Mesoscopic entanglement through central potential interactions
MMesoscopic entanglement through central-potentialinteractions
Sofia Qvarfort , , Sougato Bose , Alessio Serafini QOLS, Blackett Laboratory, Imperial College London, SW7 2AZ London, UnitedKingdom Department of Physics and Astronomy, University College London, Gower Street, WC1E6BT London, United KingdomE-mail: [email protected]
Abstract.
The generation and detection of entanglement between mesoscopic systemswould have major fundamental and applicative implications. In this work, we demonstratethe utility of continuous variable tools to evaluate the Gaussian entanglement arisingbetween two homogeneous levitated nanobeads interacting through a central potential. Wecompute the entanglement for the steady state and determine the measurement precisionrequired to detect the entanglement in the laboratory.
1. Introduction
The mastery over levitated optomechanical systems in the laboratory is becomingincreasingly refined. With advances in cooling a small levitated sphere to the ground-state [1, 2, 3, 4, 5] and in the preparation of squeezed states [6], compounded by theability to control and implant charges into levitated nanobeads [7], setups are reachingunprecedented levels of control. Furthermore, optomechanical systems have shown significantpotential for sensing applications [8, 9], especially with regards to measuring gravitationalparameters [10, 11, 12]. As a result, the engineering of schemes to entangle multiple levitatedoscillators and, more broadly, mesoscopic systems, is being identified as a major medium-term milestone of the field [13]. In fact, the entanglement for a number of mesoscopic systemshas already been demonstrated experimentally [14, 15, 16], although a more thorough andexhaustive understanding of the conditions under which entanglement may be generated isrequired to move on to applications.A key property of entanglement is that it acts as an unambiguous hallmark of non-classicality. As a result, entanglement between mesoscopic systems can aid the quest ofmapping the transition from the quantum to the classical scale [17, 18]. Furthermore,recent proposals concerning the fundamental nature of gravity have considered quantumentanglement as generated by a Newtonian potential between two massive quantumsystems [19, 20]. The Newtonian potential can be seen as an effective potential that a r X i v : . [ qu a n t - ph ] S e p esoscopic entanglement through central-potential interactions Two nanobeads in a laser trap suspended a distance r apart. The beads are allowed tointeract via a central potential, such as the Coulomb potential or the Newtonian potential. arises form an underlying fully quantum field theory. Successfully detecting gravitationalentanglement would indeed be a strong indication of the quantisation of gravity [21] and, ingeneral, the detection of entanglement as generated by gravity has significant ramifications,in the attempt to feed theories of quantum gravity with new empirical evidence at lowenergies [22].The experimental setup envisioned in [19] is challenging since it requires the generationof highly localised spatial superpositions with large separation for two levitated systems.Gaussian states, on the other hand, are more well-understood, and they can be straight-forwardly prepared in the laboratory. As such, we wish to explore whether the questionsposed in [19] can be tested with Gaussian states only. Furthermore, to detect entanglementfrom the Newtonian potential, it is reasonable to first consider an analogous case with astronger potential, such as the Coulomb potential.Our main question in this work is therefore: Is it possible to detect entanglementbetween two mesosopic systems as generated by a central potential interaction with onlyGaussian resources? To address this question, we consider a fundamental or effectivecentral potential of the form 1 /r n , for integer n and where r = | r − r | with positionvectors r and r acting between two spherically symmetric quantum systems. Crucially,we ask whether minimal initial state preparation, such as squeezing, as opposed to severerequirements of preparing highly non-Gaussian states (as, for example, proposed in [19]) isenough to generate detectable entanglement, and whether the witnessing of the generatedentanglement is possible simply by measuring position–momentum correlations. We thenquantify the leading-order contribution to entanglement within the continuous-variable (CV)framework for both the dynamical generation of entanglement from initial squeezed statesof the interacting oscillators and for the system’s steady-state in the presence of noise. esoscopic entanglement through central-potential interactions
2. Dynamics
We begin by considering two spheres trapped next to each other as per Figure 1. The non-interacting system Hamiltonian ˆ H that describes the harmonic motion for both spheres inthe local trap is given byˆ H = 12 m ω ˆ x + 12 m ω ˆ x + ˆ p m + ˆ p m , (1)where m and m are the system masses, ω and ω are the respective mechanical trappingfrequencies for each sphere, and ˆ x i and ˆ p i with i = 1 , α/ | r − r | n , where α is thecoupling constant and r and r are position vectors. In all cases considered here, thispotential is the lowest-order approximation to a description that includes the full field theory.We proceed to derive the interaction from the generic central potential to second order. Thequadratic terms will capture the largest contribution to the central-potential entanglementand retain the quadratic interaction, which maps input Gaussian states to output Gaussianstates. The last effect allows us to model the system completely within the covariance matrixformalism [23].To derive the Hamiltonian interaction term, we assume that the movement of the spheresis constrained in all but the x -direction, which is the axis along which the two systemsare trapped. We consider small perturbations to r and r , such that r → r − x and r → r − x , with x (cid:28) r and x (cid:28) r . By denoting r = r − r and ∆ x = x − x , weTaylor-expand the interaction to second order in ∆ x to find:1( r − ∆ x ) n = 1 r n + n ∆ xr n +1 + n ( n + 1)2 (∆ x ) r n +2 + . . . , (2)where we have left out the dimensional prefactor α and ignored all terms of order O [(∆ x ) ].We then quantise the positions of the two masses around their equilibrium positions, whichare taken to be a distance r apart from each other, by promoting the position coordinatesto operators: x i → ˆ x i . For gravity, this step incorporates the assumption that gravity is aquantum force that can path-entangle two quantum systems [19].To arrive at the entangling interaction term, we discard the constant term, which isthe first term on the left-hand side in Eq. (2), as its only contribution is a static energyshift. Secondly, since a displacement term in a quadratic Hamiltonian does not affect theentanglement [23], we can also discard the term linear in position, which is the second termon the left-hand side in Eq. (2). The final term, however, contains a mixing of the positionoperators in the form ˆ x ˆ x , which will generate entanglement between the two states. Theinteraction term in the Hamiltonian thus becomesˆ H I = α n ( n + 1)2 r n +2 (ˆ x − ˆ x ) . (3) esoscopic entanglement through central-potential interactions H = 12 m ω ˆ x + 12 m ω ˆ x + ˆ p m + ˆ p m + α n ( n + 1)2 r n +2 (ˆ x − ˆ x ) . (4)Since ˆ H is quadratic in the canonical operators, all initial Gaussian states exclusively evolveinto Gaussian states. Furthermore, all Gaussian states are uniquely defined by their firstand second moments, which allows us to model this system completely within the covariancematrix framework [23]. We introduce the 4 × σ , which consistsof all second moments of the Gaussian state ˆ ρ G ( t ). It is defined as σ ( t ) = Tr (cid:2) { ˆ r , ˆ r T } ˆ ρ G ( t ) (cid:3) , (5)where the vector of operators is given by ˆ r = (ˆ x , ˆ p , ˆ x , ˆ p ) T , and where the bracket {· , ·} denotes the symmetrised outer product, in the sense that { ˆ r , ˆ r T } = ˆ r ˆ r T + ( ˆ r ˆ r T ) T [23].The evolution of the second moments under the Hamiltonian in Eq. (4) can be encodedthrough the symplectic matrix S = e Ω H t/ (cid:126) , where H is the Hamiltonian matrix, defined asˆ H = 12 ˆ r T H ˆ r + ˆ r T r , (6)and where Ω is the symplectic form defined in this basis asΩ = n (cid:77) j =1 ω , with ω = (cid:32) − (cid:33) , (7)for a total of n modes. For a bipartite system like the one considered here, n = 2, whichmeans that Ω, σ , H , and S are all 4 × σ then evolves as σ ( t ) = S σ S T , where σ encodes the second moments of the initial state.To determine the Hamiltonian matrix H that arises from Eq. (4), we define thedimensionless operators ˆ x (cid:48) i and ˆ p (cid:48) i as ˆ x i = (cid:112) (cid:126) / ( m i ω i ) ˆ x (cid:48) i and ˆ p i = √ (cid:126) m i ω i ˆ p (cid:48) i . For notationalsimplicity, we then assume that m = m = m , and ω = ω = ω m . The Hamiltonian inEq. (4) therefore becomesˆ H = (cid:126) ω m (cid:0) ˆ x (cid:48) + ˆ p (cid:48) (cid:1) + (cid:126) ω m (cid:0) ˆ x (cid:48) + ˆ p (cid:48) (cid:1) + α (cid:126) ω m m n ( n + 1)2 r n +2 (ˆ x (cid:48) − ˆ x (cid:48) ) . (8)In what follows, we will rescale the laboratory–time t by ω m to obtain the dimensionless timeparameter τ = ω m t . We later consider open-system dynamics where κ denotes a mechanicaldecoherence rate. Here, we also rescale κ to ˜ κ = κ/ω m .This yields the Hamiltonian matrix ˜ H = H/ ( (cid:126) ω m ):˜ H = H + ˜ α H (1) I − ˜ α H (2) I , (9)where for convenience we have defined the dimensionless coupling˜ α = α n ( n + 1) ω m r n +2 , (10) esoscopic entanglement through central-potential interactions H = is the 4 × H ( i ) I denotes the Hamiltonian matrices responsible for the interaction. They are given by H (1) I = , H (2) I = , (11)where specifically H (2) I will generate the entanglement. We note that H (2) I is of the form oftwo-mode squeezing, which implies that the corresponding closed system will not displayperiodic behaviour. With these rescaled quantities, the evolution is now encoded as S = e Ω ˜
H τ .
3. Computing the entanglement
To compute the entanglement that arises from the central-potential interaction, we make useof the logarithmic negativity [24, 25, 26], a well-known monotone that quantifies the degreeof violation of the positive-partial-transpose (PPT) criterion [27, 28]. The latter is inspiredby the by the fact that, given a separable state ˆ ρ = ˆ ρ A ⊗ ˆ ρ B , the partial transpose withrespect to one of the subsystems leaves the state with positive eigenvalues: ˆ ρ Tp ≥
0. Hence,should we find that ˆ ρ Tp <
0, the state is entangled. Notice that this criterion turns out tobe necessary and sufficient for Gaussian states [29, 30].The PPT criterion in the CV framework can be explicitly computed by dividing σ intosubmatrices σ A , σ B and σ AB as such [31]: σ = (cid:32) σ A σ AB σ AB σ B (cid:33) . (12)We define the symplectic invariant quantity ∆ = det σ A + det σ B + 2 det σ AB . In this basis,the partial transpose is equivalent to setting ˆ p i → − ˆ p i for one subsystem, which implies∆ → ˜∆ = det σ A + det σ B − σ AB . The positive partial transpose (PPT) criterion fortwo-mode Gaussian states can thus be compactly expressed as det σ − ˜∆ + 1 ≥ E N of the state,defined by E N ( σ ) = max (0 , − log ˜ ν − ), where ˜ ν ∓ are the symplectic eigenvalues of the state,defined for bipartite systems as ˜ ν ∓ = ˜∆ ∓ (cid:112) ˜∆ − σ . (13)In this work, we consider a two-mode mechanically squeezed state σ S = diag( z, z − , z − , z )as the initial state. Note that the squeezing occurs in the opposite quadrature, which esoscopic entanglement through central-potential interactions α = - α = - α = - α = - π π π π τ E N (a) z = = =
10 z = π π π π τ E N (b) Figure 2:
Entanglement from central-potential interactions. (a)
Plot of the logarithmic negativity E N as a function of time τ for squeezing z = 1 and different values of the rescaled coupling ˜ α . (b) Plot of E N as a function of time τ for different squeezing parameters z and ˜ α = − .
4. Theoscillations are due to the rotating terms contained in the ˆ x ˆ x interaction (note however that E N never goes fully to zero). serves to increase the entanglement from this particular interaction. When z = 1, thestate is a coherent state with σ = diag(1 , , , σ S , including controlling the trap frequency or using a Duffingnon-linearity [32]. In addition, thermal optomechanical squeezing has been experimentallyrealised in [6].For the system to be thermally stable (i.e. for the Hamiltonian to be bounded frombelow), we require that the rescaled Hamiltonian matrix ˜ H in Eq. (9) satisfies ˜ H >
0, whichmeans that the eigenvalues of ˜ H must be positive [23]. For ˜ H in Eq. (9), we find that theeigenvalues λ i for i = 1 , , , λ = λ = λ = 1, and λ = 1 + 2 ˜ α . Thismeans that for a positive (repulsive) coupling ˜ α , there is no limit to the interaction strengththat will unbound the Hamiltonian. We are therefore able to obtain considerable amountsof entanglement for a repulsive potential. When ˜ α is negative (attractive), however, themaximum strength of the interaction is limited to ˜ α > − /
2. Since all three potentialsconsidered in this work can be attractive (with ˜ α < α ∼ − .
4, which is close to the limiting value.We now proceed to compute the entanglement E N . While an analytic expression for E N is available, it is too long and cumbersome to reproduce here. Instead, we plot theentanglement E N for different values of ˜ α in Fig. 2. In Fig. 2a we plot E N for coherent stateswith z = 1 as a function of time τ for various values of ˜ α . The stronger the coupling, themore entanglement is generated. Next, in Fig. 2b we plot the same general dynamics for˜ α = − . z . We set ˜ α = − . α = − .
5. Increasing z serves to amplify the already-present entanglement. esoscopic entanglement through central-potential interactions κ = κ = κ = κ = τ E N (a) z = = = = π π π π τ E N (b) Figure 3:
Open system entanglement from central-potential interactions. (a)
Plot of E N as afunction of time τ for increasingly noisy systems with decoherence rate ˜ κ at ˜ α = − . N th .( b ) Plot of E N as a function of time τ for different values of squeezing z with decoherence rate˜ κ = 0 .
05 at ˜ α = − . N th = 0.
4. Open system dynamics
All systems are subject to environmental noise that generally degrades the entanglementpresent in the system. In this work, we consider two types of noise: damping of the oscillatormotion in terms of phonon decay, which we denote ˜ κ = κ/ω m , and the number of thermalphonons N th present in the system.For Markovian dynamics, the covariance matrix σ ( τ ) evolves as˙ σ = A ( τ ) σ + σ A T ( τ ) + D, (14)where A ( τ ) = Ω H ( τ ) − ˜ κ I / H defined in Eq. (6), and D = (2 N th + 1) ˜ κ I , with ˜ κ = κ/ω m being the rescaled phonondissipation rate, N th the number of thermal phonons present in the system and I the 4 × τ as the systemsdecoheres. We plot the effects of noise on E N in Fig. 3. In Fig. 3a, we have plotted E N as afunction of time τ for a noisy environment for different ˜ κ at ˜ α = − . z = 1 and N th = 0.Similarly, in Fig. 3b, we have plotted E N as a function of rescaled time τ for differentsqueezing values z at ˜ α = − . N th = 0. We note that while increasing the squeezing z causes E N to increase at first, higher squeezing rates also makes the system more sensitiveto noise, a fact that finds extensive confirmation in the existing literature [33, 34, 35]. esoscopic entanglement through central-potential interactions After long times τ (cid:29) A σ ( ∞ ) + σ ( ∞ ) A T + D = 0to find the steady state σ ( ∞ ) , which is defined by the property ˙ σ ( ∞ ) = 0. We find thefollowing elements of σ ( ∞ ) : σ ( ∞ )11 = 6 ˜ α + ˜ κ + 48 ˜ α + ˜ κ + 4 (2 N th + 1) ,σ ( ∞ )22 = 4 ˜ α + 10 ˜ α + ˜ κ + 48 ˜ α + ˜ κ + 4 (2 N th + 1) ,σ ( ∞ )12 = σ ( ∞ )34 = − ˜ α ˜ κ α + ˜ κ + 4 (2 N th + 1) ,σ ( ∞ )14 = σ ( ∞ )23 = ˜ α ˜ κ α + ˜ κ + 4 (2 N th + 1) ,σ ( ∞ )13 = 2 ˜ α α + ˜ κ + 4 (2 N th + 1) ,σ ( ∞ )24 = − α (2 ˜ α + 1)8 ˜ α + ˜ κ + 4 (2 N th + 1) . (15)All other elements follow by symmetry from the fact that ( σ ( ∞ ) ) T = σ ( ∞ ) , and σ ( ∞ )11 = σ ( ∞ )33 .Furthermore, σ ( ∞ )22 = σ ( ∞ )44 , due to the symmetry between the two systems. We find thefollowing quantities ˜∆ ( ∞ ) = 2 4 ˜ α + 8 ˜ α + ˜ κ + 48 ˜ α + ˜ κ + 4 (2 N th + 1) , det σ ( ∞ ) = 4 ˜ α + 8 ˜ α + ˜ κ + 48 ˜ α + ˜ κ + 4 (2 N th + 1) , (16)from which it follows that(˜ ν ( ∞ ) − ) = (2 N th + 1) (cid:16) Λ − (cid:112) (Λ − (cid:17) , (17)where Λ = (4 ˜ α + 8 ˜ α + ˜ κ + 4) / (8 ˜ α + ˜ κ + 4). We note that the number of phonons N th inthe system has a strongly detrimental effect on the entanglement. For E N to be maximal,we require that ν − is small. However, since ˜ ν − ∝ N th , the entanglement decreases as N th increases. The decoherence rate ˜ κ , on the other hand, is not as influential as the phononnumber. As ˜ κ → ∞ , we find that ˜ ν − = | N th | , while as N th → ∞ , we find ν ( ∞ ) − → ∞ .We therefore conclude that reducing the number of phonons in the system takes priorityover reducing the decoherence rate. Consequently, the product N th ˜ κ which is often quotedin experimental contexts is not as enlightening as an indicator of overall noise levels here, asthe two quantities contribute differently to the entanglement.We further explore the entanglement in the steady state by plotting the logarithmicnegativity E ( ∞ ) N for noisy dynamics in Fig. 4a for a range of interaction strengths ˜ α ∈ ( − . , . κ increases, the logarithmic negativity E ( ∞ ) N decreases. The esoscopic entanglement through central-potential interactions (a) (b) Figure 4:
Steady state entanglement. (a)
Density plot of the steady state entanglement E ( ∞ ) N as afunction of ˜ α and ˜ κ at N th = 0. The slight asymmetry is a feature of the interaction. (b) Densityplot of the steady state entanglement E ( ∞ ) N as a function of N th and ˜ κ at ˜ α = − . stronger the coupling, the more resilient the system is to noise. Furthermore, all elements inEq. (15) are proportional to the term (2 N th + 1). To see how this affects the entanglement,we return to the PPT criterion. In Fig. 4b, we plot E ( ∞ ) N as a function of N th and ˜ κ . Thelow values of N th shown on the x -axis give an indication of how sensitive the system is tothe number of thermal phonons in the system.
5. Discussion
We have shown that entanglement from a central potential can be modelled to leading orderwith Gaussian states in the continuous variable framework. It remains to determine whethersuch entanglement can be detected in the laboratory.
To what precision must the entries in the covariance matrix σ be determined in order toverify the detection of entanglement? To estimate the required precision, we consider theasymptotic entanglement of the steady state σ ( ∞ ) . We then assume that each element σ ( ∞ ) ij in the covariance matrix σ ( ∞ ) can be determined to within a specific precision (cid:15) ij , which wetake to be a percentage of the total value of σ ( ∞ ) ij , such that σ ( ∞ )11 → σ ( ∞ )11 (1 + (cid:15) ). We alsoassume that the errors respect the symmetry of σ (so that, for example, (cid:15) = (cid:15) ). From esoscopic entanglement through central-potential interactions Relative errors for detecting steady state entanglement. Plot of the relative error∆ E N = δE ( ∞ ) N /E ( ∞ ) N as a function of the coupling ˜ α and the precision (cid:15) at ˜ κ = 1 and N th = 0. Fora weak coupling | ˜ α | (cid:28)
1, high precision is required to detect the entanglement. The asymmetry isdue to the higher values of E ( ∞ ) N achieved from a negative coupling. The plot does not show valuesfor which the relative error is larger than unity. this assumption, we compute the error δE ( ∞ ) N via the standard error propagation formula: δE ( ∞ ) N = (cid:118)(cid:117)(cid:117)(cid:116)(cid:88) k (cid:32) ∂E ( ∞ ) N ∂(cid:15) k (cid:15) k (cid:33) , (18)where in our case k ∈ (11 , , , , , , , , , (cid:15) ij ≡ (cid:15) for all i, j . Wethen plot the relative error ∆ E N = δE ( ∞ ) N /E ( ∞ ) N in Figure 5. From the plots, it becomesevident that detecting logarithmic negativity from small couplings | ˜ α | (cid:28) (cid:15) . We now specialise to the following potentials: the Coulomb potential and the Newtonianpotential. Given our definition of ˜ α in Eq. (10), the couplings become, respectively:˜ α Cl = q q π(cid:15) r mω , ˜ α Nw = − Gmr ω , (19)where q and q are the charges on the bipartite system (where the signs will determinewhether the potential is attractive or repulsive), (cid:15) is the vacuum permittivity, and G isNewton’s constant. esoscopic entanglement through central-potential interactions α ∼ − . We consider two optomechanical spheres, each with a single butopposite charge q = − q = e , where e is the electron charge, which results in an attractivepotential with ˜ α <
0. The number of charges on each sphere can be controlled to exquisiteprecision by using ultraviolet light [36]. Linearised Coulomb interactions are commonlyconsidered in trapped ions, and have also led to the generation of entanglement [37] but havenot yet been implemented for optomechanical systems, although a protocol to enhance cavity-mechanical entanglement through additional Coulomb interactions was proposed in [38].With the parameters in Table 1, it is possible to achieve a coupling ˜ α Cl ≈ − . α Cl ∝ m − , whichmeans that entanglement due to the Coulomb potential is suppressed for systems with alarger mass. The number of thermal phonons in the system must still be low at N th = 10 − to allow for detection of entanglement, but the decoherence damping rate can be large at˜ κ = 1 as it does not affect the system as much. If we require the error of the entanglementto be no more than 7%, we require that the quantities be determined to within 1% of theirvalue. Coulomb potentialParameter Symbol Value
Mechanical frequency ω m
100 Rad s − Charge q , q . × − COscillator mass m − kgSeparation r − mCoupling strength ˜ α Cl − . κ N th − Precision (cid:15) − Steady state entanglement E ( ∞ ) N . ± . esoscopic entanglement through central-potential interactions We first compute ˜ α for the parameters suggested in [19] tosee whether gravitational entanglement can be detected with only Gaussian states. With m = 10 − kg, r = 200 × − m and ω m = 10 Hz, we find ˜ α Nw = − . × − . Sucha weak coupling will not yield any detectable entanglement within this scheme. Even ifthese figures were more lenient, there is the added complication that the Casimir–Polderinteraction will typically dominate over gravity for most parameter choices. Nevertheless,we compute ˜ α Nw from the optimistic parameters found in Table 2. For these values, westill only find ˜ α Nw = − . × − . For entanglement to be detectable at this interactionstrength, we require that as few as 10 − thermal phonons are in the system, but we find aslightly more forgiving ˜ κ = 0 .
1. Finally, we require an extremely high precision of (cid:15) = 10 − (which we recall is the percentage of the covariance matrix elements). Newtonian potentialParameter Symbol Value
Mechanical frequency ω m
10 Rad s − Oscillator mass m − kgSeparation r − mCoupling strength ˜ α Nw − . × − Phonon decoherence ˜ κ N th − Precision (cid:15) − Steady state entanglement E ( ∞ ) N (3 . ± . × − Table 2: Values used to compute the entanglement from a Newtonian potential.For these values, we obtain an extremely small logarithmic negativity of E ( ∞ ) N =(3 . ± . × − , where the error stands at 22%. Given these numbers, we concludethat gravitational entanglement is not likely to be detectable in the near term with the useof only Gaussian resources. A crucial step in the experimental generation of entanglement is the detection stage, wherethe state is measured to verify the entanglement. Such a detection scheme has, for example,been proposed in the context of a pulsed optomechanical setup [39]. Here, the light–matterinteraction is confined to an optical pulse of a timescale much shorter than that of themechanical oscillation period [40], which means that it can be treated as a single unitaryoperator that entangles the light and mechanics. By then performing the appropriatemeasurements on the light, the mechanical state can be inferred by the pulse. In [39], a esoscopic entanglement through central-potential interactions
6. Conclusions
In this work, we computed the leading-order entanglement due to a generic central-potentialinteraction between two levitated nanobeads. We derived the Hamiltonian matrix for alinearised potential and investigated the entanglement arising between two initially squeezedstates given unitary and noisy dynamics. Furthermore, we derived an analytic expression forthe steady state of the system in the noisy setting and proposed a simple continuous-variabletest for detecting the entanglement. With these tools, we computed the entanglement froman attractive Coulomb potential and the Newtonian potential. By considering errors thatoccur when determining the covariance matrix elements, we determined the measurementprecision required in each scenario.Most importantly, we emphasise that this particular setup will not suffice for thedetection of entanglement due to gravity. Our results suggest that the inclusion of non-Gaussian resources may play a significant role here (they may, for example, explain theviability of the scheme with freely-falling systems envisioned in Ref [19], which relies on thecreation of highly non-Gaussian initial states). To prepare two trapped mesoscopic systemsin non-Gaussian states, one may utilise the nonlinear optomechanical interaction [41], whichcouples the photon number to the position of the mechanical element through the addition ofa cavity [42]. Specific schemes for generating mechanical cat-states through this interactionhave already been proposed by means of the nonlinear optomechanical evolution [43, 44], asimple optical interferometry setup [45] or within the pulsed optomechanical regime [46].We further limited our investigation to Markovian environmental noise in this work,however non-Markovian noise-sources are expected to affect optomechanical resonators atlow temperatures [47]. It is therefore imperative to investigate the effect of non-Markoviannoise on entanglement in this setting, which can be done with a non-Markovian masterequation [48]. Finally, to mitigate the small entanglement rates, one may also considerthe addition of feedback techniques, which by themselves cannot generate entanglement,but which can serve to amplify already present interactions. These questions, and furtherscenarios that take into account the noise from the trapping laser will be considered in futurework.After this work was first completed, the authors became aware of similar work byKrisanda et al . [49]. esoscopic entanglement through central-potential interactions Acknowledgments
We thank Sahar Sahebdivan, Alexander D. Plato, Marko Toroˇs, Anja Metelmann, GavinMorley, Dennis R¨atzel, Nathana¨el Bullier, Peter F. Barker, Alessio Belenchia, and MarkusAspelmeyer for fruitful discussions. SQ is supported by an EPSRC Doctoral Prize Fellowship.
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