Cooling and entanglement of multimode graphene resonators via vacuum fluctuations
CCooling and entanglement of multimode graphene resonators via vacuum fluctuations
Sofia Ribeiro and Hugo Ter¸cas Instituto de Telecomunica¸c˜oes, Lisbon, Portugal ∗ Instituto de Plasmas e Fus˜ao Nuclear, Lisbon, Portugal
Sympathetic laser cooling of a single mode graphene membrane coupled to an atomic cloud inter-acting via Casimir-Polder forces has been recently proposed. Here, we extend this study to the effectof secondary graphene membrane whose frequency may be far or close to resonance. We show thatif the two mechanical modes are close together, it is possible to simultaneously cool both modes.Conversely, if the two frequencies are set far apart, the secondary mode does not affect the coolingof the first one. We also study the entanglement properties of the steady-state using the logarithmicnegativity. We show how stationary mechanical entanglement between two graphene sheets can begenerated by means of vacuum fluctuations. Moreover, we find that, within feasible experimen-tal parameters, large steady-state acoustomechanical entanglement, i.e. entanglement between thephononic and mechanical mode, E N ≈
5, can be generated.
PACS numbers: 31.30.jh, 03.65.Ud, 42.50.Nn, 63.22.Rc
I. INTRODUCTION
In the technological push towards miniaturization, one of the ultimate goals is to build nanomechanical resonatorsthat are only one atom thick. Two-dimensional (2D) nanoresonators offer a unique platform for quantum technologiesthanks to their low mass, low stress, and high quality factors. Graphene’s extraordinary electronic and opticalproperties hold great promise for applications in photonics, electronics and optomechanical systems [1]. Lately, agreat effort has been put in harnessing the mechanical properties of graphene for mass sensing [2], studying nonlinearmechanics [3, 4], and voltage tunable oscillators [5, 6]. Building these miniaturized mechanical systems evolved to theidea of devising small structures based on graphene, where neutral atoms and graphene are held in close proximity.Such a hybrid system would consist of atoms that are manipulated by laser light, and graphene sheets that could, forinstance, be controlled by electrical currents or piezoelectrics [7, 8].Atom-graphene coupling can be achieved via vacuum forces [9]. Casimir-Polder interaction is a promising tool tomanipulate and cool quantum states of mechanical oscillators. Proposals range from shielding vacuum fluctuationswith graphene sheets [10], quantum sensing of graphene motion [11], manipulation of the atomic states to create rippleson demand [12], and passive sympathetic cooling carbon nanotubes by immersing them in cold atom clouds [13]. Ina recent study, the authors proposed a method to actively sympathetic cool graphene membranes by laser-coolingan atomic cloud placed at a few µ m distance [14]. This overcomes the important limitation of radiation-pressurecooling of graphene, a process that is known to be hindered by its broad absorption spectrum [15]. The influenceof vacuum forces in optomechanical systems has been the focus of many recent studies. In reference [16], W. Nieand co-authors studied the effect of the Casimir force between a dielectric nanosphere coupled to a movable mirrorof a Fabry-P´erot cavity. They have shown that the ground state cooling of the nanosphere is achieved for certainsphere-mirror distances and that it can be optimized by tuning the mirror oscillation frequency. In reference [17],the study is extended to consider two nanospheres trapped near the cavity mirrors by an external driving laser. Bytuning the external control parameters and the cavity-sphere distance, they found to be possible to achieve largesteady-state optomechanical entanglement. In reference [18], the authors focused on the Casimir-Polder interactionof an ensemble of quantum emitters coupled to a movable mirror inside a cavity. It is shown that vacuum forces notonly greatly enhance the effective damping rate but also lead, in the bad cavity limit, to the ground-state cooling ofthe mechanical motion.Accessing and controlling the quantum ground-state is a milestone in all optomechanical system, as it allows us toharness the quantum behaviour of a macroscopic object and to explore the quantum-classical boundary. However,up to now, most theoretical treatments of cooling have focused on a single phononic mode – single mechanical mode interaction. Here, we shall extend the treatment of cooling introduced in reference [14] by including the effect ofsecondary modes whose resonance frequency is not far from that of the mechanical mode of interest. This secondarymode can be considered on a single membrane device or in a multi-mode membrane device, as illustrated in figure 1. ∗ Electronic address: sofi[email protected] a r X i v : . [ qu a n t - ph ] J un Atomic Cloud z A FIG. 1: (Colour online) Schematic representation of the experimental setup. A cold atomic gas is placed at a distance z A froma system with two graphene membranes suspended on a substrate. Quantum excitations in the atomic cloud (phonons) arecoupled to the flexural (out-of-plane) modes of graphene via vacuum forces. The cooling and entanglement of the phonons ofthe gas can be done with the help of the cooling laser with Rabi frequency Ω. For our calculations, we have chosen an atomiccloud of Rb.
In addition, we will also study the steady-state entanglement between the different mechanical modes. Entanglementis a typical property of the quantum world, non existent in the classical realm. However, there is nothing in thequantum mechanical principles that prevents macroscopic systems to be entangled. In fact, entanglement has beenexperimentally achieved in microscopic quantum systems such as photons [19–22], ions [23], electrons [24], buckyballs[25, 26], and in macroscopic systems such as diamonds [27]. Beside the inherent fundamental interest, the ability tocreate entangled states as also useful applications, as in high precision and metrology applications where entangledstates represent a very sensitive probe [28] and can have profound implications to optical information science andquantum computing [29, 30].In this work, we investigate the macroscopic mechanical entanglement generation in two or more coupled graphenenanoresonators. As we are about to show, entanglement is achieved via vacuum forces that couple the differentmembranes through the elementary excitations (phonons) of an cold atomic cloud. We propose experimentally feasibleschemes to create and probe acoustomechanical entanglement, i.e. entanglement between the phonon mode (hereplaying the role of a cavity photon) and the graphene flexural mode.This paper is organized as follows. In Sec. II, the theoretical model of the optomechanical system with multi-modesflexurons coupled via electromagnetic vacuum fluctuations to a single atomic cloud is introduced. In Sec. III, we studythe simultaneous cooling of flexuron modes in near by membranes. We extend our study also to the acoustomechanicaland mechanical entanglement in different configurations. Finally, in Sec. V, we provide some concluding remarks.
II. THEORETICAL MODEL FOR THE EXPERIMENTAL SETUPA. Description of vibrating graphene
Although classic elastic theory focus on very large systems, it has been shown that the elastic continuum theoryis still valid for a small graphene flake [31]. Due to thermal fluctuations, the graphene sheet undergoes mechanicalout-of-plane vibrations (flexural phonons), which can be well described within the Kirchhoff-plate theory of elasticity[32]. In what follows, we will consider a squared graphene flake with both edges clamped to a substrate subject toa restoring force. Having determined the eigenmodes of the graphene membrane, the Hamiltonian for the flexuralmodes easily follows from the canonical quantization of the dynamics of the out-of-plane vibrations. Thus, we writethe Hamiltonian as ˆ H F = (cid:82) d r (cid:2) D ∇ u ( r , t ) + h ¨ u ( r , t ) + 2 t cl ∇ u ( r , t ) (cid:3) . (1)We then express the out-of-plane displacement in the formˆ u ( r ) = 1 √ (cid:88) k ,σ φ k ,σ ( r ) e σ (cid:16) ˆ f k ,σ + ˆ f † k ,σ (cid:17) (2)with two polarizations σ = ( x, y ) and satisfying the normalization condition (cid:104) φ k , φ k (cid:48) (cid:105) = (cid:126) / ( M ν k ) δ kk (cid:48) , where M is the membrane mass and ν k the vibration frequency of the flexural mode ν = (cid:113) Dρ k + t cl ρ k , with ρ areal massdensity, D = Y h / (cid:0) − υ (cid:1) the bending modulus, Y ∼ υ = 0 .
17 the Poisson ratio, h = 3 .
35 ˚A the thickness of the plate and t cl the clamping tension (for simplicity, we consider it to be equal along both x and y directions). The flexural operators obey the usual bosonic commutation relation (cid:104) ˆ f k ,σ , ˆ f † k (cid:48) ,σ (cid:48) (cid:105) = δ kk (cid:48) δ σ,σ (cid:48) .Thus, the Hamiltonian of the vibrating graphene is simply given as followsˆ H flex = (cid:88) k ,σ (cid:126) ν k ,σ ˆ f † k ,σ ˆ f k ,σ . (3) B. Description of the total Hamiltonian
As described above, our system of interest consists of a laser-cooled two-dimensional cloud of atoms that is placeda few micrometres from one or more graphene membranes (see figure 1). Due to the tight confinement in theperpendicular direction, the (transverse) phonon modes of the atomic cloud are quantized. The initial Hamiltoniancontains five terms: i) the energy of the electronic states of the atoms; ii) the energy associated to the quantizedatomic motion; iii) the quantized modes of the membranes; iv) the coupling between the laser and the atomic motionand v) Casimir-Polder (CP) interaction between the atoms and the membrane (for a detailed derivation, please referto reference[14]). We first proceed to the adiabatic elimination of the excited electronic states and assume that theatoms are cooled enough to be in the Lamb-Dicke regime. This means that we are in a situation where the differencebetween the atomic phonon modes is much larger than the difference of the flexural modes in graphene, such that wecan safely retain the lowest phononic mode only. Finally, we obtain the effective Hamiltonian, which is nonlinear inthe phonon operator ˆ a , and the effective Lindblad operator as [14]ˆ H eff = (cid:126) ω ˆ a † ˆ a + (cid:126) (cid:80) j ν j ˆ f † j ˆ f j + i (cid:126) (cid:80) j g j ˆ a † ˆ a (cid:16) ˆ f j + ˆ f † j (cid:17) + i (cid:126) ξ (cid:0) ˆ a † + ˆ a (cid:1) (4) L eff (cid:16) ˆ O (cid:17) = γ (cid:16) a † ˆ O ˆ a − ˆ O ˆ a † ˆ a − ˆ a † ˆ a ˆ O (cid:17) . (5)Here, we have defined the reduced quantities ω = ω ph − η (cid:126) Ω ∆4∆ + Γ + (cid:88) j ω CP j , ξ = η Ω ∆4∆ + Γ , γ = Γ η Ω ) , where ω ph is the energy of the phonon excitation, η the Lamb-Dicke parameter, ∆ the detuning between the laser andthe electronic transition, Γ the atomic spontaneous emission rate and Ω the Rabi frequency. The coupling strengthbetween the graphene sheet and the atomic cloud is given by [33] g j = 2 q j (cid:115) (cid:126) mν j n ω CP j (6)with n being the atomic density. In the non-retarded limit, that is, when the atom-surface distance z A is smallwhen compared to the effective atomic transition wavelength z A (cid:28) c/ω eg , the Casimir-Polder potential becomes U CP = C /z A . For Rubidium atom near a graphene sheet, one finds C = 215 .
65 Hz µ m [34]. After performing aFourier transformation, the fundamental Casimir-Polder frequency reads [14] ω CP1 = 2 πC e − q z A z A . (7) C. Heisenberg-Langevin equations of motion
An appropriate treatment of the problem requires including other different effects, the main one being the lossesin the flexural modes which are quantified by the energy dissipation rate κ j = ν j /Q j , where Q j is the mechanicalquality factor. In reference [35], the authors demonstrated coupling between a multilayer graphene resonator withquality factors up to 2.2 × , which results in dissipative rates of the orders of a few tens of Hz or lower. Therefore,by defining the dimensionless position and momentum operator operatorsˆ q j = i (cid:16) ˆ f † j + ˆ f j (cid:17) √ , ˆ p j = ˆ f † j − ˆ f j √ , (8)with [ δ ˆ q k , δ ˆ p j ] = iδ kj , and adopting the formalism of quantum Langevin equations, we find˙ˆ a = − iω ˆ a − i (cid:88) j g j √ a ˆ q j + ξ − γ ˆ a, (9)˙ˆ q j = − ν j ˆ p j , (10)˙ˆ p j = ν j ˆ q j − κ j ˆ p j − √ g j ˆ a † ˆ a + ζ, (11)where the mechanical modes of graphene are affected by a viscous force with damping rate κ and by a Brownianstochastic force with zero mean value ζ , with correlation function [36] (cid:104) ζ ( t ) ζ ( t (cid:48) ) (cid:105) = κ j ν j (cid:90) dω π e − iω ( t − t (cid:48) ) ω (cid:20) coth (cid:18) (cid:126) ω k B T (cid:19) + 1 (cid:21) , (12)where k B is the Boltzmann constant and T is the graphene temperature. ζ ( t ) is a Gaussian quantum stochasticprocess and non-Markovian, i.e., neither its correlation function or its commutator are proportional to a Dirac delta.However, we can simplify the thermal noise contribution. k B T / (cid:126) ∼ s − even at cryogenic temperatures [37, 38],as so, it is always much larger than all the other parameters. At these higher values of frequency the position spectrumis negligible and therefore one can safely approximate the integral [39] κ j ων j coth (cid:18) (cid:126) ω k B T (cid:19) (cid:39) κ j k B T (cid:126) ν j (cid:39) κ j (2 m j + 1) , (13)where m j = (exp ( (cid:126) ν j /k B T ) − − is the mean thermal number of the mode j .Following ref. [38–40], we will arrive at a system of linearised quantum Langevin equations (see more details in A) δ ˙ˆ q j = − ν j δ ˆ p j , (14) δ ˙ˆ p j = ν j δ ˆ q j − κ j δ ˆ p j − g j | α | δ ˆ X + ζ, (15) δ ˙ˆ X = ϑ ( N ) δ ˆ Y − γδ ˆ X, (16) δ ˙ˆ Y = − ϑ ( N ) δ ˆ X − γδ ˆ Y + (cid:88) j | α | g j δ ˆ q j , (17)where we have defined ϑ ( N ) = ω + (cid:80) j g j | α | /ν j , with N being the total number of modes considered. These canbe written in a compact form as ˙ u ( t ) = Au ( t ) + n ( t ) , (18)where we defined the fluctuation vector u ( t ) and the noise vector n ( t ) and A is the drift matrix that governs thedynamics of the expectation values. Since the dynamics is linearised, the quantum steady-state of fluctuations is azero-mean multipartite Gaussian state, fully characterized by its correlation matrix V that can be find by solving A V + V A T = − D. (19)where D = Diag [0 , κ j (2 m j + 1) , ,
0] is the diagonal matrix determined by the noie correlation function (for thecomplete derivation please see A) This equation is linear for V and can be straight-forwardly solved. The stationaryvariances of the mechanical modes are given by the diagonal terms of V , (cid:10) δ ˆ q (cid:11) = V , (cid:10) δ ˆ p (cid:11) = V , (cid:10) δ ˆ q (cid:11) = V , (cid:10) δ ˆ p (cid:11) = V , . . . , (cid:68) δ ˆ X (cid:69) = V N − ,N − , (cid:68) δ ˆ Y (cid:69) = V NN . ����� ����� ����� ����� ����� ����� ������������ ϑ ( � ) / ν � � �� ����� ����� �������������������� ϑ ( � ) / ν τ � �� ( � ) FIG. 2: (Colour online) Steady-state flexural number m eff (left) and decay rate of one flexural mode of a single graphene sheetversus normalized detuning ϑ (1) /ν . The atomic and mechanical parameters are Γ = 6 . ω ph = 477 Hz, Ω = 12 MHz,∆ = 45 MHz, η = 0 . κ = 2 Hz and ν = 2 MHz and T = 0 .
01 K, that corresponds to an initial occupancy m = 10 . Theyellow solid line corresponds to a coupling strength g = − . T = 0 . m = 10 ), the blue solid line to g = − g = 0. ����� ����� ����� ����� ����� ����� ����������������������������� ϑ ( � ) / ν � η - ����� ����� ����� ����� ����� ����� ������������ ϑ ( � ) / ν � � � FIG. 3: (Colour online) Acoustomechanical entanglement η − (left) and logaritmic negativity E N (right) for a single graphenesheet versus normalized detuning ϑ (1) /ν . The atomic and mechanical parameters are Γ = 6 . ω ph = 477 Hz, Ω = 12 MHz,∆ = 45 MHz, η = 0 . κ = 2 Hz and ν = 2 MHz. The solid lines are for T = 0 .
01 K and the dashed line to T = 0 . g = − . g = − III. STEADY-STATE AND ENTANGLEMENT OF MULTIMODE GRAPHENE
At the steady-state, the energy of each mechanical mode can be written in the terms of the variances of thecorresponding position and momentum operators U j = (cid:126) ν j m j eff = − (cid:126) ν j (cid:2)(cid:10) δq j (cid:11) + (cid:10) δp j (cid:11) + 1 (cid:3) , (20)where m j eff = − (cid:0)(cid:10) δq j (cid:11) + (cid:10) δp j (cid:11) + 1 (cid:1) / j th mode. A. Single mechanical mode
If only one mechanical mode is considered, the drift matrix assumes the following explicit form A (1) = − ν ν − κ − | α | g
00 0 − γ ϑ (1) − | α | g − ϑ (1) − γ . (21)The stability conditions can be determined by applying the Routh-Hurwitz criterion [41] ν (cid:16) ϑ + γ (cid:17) − ν | α | g ϑ (1) > , (22)2 γκ (cid:104) ϑ + ϑ (cid:0) κ + 2 κγ + 2 γ − ν (cid:1) + (cid:0) κγ + γ + ν (cid:1) (cid:105) + 4 ν | α | g ϑ (1) ( κ + 2 γ ) > . (23)In agreement with the results of ref. [14], with an appropriate choice of g and ϑ (1) , it is possible to have effectivesteady-states with a low number of flexural modes, (typically m eff < Q microwavecavity. Our cooling scheme could be implemented in combination with these other optomechanical cooling protocols;the phonon assisted cooling via vacuum interactions would then be used to cool the membrane further to the groundstate. Based on that, we from now on we will choose a different set of initial graphene temperatures, T = 0 . , . g and initial temperature, we observethat the minimum value of the cooling occurs for ϑ (1) (cid:16) = ω + 2 g | α | /ν (cid:17) (cid:39) ν . In fact, optimal cooling occurs in anarrow interval around ϑ (1) /ν = 1 (see the left panel of figure 2 for illustration). The eigenvalues of A determine therelaxation time, which is given by the inverse of that having the smallest real part. For g = 0, the relaxation time isgiven by the mechanical relaxation time κ − ( τ ∼ . g (cid:54) = 0, we can obtain much larger decay rates(see the right panel of figure 2 for illustration). Acoustomechanical entanglement in a single mode membrane
In order to establish the conditions under which the flexural and the phononic modes are entangled, we considerthe logarithmic negativity E N [42, 43] defined as E N = max (cid:2) , − ln 2 η − ( V bip ) (cid:3) , (24)where V bip is a generic 4 × V bip = (cid:18) A CC T B (cid:19) , (25)and η − ( V bip ) is given by η − ( V bip ) ≡ √ (cid:114) Σ ( V bip ) − (cid:113) Σ ( V bip ) − V bip (26)with Σ ( V bip ) ≡ det A + det B − C . (27)A Gaussian state is entangled if and only if η − ( V bip ) < /
2, which is equivalent to the positive partial transposecriterion, a necessary and sufficient condition for Gaussian states [44]. For a single mechanical mode, V bip ≡ V definedby equation (19). figure 3 shows η − and the logarithmic negativity E N versus the normalized detuning ϑ (1) /ν fortwo different temperatures and coupling strengths. One can see that, in all the cases, there is acoustomechanicalentanglement, and this entanglement increases around the resonance condition ϑ (1) ∼ T = 0 . E N ≈
5, is shown to be possible via Casimir-Polder interactions, in agreement with ref. [17]. � � = � ��� � �� ) ����� ����� ����� ����� ����������������������� ϑ ( � ) / ν � � � �� � ) ����� ����� ����� ����� ����������������������������� ϑ ( � ) / ν � η - � = � ���� � �� ) ����� ����� ����� ����� ����������������������� ϑ ( � ) / ν � � � �� � ) ����� ����� ����� ����� ����������������������������� ϑ ( � ) / ν � η - FIG. 4: (Colour online) a) and c) mean effective flexuron number of the modes j = 1 (yellow) and j = 2 (blue). b) andd) mechanical entanglement versus normalized detuning ϑ (2) /ν . The atomic and mechanical parameters are Γ = 6 . ω ph = 477 Hz, ∆ = 45 MHz, η = 0 . κ = 2 Hz, ν = 2 MHz and ν = 0 . ν . On the top, for a) and b), T = 0 . . g ≈ g ≈
43 kHz; on the bottom, for c) and d), T = 0 .
01 K, Ω = 12 MHz and g ≈ g ≈
40 kHz.
IV. SIDE-BY-SIDE MEMBRANES: SIMULTANEOUS COOLING AND ENTANGLEMENTPROPERTIES
We now consider two spatially separated side-by-side membranes placed near a single atomic cloud. There is nodirect interaction between the membranes, but each membrane is coupled via Casimir-Polder forces to the atomiccloud. It is assumed that each membrane is restricted to a single flexural mode therefore, the dynamics of the systemare therefore described by the drift matrix A (2) = − ν ν − κ − | α | g
00 0 0 − ν ν − κ − | α | g
00 0 0 0 − γ ω − | α | g − | α | g − ω − γ . (28)There are two distinct situations, depending on the difference between the two flexural frequencies. If they are verydifferent, the cooling of the mode (1) is not perturbed by the presence of the mode (2), like the case of figure 7. Onthe other hand, if the frequencies of the two modes are similar, both modes are simultaneously cooled close to theirground state. As so, we set ν = ν + δ , with δ being small. For our numerical results, we considered a difference of1% between the two frequencies. The two modes are optimally cooled at two well-distinct values of ϑ (2) and one canefficiently cool both modes if one fixes the detuning within a very narrow interval halfway between the two mechanicalresonances, ϑ (2) ≈ ( ν + ν ) /
2. The value of γ can be tuned by changing Ω and, as a result, one can tune the valueof m eff to be lower than 1, thus entering in the quantum regime (see figure 4).For this case, we would also like to study the steady-state acoustomechanical and mechanical entanglement. Thesteady-state correlation matrix V for two mechanical modes is a 6 × × V = A C D C T A D D T D T B . (29)To check if the two mechanical modes are entangled at the steady-state, one eliminates the entries in V that correspond � � = � ���� � �� ) ����� ����� ����� ����� ����������������������� ϑ ( � ) / ν � � � �� � ) ����� ����� ����� ����� ����������������������� ϑ ( � ) / ν � η - FIG. 5: (Colour online) a) Mean effective flexuron number of the modes j = 1 (yellow), 2 (blue), 3 (green) of a three side-by-sidegrahene sheets and b) mechanical entanglement between membrane 1 and 2 (green), 1 and 3 (dashed yellow) and 2 and 3 (blue)versus normalized detuning ϑ (3) /n for T = 0 .
01 K. The atomic and mechanical parameters are Γ = 6 . ω ph = 477 Hz,Ω = 17 . η = 0 . κ = 2 Hz and ν = 2 MHz, ν = 0 . ν and ν = 1 . ν and g , , ≈ − . to the atomic phonon field, to get V bip = (cid:18) A C C T A (cid:19) . (30)Alternatively, in the case we would like to analyse the entanglement between one of the mechanical modes and thephonon modes, it suffices to eliminate the rows and columns that correspond to the other mechanical mode from thematrix V V bip = (cid:18) A , D , D T , B (cid:19) . (31)The entanglement properties of the mechanical steady-state of the system will again strongly depend on the ex-perimental situations. Although these two modes are not directly interacting, they can become entangled at thesteady-state via the Casimir-Polder interaction with the atomic cloud. As it has been previously seen, when the twomodes are well separated, either simultaneous cooling of the membranes or (purely) mechanical entanglement is notachievable. The situation is drastically different when two mechanical modes become very close in frequency ν ≈ ν .For this situation, when ϑ (2) (cid:54) = ( ν + ν ) /
2, although we do not have simultaneous cooling, the membranes may beentangled for a certain set of parameters. In figure 4, we compare two cases for two different temperatures. In bothcases, when we have optimal cooling of the membranes to m eff <
1, there is no mechanical entanglement betweenthem. However, there might be other interesting states, such as the Fock states | n, (cid:105) , | , n (cid:105) , where both membranescan be mechanically entangled (we show, for instance, the particular case where n = 2 for both temperatures). More-over, although at first sight the results depicted in figure 4, show that lower temperatures will give rise to weakermechanical entanglement strengths, in fact, if compared in the same parameters range, we observe that entanglementis very fragile with respect to temperature. This means, that if we choose the same parameters as figure 4 d), but atemperature sightly higher, for instance T = 0 .
02 K, the two graphene membranes are no longer entangled.
Three modes: Simultaneous cooling
Finally, we would like to analyse the case of three side-by-side membranes (see figure 1). In this situation, the driftmatrix becomes a 8 × T = 0 .
01 K (see figure 5). Wehave also studied the entanglement of the bipartite system composed by membrane 1 and 2 (green), 2 and 3 (dashedyellow) and 3 and 4 (blue). Although we find no bipartite entanglement between the membranes for a reasonable setof parameters, it is possible to cool all membranes down to the quantum regime m eff < V. CONCLUSIONS
In this paper, we analysed the effect of the presence of one or more secondary modes in the sympathetic lasercooling of a graphene sheet coupled to an atomic cloud via Casimir-Polder interactions. We have seen that thesimultaneous cooling of two modes crucially depends on the difference between their frequencies. We have shownthat, for a single graphene sheet, the frequency of the fundamental and first excited flexural (out-of-plane) modesare too separated, such that the excited mode does not affect cooling. In fact, we observed that there are differentexperimental parameters that would allow us to cool one mode without affecting the other. Considering a multiplemembrane system, where different non-interacting graphene sheets are placed side-by-side, we have shown that themodes are optimally cooled at well-distinct values of ϑ ( N ) and one can efficiently cool both modes by setting thedetuning within a very narrow interval halfway between the mechanical resonances.Under the same conditions, we have also studied the acoustomechanical and mechanical entanglement consideringmultiple flexural modes. Large acoustomechanical entanglement can be achieved for single or multimode case, con-firming previous results that indicated that vacuum forces enable steady-state acoustomechanical entanglement [17].On the other hand, we demonstrated that the mechanical entanglement is very fragile and strongly depends on ϑ ( N ) .However, we are still able to prove that the mechanical states can become entangled thanks to the common interactionwith the quantum gas. Acknowledgements
The authors acknowledge the Security of Quantum Information Group for the hospitality and for providing theworking conditions. The authors would like to thank the support from Funda¸c˜ao para a Ciˆencia e a Tecnologia(Portugal), namely H.T. through scholarship SFRH/BPD/110059/2015 and S.R. via UID/EEA/50008/2013 project.
Appendix A: Linearisation procedure
When the ground-state cooling is achieved and if the system is stable, the system is characterized by a semiclassicalsteady-state where the phonons in the atomic system can be rewritten as a displacement transformation with anaverage amplitude α and a fluctuating part δ ˆ a . The steady-state values of the position and momentum operators, asobtained from the stationarity of equation (11) and (10), are given by p sj = 0 , (A1) q sj = √ g j | α | ν j , (A2) α = ξiϑ ( N ) + γ , (A3)where we have defined ϑ ( N ) = ω + (cid:80) j g j | α | /ν j , with N being the total number of modes considered. Then, wecan linearise Eqs. (9)-(11) around the steady-state values by setting ˆ a → α + δ ˆ a , ˆ q j → q sj + δ ˆ q j and ˆ p j p sj → δ ˆ p j . Forsmall fluctuations, the second-order terms δ ˆ a † δ ˆ a and δ ˆ aδ ˆ q j are ruled out of the dynamics. Introducing the phononquadratures δ ˆ X = αδ ˆ a † + α ∗ δ ˆ a (cid:113) | α | , (A4) δ ˆ Y = i (cid:0) αδ ˆ a † − α ∗ δ ˆ a (cid:1)(cid:113) | α | , (A5)with (cid:104) δ ˆ X, δ ˆ Y (cid:105) = i , we arrive at a system of linearised quantum Langevin equations δ ˙ˆ q j = − ν j δ ˆ p j , (A6) δ ˙ˆ p j = ν j δ ˆ q j − κ j δ ˆ p j − g j | α | δ ˆ X + ζ, (A7) δ ˙ˆ X = ϑ ( N ) δ ˆ Y − γδ ˆ X, (A8) δ ˙ˆ Y = − ϑ ( N ) δ ˆ X − γδ ˆ Y + (cid:88) j | α | g j δ ˆ q j . (A9)0These can be written in a compact form as ˙ u ( t ) = Au ( t ) + n ( t ) , (A10)where we defined the fluctuation vector u ( t ) = (cid:16) δ ˆ q ( t ) , δ ˆ p ( t ) , . . . , δ ˆ q j ( t ) , δ ˆ p j ( t ) , . . . , δ ˆ X ( t ) , δ ˆ Y ( t ) (cid:17) T , (A11)the noise vector n ( t ) = (0 , ζ ( t ) , . . . , , ζ j ( t ) , . . . , , T , (A12)and A is the drift matrix that governs the dynamics of the expectation values. The solution of equation (A10) is givenby u ( t ) = M ( t ) u (0) + (cid:90) t dτ M ( τ ) n ( t − τ ) , (A13)where M ( t ) = exp ( At ). The system is stable and reaches its steady-state when all of the eigenvalues of A havenegative real parts, such that M ( ∞ ) = 0. Since the dynamics is linearised, the quantum steady-state of fluctuationsis a zero-mean multipartite Gaussian state, fully characterized by its correlation matrix V whose elements read V lm = (cid:104) u l ( ∞ ) u m ( ∞ ) + u m ( ∞ ) u l ( ∞ ) (cid:105) . (A14)When the system is stable, one gets V lm = (cid:88) k,l (cid:90) ∞ dτ (cid:90) ∞ dτ (cid:48) M ik ( τ ) M jl ( τ (cid:48) )Φ kl ( τ − τ (cid:48) ) , (A15)where Φ kl ( τ − τ (cid:48) ) = (cid:104) n k ( τ ) n l ( τ (cid:48) ) + n l ( τ (cid:48) ) n k ( τ ) (cid:105) / ζ ( t ) is not delta-correlated and therefore does not describe a Markovianprocess [36, 39]. Quantum effects are achievable only for oscillators with a large mechanical quality factor Q j = ν j /κ j (cid:29)
1, such as the case of graphene [35]. In ref. [45, 46], it has been shown that if a process is purely Gaussianrandom and if we can treat the dynamical system quantum mechanically and interpret the canonical distribution ofthe heat bath also quantum mechanically, then, in this limit, one can recover a Markovian process and ζ ( t ) satisfies[40] (cid:104) ζ ( τ ) ζ ( τ (cid:48) ) + ζ ( τ (cid:48) ) ζ ( τ ) (cid:105) ≈ κ j (2 m j + 1) δ ( τ − τ (cid:48) ) . (A16)Since the components of n ( t ) are now uncorrelated, we getΦ kl ( τ − τ (cid:48) ) = D kl δ ( τ − τ (cid:48) ) , (A17)where D = Diag [0 , κ j (2 m j + 1) , ,
0] is the diagonal diffusion matrix determined by the noise correlation functions.Thus, we find V = (cid:90) ∞ dτ M ( τ ) DM T ( τ ) . (A18)When the stability condition M ( ∞ ) = 0 is satisfied, one gets the following equation for the steady-state correlationmatrix A V + V A T = − D. (A19) Appendix B: Results single mode
To make a comparison with the results of reference [14], in figure 6, we have plotted m eff in terms of the tunableexperimental parameters ϑ (1) /ν and coupling strength g . Since a more realist system must include losses in thevibrational motion of the membrane, the final m eff is temperature dependent, see differences between figure 6a) for T = 100 K, b) T = 10 K, c) T = 0 . T = 0 .
01 K, which corresponds to initial flexuron occupation numbersof 10 , 10 , 10 and 10 , respectively. We show that these results agree with our previous ones, that is, with anappropriate choice of g and ϑ (1) , it is possible to have effective steady-states with a low number of flexural modes.1 � ��� ���������� � ��� ���������� � ��� ���������� � ��� ���������� FIG. 6: (Colour online) Density plot of the stationary state flexural mode number m eff of a single graphene sheet with a singlevibrational mode versus normalized detuning ϑ (1) /ν and coupling parameter g . The atomic and mechanical parameters areΓ = 6 . ω ph = 477 Hz, Ω = 12 MHz, ∆ = 45 MHz, η = 0 . κ = 2 Hz and ν = 2 MHz. a) corresponds to an initialtemperature of T = 100 K, b) to T = 10 K, c) to T = 0 . T = 0 .
01 K. The dashed yellow lines correspond to acoupling strength g = − . g = − Two Flexural modes in a single membrane: Simultaneous cooling
In order to see if the presence of a secondary vibrational mode in a same membrane affects the ground-state cooling,we can exactly solve equation (19) and analyse the stationary position and momentum variances of the two mechanicalmodes in order to calculate the effective flexuron number m ( ν i )eff . To do so, we choose a parameter regime close to thatof optimal cooling for a single mode. Our results indicate that when the two mechanical modes in a graphene sheetare well separated, ν (cid:39) . ν , the secondary mode does not disturb the cooling of the mechanical mode of interest,as we can see by the overlap of the curves for the single- and two-modes cases (see figure 7). Furthermore, we observethat both modes are optimally cooled at two well distinct values of ϑ (2) : by cooling one, the other remains unaffected.We would also like to study the steady-state acoustomechanical and mechanical entanglement. Our results showthat the presence of the second mode does not affect significantly the entanglement between the first mode and thephonon, see (yellow line) figure 8. Furthermore, we verify that the second mode is also entangled with the phononicmode (see blue line). However, we find that the purely mechanical entanglement between the first and second modescannot be generated for this case, see figure 9.2 ����� ����� ����� ����� ����� ����� ���������� ϑ ( � ) / ν � � � �� ( ν � ) ����� ����� ����� ����� ����� ����� ������������� ϑ ( � ) / ν � � � �� ( ν � ) FIG. 7: (Colour online) Mean effective flexuron number of the modes j = 1 (left) and j = 2 (right) of a single graphene sheetversus normalized detuning ϑ (2) /ν . The atomic and mechanical parameters are Γ = 6 . ω ph = 477 Hz, Ω = 12 MHz,∆ = 45 MHz, η = 0 . κ = 2 Hz and ν = 2 MHz, ν = 1 . ν , with T = 0 .
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