Clauser-Horne-Shimony-Holt Bell inequality test in an optomechanical device
Juuso Manninen, Muhammad Asjad, Risto Ojajärvi, Petri Kuusela, Francesco Massel
CClauser-Horne-Shimony-Holt Bell inequality test in an optomechanical device
Juuso Manninen, Muhammad Asjad, Risto Ojaj¨arvi, Petri Kuusela, and Francesco Massel ∗ Department of Applied Physics, Low Temperature Laboratory,Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland Department of Physics and Nanoscience Center, University of Jyv¨askyl¨a,P.O. Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland
We propose here a scheme, based on the measurement of quadrature phase coherence, aimed at testing theClauser-Horne-Shimony-Holt Bell inequality in an optomechanical setting. Our setup is constituted by twooptical cavities dispersively coupled to a common mechanical resonator. We show that it is possible to generateEPR-like correlations between the quadratures of the output fields of the two cavities, and, depending on thesystem parameters, to observe the violation of the Clauser-Horne-Shimony-Holt inequality.
I. INTRODUCTION
In his seminal work, motivated by the work by Einstein,Podolsky and Rosen, [1], Bell showed that theories relying onlocal (possibly hidden) variables, which are bound to satisfycertain inequalities, cannot describe all quantum mechanicalpredictions [2]. From the point of view of quantum theory,a violation of these Bell inequalities (BIs) necessarily impliesentanglement between spatially separated subsystems [3]. Be-yond their intrinsic conceptual relevance, BI tests have poten-tially important technological repercussions, allowing to cer-tify the security of quantum cryptographic schemes [4], mak-ing it relevant to explore the possibility of performing suchtest in di ff erent setups and for di ff erent physical systems.Since the work of Bell, multiple experimental realizationsof BI tests have been conducted [5–18], the first one beingperformed by Freedman and Clauser [5]. However, the confir-mation that, without any additional assumptions –i.e., closingall loopholes–, predictions o ff ered by locally realistic theoriescannot reproduce the experimental results has been obtainedonly in the last few years [15–17]. Even more recently, basedon an early theoretical proposal [19] and resorting to an ex-perimental setup similar to the employed in the Bell test per-formed by Ou and Mandel [20], a BI test relying on continu-ous variable measurement has been performed [18].Owing to the recent progresses in the concomitant manip-ulation of mechanical and optical degrees of freedom at thequantum level [21, 22], cavity optomechanical systems repre-sent one of the cornerstones for future quantum informationand communication technologies. On a more fundamentallevel, these systems represent one of the most promising plat-forms for experimental verification of physical theories, withapplications ranging from gravitational wave detection [23]to the potential observation of quantum gravitational e ff ects[24] and entanglement between nearly-macroscopic mechani-cal objects [25–29] .In this spirit, in this article, we investigate the test of theClauser-Horne-Shimony-Holt (CHSH) [30] BI in an optome-chanical system. Our main focus is a two-cavities optome-chanical setup, either in the microwave or in the visible-light ∗ francesco.p.massel@jyu.fi regime, allowing for unrivaled flexibility in the choice of de-tectors and transmission lines for loophole-free tests. In ad-dition, the nature of the optomechanical interaction charac-terizing our proposal opens up the possibility for BI tests inmixed microwave / optical settings [31] . The two cavities / onemechanics setup, which we consider here for the BI test, wasdiscussed in the past in connection with entanglement prop-erties of optomechanical systems [25, 31, 32] and was exper-imentally realized in the context of multimode quantum sig-nal amplification of microwaves [33]. While other ideas fortesting BIs in an optomechanical setting have recently beenproposed[34, 35], they are based on a rather di ff erent setupthan the one discussed here, for which, due to the sequentialnature of the pusling scheme, closing all loopholes, in par-ticular the locality loophole, requires to address extra tech-nical di ffi culties as discussed in the supplemental material ofRef. [34] which are not present in the setup discussed here.On more general grounds, it is worth mentioning that clos-ing the locality loophole in a microwave setting represents aformidable challenge due to the necessity of the noiseless dis-tribution of microwave signals. In this sense an all-opticalrealization of our proposal would thus seems favorable. In thefollowing, however, in order to underline the relation to thepresent state-of-the-art experimental capabilities, we mainlyfocus on the experimental parameters of the microwave setupdiscussed in Ref. [33].While the previous BI tests mentioned above rely eitheron the polarization degree of freedom of optical photons [5–9, 13, 14, 16, 17], or on di ff erent realizations of two-level sys-tems in a condensed-matter context [10–12, 15], our proposalfollows the ideas suggested by Tan et al. [36, 37], and consid-ers the possibility of a CHSH BI violation through the detec-tion of the quadrature phases, in our case, in an optomechani-cal setting.The paper is organized as follows. In Sec. II we introducethe model and discuss the conditions for the violation of theCHSH BI. In Sec. III we describe the numerical results for theviolation of the BI and we show its sensitivity to variationsof other system parameters. Lastly, we discuss the e ff ect ofvarious noise sources on the violation of the inequality in oursetup. a r X i v : . [ qu a n t - ph ] O c t II. MODEL AND EQUATION OF MOTION
The setup considered here is constituted by two electromag-netic resonant cavities (A and C, respectively) –either in theoptical or microwave regime– dispersively coupled to a me-chanical resonator. Following the standard description of op-tomechanical systems [22, 38–41], the Hamiltonian for the
FIG. 1. Schematic of the detection scheme. Outputs of the cavi-ties are directed to di ff erent beam splitters, where they are mixedwith local oscillators (LOs) fields. The mixed signals are sent tophotodetectors D1, E1, D2, and E2, characterized by fields d , e , d , e , respectively. Unlike the case of (balanced) homodyne detec-tion schemes, where the signals emerging from the two branches ofeach beam splitter –in our case directed towards detectors D1 / E1 andD2 / E2– are combined, we keep track of all four signals and theirintensity correlations described by Eqs.(9a-9d). system can be written as H = ω a a † a + ω c c † c + ω m b † b + (cid:16) g a a † a + g c c † c (cid:17) (cid:16) b † + b (cid:17) , (1)where a , c and b represent the lowering operators associatedwith cavity A and C and the mechanical modes, respectively; ω a , ω c , ω m are their resonant frequencies and g a and g c are thesingle-photon radiation pressure couplings for modes a and c with the mechanical mode.Along the lines of the experiment discussed in [33], weassume that each cavity is driven by a strong coherent tone α in , a and α in , c (for cavity A and C, respectively). We con-sider that driving of each cavity is detuned from the cavityresonance: we assume cavity A to be driven with a frequency ω d , a = ω a + ω m (blue mechanical sideband) and cavity C witha frequency ω d , c = ω c − ω m (red mechanical sideband). Inour analysis, we employ the usual description of the systemin terms of quantum Langevin equations [42] for the fluctu-ations around the cavity fields induced by the drives. In thisscenario, we consider the linearized dynamics of the fluctu-ations around the pump tones and replace a → a + α A and c → c + α c (see Appendix A). Moving to a frame rotatingat ω d , a and ω d , c for modes a and c respectively and, defining ∆ x = ω d , x − ω x ( x = a , c ), we obtain the following equations of motion for the fluctuations˙ a = (cid:18) − i ∆ a − κ a (cid:19) a − iG + (cid:16) b † + b (cid:17) + √ κ e , a a i + √ κ i , a a I , (2a)˙ c = (cid:18) − i ∆ c − κ c (cid:19) c − iG − (cid:16) b † + b (cid:17) + √ κ e , c c i + √ κ i , c c I , (2b)˙ b = (cid:18) − ω m − γ (cid:19) b − iG + (cid:16) a † + a (cid:17) − iG − (cid:16) c † + c (cid:17) + √ γ b i , (2c)where G + = g a α A and G − = g c α C are the linearized optome-chanical couplings, and κ a , κ c and γ are the linewidths of thecavities A, C and the mechanical resonator. Moreover, wehave defined a i , a I , c i , c I , b i to be the input operators associ-ated to the external input and internal fields respectively ( i and I ) for cavities A and C and the mechanics, respectively.It is possible to obtain the expression of the cavity fieldsin frequency space by Fourier transforming Eqs. (2a-2c). Thetransformation leads to the following set of linear algebraicequations − i ω a = (cid:18) − i ∆ a − κ a (cid:19) a − iG + (cid:16) b † + b (cid:17) + √ κ e , a a i + √ κ i , a a I , (3a) − i ω c = (cid:18) − i ∆ c − κ c (cid:19) c − iG − (cid:16) b † + b (cid:17) + √ κ e , c c i + √ κ i , c c I , (3b) − i ω b = − γ b − iG + (cid:16) a † + a (cid:17) − iG − (cid:16) c † + c (cid:17) + √ γ b i , (3c)which can be solved through standard techniques. Further-more, according to the input-output theory [42], the opera-tors for the output fields of cavity A are related to the cav-ity operators and to the input noise operators by the relation a o = √ κ e , a a − a i where κ e , a is the external coupling rate forcavity A – and analogously for cavity C.These relations, combined with the solution of Eqs. (3a -3c), allow us to map the the input cavity modes to the outputfields a o , c o in the frequency domain as a o = A d a i + A x c † i + N a , (4a) c o = C d c i + C x a † i + N c . (4b)where the operators N a ( N c ) account for the noise associ-ated with the mechanical resonator and the internal losses ofthe cavity. In addition to these noise sources, we considerthat the external ports of the device represent potential fur-ther noise sources (see Appendix B). While the direct solu-tion of Eqs. (3a-3c) outlined above is su ffi cient to determinethe value of the coe ffi cients in Eqs. (4a), a deeper physicalintuition into the mechanism leading to the quantum corre-lations among the modes –required for the violation of theBI– can be obtained by resorting to the rotating-wave approx-imation (RWA): the full derivation of the expressions for thecoe ffi cients given in Eq. (4a) within the RWA is given in Ap-pendix B, where we also compare RWA results with the fullsolution of Eqs. (3a-3c), which shows that, as it is usually thecase RWA and full results coincide in the so-called sidebandresolved regime( ω m /κ (cid:29) a = − κ a a − iG + (cid:16) b † + b exp [ − i ω m t ] (cid:17) + √ κ e , a a i + √ κ i , a a I , (5a)˙ c = − κ c c − iG − (cid:16) b † exp [2 i ω m t ] + b (cid:17) + √ κ e , c c i + √ κ i , c c I , (5b)˙ b = − γ b − iG + (cid:16) a † + a exp [ − i ω m t ] (cid:17) − iG − (cid:16) c + c † exp [2 i ω m t ] (cid:17) + √ γ b i , (5c)the RWA approximation consists in neglecting the (fast-rotating) time-dependent terms in Eqs. (5a-5c), leading to thefollowing simplified EOMs˙ a = − κ a a − iG + b † + √ κ e , a a i + √ κ i , a a I , (6a)˙ c = − κ c c − iG − b + √ κ e , c c i + √ κ i , c c I , (6b)˙ b = − γ b − iG + a † − iG − c + √ γ b i . (6c)We rewrite Eqs. (6a-6c) in terms of two Bogolyubov operators η a = cosh ξ c + sinh ξ a † , (7a) η c = cosh ξ a + sinh ξ c † , (7b)where cosh ξ = G − / G , sinh ξ = G + / G with G = (cid:112) G − − G + and rewrite Eq. (6a-6c) in terms of the Bogolyubov modes η a and η c as ˙ η a = − κ η a − i G b + √ κ e η a , i + √ κ i η a , I , (8a)˙ η c = − κ η c + √ κ e η c , i + √ κ i η c , I , (8b)˙ b = − γ b − i G η a + √ γ b i . (8c)where η a , i = cosh ξ c i + sinh ξ a † i , η c , i = cosh ξ a i + sinh ξ c † i .Eqs. (8a-8c) thus show that it is possible to recast the prob-lem in terms of the dynamics of two operators ( η a and η c )resulting from the action of a two-mode squeezing opera-tor on the original field operators, suggesting that the out-put modes of the field are entangled and therefore that, po-tentially, nonlocal correlations are present. For an incom-ing signal at the resonance frequency of either cavity, theRWA analysis of the problem allows us to establish that inthe limit of large cooperativity ( C − = G − /κγ (cid:29)
1) wehave that A d = r e / (1 − r ) − , C d = − r e r / (1 − r ) − A x = − C x = r e r / (1 − r ), where r = G + / G − and r e = κ e /κ isthe ratio between the external coupling rate to the total lossesof the cavities.Nevertheless, in our analysis, unless explicitly stated, weshow the results for the full solution of Eqs. (3a-3c) (i.e. with-out resorting to the RWA) and we assume that both cavitieshave the same environment coupling properties.In our discussion, we will consider that, in addition to thestrong coherent tone α A and α C , cavity A and cavity C are also driven by small coherent input fields α i and χ i , respectively.In this scenario, the relation between input and output fieldsgiven by Eq. (4a) allows us to evaluate the response at theoutput of each cavity to the fields α i and χ i . The correlationsbetween a o and c o introduced by the combined dynamics ofthe two cavities and of the mechanical resonator represent thekey ingredient for the generation of the correlations requiredto violate the CHSH BI.As anticipated, the protocol that we have in mind is basedon the measurement of the field intensity at two pairs of de-tectors D1 / E1, D2 / E2 corresponding to the photodetectionscheme of the Ref. [36] after mixing the signals a o and c o emerging from the optomechanical device with two local os-cillators (LOs). This detection scheme is closely related toa balanced homodyne detection setup, in the case discussedhere, however, both signals originating from the beam split-ters are recorded in order to measure the required correlations.More specifically, the outputs a o and c o of the cavities are di-rected to two detectors, constituted by a beam splitter and twophotodetectors each (see Fig. 1). At each detector the signalfield is mixed with a coherent field of a LO β , by a 50:50beam splitter. The signals originating form the beam splittersare then measured at the photodetectors D1, E1, D2, and E2.In order to evaluate the correlations needed for the verificationof the violation of the CHSH inequality, we define the correla-tions pairs D / E D / E ff erent phases of the LOsas R + + ( θ, φ ) = (cid:68) d † d † d d (cid:69) , (9a) R + − ( θ, φ ) = (cid:68) d † e † e d (cid:69) , (9b) R − + ( θ, φ ) = (cid:68) e † d † d e (cid:69) , (9c) R − − ( θ, φ ) = (cid:68) e † e † e e (cid:69) , (9d)where d / d , e / e are the fields associated with each of pairof photodetectors, and θ and φ represent the coherent fieldphases of each LO. In the language of quantum optics, R i j ( i , j = ± ) represent the intensity correlations among photocur-rents in the 4 detectors, e.g. R + − measures correlations be-tween the photocurrent in D and the one in E . The setupwe are discussing here is analogous to the more conventionalpolarization experiments [5–9, 13, 14, 16, 17]: in these ex-periments each channel D1 / E1, D2 / E2 is selected by adjust-ing the angle of a polarizer at each detection branch. Theparallel with the polarization experiments, is represented bythe fact that, by changing the phase of the LO, we are se-lecting the detection channel, essentially performing a quadra-ture measurement of the output fields originating from of theoptomechanical system, since it is possible to relate R i j inEqs. (9a-9d) to the quadratures X a ( θ ) = (cid:16) a † o e i θ + a o e − i θ (cid:17) / √ X c ( φ ) = (cid:16) c † o e i φ + c o e − i φ (cid:17) / √ d and e asthe result of the mixing between the LO field b LO1 and a o , theoutput field of cavity A as d = √ η a o + i (cid:112) − η b LO1 , (10a) e = √ η b LO1 + i (cid:112) − η a o , (10b)where η is the transmissivity of the beam splitter associatedwith the lhs detector of Fig. 1. Therefore, as discussed morein detail in Appendix C, we can express the correlators inEqs. (9a-9d) in terms of a o and c o .Regardless of the physical implementation, either in the op-tical of the microwave frequency range, the original formula-tion of the CHSH inequality is given by the following relation | S | = | E ( θ , φ ) + E ( θ , φ ) + E ( θ , φ ) − E ( θ , φ ) | ≤ , (11)where, in our case, we have E ( θ, φ ) = R + + + R − − − R − + − R + − R + + + R − − + R − + + R + − . (12)In terms of correlations of the original optomechanical fields a o and c o , Eq. (12) can be written as E = C cos (cid:104) ¯ θ − ¯ φ (cid:105) + D cos (cid:104) ¯ θ + ¯ φ (cid:105) , (13)where C = |(cid:104) a † o c o (cid:105)| / ZD = |(cid:104) a o c o (cid:105)| / Z with Z = (cid:113) (cid:104) a † o c † o c o a o (cid:105) + (cid:104) a † o a o + c † o c o (cid:105) and we have absorbedthe phases of (cid:104) a o c o (cid:105) and (cid:104) a † o c o (cid:105) in the definitions of ¯ θ and ¯ φ ,and | β | = | β | = | β | = (cid:104) a † o c † o c o a o (cid:105) / . It can be shown thatthe latter condition maximizes the violation of the inequalitygiven in Eq. (11) – see Appendix C.The maxima of S occur when ¯ θ =
0, ¯ φ = − ζ , ¯ θ (cid:48) = − π/ φ (cid:48) = ζ and with a maximum value is given by S = √ √ C + D sin( ζ − ζ ) , (14)where tan( ζ ) = ( C + D ) / ( C − D ). It is clear that the CHSH in-equality given in Eq. (11), can be translated into the condition[36] F = C + D < . (15)The BI test in the optomechanical setting described byEq. (15) can be straightforwardly evaluated considering thedefinitions of C and D , and the input-output relations givenby Eqs. (2a - 2c). III. RESULTS AND DISCUSSION
In Fig. 2 we have plotted the value of F as a function of theratio between the linearized pump strengths r = G + / G − andthe coherent inputs α i and χ i in the absence of noise sources (b)(a) 0 0.1 0.2 0.3 0.400.050.10.150.2 FIG. 2. a. Value of F as a function of α i and r = G + / G − . Param-eters: κ a = κ c = κ = . κ e = . κ , γ = × − – all energiesexpressed in units of ω m ( (cid:126) =
1) throughout the manuscript. Thedashed curve corresponds to the exact boundary region F = / b. Boundary F = / ff erent values of r e = κ e /κ . Smaller regions are associated withsmaller values of r e . For α i (cid:39) . ff erent values of r e , hinting a non-trivial relation between entanglement and violation of the CHSH BI(see text). The solid line corresponds to the exact boundary as in a.,the dashed line correspond to the expression given in Eq. (16). for parameters compatible with present-day experimental ca-pabilities. Form this figure one can see that there is a finiteparameters region for which the inequality is violated. In thelimit of large cooperativity ( C − (cid:29) r leading to a violation of the BI is obtained for α i , χ i → r = (15 + √ − / . Furthermore the maximum vi-olation of the BI F = α i , χ i → r → + .More specifically, for large cooperativity ( C − (cid:29) F exhibits a discontinuity at α in ( = χ in ) = r =
0. As ex-pected, for G + = r =
0) modes a o and c o are not entangledand F = r − dependence of the function F iscontrasted by the r − dependence of entanglement. From thedefinition of the parameters A d and A x , following Eqs. (4a),it is possible to see that, since the squeezing parameter z = arctanh [ A x / A d ] → ∞ for r → − , one obtains an infinitelysqueezed state in this regime. This seemingly contradictoryconclusion, analogous to the one derived in [36, 37], is how-ever corroborated by observing that, for mixed states, the re-lation between entanglement and nonlocality exhibits aspectsthat are still not fully understood [3]: in particular it can beshown that maximally entangled states ( r → − , in our case)do not necessarily violate locality constraints, which, con-versely, can be violated by less entangled states [43–45]. Inour setup, this complex interplay between entanglement andnonlocality is further exemplified by the crossing between the F = / ff erent values of r e : as it ispossible to see in Fig. 2, for intermediate values of the co-herent drive ( α i (cid:39) . − . r e lead to a reduction of the value of r for which the violation isobserved.It is clear that a violation of the CHSH inequality is possi-ble only for small values of the input fields α i and χ i , and forsmall values of r implying | A d | = | C d | ≈ | A x | = | C x | (cid:28) (cid:104) a † o a o (cid:105) (cid:39) | A d | (cid:104) a † i a i (cid:105) ≈ . (cid:104) c † o c o (cid:105) (cid:39) | C d | (cid:104) c † i c i (cid:105) ≈ .
1. This condition combines the concomitant requirementsthat the value of F and the output signals have to be maxi-mized. In order to gain better insight on the range of physicalparameters for which the BI inequality is violated, we can es-tablish an approximate analytical expression for the maximumvalue of α i violating the inequality as α i = (cid:113) r e ¯ r (cid:0) − r − r −
12 ¯ r (cid:1) / (cid:0) K ¯ r + K ¯ r + K (cid:1) , (16)where K = r , K = − r e + r ) and K = − r e ) .Eq. (16) is obtained as a second-order expansion of F in theinput field intensity α evaluated here for the RWA solution ofthe problem.So far, the discussion has focused on the ideal situation forwhich the e ff ect of noise is negligible. In the following, weaddress the role played by the di ff erent environmental noisesources. In particular, we take into account the presence of athermal environment for the mechanical resonator (¯ n m , “me-chanical noise”), for the two resonant cavities (¯ n i , “internalnoise”) and to the noise associated with the coupling of thetwo resonant cavities to the input and output ports (¯ n e , “exter-nal noise”). Without loss of generality, in Eq. (17) we have as-sumed that the noise temperature for the two cavities is equaland that all noise sources are independent. If we consider thee ff ect of the noise on F to the first order, we can write F = F − F m ¯ n m − F e ¯ n e − F i ¯ n i , (17)where F is the quantity previously considered for the viola-tion of the BI, the second term represents the contribution as-sociated with the mechanical noise, and the third (fourth) termdescribes the external (internal) noise contribution due to thethermal environment associated with the cavity modes. Thesensitivity of the BI violation to the noise terms is encoded inthe coe ffi cients F e , F i and F m : the larger the coe ffi cients, themore each noise term contributes to the reduction of the valueof F and, therefore, to the reduction of the region for whichthe BI is violated. An approximate expression for the factors (a)(b) 0 0.5 1 1.5 200.20.40.60.80 0.05 0.1 0.15 0.2 0.2500.20.40.60.8 FIG. 3. Dependence of the value of F (for r → r opt and α i , χ i → n m , blue –flattest– curve), internal noise (¯ n i , red –intermediate–curve) and external noise (¯ n e , black –steepest– curve). Solid linescorrespond to the exact solution from the equations of motion witheach noise source considered independently. Dashed lines are theapproximations given in Eq. (7a) and Eqs.(18a-18d). Values of F above the horizontal dashed line at F = / a. κ = . κ e = . κ , G − = . γ = · − . b. κ e = . κ , all other parameters as in a. . appearing in Eq. (17) can be obtained expanding the RWA ap-proximation for F , F m , F e , F i , to the lowest order in 1 / C − F = (2 r − + r (18a) F m = − r − + r (10 r − C − (18b) F e = − (2 r − + r (16 r − r r + r r e (18c) F i = − (2 r − + r (16 r − r r i (18d)The portion of the noise associated with the mechanics anddescribed in the linear approximation by F m –see Eq. (18b)–can be modified by tuning the parameter C − . This dependencecan be understood as the result of a sideband cooling processoperated by the drive of cavity C, which is driven on the redsideband. In addition, F i can be reduced by minimizing thecontribution of internal losses –see Fig. 3–, whereas F e cannotbe altered significantly and thus represents the most criticalparameter. (b) 0 0.1 0.2 0.3 0.400.050.10.150.20 0.1 0.2 0.3 0.400.050.10.150.2(a)(c) 0 0.1 0.2 0.3 0.400.050.10.150.2 FIG. 4. Noise-dependence of the F = / This conclusion is corroborated by Fig. 4, where we havedepicted the separate e ff ects of di ff erent noise sources on thevalue of F , it is clear that the input noise ¯ n e represents themost sensitive parameter in the violation of the CHSH in-equality. In this perspective, we thus select a value of r that,whilst representing a sub-optimal choice (i.e. F <
1) for thenoiseless case, allows for the largest possible value of n e and n i compatible with the violation of the BI given in Eq. (15). Inthe linearized regime described by Eq. (17), in the presence ofcavity (external and internal) noise only, the relation describ-ing the boundary for the violation of the BI can be expressed as F ( r ) + F e ( r ) n e + F i ( r ) n i =
12 (19)where we have supposed that r e is held fixed. From Eqs. (18a-18d) we can write Eq. (19) as F ( r ) − + F T ( r ) n T = F i / r i = F i r e / (cid:16) r + r (cid:17) = F T and n T = (cid:16) r + r (cid:17) / r e n e + r i n i . From Eq. (20) n T = [1 / − F ( r )] / F T ( r ) can be straight-forwardly maximized yielding the optimal value for r = r opt .We would like to stress however that the contribution asso-ciated with n e assumes that the baths for the cavities are uncor-related with each other, which represents a somewhat worst-case scenario. The potential presence of correlated noise canbe considered, from the perspective of the BI violation, as acontribution to the input signals α i and χ i .For a microwave setting, we can assume that the cav-ity internal and external thermal populations are set by thebase temperature of the dilution fridge ( T = ω c = π
10 GHz) corresponding to n i = n e (cid:39) . T =
300 K, ω c = π
500 THz) we have n i = n e (cid:39) .
02. While in both cases thedeviation from ideality is significant, the BI is still clearly vi-olated both for the microwave setting (F (cid:39) .
56, for r e = . (cid:39) .
58, for r e = .
99) and for the optical case (
F (cid:39) . r e = . F (cid:39) .
60, for r e = . F . This e ff ect isclosely related to the physics of the quantum-limited amplifierdiscussed in Ref. [33]: in both cases the mechanics, while me-diating the interaction required to generate the output fields, isconcomitantly cooled by the pumping tones. IV. CONCLUSION
We have discussed here a potential CHSH Bell inequalitytest based on a quadrature phase coherence measurement inan optomechanical setting. We have shown that it is possi-ble to violate the CHSH Bell inequality in an optomechanicalsetting by weakly driving two cavity / one mechanics device.Furthermore, we have demonstrated that, while the thermalnoise associated with cavities and mechanical degrees of free-dom degrades the performances of the device proposed here,the latter is naturally suppressed by the working principle ofour device. We hypothesize that our proposal could be imple-mented either in an optical or in a circuit QED setting. ACKNOWLEDGMENTS
We thank Elli Selenius, Mika Sillanp¨a¨a and Caspar F.Ockeloen-Korppi for useful discussions. This work was sup-ported by the Academy of Finland (Contract No. 275245) andthe European Research Council (Grant No. 670743).
Appendix A: Equations of motion
We derive here the equations of motion for the 2 cavities / a = − ( i ω a + κ a a − ig a a ( b + b † ) + √ κ e , a a i + √ κ i , a a I , (A1a)˙ c = − ( i ω c + κ c c − ig c c ( b + b † ) + √ κ e , c c i + √ κ i , c c I , (A1b)˙ b = − ( i ω m + γ b − ig a a † a − ig c c † c + √ γ b i , (A1c)where κ a = κ e , a + κ i , a is the total cavity decay rate where κ i , a and κ e , a are the internal and external cavity decay rates, (analogousrelations hold for cavity C). The fields a i , c i , b i , represent the input fields driving the cavities and the mechanical resonator,whereas a I and c I describe the contributions from the internal noise for cavity A and cavity C, respectively. In the main text weconsider the case of a strong drive for both cavities ( with amplitudes α in , A and α in , C , at frequencies ω d , A and ω d , C , respectively).In this case, the quantum Langevin equations given in Eqs. (A1a-A1c) can be linearized around the the cavity fields induced bythe pump tones, leading to the following expression for the steady state for the cavity fields¯ α A = α in , A κ a + i ( ω a − g a α A ( b s + b ∗ s )) e − i ω d , A t = α A e − i ω d , A t , (A2a)¯ α C = α in , C κ c + i ( ω c − g c α C ( b s + b ∗ s )) e − i ω d , C t = α C e − i ω d , C t (A2b)while the equations for the fluctuations around the steady-state values are given by˙ a = − ( i ω a + κ a a − ig a ¯ α A ( b + b † ) + √ κ e , a a i + √ κ i , a a I , (A3a)˙ c = − ( i ω c + κ c c − ig c ¯ α C ( b + b † ) + √ κ e , c c i + √ κ i , c c I , (A3b)˙ b = − ( i ω m + γ b − ig a ¯ α A ( a + a † ) − ig c ¯ α C ( c + c † ) + √ γ b i . (A3c)Moving to a frame rotating at ( ω d , a , ω d , c and ω m for cavity A, cavity C and mechanics respectively), by substituting the valuesof ¯ α A and ¯ α C in Eqs. (A3a-A3c), the corresponding linearized quantum Langevin equations for the fluctuations around thestationary values induced by the pumps (Eqs. (2a,2c) of the main text), are˙ a = (cid:18) − i ∆ a − κ a (cid:19) a − iG + (cid:16) b † + b (cid:17) + √ κ e , a a i + √ κ i , a a I , (A4a)˙ c = (cid:18) − i ∆ c − κ c (cid:19) c − iG − (cid:16) b † + b (cid:17) + √ κ e , c c i + √ κ i , c c I , (A4b)˙ b = (cid:18) − i ω m − γ (cid:19) b − iG + (cid:16) a † + a (cid:17) − iG − (cid:16) c † + c (cid:17) + √ γ b i , (A4c)where G + = g a α A and G − = g c α C are the e ff ective linearized couplings (without loss of generality, hereafter we assume that κ a = κ c = κ ). Appendix B: Input / output equations in the rotating-wave approximation While the coe ffi cients A d , A x , C d , C x –and therefore the condition expressing the violation of the BI–given in Eqs. (4a,4b) ofthe main text can be obtained without resorting to RWA, in order to outline the essential physical process behind our proposal,we determine here the explicit analytical expression for these coe ffi cients within the RWA.In Fig. 5, it is possible to see how the validity of the RWA in the determination of the BI violation relies on the condition ω m (cid:28) / O coe ffi cients A d , A x , C d , C x within the RWA, we define a Bogolyubov unitary transformation of FIG. 5. Comparison between the value of F calculated from the full solution of the equations of motion (full lines, κ = . , . , .
1, largervalues correspond to smaller regions for which F > / the optical modes operator as η a = cosh ξ c + sinh ξ a † , (B1a) η c = cosh ξ a + sinh ξ c † , (B1b)where cosh ξ = G − / G , sinh ξ = G + / G with G = (cid:112) G − − G + and rewrite Eq. (A4a-A4c) in terms of the Bogolyubov modes η a and η c as ˙ η a = − κ η a − i G b + √ κ e η a , i + √ κ i η a , I , (B2a)˙ η c = − κ η c + √ κ e η c , i + √ κ i η c , I , (B2b)˙ b = − γ b − i G η a + √ γ b i . (B2c)where η a , i = cosh ξ c i + sinh ξ a † i , η c , i = cosh ξ a i + sinh ξ c † i . We then transform the quantum Langevin equations of the twoBogolyubov modes η a and η c to Fourier domain η a = χ a + χ m χ a G ( √ κ e η a , i + √ κ i η a , I ) − i χ m χ a G + χ m χ a G √ γ b i , (B3a) η c = χ a ( √ κ e η c , i + √ κ i η c , I ) , (B3b)where χ m = (cid:18) γ − i ω (cid:19) − and χ a = (cid:18) κ − i ω (cid:19) − . Since, according to the input-output theory [42], the operator for the output field isrelated to the cavity and to the input noise operator by the relation a o = √ κ e a − a i and c o = √ κ e c − c i by using the transformation a = cosh ξη c − sinh ξη † a and c = cosh ξη a − sinh ξη † c , the outputs of the two cavity modes can be written as a o = ( κ e A aa − a i + κ e A ac c † i + √ κ i κ e A aa a I + √ κ i κ e A ac c † I + i √ γκ e G + ( χ a χ m ) − + G b † i , (B4) c o = ( κ e A cc − c i + κ e A ca a † i + √ κ i κ e A cc c I + √ κ i κ e A ca a † I − i √ γκ e G − ( χ a χ m ) − + G b i , (B5)where A aa = χ a cosh ξ − χ ea sinh ξ, A cc = χ e a cosh ξ − χ a sinh ξ, (B6a) A ac = ( χ a − χ e a ) cosh ξ sinh ξ, A ca = ( χ ea − χ a ) cosh ξ sinh ξ, (B6b)and χ ea = χ a (cid:16) + G χ a χ m (cid:17) − represents the e ff ective cavity response in presence of the two-tone optomechanical drive. It ispossible to write Eq. (B4-B5) in more compact form as given in Eqs. (4a,4b) of the main text as a o = A d a i + A x c † i + N a (B7a) c o = C d c i + C x a † i + N c (B7b)where N a = A d , I a I + A x , I c † I + A m b † i , N c = C d , I c I + C x , I a † I + C m b i represent the operators associated with the mechanical and cavity internal noise. Furthermore, the coe ffi cients relating input andnoise operators to the output are given by A d = κ e A aa − , C d = κ e A cc − , A m = + i √ γκ e G + χ ea /χ a A x = κ e A ac , C x = κ e A ca , C m = − i √ γκ e G − χ ea /χ a A d , I = √ κ i κ e A aa , C d , I = √ κ i κ e A cc , A x , I = √ κ i κ e A ac , C x , I = √ κ i κ e A ca . In the limit of large cooperativity C − = G /κγ (cid:29) ffi cients can be written as A d = r e − r − , C d = − r e r − r − , A x = rr e − r = − C x , A m = − i r √ r e √ C − ( − r ) = rC m , A d , I = √ r e r i − r , C d , I = − r A d , I , A x , I = r √ r e r i − r = − C x , I , where r = G + / G − , r e = κ e /κ and r i = κ i /κ . Appendix C: CHSH violation
We derive here the relation between the usual condition for the violation of CHSH inequality expressed by Eq. (11), andEq. (15) of the main text. To this end, we evaluate the quantity defined in Eq. (9a-9d) of the main text in terms of the outputcorrelators of the optomechanical system. For beam splitters of transmissivity given by η and η , the detected fields are givenby d = √ η a o + i (cid:112) − η b LO1 , (C1a) d = √ η c o + i (cid:112) − η b LO2 , (C1b) e = √ η b LO1 + i (cid:112) − η a o , (C1c) e = √ η b LO2 + i (cid:112) − η c o , (C1d)where b , are the fields of the local oscillators. With the definitions given by Eq. (C1a - C1d) and assuming that the LO state isdescribed by a coherent state (cid:104) b | LO | = (cid:105) β exp [ i θ ], we can calculate (cid:104) d † d (cid:105) = (1 − η ) (cid:104) b † LO1 b LO1 (cid:105) + η (cid:104) a † o a o (cid:105) − i (cid:112) η (1 − η ) (cid:104) (cid:104) b † LO1 a o (cid:105) − (cid:104) a † o b LO1 (cid:105) (cid:105) = (1 − η ) | β | + η (cid:104) a † o a o (cid:105) + (cid:112) η (1 − η ) | β | (cid:104) X θ a (cid:105) , (C2)where X θ a = X a ( θ + π/ = − i (cid:16) a o exp [ − i θ ] − a † o exp [ i θ ] (cid:17) . (cid:104) e † e (cid:105) = η | β | + (1 − η ) (cid:104) a † o a o (cid:105) − (cid:112) η (1 − η ) | β | (cid:104) X θ a (cid:105) , (C3)and analogously for detector 2.In addition to the intensities at the detectors D1, D2, E1, E2 we have to evaluate the correlations among them. To this end weevaluate he full expression for (cid:104) d † d † d d (cid:105) which is given by R + + ( θ, φ ) = (cid:104) d † d † d d (cid:105) = (1 − η ) (1 − η ) (cid:104) b † LO1 b † LO2 b LO2 b LO1 (cid:105) + i (cid:112) η (1 − η )(1 − η ) (cid:16) (cid:104) a † o b † LO2 b LO2 b LO1 (cid:105) − (cid:104) b † LO1 b † LO2 b LO2 a o (cid:105) (cid:17) + i (cid:112) η (1 − η )(1 − η ) (cid:16) (cid:104) b † LO1 c † o b LO2 b LO1 (cid:105) − (cid:104) b † LO1 b † LO2 c o b LO1 (cid:105) (cid:17) + η (1 − η ) (cid:104) a † o b † LO2 b LO2 a o (cid:105) + η (1 − η ) (cid:104) b † LO1 c † o c o b LO1 (cid:105)− √ η η (cid:112) (1 − η ) (1 − η ) (cid:16) (cid:104) b † LO1 b † LO2 c o a o (cid:105) + (cid:104) a † o c † o b LO2 b LO1 (cid:105) − (cid:104) b † LO1 c † o b LO2 a o (cid:105) − (cid:104) a † o b † LO2 c o b LO1 (cid:105) (cid:17) + i (cid:112) η (1 − η ) η (cid:16) (cid:104) a † o c † o c o b LO1 (cid:105) − (cid:104) b † LO1 c † o c o a o (cid:105) (cid:17) + i (cid:112) η (1 − η ) η (cid:16) (cid:104) a † o c † o b LO2 a o (cid:105) − (cid:104) a † o b † LO2 c o a o (cid:105) (cid:17) + η η (cid:104) a † o c † o c o a o (cid:105) (C4)and, since we assume the LO to be in a coherent state, we have that b LO1 → | β | exp [ i θ ], b LO2 → | β | exp (cid:2) i φ (cid:3) , we get R + + ( θ, φ ) = (cid:104) d † d † d d (cid:105) = (1 − η ) (1 − η ) | β β | + (1 − η ) (cid:112) η (1 − η ) | β | | β | (cid:104) X θ a (cid:105) + (1 − η ) (cid:112) η (1 − η ) | β | | β | (cid:104) X φ c (cid:105) + √ η η (cid:112) (1 − η ) (1 − η ) | β β | (cid:104) : X θ a X φ c : (cid:105) + η (1 − η ) | β | (cid:104) a † o a o (cid:105) + η (1 − η ) | β | (cid:104) c † o c o (cid:105) + η (cid:112) η (1 − η ) | β | (cid:104) : X θ a c † o c o : (cid:105) + η (cid:112) η (1 − η ) | β | (cid:104) : X φ c a † o a o : (cid:105) + η η (cid:104) a † o c † o c o a o (cid:105) , (C5)where with (cid:104) :: (cid:105) we denote normal ordering, i.e. (cid:68) : X θ a X φ c : (cid:69) = − (cid:68) a † o c † o exp (cid:2) i ( θ + φ ) (cid:3) + c o a o exp (cid:2) − i ( θ + φ ) (cid:3) − c † o a o exp (cid:2) − i ( θ − φ ) (cid:3) − a † o c o exp (cid:2) i ( θ − φ ) (cid:3)(cid:69) . (C6)The other terms are obtained replacing (where appropriate) √ η → i (cid:112) − η i and (cid:112) − η i → − i √ η in Eqs. (C4) and (C5).Using the expression of R ± ± ( θ, φ ) given by Eq. (C5) and assuming 50:50 beam splitters, i.e. η = η = /
2, the correlationcoe ffi cient E ( θ, φ ) in Eq. (12) of the main text can be written as E ( θ, φ ) = | β β | (cid:68) : X θ a X φ c : (cid:69) | β | | β | + | β | (cid:68) c † o c o (cid:69) + | β | (cid:68) a † o a o (cid:69) + (cid:68) a † o c † o c o a o (cid:69) . (C7)In addition, it is possible to show [36] that the optimal value of the local oscillators for the violation of the Bell inequality isgiven by β = β = (cid:104) a † o c † o c o a o (cid:105) / . At this point, with the expression of the correlators given in Eqs. (C2-C6), we are in theposition to express the correlation function E ( θ, φ ) as E ( θ, φ ) = C cos(¯ θ − ¯ φ ) + D cos(¯ θ + ¯ φ ) , (C8)where ¯ θ − ¯ φ = θ − φ − arg (cid:104) a † o c o (cid:105) , ¯ θ + ¯ φ = θ + φ − arg (cid:104) a † o c † o (cid:105) .The maxima of S occur when ¯ θ =
0, ¯ φ = − ζ , ¯ θ (cid:48) = − π/ φ (cid:48) = ζ and with a maximum value is given by S = √ √ C + D sin( ζ − ζ ) , (C9)where tan( ζ ) = ( C + D ) / ( C − D ). The CHSH inequality, as expressed in Eq. (11), can be written as F = C + D <
12 (C10)given in Eq. (15) of the main text.1
Appendix D: Output field correlators
In order to verify the violation of the CHSH inequality in the setup described in the text, we evaluate C = (cid:12)(cid:12)(cid:12)(cid:12)(cid:68) a † o c o (cid:69)(cid:12)(cid:12)(cid:12)(cid:12) (cid:113)(cid:68) a † o c † o c o a o (cid:69) + (cid:68) c † o c o (cid:69) + (cid:68) a † o a o (cid:69) , (D1) D = | (cid:104) a o c o (cid:105) | (cid:113)(cid:68) a † o c † o c o a o (cid:69) + (cid:68) c † o c o (cid:69) + (cid:68) a † o a o (cid:69) . (D2)in the presence of two weak coherent drives for each cavity. In addition we consider the possibility of the presence of thermalnoise for the mechanics and both cavities. The latter can be divided in ”external” i.e. incoming through the driving ports, orinternal. In this case, we can write the input fields as a i = α i + a E and c i = χ i + c E , where χ i and α i represent the weak coherentdrives, while a E and c E are the operators associated to the ”external” thermal noise.In this framework, the correlations required to evaluate the CHSH inequality are given by (cid:68) a † o a o (cid:69) = | A d | (cid:16) | α i | + ¯ n e , a (cid:17) + | A x | (cid:16) | χ i | + ¯ n e , c + (cid:17) + A ∗ d A x α ∗ i χ ∗ i + A ∗ x A d α i χ i (D3) + (cid:12)(cid:12)(cid:12) A d , I (cid:12)(cid:12)(cid:12) ¯ n i , a + (cid:12)(cid:12)(cid:12) A x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , c + (cid:1) + | A m | (¯ n m + , (cid:68) c † o c o (cid:69) = | C d | (cid:16) | χ i | + ¯ n e , c (cid:17) + | C x | (cid:16) | α i | + ¯ n e , a + (cid:17) + C ∗ d C x α ∗ i χ ∗ i + C ∗ x C d α i χ i (D4) + (cid:12)(cid:12)(cid:12) C d , I (cid:12)(cid:12)(cid:12) ¯ n i , c + (cid:12)(cid:12)(cid:12) C x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , a + (cid:1) + | C m | ¯ n m , (cid:68) a † o c o (cid:69) = A ∗ d C x α ∗ + (cid:16) A ∗ d C d + A ∗ x C x (cid:17) α ∗ i χ i + A ∗ x C d χ , (D5) (cid:104) a o c o (cid:105) = A d C x (cid:16) | α i | + ¯ n e , a + (cid:17) + A x C d (cid:16) | χ i | + ¯ n e , c (cid:17) + A d C d α i χ i + A x C x α ∗ i χ ∗ i (D6) + A d , I C x , I (cid:0) ¯ n i , a + (cid:1) + A x , I C d , I ¯ n i , c + A m C m ¯ n m . (cid:68) a † o c † o a o c o (cid:69) = | A d C x | (cid:16) | α i | + | α i | + | α i | ¯ n e , a + n , a (cid:17) (D7) + | A x C d | (cid:16) | χ i | + | χ i | + | χ i | ¯ n e , c + n , c + n e , c + (cid:17) + | A d | (cid:0) | α i | + ¯ n e , a (cid:1) (cid:20) | C d | (cid:16) | χ i | + ¯ n e , c (cid:17) + (cid:12)(cid:12)(cid:12) C d , I (cid:12)(cid:12)(cid:12) ¯ n i , c + (cid:12)(cid:12)(cid:12) C x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , a + (cid:1) + | C m | ¯ n m (cid:21) + | A x | (cid:0) | χ i | + ¯ n e , c + (cid:1) (cid:20) | C x | (cid:16) | α i | + ¯ n e , a + (cid:17) + (cid:12)(cid:12)(cid:12) C d , I (cid:12)(cid:12)(cid:12) ¯ n i , c + (cid:12)(cid:12)(cid:12) C x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , a + (cid:1) + | C m | ¯ n m (cid:21) + (cid:12)(cid:12)(cid:12) A d , I (cid:12)(cid:12)(cid:12) ¯ n i , a (cid:20) | C d | (cid:16) | χ i | + ¯ n e , c (cid:17) + | C x | (cid:16) | α i | + ¯ n e , a + (cid:17) + (cid:12)(cid:12)(cid:12) C d , I (cid:12)(cid:12)(cid:12) ¯ n i , c + | C m | ¯ n m (cid:21) + (cid:12)(cid:12)(cid:12) A x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , c + (cid:1) (cid:20) | C d | (cid:16) | χ i | + ¯ n e , c (cid:17) + | C x | (cid:16) | α i | + ¯ n e , a + (cid:17) + (cid:12)(cid:12)(cid:12) C x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , a + (cid:1) + | C m | ¯ n m (cid:21) + | A m | (¯ n m + (cid:20) | C d | (cid:16) | χ i | + ¯ n e , c (cid:17) + | C x | (cid:16) | α i | + ¯ n e , a + (cid:17) + (cid:12)(cid:12)(cid:12) C d , I (cid:12)(cid:12)(cid:12) ¯ n i , c + (cid:12)(cid:12)(cid:12) C x , I (cid:12)(cid:12)(cid:12) (cid:0) ¯ n i , a + (cid:1)(cid:21) + (cid:12)(cid:12)(cid:12) A d , I C x , I (cid:12)(cid:12)(cid:12) n I2a + (cid:12)(cid:12)(cid:12) A x , I C d , I (cid:12)(cid:12)(cid:12) (cid:16) n I2c + (cid:17) + | A m C m | (cid:16) ¯ n + n m + (cid:17) + A ∗ d C ∗ d α ∗ i χ ∗ i (cid:104) A x C x α ∗ i χ ∗ i + A d , I C x , I ¯ n i , a + A x , I C d , I (cid:0) ¯ n i , c + (cid:1) + A m C m (¯ n m + (cid:105) + A ∗ d C ∗ x (cid:16) | α i | + ¯ n e , a (cid:17) (cid:104) A x C d (cid:16) | χ i | + ¯ n e , c + (cid:17) + A d , I C x , I ¯ n i , a + A x , I C d , I (cid:0) ¯ n i , c + (cid:1) + A m C m (¯ n m + (cid:105) + A ∗ x C ∗ d (cid:16) | χ i | + ¯ n e , c + (cid:17) (cid:104) A d C x (cid:16) | α i | + ¯ n e , a (cid:17) + A d , I C x , I ¯ n i , a + A x , I C d , I (cid:0) ¯ n i , c + (cid:1) + A m C m (¯ n m + (cid:105) + A ∗ x C ∗ x α i χ i (cid:2) A d C d α i χ i + A d , I C x , I ¯ n i , a + A x , I C d , I (cid:0) ¯ n i , c + (cid:1) + A m C m (¯ n m + (cid:3) + A ∗ d , I C ∗ x , I ¯ n i , a (cid:104) A d C d α i χ i + A d C x (cid:16) | α i | + ¯ n e , a (cid:17) + A x C d (cid:16) | χ i | + ¯ n e , c + (cid:17) + A x C x α ∗ i χ ∗ i ++ A x , I C d , I (cid:0) ¯ n i , c + (cid:1) + A m C m (¯ n m + (cid:105) + A ∗ x , I C ∗ d , I (cid:0) ¯ n i , c + (cid:1) (cid:104) A d C d α i χ i + A d C x (cid:16) | α i | + ¯ n e , a (cid:17) + A x C d (cid:16) | χ i | + ¯ n e , c + (cid:17) + A x C x α ∗ i χ ∗ i ++ A d , I C x , I ¯ n i , a + A m C m (¯ n m + (cid:105) + A ∗ m C ∗ m (¯ n m + (cid:104) A d C d α i χ i + A d C x (cid:16) | α i | + ¯ n e , a (cid:17) + A x C d (cid:16) | χ i | + ¯ n e , c + (cid:17) + A x C x α ∗ i χ ∗ i ++ A d , I C x , I ¯ n i , a + A x , I C d , I (cid:0) ¯ n i , c + (cid:1) (cid:105) + | A d | C ∗ d C x χ ∗ i (cid:16) α ∗ i | α i | + α ∗ i ¯ n e , a (cid:17) + | A d | C d C ∗ x χ i (cid:16) α i | α i | + α i ¯ n e , a (cid:17) + | A x | C ∗ d C x α ∗ i (cid:16) χ ∗ i | χ i | + χ ∗ i ¯ n e , c + χ ∗ i (cid:17) + | A x | C d C ∗ x α i (cid:16) χ i | χ i | + χ i ¯ n e , c + χ i (cid:17) + A ∗ d A x | C d | α ∗ i (cid:16) χ ∗ i | χ i | + χ ∗ i ¯ n e , c + χ ∗ i (cid:17) + A d A ∗ x | C d | α i (cid:16) χ i | χ i | + χ i ¯ n e , c + χ i (cid:17) + A ∗ d A x | C x | χ ∗ i (cid:16) α ∗ i | α i | + α ∗ i ¯ n e , a + α ∗ i (cid:17) + A d A ∗ x | C x | χ i (cid:16) α i | α i | + α i ¯ n e , a + α i (cid:17) . 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