# Achieving steady-state entanglement of remote micromechanical oscillators by cascaded cavity coupling

## Abstract

In this paper, we propose a scheme for generating steady-state entanglement of remote micromechanical oscillators in unidirectionally-coupled cavities. For the system of two mechanical oscillators, we show that when two cavity modes in each cavity are driven at red- and blue-detuned sidebands, respectively, a stationary two-mode squeezed vacuum state of the two mechanical oscillators can be generated with the help of the cavity dissipation. The degree of squeezing is controllable by adjusting the relative strength of the pump lasers. Our calculations also show that the achieved mechanical entanglement is robust against thermal fluctuations of phononic environments. For the case of multiple mechanical oscillators, we find that the steady-state genuine multipartite entanglement can also be built up among the remote mechanical oscillators by the cavity dissipation. The present scheme does not require nonclassical light input or conditional quantum measurements, and it can be realized with current experimental technology.

## I Introduction

Besides fundamental research interests in quantum physics (1), realizing quantum effects of macroscopic objects is crucial for potential applications in ultrahigh precision measurements and quantum information processing (2); (3); (4). Thanks to the recent achievements in ground-state cooling of micromechanical oscillators via optomechanical coupling (5); (6); (7); (8), the emerging field of cavity optomechanics as an interface between mechanical systems and optical field has become a unique platform to study quantum behavior of macroscopic mechanical systems (9); (10); (11); (12); (13); (14). Using well-established quantum optical techniques, optomechanics holds the promise to effectively prepare and manipulate nonclassical mechanical states.

Several schemes have been proposed to establish entanglement between a mechanical element and the driven cavity field (15) or between vibrating membranes or end mirrors (16); (17); (18); (19) by optomechanics. Apart from short-distance mechanical entanglement, remote entanglement between two micromechanical oscillators in separated cavities can also be entangled via injecting squeezed light or conditional quantum measurements (20); (21); (22). It was also showed that weak mechanical entanglement between two distant optomechanical oscillators can be possibly achieved merely by optomechanical coupling (23). The entanglement of remote mechanical elements is of importance for constructing long-distance quantum communication networks (24).

On the other hand, generating quantum states by quantum-reservoir engineering has attracted a lot of attention recently. In this approach, the interaction between system and environment is engineered in such a way that the system relaxes into a desired state. The resulting quantum states are steady, independent of initial conditions, and most importantly robust against incoherent noise. To date, several schemes have been proposed to prepare entangled states of atomic systems by quantum dissipation (25); (26); (27); (28); (29); (30); (31) and the dissipative creation of steady-state entanglement between two separated cold atomic ensembles has been experimentally realized (32).

In this paper, we consider the generation of steady-state entanglement of remote micromechanical oscillators (membranes) by cavity dissipation. We at first investigate the entanglement between two micromechanical membrane oscillators in a cascaded cavity system. In each cavity, a membrane oscillator is coupled to two nondegenerate cavity modes via parametric and beam-splitter-like interactions by driving the relevant cavity modes on blue- and red-detuned sidebands respectively. For negligible mechanical damping, we find that the cavity dissipation can pull the two distant mechanical oscillators into a stationary two-mode squeezed vacuum. It is also shown that the two-mode entanglement is robust against thermal fluctuations when one takes into account the mechanical damping. We then extend the two-mode mechanical model to the case of multiple mechanical oscillators in an array of cascaded cavities. We show that in this system genuine multipartite steady-state entanglement can be built up among the remote mechanical oscillators with the help of the cavity dissipation.

The reminder of this paper is arranged as follows. In Sec.II, the model of two cascaded optomechanical system is introduced and the steady-state entanglement between the mechanical oscillators is investigated in detail. In Sec.III, we extend the previous model to the case of multiple mechanical oscillators in an array of unidirectionally-coupled cavities and discuss the generation of multipartite entanglement among multiple mechanical oscillators. At last, we give the conclusion in Sec.IV.

## Ii Entanglement of two mechanical oscillators

### ii.1 Model and equations

As schematically shown in Fig.1, we investigate a system consisting of two identical optical cavities connected by unidirectional coupling (33). In each cavity, two driven cavity modes are coupled to a vibrating membrane via radiation pressure (34); (35). The role of the membranes could also be played by other mechanical systems such as trapped clouds of ultracold atoms (36). After removing the carrier photons with filters, the output quantum fluctuations from the first cavity are directed to the second cavity to drive the corresponding cavity modes. With the light fields rotating at their driving frequencies, the Hamiltonian of the system is given by

 H/ℏ =∑j=1,2[δaja†jaj+δbjb†jbj+ωmjc†jcj +(~gaja†jaj+~gbjb†jbj)(cj+c†j) +i(Eaja†j−E∗ajaj)+i(Ebja†j−E∗bjbj)], (1)

where and are annihilation (creation) operators for the cavity modes and for the mechanical modes of the vibrating membranes in each cavity. The cavity-laser detunings , with being the cavity resonant frequencies and the corresponding driving frequencies. The mechanical frequencies of the membranes are denoted by and the optomechanical coupling , with being the cavity length and the effective mass of the membranes. The amplitudes of the driving lasers , where are the powers of the pump lasers and the cavity loss rates of the left cavity mirrors.

We expand the quantum operators as , where are the steady-state classical amplitudes and the corresponding quantum fluctuation operators. By taking into account cavity losses and mechanical damping, the classical amplitudes are obtained as and , where , are the cavity loss rates from the output mirrors on the right of the cavities, and are the mechanical damping rates. Note that here we have assumed the cavity loss rates such that losses from the left cavity mirrors can be neglected. For intense driving fields we have and the Hamiltonian (1) can be linearized. Then, by dropping the symbol in the fluctuation operators for simplicity of notation, the resulting Langevin equations of motion for the quantum fluctuations of the cavity and mechanical modes are obtained as

 ˙aj= −(κaj+iΔaj)aj−igaj(cj+c†j)+√2κajainj(t), ˙bj= −(κbj+iΔbj)bj−igbj(cj+c†j)+√2κbjbinj(t), ˙cj= −(γmj+iωmj)cj−igaj(aj+a†j)−igbj(bj+b†j) +√2γmjcinj(t), (2)

where the effective optomechanical coupling . The noise operators and describe vacuum inputs to the first cavity and satisfy nonzero correlations and . The input noise of the second cavity, characterized by the operators and , are from the output fluctuations of the first cavity and transmission losses in the coupling. When the output quantum field of the cavity mode is used to drive the cavity mode , then one has

 ain2(t) =√ηa[ain1(t)−√2κa1a1(t)]e−i(νa1−νa2)t +√(1−ηa)~ain2(t), (3a) bin2(t) =√ηb[bin1(t)−√2κb1b1(t)]e−i(νb1−νb2)t +√(1−ηb)~bin2(t), (3b)

where accounts for the imperfect couplings between the two cavities. The operators and denote the local vacuum noise input to the second cavity. The parameter corresponds to a lossless unidirectional coupling between the two cavities, whereas describes two independent cavities. Note here that the exponential factors in the above equations result from the differences between the frequencies of the relevant pump lasers. In addition, are noise operators of the mechanical oscillators which have nonzero correlations and , where the mean thermal phonon numbers at temperature is given by , with the Boltzmann constant.

Now we choose the detunings

 Δa1=−Δb1=−ωm1, Δa2=−Δb2=ωm2, (4)

i.e., the cavity modes and are pumped by lasers which are blue-detuned from their resonance frequencies by the mechanical frequencies, while the modes and are driven by pump lasers which are red detuned by the same amount, as illustrated in Fig. 1(b). Therefore, the pump frequencies should satisfy

 νa1−νa2 =(ωm1+ωm2), (5a) νb1−νb2 =−(ωm1+ωm2). (5b)

With the above choices of detunings, by performing the transformations , , , and , and neglecting fast oscillating terms proportional to , the Langevin equations (2) reduce to

 ˙a1=−κa1a1−iga1c†1+√2κa1ain1(t), (6a) ˙b1=−κb1b1−igb1c1+√2κb1bin1(t), (6b) ˙a2=−κa2a2−iga2c2−2√ηaκa1κa2a1+√2ηaκa2a% in1(t) +√2(1−ηa)κa2~ain2(t), (6c) ˙b2=−κb2b2−igb2c†2−2√ηbκb1κb2b1+√2ηbκb2bin1(t) +√2(1−ηb)κb2~bin2(t), (6d) ˙c1=−γm1c1−iga1a†1−igb1b1+√2γm1cin1(t), (6e) ˙c2=−γm2c2−iga2a2−igb2b†2+√2γm2cin2(t). (6f)

It should be noted that for our approximations to be valid, we require our system to be in the resolved sideband regime, , as well as to satisfy . The above equations show that in each cavity, the mechanical mode is coupled to the cavity modes via effective parametric amplification as well as beam-splitter-like mixing. While the former interaction leads to photon-phonon entanglement and optical amplification, the latter is damping the mechanical modes. If the coupling strengths satisfy and , optical damping is dominant over amplification and both mechanical oscillators are cooled.

### ii.2 Two-mode mechanical entanglement

We can equivalently reexpress Eqs. (6) as , with the vector , in terms of the quadrature operators defined as and , while contains the corresponding noise operator contributions. The entanglement between the mechanical systems is contained in the correlation matrix given by . In steady-steady state, it satisfies , where is the noise matrix . Since we are only interested in the entanglement between the two mechanical modes, it is enough to consider the reduced correlation matrix related to the two-mode mechanical states. It has the simple structure , where , , and are matrices containing the autocorrelations of the two systems and their cross-correlations respectively. The entanglement between the two mechanical modes can be quantified with the logarithmic negativity (37), which is defined as

 E12=max[0,−ln(2ζ12)], (7)

where is given in terms or the reduced correlation matrix

 ζ12=2−1/2√Σ(σ12)−√Σ(σ12)−4% detσ12, (8)

with .

Solving Eqs.(6) numerically and using Eq. (7) we can investigate the mechanical entanglement in the system. Let us first, however, turn to a regime where we can obtain analytical results. To this end, we consider the cavity dissipation rates , the perfect cavity couplings , and the effective optomechanical couplings

 ga1=gb2=g1,  ga2=gb1=g2. (9)

If the cavity dissipation rate is dominating the dynamics of the system, i.e. , the cavity modes follow changes of the mechanical oscillators adiabatically for times . In this case we can eliminate the cavity modes and find the simple equations of motion for the mechanical modes

 ˙c1(t)=−(γm1+~γm)c1(t)+√2γm1cin1(t)+~cin1(t), (10a) ˙c2(t)=−(γm2+~γm)c2(t)+√2γm2cin2(t)+~cin2(t), (10b)

where is the net optomechanical damping rate. The noise operators are given by

 ~cin1(t)=−√2ig1√κain†1(t)−√2ig2√κbin1(t), (11a) ~cin2(t)=√2ig2√κain1(t)+√2ig1√κbin†1(t), (11b)

and have non-vanishing correlations

 ⟨~cin†j(t)~c% inj(t′)⟩=2~γmNmδ(t−t′), (12a) ⟨~cinj(t)~cin†j(t′)⟩=2~γm(Nm+1)δ(t−t′), (12b) ⟨~cin1(t)~cin2(t′)⟩=2~γm√Nm(Nm+1)δ(t−t′), (12c)

with . The above correlations indicate that the two mechanical oscillators are effectively coupled to a broadband quantum reservoir in a two-mode squeezed vacuum state (38). In the absence of the mechanical damping (), the mechanical oscillators will reduce to the state of the reservoir in the long-time limit, i.e., the two-mode squeezed vacuum

 |ψ⟩ss12=exp(−rc†1c†2+rc1c2)|0c1,0c2⟩, (13)

with the squeezing parameter only dependent on the relative strengths of the two pump lasers. Therefore, the strong mechanical entanglement can be built up in principle just by controlling the ratio of the strengths of the pump lasers. It should be pointed that our scheme is quite different from that in Ref.(20) which discussed the establishment of the stationary entanglement between two mechanical oscillators by injecting externally squeezed light into the cavities. Here, instead of creating entanglement in an external source, the entanglement between the mechanical oscillator and blue-detuned cavity mode is created via the parametric interaction in each cavity, and the photon-phonon entanglement is then transferred to the mechanical oscillators with the help of the beam-splitter interaction.

By taking into account mechanical damping, from Eq.(10) we have the steady-state values and , where we have assumed and for simplicity. It is easy to the entanglement parameter

 ζ12=12−g1(g2−g1)−κγm¯nthκγm+g22−g21. (14)

Clearly, steady-state mechanical entanglement can be achieved at non-zero temperature, provided that the mean number of thermal phonons satisfy

 ¯nth

Given that the couplings are tunable through the pump lasers, this condition demonstrates the robustness of steady-state entanglement against thermal noise in the mechanical systems.

We next turn to the numerical results from solving Eqs.(6), which allows us to investigate the entanglement property in the regime where the adiabatical elimination of the cavity modes is invalid. In Fig.2 the dependence of steady-state mechanical entanglement on the cavity decay rate is plotted for different values of and the mechanical damping . Consistently with our analytic results we observe for large cavity decay , the entanglement becomes saturated and independent of . The increase of the entanglement with decreasing coupling ratios is also evident in this regime. Furthermore, for the converse situation , we also observe the steady-state entanglement, although to a smaller degree than in the adiabatic regime. The behavior of the steady-state entanglement in the presence of mechanical damping is demonstrated in Fig.3. We see that in this case the optimal entanglement does not occur in the adiabatical regime. With increasing thermal phonon number stronger coupling strengths are needed to achieve the maximum entanglement. However, the robustness of the generated entanglement is obvious, as it can still be maintained for a relatively high mean thermal phonon number . Reaching the quantum ground-state of the vibrational modes is therefore not a prerequisite of the present scheme, which reduces experimental difficulties considerably.

## Iii Multipartite mechanical entanglement

In this section, we generalize the previous two-cavity model to a system of mechanical oscillators in coupled cavities, as illustrated in Fig. 4 and proceed to discuss the generation of multipartite mechanical entanglement. Staying consistent with the previous model, we introduce the convention that the cavity modes and are pumped by lasers which are blue-detuned from their resonances, while and are driven by red detuned pumps. Therefore, the red and blue detuned modes are and respectively for odd cavities, while they are and for even cavities, see Fig. 1.(b). Assuming identical mechanical frequencies for all oscillators () and identical cavities, we thus find the effective detunings

 Δa2n−1=−Δb2n−1=−ωm,  Δa2n=−Δb2n=ωm, (16)

and for the driving frequencies

 νa2n−νa2n−1=−2ωm,  νb2n−νb2n−1=2ωm. (17)

With the same procedures and approximations as before, the Langevin equations of motion for the cavity modes and the mechanical modes can be obtained and read

 ˙zj= −κzzj−igzczj−2κzj−1∑s=1(√ηz)j−szs +√2κzj∑s=2√ηj−sz(1−ηz)~zins(t) +√2κz(√ηz)j−1zin1(t), (18a) ˙cj= −γmcj−igaaxj−igbbxj+√2γmcinj(t), (18b)

where the symbols are

 (ga,gb,caj,cbj,axj,bxj)=⎧⎨⎩(g1,g2,c†j,cj,a†j,bj)for j odd(g2,g1,cj,c†j,aj,b†j)for j even

with matched optomechanical couplings and , local vacuum noise operators , optical and mechanical loss-rates designated by and respectively and finally the coupling efficiencies . Before we turn to the numerical solutions, let us first consider the situation that , which allows us to adiabatically eliminate the cavity modes. For the perfect intercavity couplings and identical cavity loss rates , the equations of motion for the odd and even mechanical oscillators are

 ˙c2n−1(t) =−(γm+~γm)c2n−1(t)−2~γmn−1∑s=1c2s−1(t) +√2γmcin2n−1(t)+~cin1(t), (19a) ˙c2n(t) =−(γm+~γm)c2n(t)−2~γmn−1∑s=1c2s(t) +√2γmcin2n(t)+~cin2(t). (19b)

From the above equations we see that the odd and even mechanical oscillators are coupled to the noise operators and , respectively. Therefore, the entanglement may be established between any odd and even mechanical oscillators with the nonclassical correlations between the noises and given in Eq.(12). However, between oscillators with same parity, quantum entanglement cannot be established. This because that the source of entanglement in this scheme results from the coupling of the red sideband output into the blue sideband input and vice versa. For two even or odd oscillators, the cavity modes coupled to these two oscillators have the same detunings from the pump lasers, which leads them not to being entangled but to the mode coupling through an incoherent exchange interaction with rate . These results are verified in the following via numerical solution of Eqs.(18).

For , we plot in Fig.5 (a) and (b) the bipartite entanglement between the mechanical modes and and the bipartite entanglement of the modes and , respectively. We see that the entanglement for the same parameters. As predicted above, bipartite entanglement between the mechanical modes and is absent. Nevertheless, as demonstrated in Fig.5 (d), full inseparable (genuine) tripartite entanglement can be established among the three remote mechanical oscillators. Fig.5 (d) depicts the negative eigenvalues of the partially transposed three-mode correlation matrix with respect to the -th mode. The appearance of negative eigenvalue confirms bipartite entanglement between the transposed mode and the subsystem of the remaining modes and , and fully inseparable (genuine) multipartite entanglement is demonstrated in the regime where the negative eigenvalues simultaneously exist for (39). Also, Fig.5 (d) shows that bipartite entanglement between the mode and the remaining two modes and is largest, since it is the only mode which is simultaneously entangled to the two other subsystems and . Finally, the entanglement between the mode and the subsystem including and is smallest, since the bipartite entanglement satisfies and . Therefore see that the negativities will satisfy .

When extending the above three-mode mechanical system to the four-mode case, i.e. , it is not difficult to see from Eq.(18) that the reduced bipartite entanglement and are not affected due to the unidirectional cavity coupling. Therefore, the entanglement and are the the same as in the case plotted in Fig.5. Furthermore, it can be inferred from Eq.(19) that the bipartite entanglements will satisfy in the bad-cavity limit. In Fig.6 (a), we plot the bipartite entanglement between the mechanical modes and , and it is obvious that it exhibits similar behavior as the entanglement between modes and ( see Fig.5). We therefore see that quadripartite square graph-state entanglement among four remote mechanical oscillators can be achieved via cascaded cavity couplings. This kind of multipartite entanglement is useful in the field of long-distance quantum communication. The effects of imperfect cavity couplings are illustrated in Fig.6 (c). We see that for the coupling efficiency as low as , the genuine quadripartite entanglement of four distant mechanical oscillators can still be achieved.

## Iv Conclusion

In conclusion, we propose a scheme to generate steady-state entanglement of remote mechanical oscillators in unidirectionally coupled cavities in the cascaded way. We note here that while the present model assumes the membranes as mechanical oscillators, the role of mechanical elements can also be played by momentum modes of clouds of ultracold atoms. By choosing the detuning of the pump lasers, in each cavity the mechanical oscillator is coupled to the two cavity modes via parametric and beam-splitter-like interactions. The output quantum fluctuating field of the first cavity is subsequently driving the second cavity with reversed detunings. For the case of two mechanical oscillators in cascaded cavities, the cavity dissipation can pull the two mechanical oscillators into a stationary two-mode squeezed vacuum state for negligible mechanical damping. The two-mode mechanical entanglement depends on the relative strength of the pump lasers and is robust to thermal fluctuations. For multiple mechanical oscillators in multiple cascaded cavities, it is found that the steady-state bipartite entanglement can be established between the odd and even oscillators, whereas odd and even oscillators do not become entangled. We show that using this scheme the genuine multipartite entanglement can be achieved among remote mechanical oscillators by cavity dissipation. This kind of remote multipartite macroscopic entanglement is a useful resource in the construction of long-distance quantum communication networks.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11274134, 11074087, 61275123), the National Basic Research Program of China (Grant No. 2012CB921602), the Natural Science Foundation of Hubei Province (Grant No. 2010CDA075), and the Natural Science Foundation of Wuhan City (Grant No. 201150530149). THT acknowledges the support from the CSC. BFB and HS acknowledge Pierre Meystre for his ongoing support.

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