resulting inΓ gas ,j = Γ gas ∀ j . Furthermore, for the sake of brevity, we can make Γ recoil ,j ≈ Γ recoil ∀ j if all the particles are of thesame material and size, and the tweezers have approximately the same intensity at their focus.Ref. [6] observed Γ gas = 2 π × . recoil = 2 π × R = 100 nm whichresults in a greater scattering cross section for the tweezer’s photons and a inversion of the dominant decoherencemechanism as now recoil heating Γ recoil = 2 π × . gas = 2 π × .
49 kHz. Oneshould note, though, that a bigger recoil heating rate still does not signify we must take into account interaction withthe free field as the total expected coherence time τ = 1 / (cid:0) Γ recoil + Γ gas (cid:1) (cid:39) . µµ