Unconditional Wigner-negative mechanical entanglement with linear-and-quadratic optomechanical interactions

The generation of entangled states that display negative values of the Wigner function in the quantum phase space is a challenging task, particularly elusive for massive, and possibly macroscopic, systems such as mechanical resonators. In this work, we propose two schemes based on reservoir engineering for generating Wigner-negative entangled states unconditionally. We consider two non-interacting mechanical resonators that are radiation-pressure coupled to either one or two common cavity fields; the optomechanical coupling with the field(s) features both a linear and quadratic part in the mechanical displacement and the cavity is driven at multiple frequencies. We show analytically that both schemes stabilize a Wigner-negative entangled state that combines the entanglement of a two-mode squeezed vacuum with a cubic nonlinearity, which we dub cubic-phase entangled (CPE) state. We then perform extensive numerical simulations to test the robustness of Wigner-negative entanglement attained by approximate CPE states stabilized in the presence of thermal decoherence.


I. INTRODUCTION
The theoretical identification and experimental verification of the specific traits that characterize quantum mechanics is fundamental to refine our description of physical systems.Two pivotal features that mark the departure of the quantum model of physical reality with respect to its classical counterpart are quantum entanglement and contextuality.
Entangled quantum systems [1] display correlations that cannot be established by merely communicating classical information between non-interacting but otherwise locally controllable quantum systems.In fact, from an applicative viewpoint, entanglement underpins the majority of quantum communication protocols, for which it represents a rigorously quantifiable resource [2][3][4].
Quantum contextuality [5] seeks to capture the nonexistence of a classical probability distribution able to describe the outcomes of the measurements on a quantum system.More specifically, it is defined as the non-existence of classical models reproducing quantum mechanical measurement outcomes from predetermined assignments to observables, independently of the specific measurement context.Contextuality has been identified as an essential ingredient for quantum advantage in various tasks, including universal quantum computing for certain models [6].For continuous-variable quantum systems [7][8][9], it has been recently shown [10,11] that the notion of quantum contextuality, with respect to generalised position and momentum measurements, is equivalent to the presence of negative values of the Wigner function [12] associated to the system at hand when described in the quantum phase space, or Wigner negativity for short.
Hereafter we will refer to the simultaneous presence of these two hallmarks of quantum mechanics as Wignernegative entanglement.The generation and verification of Wigner-negative entangled states in a macroscopic system (involving a large number of constituents, such as photons, atoms, or ions) represents a remarkable task, especially elusive if the system components are massive, such as for mechanical resonators.Optomechanical systems [13] represent an especially promising candidate to tackle this challenge.At the single-phonon level, the generation of a Wigner-negative entangled state of two mechanical resonators has been demonstrated, both probabilistically [14] and deterministically [15].It has further been the subject of a number of theoretical studies [16], mainly pointing towards conditional schemes [17][18][19][20][21][22] or transient dynamics [23][24][25].However, the intrinsic single-phonon nature of such settings make them fragile to environmental noise and call for more robust demonstrations.In particular, unconditional Wigner-negative entanglement involving many phonons of massive systems is desirable.
In this work we present a scheme to stabilize Wignernegative entanglement over continuous quantum degrees of freedom of two massive mechanical modes.Our scheme does not require engineering strong mechanical non-linearity or direct interactions between resonators, but instead exploits higher-than-linear optomechanical interactions between each mechanical mode and a common auxiliary field.In particular, it relies on a tunable optomechanical-like coupling featuring both a linear and a quadratic part in the mechanical position [26].We present two variants of our scheme, where the mechanical resonators interact with either a single mode or two independent modes, as sketched in Fig. 1 (a) and (b).The main advantage of such protocols is that Wigner-negative entanglement is generated unconditionally at the steady state of a driven-dissipative dynamics.We first introduce the ideal target state stabilized by our schemes, which we dub cubicphase entangled (CPE) state, and characterize it analytically.We then perform extensive numerical simulations to test the arXiv:2302.03702v1[quant-ph] 7 Feb 2023 < l a t e x i t s h a 1 _ b a s e 6 4 = " u w Q P 4 a b t W 9 Z L S O k s n r i b G j X 2 W l 4 = " > A A A B 9 X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B C 8 G H a D q H g K e P E Y w T w g 2 Y T Z y S Q Z M j u 7 z P Q q Y c l / e P G g i F f / x Z t / 4 y T Z g y Y W N B R V 3 X R 3 B b E U B l 3 3 2 1 l Z X V v f 2 M x t 5 b d 3 d v f 2 C w e H d R M l m v E a i 2 S k m w E 1 X A r F a y h Q 8 m a s O Q 0 D y R v B 6 H b q N x 6 5 N i J S D z i O u R / S g R J 9 w S h a q d N G E X J D P L e T n p c n 3 U L R L b k z k G X i Z a Q I G a r d w l e 7 F 7 E k 5 A q Z p M a 0 P D d G P 6 U a B Z N 8 k m 8 n h s e U j e i A t y x V 1 C 7 z 0 9 n V E 3 J q l R 7 p R 9 q W Q j J T f 0 + k N D R m H A a 2 M 6 Q 4 N I v e V P z P a y X Y v / Z T o e I E u W L z R f 1 E E o z I N A L S E 5 o z l G N L K N P C 3 k r Y k G r K 0 A a V t y F 4 i y 8 v k 3 q 5 5 F 2 W v P u L Y u U m i y M H x 3 A C Z + D B F V T g D q p Q A w Y a n u E V 3 p w n 5 8 V 5 d z 7 m r S t O N n M E f + B 8 / g B R 4 Z G 4 < / l a t e x i t > ⇥10 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " z A F 7 6 q h I E N t g K G 1 O Y p R F K w w T V u u u 5 i f E z q g x n A q e l X q o x o W x M h 9 i q m j E j Z / N 7 5 2 S M 6 v 0 S R h r W w r J X P 0 9 k d H I m E k U 2 M 6 I 4 t A s e z P x P 6 + T Y n j j Z 0 I l K X L F F o v C V B K M y e x 5 0 h e a M 5 Q T S y j T w t 5 K 2 J B q y t B G V L I h e M s v r 5 L m R d W 7 q n o P l 5 X a b R 5 H E U 7 g F M 7 B g 2 u o w T 3 U o Q E M J D z D K 7 w 5 Y + f F e X c + F q 0 F J 5 8 5 h j 9 w P n 8 A R / K P c w = = < / l a t e x i t > b1 < l a t e x i t s h a 1 _ b a s e 6 4 = " s L k L B a W a N 7 x / d d R 4 X t e L V W k A r b Y = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m K q H g q e P F Y w X 5 A G 8 p m u 2 m X b j Z x d y K U 0 D / h x Y M i X v 0 7 < l a t e x i t s h a 1 _ b a s e 6 4 = " N Y r 6 n j g w R y o X a G L f e 9 a / X n L a e g 8 = " > A q m j E j Z / N 7 5 2 S M 6 v 0 S R h r W w r J X P 0 9 k d H I m E k U 2 M 6 I 4 t A s e z P x P 6 + T Y n j j Z 0 I l K X L F F o v C V B K M y e x 5 0 h e a M 5 Q T S y j T w t 5 K 2 J B q y t B G V L I h e M s v r 5 L m R d W 7 q n o P l 5 X a b R 5 H E U 7 g F M 7 B g 2 u o w T 3 U o Q E M J D z D K 7 w 5 Y + f F e X c + F q 0 F J 5 8 5 h j 9 w P n 8 A R / K P c w = = < / l a t e x i t > b1 < l a t e x i t s h a 1 _ b a s e 6 4 = " t u / c J l l N A b j q D j T n D u l X 6 A H H Y I 4 = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m K q H g q e P F Y w M s l s d + t j l q y i p k q + A P r w f G P Z Q y < / l a t e x i t > / q2 j < l a t e x i t s h a 1 _ b a s e 6 4 = " T A K c N S 7 d 9   (q 1 + q 2 )/ p 2 FIG.1: (a) Sketch of an optomechanical system made of two non-interacting mechanical oscillators, with modes b1 and b2 , coupled to a common cavity mode â.Both modes feature a coupling term proportional to the position q j (solid black line) while only one mode, here b1 , features also a quadratic coupling proportional to q2 1 (dotted black line).The cavity, which is driven by a multi-tone drive and dissipates into the vacuum at rate κ, provides an engineered reservoir that steers the mechanical modes toward the desired Wigner-negative entangled steady state.(b) Sketch of a similar system (same graphical notation applies) where the two non-interacting mechanical oscillators are instead coupled to two cavity modes â1 and â2 .In the absence of thermal environments, both (a) and (b) stabilize the cubic-phase entangled (CPE) state in Eq. ( 3), while the presence of thermal environments with occupation n th,1 and n th,2 determine a partial loss of fidelity (see main text).The configuration (a) -which we call Hamiltonian-switching scheme -requires two consecutive steps to stabilize the (approximate) target CPE state, while configuration (b) -which we call two-dissipator scheme -exploits an additional cavity mode to achieve the target state in a single step.(c-d) Plots of the Wigner distribution of the ideal CPE state [see Appendix C for the full analytical expression] in terms of the EPR-like variables (q 1 + q 2 )/ √ 2 and (p 1 − p 2 )/ √ 2, for fixed value of (q 1 − q 2 ) = 0 and, respectively, (p 1 + p 2 ) = 0 (c) and (p 1 + p 2 ) = √ 2 (d).Other parameters are γ = 0.3, s 1 = 1/s 2 = 2 (≈ 6 dB squeezing).The dashed circle represents the extent of vacuum fluctuations for each of the two cuts.attainability of Wigner-negative entanglement by approximate CPE states stabilized by a realistic protocol in the presence of thermal decoherence.
For continuous position and momentum quantum variables, Gaussian [27][28][29] mechanical entangled states have been unconditionally generated via reservoir engineering in microwave optomechanics [30,31], following a theory proposal to stabilise entanglement by means of a single engineered reservoir [32].A direct observation of two-mode entanglement was also reported for two mechanical oscillators driven by a microwave cavity [33].The schemes we consider in this work are also based on reservoir engineering [34] but, crucially, we gain access to the Wigner-negative regime by exploiting higher-than-linear optomechanical interactions.Notice that, in contrast with the proposal for unconditional generation put forward in Ref. [35], we do not require the strong single-photon optomechanical regime.We build in fact on recently proposed schemes for the generation of singleoscillator Wigner-negative states [26,36] and non-Gaussian gates for quantum computation [37].
The remainder of this manuscript is organized as follows.In Sec.II, we introduce the model for the system under investigation.In Sec.III, we first define our target state, the CPE state, and discuss its key characteristics.We then present two generation schemes in Sec.III A and Sec.III B, respectively: the Hamiltonian-switching scheme and the two-dissipator scheme.In Sec.III C, we describe the numerical methods employed to obtain our results.In Sec.IV, we first establish the key figures of merit used to characterize our results, and then present them considering both the ideal and noisy case scenarios.The experimental feasibility of these schemes in various platforms is discussed in Sec.V. Finally, we conclude with a summary of our findings and future directions in Sec.VI.The appendixes collect further details on the derivation of the system Hamiltonian (App.A), the transformation of the fields operators leading to the CPE state (App.B) and its Wigner function (App.C), the effective dynamics (App.D) used for our numerical simulations and some details on the latter (App.E).

II. SYSTEM MODEL AND DYNAMICS
We consider an optomechanical system featuring two noninteracting mechanical modes, with annihilation operator b1 and b2 , radiation-pressure coupled to either one or two orthogonal cavity modes, respectively denoted as â or â1,2 , see Fig. 1 (a) and (b).We retain both the linear and the quadratic contributions to the optomechanical coupling, i.e., terms proportional to both q j = 1 √ 2 ( b j + b † j ) and q2 j , j = 1, 2, and assume a multi-tone drive for the cavity mode(s).Under the appropriate conditions (see Appendix.A for details and approximations) we can perform standard linearization and cast the Hamiltonian of the system in the following form ( = 1) We note that, following the linearization, â and b j describe the cavity and mechanical fluctuations around their respective steady-state values, and the complex parameters g ( j) ,k are the drive-enhanced optomechanical couplings.The above expression of the Hamiltonian is valid when the system parameters are set in a regime where the mechanical modes are individually addressed, which requires non-overlapping mechanical frequencies.Additionally, the linearization with respect to the cavity modes requires strong driving fields but within a weak coupling regime (see Appendix.A for more details).The implementation of the most general Hamiltonian shown in Eq. ( 1) necessitates each cavity (a ) to be driven at both red-and blue-detuned mechanical sidebands and second mechanical sidebands, in order to arbitrarily set all the effective couplings (g (1)  ,1 . . .g (1)  ,5 and g (2)  ,1 . . .g (2)  ,5 ).This results in a total of ten (twenty) transition frequencies to be driven, for a protocol utilizing one (two) cavity mode(s).However, for the cases we will be interested in, the number of drives required may be significantly reduced, as we will discuss in the subsequent sections.
In order to address realistic settings, besides considering that the cavity modes are coupled to vacuum reservoirs, we impose that each mechanical oscillator is in contact with a thermal bath.The dynamics of the open system therefore obeys a master equation of the following form with κ and γ j are respectively the cavity and mechanical dissipation rates, n th, j are the thermal phonon numbers for mechanical mode j, and D[•] is the superoperator defined for an operator ĉ and density operator ρ by In Sec.V we will comment with some details on the experimental feasibility of the proposed model.Here, let us only stress that the second-order terms in the operators b j appearing in Hamiltonian (1) are the key ingredients of the proposed model, since they allow to access Wigner-negative entanglement.In this sense, they are also a minimal extension to the reservoir-engineering schemes implemented experimentally in Refs.[30,31].

III. UNCONDITIONAL PREPARATION OF THE CUBIC-PHASE ENTANGLED STATE
Our goal is to generate a robust non-Gaussian entangled state of the two mechanical modes, capable of retaining some Wigner negativity even for moderate thermal noise.We begin our analysis by introducing the ideal target state stabilized by our protocol in the absence of any noise termi.e., when the only dissipation channel is provided by the engineered reservoir -and illustrating its properties.In this limit, the mechanical steady state is pure and the requirement of Wignernegative entanglement reduces to the state being entangled and non-Gaussian [38].
As already mentioned, we call CPE state the ideal target state of our scheme.The CPE state is a two-mode pure non-Gaussian state obtained by coupling two squeezed vacuum modes via a beam-splitter interaction, and then a cubic-phase gate is applied to one of the modes.We denote the CPE state by where |00 is the vacuum state of the two mechanical modes, Ŝ (s 1 , s 2 ) = Ŝ 1 (s 1 ) ⊗ Ŝ 2 (s 2 ) is the product of two single-mode squeezing operators with j = 1, 2, corresponding to an amount of 20 log 10 s j decibel (dB) of squeezing, BBS is the beam-splitter operator and Λ(λ) is the cubic-phase gate with cubicity parameter λ acting on mode b1 , which is given by The cubic phase gate is a well-known resource in continuousvariable quantum computation [39], where it enables the simulation of any Hamiltonian with arbitrary precision [40].Its action on a single-mode momentum squeezed state results in the so-called cubic phase state, which can be used to generate a non-Gaussian cluster state, e.g. to be employed for universal measurement-based quantum computation [28,41].
In the remaining of the paper, we set the beam-splitter coupling to θ = π 4 , the squeezing parameters to s 1 = 1 s 2 ≡ s and denote the target state by s, 1 s , λ, π

4
≡ |s, λ .Since BBS (π/4) Ŝ (s, s −1 ) |00 acting on the vacuum returns a twomode squeezed vacuum state, the CPE state can be seen as a minimal non-Gaussian extension of a two-mode squeezed vacuum; minimal in the sense that a single local operator is responsible for the non-Gaussian nature of the state.We derived the analytical expression of the the Wigner distribution W(q 1 , p 1 ; q 2 , p 2 ) of the CPE state |s, λ (the full expression and the details of the derivation are reported in Appendix.C).In Fig. 1 (c) and (d) we show two cuts of the Wigner distribution as a function of two EPR-like mechanical variables, namely the mean position (q 1 + q 2 )/ √ 2 and the relative momentum (p 1 − p 2 )/ √ 2, the latter being the eigenvalues of the momentum operator p j = i √ 2 ( b † j − b j ).From the plots we can appreciate the two distinctive features of Wigner-negative entanglement.First, most of the quasi-probability density is concentrated in a region smaller than the zero point fluctuation (marked by the dashed circle).The joint reduction of the variances of both EPR-like variables below that of the vacuum marks the presence of entanglement, as e.g.expressed quantitatively by the Duan criterion [42].Second, regions of phase space with negative Wigner density are clearly visible.
From a theoretical point of view (irrespective of the specific implementation at hand) the CPE state represents, to the best of our knowledge, a new instance of non-Gaussian entangled states of continuous variables, to be contrasted e.g. with two-mode extensions of Schrodinger's cat states (also known as entangled coherent states), which have been the subject of several theoretical [43][44][45][46] and experimental [47] works.
In the following we propose two methods to generate the CPE state.In one approach we consider two cavity modes, while in the other one we consider only one cavity mode.We also notice that, although we focus on the least demanding case of local non-linearity acting on mode b1 , our analysis can be readily extended to the case of two local cubic phase gates.

A. Single dissipator approach: Hamiltonian switching
This approach, sketched in Fig. 1 (a), involves only one cavity mode but requires two consecutive steps to generate the target CPE state.It represents an extension of the so-called Hamiltonian-switching scheme [48,49] to a non-Gaussian target state.In the following, all subscripts corresponding to the cavity mode are dropped, i.e., a 1 ≡ a, κ 1 ≡ κ and g ( j) 1,k ≡ g ( j) k .The Hamiltonian (1), with one cavity mode, may be put in the form Consider now the transformation of the mechanical modes f j = Û b j Û † induced by the target unitary in Eq. ( 3).The explicit expression of these transformed modes, which we will refer to as engineered modes in the following, is given by: where we set s ± = 1 2 √ 2 s ± 1 s .A full derivation of Eq. ( 8) is provided in App.B, from which it is also clear that the vacuum state corresponding to the field operators f1 and f2 (namely, the simultaneous ground state of both operators) is the CPE state.
If we now set the coupling parameters appearing in Hamiltonian (7) to match the transformation f1 , we obtain the fol-lowing relations where g is a positive real parameter playing the role of a unit for all couplings; notice that the quadratic optomechanical coupling is only present between the cavity and mode b1 .The Hamiltonian then becomes Similarly, by suitably choosing the couplings to implement the transformation f2 , the Hamiltonian can be put in the form Therefore, in this scheme the target CPE state |s, λ is obtained in two steps: in step (i) we set coupling parameters such that Hamiltonian H 1 is implemented.Including photon losses but neglecting for the moment any form of mechanical noise, the system evolves according to the master equation Crucially, this dynamics leads to a factorized steady state where both the cavity mode â and the engineered mode f1 reach their respective vacuum states.In other words, the combined action of cavity dissipation and excitation swapping results in an entropy sink for the engineered mode f1 , similarly to the dissipation engineering schemes developed in Refs.[26,32,36].
After the system reaches the steady state, we implement step (ii) by switching the coupling parameters (via switching the driving fields) to obtain Hamiltonian Ĥ2 .The master equation becomes Again, the steady state is a factorized state, now of the vacua for modes â and f2 .Therefore, the steady state of the mechanical modes after the second step is the simultaneous vacuum of modes f1 and f2 [49] which, by construction, is our target CPE state.An advantage of this approach is that it less onerous in terms of the number of frequencies that need to be driven; it follows from Eqs. (9-12) that seven tones are required in total at any instant of time: five to address the mechanical oscillator featuring both the linear and the quadratic coupling term (resonator 1), and two for the other oscillator.
We stress that the above result is valid when the two mechanical oscillators are decoupled from their respective baths, i.e. there is no mechanical noise.Any interaction between mechanical modes and their environment will negatively affect the target state.An extensive numerical validation of our results in the presence of mechanical noise will be dealt with in more detail in Sec.III D.

B. Two-dissipator approach
In contrast to the switching approach given above, where two steps are needed to produce the target state, here the CPE state is reached in one step only, but we need two cavity modes.We set the coupling parameters in Hamiltonian (1) such that it is written as with f1 and f2 are defined in Eq. ( 8), and g is a unit for the couplings, as explained previously.Again, when neglecting thermal noise, the system's state evolves according to the master equation For the same reasons outlined in Sec.III A, the target CPE state is obtained in the long-time limit, when both the engineered modes f1 and f2 reach their common ground state.Again, numerical validations of this result in the presence of mechanical noise will be given in Sec.III D. Compared to the switching scheme discussed earlier, the two-dissipator approach requires a larger number of driving fields since there are two cavity modes coupled to the mechanical oscillators.For each cavity mode seven transitions need to be driven (as before, five fields for the linearly-andquadratically coupled oscillator, two fields for the linearly coupled one), and hence fourteen drives are needed in total to implement the two-dissipator approach.

C. Effective dynamics
The numerical simulations required to study the system in the presence of mechanical noise are computationally expensive, given the relatively large Hilbert space needed to accurately describe the full system.To address this issue, we employ the adiabatic elimination technique outlined in Ref. [50] to eliminate the cavity mode(s).The system under consideration exhibits two distinct time scales: one for the rapidly varying cavity modes and another for the slower mechanical modes.By adiabatically eliminating the cavity modes, we obtain a simplified set of dynamical equations that depend solely on the mechanical modes.We refer to this simplified dynamics as the effective dynamics.It is worth noting that our state generation methods do not require this step of adiabatic elimination.However, it proves beneficial in reducing the computational cost and enabling the numerical study of states with higher non-linearity and squeezing, as well as states with higher degree of entanglement and Wigner negativity.
Starting with the Hamiltonian-switching scheme, Eqs. ( 15) and ( 16) -after adiabatically eliminating the cavity mode â (see Appendix.D for details) -respectively lead to the effective dynamics described with the master equation for step j ( j = 1, 2), where ρ b is the density matrix that describes the quantum state of the mechanical system alone.
Here the effective decay rate κ 0 is given in terms of the original cavity and mechanical decay rates as κ 0 = 4g 2 κ , where g and κ appear in Hamiltonians (13)(14) and master equations (15)(16).
Similarly, in the two-dissipator approach we adiabatically eliminate the two cavity modes with annihilation operators â1 and â2 from the dynamics described with Hamiltonian (17) and master equation (18) and we obtain an effective master equation of the form (again, see Appendix.D for details) where again κ 0 = 4g 2 κ 1,2 is the new effective decay rate with g and κ ( = 1, 2) are given in Hamiltonian (17) and master equation (18).In the following, we will consider the case κ 1 = κ 2 .Notice that Eq. ( 20) clarifies the reason why we referred to this approach as the two-dissipator method: in fact, two collective dissipative channels are obtained via the adiabatic elimination procedure.
Including standard mechanical noise for the two massive oscillators with dissipation rates γ j and thermal phonon numbers n th, j , the effective master equations above become for the Hamiltonian-switching scheme, and for the two-dissipator approach.
The following numerical simulations will be generated from these effective master equations, although our methods are still valid for the evolutions of the full system.

D. Numerical results in the presence of mechanical noise
Here we numerically analyze the effects of mechanical noise on the results derived above.For the sake of providing a relevant example, in the following we set the non-linearity of the target state to λ = 0.175 and the squeezing to s = 1.26 (2 dB).We numerically confirmed that the results presented below remain valid for other values of the CPE state parameters.We also set the evolution time per step to t f = 10 κ , enough to reach the steady state with good accuracy.
In the following we will use the fidelity [51] as a figure of merit to assess the similarity between the state obtained dynamically with our two schemes and the target CPE state.The explicit expression of the fidelity between the mechanical subsystem's steady state ρ ss and the target state ρ s,λ ≡ |s, λ s, λ| is given by: The closer the fidelity is to one the more similar the states are.
Notice that F will denote the final fidelity, namely the fidelity between the target state and the system's steady state at second step for the case of Hamiltonian-switching scheme and at the end of the single step for the case of two-dissipator approach.
We plot the final fidelity as a function of the mechanical decay rates γ 1 and γ 2 for different temperatures (given by average thermal phonon numbers n th,1 and n th,2 ).As said, for the sake of simplicity, we consider identical bath for both mechanical oscillators by setting γ 1 = γ 2 = γ and n th,1 = n th,2 = n th .The cases of Hamiltonian-switching and two-dissipator approach are shown in Fig. 2 (a) and (b), respectively.As expected, the system evolves to the target state in both approaches when it is completely decoupled from the thermal bath (i.e.γ = 0).The detrimental effects of mechanical noise become apparent for larger values of γ, even if the decay is mainly linear in γ.In particular, we note that the twodissipator case is more robust than the Hamiltonian-switching scheme to the effect of noise.This is related to the fact that the time needed to the system to reach the steady state in the Hamiltonian-switching approach is longer (in our simulation, double) than what needed for the two-dissipator scheme.Therefore, the detrimental effects of thermal mechanical noise act for a longer time causing, in turn, a larger loss of fidelity.
In Fig. 2, as well as in the rest of the figures in this work, we have included a trend line for each set of numerical results to provide a visual guide.These lines are obtained by fitting the numerical results with the function y = Ae −Bx + C, where the parameters A, B, and C are determined through least-square minimization.The sole purpose of these trend lines is to aid in visualizing the data.
The purity of the steady state is a second key metric for evaluating in general the robustness of generation schemes against mechanical noise.As confirmed in Fig. 3, when γ = 0, the steady state is a pure CPE state.However, as mechanical thermal noise is introduced, the purity of the state is diminished, even at relatively low values of γ and n th .Despite this, the key properties of the CPE state that we are interested in are not significantly affected, as we will demonstrate in the next section.

IV. NON-GAUSSIAN ENTANGLEMENT
We will now concentrate on the two most important characteristics of the CPE state that we are interested in, which are entanglement and Wigner-negativity.
In particular, we evaluate the entanglement of the generated states using the logarithmic negativity of entanglement [52].For a given bipartite quantum state ρ, composed of two subsystems A and B, the latter is defined as FIG.2: Effect of mechanical thermal noise on the final fidelity for Hamiltonian-switching scheme (a) and the twodissipator scheme (b).The target state is set to λ = 0.175 and 20 log 10 s j = 2 dB of squeezing.Both mechanical oscillators are subject to a thermal environment with dissipation rate γ and thermal phonon number n th .The time evolution is terminated at 10 κ 0 for each step of the Hamiltonian-switching scheme and for the full evolution in the two-dissipator case.We set n th = [0, 1, 2, 3] in the Hamiltonian-switching scheme and n th = [0, 5,10,15] in the two-dissipator scheme.The mechanical dissipation γ κ 0 ranges between 0 and 0.01 for both methods.
Here Γ A is the partial transpose with respect to sub-system A, and • 1 denotes the trace norm.In our case, system A refers to oscillator b For what concerns the Wigner negativity, let us first recall that, while for pure states non-Gaussianity is equivalent to Wigner-negativity, this is no longer the case for mixed states.In fact, Wigner-negativity is a more stringent requirement [53].In order to confirm the presence of negative values of the Wigner function, we report here the value of the most negative point of the Wigner cross-section corresponding to the first mode's momentum axis [specifically, W min (ρ) = min p 1 W(0, p 1 , 0, 0)].We choose the latter since it aligns with the most negative point of the entire Wigner function of the produced state, as it can be intuitively understood from the known properties of the single-mode cubic-phase state.
Clearly, these two figures of merit E N (ρ) and W min (ρ) are sufficient to confirm Wigner-negative entanglement even in the presence of noise.For illustration purposes, we once again set as target-state parameters λ = 0.175 and 2 dB of squeezing.Due to the two-dissipator scheme being more robust to noise compared to switching scheme, as shown in Sec.III D, the noise parameters will be set differently in each case.
Our results are reported in Figs.4(a) and 5(a), for what concerns the Hamiltonian-switching scheme, and Figs.4(b) and 5(b) for the two-dissipator approach.Crucially, one can observe that both schemes retain Wigner-negative entanglement in realistic conditions, even in regimes where the purity of the state is low (see Fig. 3).Notice that, also with respect to these figures of merit, it is clear that the two-dissipator scheme is more robust to noise than the Hamiltonian-switching scheme.As already noted, this is due to the longer time needed to the system to reach the steady state in the Hamiltonian-switching approach.Notwithstanding this, experimental considerations might still induce to prefer it with respect to the two-dissipator scheme, since the latter is more demanding, in that it requires to control an additional cavity mode.Let us mention here that, in our numerical simulations, we have initialized the mechanical system in the vacuum state.This assumption has no practical consequences for the case of the two-dissipator scheme, as the latter possesses a unique steady state, independently of the initial state.However, that is not true anymore for the Hamiltonian-switching scheme, where small deviations from the vacuum initial state will result in a slightly different steady state with respect to what we report here.In this case, our assumption therefore requires in practice a precooling stage, that cools the initial state of a All the parameters are set as in Fig. 2.
realistic setting to a state close to the vacuum.This point is addressed in more details in Appendix.E.

V. EXPERIMENTAL FEASIBILITY
The implementation of our ideas requires the ability to engineer an optomechanical coupling with both a linear and a quadratic term in the mechanical displacement and the ability to couple a cavity field to two mechanical modes.
Quadratic coupling in optomechanics has been intensively investigated, especially in connection with quantum nondemolition measurement of the phonon number.For this purpose, optomechanical systems in the so-called membranein-the-middle configuration are among the most promising candidates [54].There, an enhanced quadratic coupling can arise due to the avoided crossing between two coupled optical modes, each linearly coupled to the mechanical mode [55].Experimental demonstrations of quadratic coupling have been reported in membrane-in-the-middle related configurations, featuring actual dielectric membrane resonators [54,56,57], cold atoms [58], levitated nano-particles [59] and photonic crystal cavities [60,61].Recently, the enhanced nonlinear optomechanical measurement of mechanical motion has been demonstrated in a photonic crystal device [62].Other systems where quadratic coupling can be engineered are microdisk resonators [63] and paddle nano cavities [64], where quadratic coupling emerges due to shared symmetries between the optics and mechanical motion [65].
We stress that the requirements for implementing our scheme are less stringent than those imposed by the still elusive regime of nonlinear phonon number measurement.Indeed, the latter requires a purely quadratic coupling and is corrupted by parasitic linear coupling, which are known to arise in membrane-in-the-middle setups, while in our case the presence of a spurious linear coupling is not detrimental.For a detailed discussion on how to engineer tunable linear-andquadratic optomechanical interactions in a photonic crystal cavity, based on a double slotted geometry [60,61], we refer to Refs.[26,36].
Radiation-pressure coupling several mechanical resonances to a common cavity mode is the main goal of multimode optomechanics.Focusing on the generation of mechanical entanglement via reservoir engineering related schemes [32,66], the preparation and verification of entanglement between the centre-of-mass motion of two micromechanical oscillators via engineered dissipation has been recently demonstrated in microwave optomechanics [30].A direct observation of twomode entanglement was also reported for two mechanical oscillators driven by a microwave cavity [33].
This shows that two main ingredients of our proposal are attainable with current optomechanical technology.Of course, a detailed account on how to combine these ingredients and a quantitative assessment of linear and quadratic terms and the role of non-idealities and limitations, requires to single out a specific platform.This goes beyond the scope of the present study, and will be addressed in future works.

VI. CONCLUSIONS
In this work, we presented two methods for stabilizing entangled states of two massive oscillators, which exhibit negative values in their phase-space representation as captured by the Wigner function.The non-linearity required for this purpose is of the minimal possible order (third order in the canonical operators) and it is effectively achieved through a drivendissipative dynamics between the mechanical oscillators of interest and auxiliary cavity modes.Furthermore, this approach intrinsically endows the generated states with robustness against common sources of noise, as we demonstrated through numerical simulations.
The cubic-phase entangled (CPE) state stabilized by our schemes can be utilized in various quantum information processing tasks.In particular, incorporating CPE states into a Gaussian cluster state allows for quantum computational universality through gate teleportation [67,68].More generally, there has been significant interest in accessing the genuine quantum regime in massive systems for various applications [13] -such as quantum non-demolition detection and sensing -as well as for testing foundational principles of quantum theory -such as collapse models [69] and macrorealism [70,71].The ability to access the Wigner-negative regime, as proposed in this work, could further enhance the applicability of these systems in such contexts.In fact, it is known that non-Gaussian states can have larger metrological power with respect to their Gaussian counterpart [72][73][74] and are espe-cially sensitive to decoherence effects [75].Possible applications of our findings along these directions are left for future research.
4 Z 6 W L d O b 3 u u s = " > A A A B 6 H i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 K k k p 6 r H g x W M L 9 g P a U D b b S b t 2 s w m 7 G 6 G E / g I v H h T x 6 k / y 5 r 9 x 2 + a g r Q 8 G H u / N M D M v S A T X x n W / n Y 3 N r e 2 d 3 c J e c f / g 8O i 4 d H L a 1 n G q G L Z Y L G L V D a h G w S W 2 D D c C u 4 l C G g U C O 8 H k b u 5 3 n l B p H s s H M 0 3 Q j + h I 8 p A z a q z U r A 5 K Z b f i L k D W i Z e T M u R o D E p f / W H M 0 g i l Y Y J q 3 f P c x P g Z V Y Y z g b N i P 9 W Y U D a h I + x Z K m m E 2 s 8 W h 8 7 I p V W G J I y V L W n I Q v 0 9 k d F I 6 2 k U 2 M 6 I m r F e 9 e b i f 1 4 v N e G t n 3 G Z p A Y l W y 4 K U 0 F M T O Z f k y F X y I y Y W k K Z 4 v Z W w s Z U U W Z s N k U b gr f 6 8 j p p V y v e d c V r 1 s r 1 W h 5 H A c 7 h A q 7 A g x u o w z 0 0 o A U M E J 7 h F d 6 c R + f F e X c + l q 0 b T j 5 z B n / g f P 4 A e e O M r g = = < / l a t e x i t > 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " J 7 D 7 w 7 T 8 6 r 8 7 F q L D n F x B n 8 g f P 5 A x K Z i 4 Y = < / l a t e x i t > 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " B v E f V K 9 P G e T W b n E o + p c 7 5 S H P r c w = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E q s e C F 4 8 t 2 A 9 o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g o b m 1 v b O 8 X d 0 t 7 + w e A H e 6 h A S 1 g g P A M r / D m P D o v z r v z s W w t O P n M K f y B 8 / k D f / O M s g = = < / l a t e x i t > 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " T y 9 u K x r L u e E i x 1 h p s T Q q 8 r x X + 1 M = " > A A A B 6 X i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9L B b B i y W R U j 0 W v H i s Y m 2 h D W W z n b R L N 5 u w u x F K 6 D / w 4 k E Rr / 4 j b / 4 b t 2 0 O 2 v p g 4 P H e D D P z g k R w b V z 3 2 y m s r W 9 s b h W 3 S z u 7 e / s H 5 c O j R x 2 n i m G L x S J W n Y B q F F x i y 3 A j s J M o p F E g s B 2 M b 2 Z + + w m V 5 r F 8 M J M E / Y g O J Q 8 5 o 8 Z K 9 x f 1 f r n i V t 0 5 y C r x c l K B H M 1 + + a s 3 i F k a o T R M U K 2 7 n p s Y P 6 P K c C Z w W u q l G h P K x n S I X U s l j V D 7 2 f z S K T m z y o C E s b I l D Z m r v y c y G m k 9 i Q L b G V E z 0 s v e T P z P 6 6 Y m v P Y z L p P U o G S L R W E q i I n J 7 G 0 y 4 A q Z E R N L K F P c 3 k r Y i C r K j A 2 n Z E P w l l 9 e J Y + X V a 9 e 9 e 5 q l U Y t j 6 M I J 3 A K 5 + D B F T T g F p r Q A g Y h P M M r v D l j 5 8 V 5 d z 4 W r Q U n n z m G P 3 A + f w D p O I z p < / l a t e x i t > 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " q u w o z 1 c i F I Y C H f 3 7 0 M B O 6 E s q u 5 y b G z 6 g y n A m c l n q p x o S y M R 1 i 1 1 J J I 9 R + N r 9 0 S s 6 s M i B h r G x J Q + b q 7 4 m M R l p P o s B 2 R t S M 9 L I 3 E / / z u q k J b / y M y y Q 1 K N l i U Z g K Y m I y e 5 s M u E J m x M Q S y h S 3 t x I 2 o o o y Y 8 M p 2 R C 8 5 Z d X S e u y 6 l 1 V v f t a p V 7 L 4 y j C C Z z C O X h w D X W 4 g w Y 0 g U E I z / A K b 8 7 Y e X H e n Y 9 F a 8 H J Z 4 7 h D 5 z P H + Y w j O c = < / l a t e x i t > 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " z r y 4 J I A K P K F T P J n z o c J w J P h I 2 L Y = " > A A A B 6 X i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S J 4 s S S l q M e C F 4 9 V 7 A e 0 o W y 2 m 3 b p Z h N 2 J 0 I J / Q d e P C j i 1 X / k z X / j t s 1 B W x 8 M P N 6 b Y y e 5 s M h O Y M 5 c Q S y r S w t x I 2 o p o y t O E U b Q j e 8 s u r p F W t e F c V 7 7 5 W r t f y O A p w C m d w A R 5 c Q x 3 u o A F N Y B D C M 7 z C m z N 2 X p x 3 5 2 P R u u b k M y f w B 8 7 n D + M o j O U = < / l a t e x i t > 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " Z w W p + u / E t l 2 j k v e l S 6 I W s a b m 7 m k = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 t 2 A 9 o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g o b m 1 v b O 8 X d 0 t 7 + w e F R + f i k r e N U M W y D K 7 w 5 j 8 6 L 8 + 5 8 L F s L T j 5 z C n / g f P 4 A d t u M r A = = < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " B Y t X W G N z n F S E 8 F / 4 / o 3 r A G B 6 T E 8 = " > A A A B 6 n i c b c n p j S P 5 Y O Z J M y P c C h 5 y C k a K 9 1 X 8 b x f r r g 1 d w 6 y S r y c V C B H s 1 / + 6 g 1 i m k Z M G i p Q 6 6 7 n J s b P U B l O B Z u W e q l m C d I x D l n X U o k R 0 3 4 2 P 3 V K z q w y I G G s b E l D 5 u r v i Q w j r S d R Y D s j N C O 9 7 M 3 E / 7 x u a s I r P H N E c 6 L 8 + 5 8 L F o L T j 5 z D H / g f P 4 A h 5 u N R g = = < / l a t e x i t > (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " w D P A M Y D h s s K s 9 p J m J f 3 S M A W S h n E = " > A A A B 6 n i c b 1 V N I I t Z / N T 5 2 S M 6 s M S B g r W 9 K Q u f p 7 I q O R 1 p M o s J 0 R N S O 9 7 M 3 E / 7 x u a s I r P + M y S Q 1 K t l g U p o K Y m M z + J g O u k B k x s Y Q y x e 2 t h I 2 o o s z Y d E o 2 B G / 5 5 V X S r t e 8 i 5 p 3 V 6 8 0 r v M 4 i n A C p 1 A F D y 6 h A b f Q h B Y w G M I z v M K b I 5 w X 5 9 3 5 W L Q W n H z m G P 7 A + f w B i S C N R w = = < / l a t e x i t > (b) â b < l a t e x i t s h a 1 _ b a s e 6 4 = " u U r U k V f A 5 E / K y b E 7 g g 8 Q O 4 y i j 6 8 = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E V D w V v H i s Y D + g D W W z 3 b R L N 5 t 0 d y K U 0 D / h x Y M i X v 0 7 3 v w 3 b t s c t P X B w O O 9 G W b m B Y k U B l 3 3 2 y m s r W 9 s b h W 3 S z u 7 e / s H 5 c O j p o l T z 3 v w 3 b t s c t P X B w O O 9 G W b m B Y k U B l 3 3 2 y m s r W 9 s b h W 3 S z u 7 e / s H 5 c O j l o l T z r H r 3 F 5 X 6 T R 5 H E U 7 g F M 7 B g y u o w x 0 0 o A k M J D z D K 7 w 5 j 8 6 L 8 + 5 8 L F o L T j 5 z D H / g f P 4 A S X a P d A = = < / l a t e x i t > b2 < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 x a / I W r 2 N Q U I c 1 R g X P k w m 0 p Y 3 M E = " > A A A B 7 X i c b V D L S g N B E O z 1 G e M r 6 t H L Y B A 8 h V 0 R F U 8 B L x 4 j m A c k S 5 i d z C Z j Z m e W m V 4 h h P y D F w + K e P V / v P k 3 T p I 9 a G B D e r A 4 B G e 4 R X e P O 2 9 e O / e x 7 x 1 x c t n j u A P v M 8 f H a y O z g = = < / l a t e x i t > â < l a t e x i t s h a 1 _ b a s e 6 4 = " h J 2 P u + 8 e M c h D s v z Y n w B p o r H r 3 F 5 X 6 T R 5 H E U 7 g F M 7 B g y u o w x 0 0 o A k M J D z D K 7 w 5 j 8 6 L 8 + 5 8 L F o L T j 5 z D H / g f P 4 A S X a P d A = = < / l a t e x i t > b2 < l a t e x i t s h a 1 _ b a s e 6 4 = " u U r U k V f A 5 E / K y b E 7 g g 8 Q O 4 y i j 6 8 = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E V D w V v H i s Y D + g D W W z 3 b R L N 5 t 0 d y K U 0 D / h x Y M i X v 0 7 3 v w 3 b t s c t P X B w O O 9 G W b m B Y k U B l 3 3 2 y m s r W 9 s b h W 3 S z u 7 e / s H 5 c O j p o l T z 6 X p u g n 5 G N Q o m + b T U S w 1 P K B v T I e 9 a q m j E j Z / N 7 5 2 S M 6 s M S B h r W w r J X P 0 9 k d H I m E k U 2 M 6 I 4 s g s e z P x P 6 + b Y n j t Z 0 I l K X L F F o v C V B K M y e x 5 M h C a M 5 Q T S y j T w t 5 K 2 I h q y t B G V L I h e M s v r 5 J W r e p d V r 3 7 i 0 r 9 J o + j C C d w C u f g w R X U 4 Q 4 a 0 A Q G E p 7 h F d 6 c R + f F e X c + F q 0 F J 5 8 5 h j 9 w P n 8 A R / C P c w = = < / l a t e x i t > â2 < l a t e x i t s h a 1 _ b a s e 6 4 = " B j g u u R o y G t L C a s x u F L + 4 V i t 7 L b Q = " > A A A B 7 3 i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e y K q H g K e P E Y w c R A s o T e y W w y Z P b h T K 8 Q Q n 7 C i w d F v P o 7 3 v w b J 8 k e N L G g o a j q p r s r S J U 0 5 L r f T m F l d W 1 9 o 7 h Z 2 t r e 2 d 0 r 7 x 8 0 T H S w C x 6 U / E / r 5 1 R e O W P Z Z x m J G I + X x R m i l H C p s + z n t S C k x p Z g l x L e y v j A 9 T I y U Z U s i F 4 i y 8 v k + Z Z 1 b u o e n f n l d p 1 H k c R j u A Y T s G D S 6 j B L d S h A R w U P M M r v D m P z o v z 7 n z M W w t O P n M I f + B 8 / g B G b I 9 y < / l a t e x i t > â1 < l a t e x i t s h a 1 _ b a s e 6 4 = " o y a 8 O d W L / 9 m y 1 R M p r 5 d l k P h 4 q e c = 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a 6 C S K x p g M c Z + 2 D R U 4 p M p L 5 5 d P 0 Z l R u q g X S V N C o 7 n 6 e y L F o V K T M D C d I d Y D t e z N x P + 8 d q J 7 t 1 7 K R J x o K s h i U S / h S E d o F g P q M k m J 5 h N D M J H M 3 I r I A E t M t A m r Y E J w l 1 9 e J Y 1 K 2 b 0 u u 4 9 X x e p d F k c e T u A U S u D C D V T h A W p Q B w J j e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w C S 5 J J P < / l a t e x i t > (q 1 + q 2 )/ p l a t e x i t s h a 1 _ b a s e 6 4 = " n M 6 Z d 1 h R 2 q H j / + 2 B E s 5 e X 7 a 4 w i w = " > A A A C C 3 i c b V D L S s N A F J 3 4 r P E V d e l m a C l U x J o U U X F V c O O y g n 1 A G 8 J k O m m H T p L p z E Q o o X s 3 / o o b F 4 q 4 9 Q f c + T d O 2 4 D a e u D C 4 Z x 7 u f c e n z M q l W 1 / G U v L K 6 t r 6 7 k N c 3 N r e 2 f X 2 t t v y D g R m N R x z G L R 8 p E k j E a k r q h i p M U F Q a H P S N M f X E / 8 5 j 0 R k s b R n R p x 4 o a o F 9 G A Y q S 0 5 F n 5 Y m n o O c d D r 3 J 0 2 p F D o S p m i X v O C f 8 R P K t g l + 0 p 4 C J x M l I A G W q e 9 d n p x j g J S a Q w Q 1 K 2 H Z s r N 0 V C U c z I 2 O w k k n C E B 6 h H 2 p p G K C T S T a e / j G F R K 1 0 Y x E J X p O B U / T 2 R o l D K U e j r z h C p v p z 3 J u J / X j t R w a W b 0 o g

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r F C 9 y u L I g U O Q B y X g g A t Q B T e g B u o A g w f w B F 7 A q / F o P B t v x v u s d c n I Z g 7 A H x g f 3 5 N 1 m D g = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " z N Y f Y P o Y + l 1 U C H s u B Q X q M p + m i + g = " > A A A B + X i c b V B N S 8 N A E J 3 U r 1 q / o h 6 9 L B a h I t S k i I q n g h e P F e w H t C F s t t t 2 6 W a T 7 m 4 K J f S f e P G g i F f / i T f / j d s 2 B 2 1 9 M P B 4 b 4 a Z e U H M m d K O 8 2 3 l 1 t Y 3 N r f y 2 4 W d 3 b 3 9 A / v w q K a 6 C S K x p g M c Z + 2 D R U 4 p M p L 5 5 d P 0 Z l R u q g X S V N C o 7 n 6 e y L F o V K T M D C d I d Y D t e z N x P + 8 d q J 7 t 1 7 K R J x o K s h i U S / h S E d o F g P q M k m J 5 h N D M J H M 3 I r I A E t M t A m r Y E J w l 1 9 e J Y 1 K 2 b 0 u u 4 9 X x e p d F k c e T u A U S u D C D V T h A W p Q B w J j e I Z X e L N S 6 8 V 6 t z 4 W r T k r m z m G P 7 A + f w C S 5 J J P < / l a t e x i t > 1 and system B to oscillator b 2 .

FIG. 3 :
FIG. 3: Effect of mechanical thermal noise on the purity of the state produced in the Hamiltonian-switching (a) and the two-dissipator (b) schemes.All the parameters are set as in Fig. 2.

FIG. 4 :
FIG.4: Effect of mechanical thermal noise on the most negative point of the first mode momentum Wigner cross-section of the state produced in the Hamiltonian-switching (a) and the two-dissipator (b) scheme.All the parameters are set as in Fig.2.

FIG. 5 :
FIG.5: Effect of mechanical thermal noise on the logarithmic negativity of entanglement of the state produced in the Hamiltonian-switching (a) and the two-dissipator (b) scheme.All the parameters are set as in Fig.2.

FIG. 6 :FIG. 7 :
FIG.6: Effects of different initial states and varying mechanical decay rate on the fidelity between the CPE state and the produced state by the two-dissipator scheme.The maximum evolution time is set to 10/κ 0 .See the text for the other parameters.