## Abstract

Experimental realization of the Kitaev model is a greatly attractive topic due to the potential applications to build robust qubits against decoherence in topological quantum computation. In this work, we investigate the charged whispering-gallery microcavity array model and simulate the normal Kitaev chain under this mechanism in the first time. We find that the system reveals profound connections with the normal Kitaev chain and its some derivatives, and the topological property of the system depends on effective optomechanical coupling strength deeply. In optomechanically induced Kitaev topologically nontrivial phase, compared to the normal Kitaev chain in the Majorana basis, the novel and distinct structure of charged whispering-gallery microcavity array model leads to controllable photonic and phononic edge localization. Furthermore, we also simulate the extended Kitaev chain and show that two topologically different nontrivial phases of the system allow one to realize more freewheeling controllable photonic and phononic edge localization. Our model offers an alternative approach to correlate with other more complicated one-dimensional noninteracting spinless topological systems relevant to the *p*-wave superconducting pairing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a prototype of one-dimensional (1D) topological superconductors, the normal Kitaev chain [1] named after the pioneering work of Kitaev, a toy description of a noninteracting spinless tight-binding model in the presence of *p*-wave superconducting pairing, has aroused enormous interest since the existence of zero-energy Majorana edge modes [2–10] fulfilling non-Abelian statistics with open boundaries, which meet with the potential applications to build robust qubits against decoherence in topological quantum computation [11,12]. Inspired by the extremely absorbing characteristic of the normal Kitaev chain, this model has been put forward to be realized by using a 1D nanowire with strong Rashba spin-orbit interaction [13–19]. Very recently, many derivatives of the normal Kitaev chain, including non-Hermitian [20–24], dimerized [25–27], generalized [28–30], kicked [31,32], and interacting [33–37] classifications, appeared and have also attracted intensive studies, which exhibit more novel and abundant physical phenomena and even change the properties of these systems drastically.

Due to the rapid developments and the crucial advancements of microfabrication and nanotechnology, on the other hand, cavity optomechanics [38–40], as an interaction interface between the optical and mechanical degrees of freedom via radiation pressure, optical gradient, or photothermal forces, has become an appealing research frontier and opened up avenues for exploring quantum behavior in macroscopic systems as well as promising applications in quantum information processing [41,42]. These progresses range from cooling [43–49], squeezing [50–54] to entanglement [55–57] and more. Moreover, optomechanical system involving the Coulomb interaction [58–60] has also drawn much attention as the external Coulomb force in this hybrid system shows remarkable effect on the research object compared with the pure optomechanical system.

Currently, there is a growing passion in the combination of topological model and optomechanical system. The simulation of two-dimensional quantum spin Hall system via 1D cavity optomechanical cells array by using diagonalization and dimensional reduction methods has been mentioned in Ref. [61]. And Wan *et al*. have studied an optomechanical Lieb lattice, where the flat-band physics of photon-phonon polaritons was demonstrated, and the controllable photon or phonon localization could be further realized by the path interference effects [62]. However, to our knowledge, although hybrid or pure optomechanical system and the normal Kitaev chain as well as its derivatives have been in the focus of their respective research fields, the relation between them has been not, as yet, concerned and revealed.

It is the purpose of this paper to provide an effective approach to investigate and simulate the normal Kitaev chain and its some derivatives. To this end, resorting to the platform of charged whispering-gallery microcavity array model, we find that the system can be equivalent to two effective bosonic chains comprised of an unadulterated photonic chain and an unadulterated phononic chain coupled by the effective optomechanical coupling strength after some approximations. The forms of real coupling coefficients we set specifically result in a direct mapping between the two effective bosonic chains and the normal Kitaev chain as well as its some derivatives. Notably, the recognizable edge distribution and tunable localization behavior of photons and phonons compared with the nontrivial Majorana fermions in the normal Kitaev chain and its some derivatives allows one to realize controllable photonic and phononic edge localization. In the situation of simulation of the normal Kitaev chain, three different types of photonic and phononic edge localization exist in the optomechanically induced Kitaev topologically nontrivial phase (TNP), that is, photonic edge localization, phononic edge localization, and photonic and phononic common but opposite edge localization. Moreover, the realistic situation is also considered to test the robustness of edge modes. It is noteworthy here that an example of simulation of the extended Kitaev chain indicates two additional approaches for controllable photonic and phononic edge localization, viz. photonic and phononic common bilateral and unilateral edge localizations. What is even more motivated is that our model can be used to potentially contact with not only 1D noninteracting spinless topological systems relevant to the *p*-wave superconducting pairing but even some topological insulators.

## 2. Model and effective Hamiltonian

The model we consider is composed of an array of charged whispering-gallery microcavities [63–65] coupled via exchange of photons and Coulomb interaction, as shown in Fig. 1. The Hamiltonian of the system is written as

with*N*is the total number of whispering-gallery microcavities,

*a*(${a}_{n}^{\u2020}$) is the annihilation (create) operator of the

_{n}*n*th cavity mode with frequency

*ω*, the vibration of the

_{c}*n*th charged whispering-gallery microcavity can be described by frequency

*ω*, effective mass

_{b}*m*

_{0}, position operator

*q*, and momentum operator

_{n}*p*,

_{n}*g*

_{0}=

*ω*/

_{c}*R*is the optomechanical coupling strength with

*R*being radius of the whispering-gallery microcavity that contains the mechanical mode, the driving field owns frequency

*ω*and amplitude Ω

_{d}*, and*

_{d}*J*is intercavity hopping strength. Moreover, the two nearest-neighbor charged whispering-gallery microcavities interact with each other by a Coulomb force with the electrostatic constant

_{m}*k*, the charge

_{e}*Q*carried by the

_{m}*m*th whispering-gallery microcavity, the equilibrium separation

*r*

_{0}between the two nearest-neighbor whispering-gallery microcavities in the absence of optomechanical and Coulomb interactions, and the tiny deviation

*q*from the equilibrium position of the

_{m}*m*th whispering-gallery microcavity originated from the optomechanical and Coulomb interactions.

In the case of *q _{m}* (

*q*

_{m+1}) ≪

*r*

_{0}in that the mechanical deviation

*q*(

_{m}*q*

_{m+1}) is relatively small compared to the equilibrium separation

*r*

_{0}, we expand the Coulomb interaction to the second-order of (

*q*−

_{m}*q*

_{m+1})/

*r*

_{0}and obtain

*m*-th and (

*m*+ 1)-th whispering-gallery microcavities. Thus we can simplify the Coulomb interaction between the two nearest-neighbor charged mechanical resonators by further ignoring the constant term as follows with ${\chi}_{m}=2{k}_{e}{Q}_{m}{Q}_{m+1}/\left(\hslash {r}_{0}^{3}\right)$. In the rotating frame with respect to the external driving frequency

*ω*, we further rewrite the Hamiltonian of the system by applying the relations $q=\sqrt{\hslash /(2{m}_{0}{\omega}_{b})}({b}^{\u2020}+b)$ and $p=i\sqrt{\hslash {m}_{0}{\omega}_{b}/2}({b}^{\u2020}-b)$ as

_{d}*∊*=

_{c}*ω*−

_{c}*ω*is the detuning of cavity mode frequency from the driving field, $g={g}_{0}\sqrt{\hslash /(2{m}_{0}{\omega}_{b})}$, and ${\lambda}_{m}={k}_{e}{Q}_{m}{Q}_{m+1}/\left({m}_{0}{\omega}_{b}{r}_{0}^{3}\right)$.

_{d}As one of the most frequently-used methods in cavity optomechanical system, the stability of the steady state of the system is determined by a linearized analysis for small perturbation around the steady state. Consequently, we can linearize the above Hamiltonian by rewriting each operator as a sum of its steady state mean value and an additional small zero mean fluctuation operator, namely *a _{n}* =

*α*+

*δa*and

_{n}*b*=

_{n}*β*+

*δb*. Substituting these expressions into Eq. (5), then we discard both the first and third order terms and further drop the notation “

_{n}*δ*” of all the fluctuation operators for simplicity, we can obtain the standard linearized Hamiltonian about the fluctuation operators as below (setting

*ħ*= 1)

*=*

_{c}*∊*− 2

_{c}*gβ*and

*G*= −

*gα*is the effective optomechanical coupling strength. Here we have assumed that

*α*and

*β*are real throughout the paper. Applying the unitary transformation, $U=\text{exp}\left[i{\sum}_{n=1}^{N}\left({\mathrm{\Delta}}_{c}{a}_{n}^{\u2020}{a}_{n}+{\omega}_{b}{b}_{n}^{\u2020}{b}_{n}\right)t\right]$, to Eq. (6) and neglecting the rapid oscillatory terms by taking advantage of red-detuned regime, viz. Δ

*=*

_{c}*ω*, the resulting effective Hamiltonian of the system is given by

_{b}*G*with one coupled via the intercavity hopping strength and the other coupled via the effective Coulomb interaction strength, as shown in Fig. 2.

As is well-known, a Dirac fermion can be thought of as composed of two Majorana fermions. Hence the normal Kitaev chain in the Majorana basis can be converted to two Majorana chains coupled by the chemical potential *μ*, and when the system is in the topologically nontrivial regime, there exist two spatially separated zero-energy Majorana bound states at ends of the chain. Hereinafter, we will map the two coupled photonic and phononic chains to the normal Kitaev chain and expect that the Kitaev phase can be exhibited by means of assigning the forms of real coupling coefficients.

## 3. Results and discussion

In this section, we mainly investigate and discuss the realization of the normal Kitaev chain by choosing the forms of real coupling coefficients. We present the energy spectrum of system and the locations of edge modes, which reveal the behavior of the unpaired Majorana bound states in the normal Kitaev chain. What is more interesting is that the system can host controllable photonic and phononic edge localization. We also test the robustness of edge modes within the realistic situation. Furthermore, the simulation of the extended Kitaev chain and a extension of controllable photonic and phononic edge localization are also given therein.

#### 3.1. Energy spectrum

To obtain the energy spectrum of the normal Kitaev chain, we adopt the following forms of real coupling coefficients,

*N*= 8. In Fig. 3 we plot the energy spectrum of system as a function of the effective optomechanical coupling strength

*G*. In the normal Kitaev chain, the chemical potential |

*μ*/

*t*| = 2 is the phase transition points and it is in the topologically nontrivial regime featured by the presence of twofold-degenerate zero-energy Majorana edge modes when |

*μ*/

*t*| < 2, while the system does not support the zero-energy Majorana edge modes in the topologically trivial regime when |

*μ*/

*t*| > 2. It is obvious that the energy spectrum of the system is analogous to the normal Kitaev chain essentially and the effective optomechanical coupling strength

*G*acts as a role of the the chemical potential

*μ*. Here we clarify that the phase transition points |

*G*/

*t*| ≠ 2 because of the smaller size of the system. However, it is exactly equal to ±2 if the size of the system becomes larger. Additionally, our scheme even can be minimally realized with the total number of whispering-gallery microcavities being

*N*= 4.

#### 3.2. Controllable photonic and phononic edge localization

As to our model made up of two coupled bosonic chains, we can make it exhibit the behavior of the unpaired Majorana bound states in the normal Kitaev chain by setting the forms of real coupling coefficients. Nevertheless, what is different from the normal Kitaev chain is that one of the two bosonic chains is an unadulterated photonic chain and the other is an unadulterated phononic chain. This naturally allows one to manipulate photonic edge localization or phononic edge localization or common edge localization depending on the specific choice of the forms of real coupling coefficients and the odevity of the number of whispering-gallery microcavities.

To illuminate the principle of controllable photonic and phononic edge localization phenomenologically, combining with the behavior of the nontrivial Majorana bound states in the normal Kitaev chain, a more intuitive diagram is given in Fig. 4. When the number of whispering-gallery microcavities is even, photonic edge localization is shown in Fig. 4(a), it is evident that two photonic edge states locate at the ends of the photonic chain. Figure 4(b) exhibits phononic edge localization showing a phononic edge state located at each end of the phononic chain. Furthermore, it is also shown in Figs. 4(c) and (d) that there are a photonic edge state at one end of the photonic chain and a phononic edge state at the other end of the phononic chain for odd number whispering-gallery microcavities, namely, photonic and phononic common edge localization.

We can achieve photonic edge localization by setting the real coupling coefficients and parameters approximately, as adopted in Sec. III A. To see it clearly, the population of eigenfunction corresponding to the magenta line in Fig. 3 as a function of the effective optomechanical coupling strength *G* and the site is shown in Fig. 5(a), in which the populations of photonic and phononic chains are individually plotted and displayed respectively with hot and gray colorbars. It is visualized that photons are well concentrated at the ends of the photonic chain when the system is in the topologically nontrivial regime, while photons are extended over the whole photonic chain in the topologically trivial regime. In contrast, phonons are delocalized all the time over the whole phononic chain whatever the system is topologically nontrivial or trivial.

Now we turn to the realization of phononic edge localization with the same parameters as adopted in Sec. III A but with the reverse forms of real coupling coefficients for each bosonic chain, namely,

On condition that the number of whispering-gallery microcavities is odd, in spite of the similar energy spectrum of the system compared with that in Fig. 3, another novel phenomenon appears showing photonic and phononic common but opposite edge localization. What’s more, we also can designate the edge for photonic or phononic localization concretely. Particularly, when the system is in the topologically nontrivial regime, if we take the forms of real coupling coefficients given by Eq. (8), photons can be well concentrated at left edge of the photonic chain, while phonons can be well localized on right edge of the phononic chain. On the other hand, we can make photons locate at the right edge of the photonic chain and make phonons concentrate at the left edge of the phononic chain by choosing the forms of real coupling coefficients of Eq. (9). As an example, we take the same forms of real coupling coefficients and parameters adopted in Sec. III A except the number of whispering-gallery microcavities being set *N* = 7. Both of the populations of eigenfunctions corresponding to the two middle eigenvalues owning zero-energy eigenvalue are displayed in Fig. 5(c). It is intuitional that the edge for photonic or phononic localization matches with our preceding analysis perfectly. What is noteworthy is that the photonic and phononic edge localization are unambiguous when the effective optomechanical coupling strength *G* = 0, the reason is that photonic and phononic chains are thoroughly decoupled from each other in this case.

Finally, we refer our system to the practical condition, in which the system subjected to a parametric imperfection introduced in the real coupling coefficients simultaneously suffer the dissipation of cavity mode and the damping of mechanical mode, to check the robustness of edge modes. The energy spectrum of the system is thus become complex and the bulk structure of its real part compared with that in Fig. 3 is also altered. We discover the presence of robust edge modes whose eigenvalues are pure imaginary in three different types of controllable photonic and phononic edge localization, respectively. As an example, we show the photonic edge localization with the dissipation of cavity mode being *κ* = 0.3*t*, the damping of mechanical mode being Γ = 0.1*t*, and a random disorder perturbation added in the intercavity hopping strength and the effective Coulomb interaction strength whose strength is given by *dJ _{m}* ∈ [−0.3

*J*, 0.3

_{m}*J*] and

_{m}*dλ*∈ [−0.3

_{m}*λ*, 0.3

_{m}*λ*], respectively. The real and imaginary parts of energy spectrum of the system are depicted in Fig. 6 and we further label the eigenvalues with the minimum absolute values of real part in magenta and cyan, of which the pure imaginary eigenvalues correspond to the energy of these robust edge modes.

_{m}#### 3.3. Extension of controllable photonic and phononic edge localization

Our model will be a good choice of simulating more complicated 1D noninteracting spinless topological system associated with *p*-wave superconducting pairing. We add an additional parameter *δ* into the Eq. (8), that is,

*δ*play a major role for

*p*-wave superconducting pairing amplitude and dimerization strength introduced in nearest-neighbor hopping amplitude, respectively.

We plot the phase diagram of the system expanded by the parameters *G*, Δ, and *δ*, as shown in Fig. 7(a). The phase boundaries are determined by the following two conditions: *G*^{2}/4 + Δ^{2} = *δ*^{2} (|*δ*/*t*| < 1) and *G*/*t* = ±2, which exhibits that the system hosts three different phases: the Su-Schrieffer-Heeger- (SSH-) like TNP, the Kitaev-like TNP, and the topologically trivial phase (TTP). Here, for the parameter region enclosed by the upper ellipsoidal cone, the system is in the SSH-like TNP; for the parameter region between the two ellipsoidal cones and the two planes *G*/*t* = ±2, the system is in the Kitaev-like TNP; while for the rest of parameter regions, including the region enclosed by the nether ellipsoidal cone and those outside the two planes *G*/*t* = ±2, the system is in the TTP.

For the sake of clarity, with the choice of the parameters *G* = Δ = 0.3*t* and *N* = 70, we further plot the energy spectrum of the system as a function of *δ*, as shown in Fig. 7(b), corresponding to the red line in Fig. 7(a). It is obvious that the red line spans through the parameter regions of three different phases and the energy spectrum of the system also exhibits three different phases. The phase transition points are $\pm \sqrt{{G}^{2}/4+{\mathrm{\Delta}}^{2}}\approx +0.335t$, which corresponds to the points of energy gap closed and reopened, as marked by the two red lines in Fig. 7(b). In the region of −1 < *δ*/*t* < −0.335, the system is in the TTP. When −0.335 < *δ*/*t* < 0.335, the system is in the Kitaev-like TNP and there are twofold-degenerate zero-energy edge modes whose populations of eigenfunctions exhibits the same behavior compared with those mentioned in Sec. III B. Therefore, controllable photonic and phononic edge localization as discussed in Sec. III B can also be realized in this region. More interestingly, when the number of whispering-gallery microcavities is even, in the region of 0.335 < *δ*/*t* < 1, the system is in the SSH-like TNP. In analogy with the two Dirac fermions located at ends of the original extended Kitaev chain, it is also characterized by the existence of two twofold-degenerate nonzero-energy edge modes. Therefore, the eigenfunctions of the four nonzero-energy edge modes correspond to two photonic edge states and two phononic edge states concentrated at ends of the photonic and phononic chains, respectively. We thus can realize photonic and phononic common bilateral edge localization in the SSH-like TNP. For odd number whispering-gallery microcavities, the energy spectrum of the system is displayed in Fig. 7(c), which shows that there are two nonzero-energy edge modes in both the regions of 0.335 < *δ*/*t* < 1 and −1 < *δ*/*t* < −0.335. This even-odd effect in the energy spectrum is a typical feature of the SSH model. Moreover, the eigenfunctions of two nonzero-energy edge modes in the region of 0.335 < *δ*/*t* < 1 correspond to a photonic edge state and a phononic edge state centralized at left edges of the photonic and phononic chains, while the eigenfunctions of two nonzero-energy edge modes in the region of −1 < *δ*/*t* < −0.335 correspond to a photonic edge state and a phononic edge state localized on right edges of the photonic and phononic chains. We also can realize photonic and phononic common unilateral edge localization and even designate the edge of photonic and phononic common unilateral edge localization hinging on the value of *δ* we choose. For a better understanding, a more intuitive diagram is given in Fig. 8.

Furthermore, it is promising that the other more complicated 1D noninteracting spinless topological systems associated with the *p*-wave superconducting pairing, such as the dimerized Kitaev chain [25,27] and the generalized Aubry-André-Harper model with *p*-wave pairing [30], can be realized. Last but not least, for the special limit of *G* = 0, on the one hand, we can simulate the commensurate Aubry-André-Harper model [66] as the model in the Majorana basis contains two identical 1D Majorana chains which are decoupled from each other, on the other hand, we also can utilize the two decoupled bosonic chains to simulate the *Z*_{2} topological insulators with two-component fermions [67] via a direct mapping of spin-up and -down atomic gases.

## 4. Conclusions

In conclusion, we have proposed a novel scheme to simulate the normal Kitaev chain and its some derivatives based on charged whispering-gallery microcavity array model. We demonstrated that the system is made up of an unadulterated photonic chain and an unadulterated phononic chain coupled by the effective optomechanical coupling strength. Direct mapping between the two effective bosonic chains and the normal Kitaev chain as well as its some derivatives can be accomplished by assigning the forms of real coupling coefficients specifically. Controllable photonic and phononic edge localization can be further captured, which is originated from the recognizable edge distribution and tunable localization behavior of photons and phonons compared with the nontrivial Majorana fermions in the normal Kitaev chain and its some derivatives. Two special examples are enumerated and discussed in detail to check the feasibility of our scheme. One is the simulation of the normal Kitaev chain. We find the existence of three different types of controllable photonic and phononic edge localization in the optomechanically induced Kitaev TNP, that is, photonic edge localization, phononic edge localization, and photonic and phononic common but opposite edge localization. The robustness of edge modes to the practical condition is also checked. The other is the simulation of the extended Kitaev chain. Besides three foregoing schemes of controllable photonic and phononic edge localization realized in the Kitaev-like TNP, the SSH-like TNP and the even-odd effect the system possesses endow the system with two additional types of controllable photonic and phononic edge localization, viz. photonic and phononic common bilateral and unilateral edge localizations. Moreover, not only the 1D noninteracting spinless topological systems relevant to the *p*-wave superconducting pairing but even some topological insulators potentially relates with our model.

## Funding

National Natural Science Foundation of China (NSFC) (11465020, 11264042, 61465013, 11564041); The Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (20160519022JH).

## References and links

**1. **A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,” Phys.-Usp. **44**, 131–136 (2001). [CrossRef]

**2. **F. Wilczek, “Majorana returns,” Nat. Phys. **5**, 614–618 (2009). [CrossRef]

**3. **S. Gangadharaiah, B. Braunecker, P. Simon, and D. Loss, “Majorana edge states in interacting one-dimensional systems,” Phys. Rev. Lett. **107**, 036801 (2011). [CrossRef] [PubMed]

**4. **E. Sela, A. Altland, and A. Rosch, “Majorana fermions in strongly interacting helical liquids,” Phys. Rev. B **84**, 085114 (2011). [CrossRef]

**5. **E. M. Stoudenmire, J. Alicea, O. A. Starykh, and M. P. A. Fisher, “Interaction effects in topological superconducting wires supporting Majorana fermions,” Phys. Rev. B **84**, 014503 (2011). [CrossRef]

**6. **R. M. Lutchyn and M. P. A. Fisher, “Interacting topological phases in multiband nanowires,” Phys. Rev. B **84**, 214528 (2011). [CrossRef]

**7. **M. Leijnse and K. Flensberg, “Introduction to topological superconductivity and Majorana fermions,” Semicond. Sci. Technol. **27**, 124003 (2012). [CrossRef]

**8. **C. W. J. Beenakker, “Search for Majorana fermions in superconductors,” Annu. Rev. Condens. Matter Phys. **4**, 113–136 (2013). [CrossRef]

**9. **Y. H. Chan, C. K. Chiu, and K. Sun, “Multiple signatures of topological transitions for interacting fermions in chain lattices,” Phys. Rev. B **92**, 104514 (2015). [CrossRef]

**10. **S. R. Elliott and M. Franz, “Majorana fermions in nuclear, particle, and solid-state physics,” Rev. Mod. Phys. **87**, 137 (2015). [CrossRef]

**11. **J. Alicea, “New directions in the pursuit of Majorana fermions in solid state systems,” Rep. Prog. Phys. **75**, 076501 (2012). [CrossRef] [PubMed]

**12. **C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation,” Rev. Mod. Phys. **80**, 1083 (2008). [CrossRef]

**13. **L. Fu and C. L. Kane, “Superconducting proximity effect and Majorana fermions at the surface of a topological insulator,” Phys. Rev. Lett. **100**, 096407 (2008). [CrossRef] [PubMed]

**14. **J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. **104**, 040502 (2010). [CrossRef] [PubMed]

**15. **J. Alicea, “Majorana fermions in a tunable semiconductor device,” Phys. Rev. B **81**, 125318 (2010). [CrossRef]

**16. **M. Sato, Y. Takahashi, and S. Fujimoto, “Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors,” Phys. Rev. B **82**, 134521 (2010). [CrossRef]

**17. **R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures,” Phys. Rev. Lett. **105**, 077001 (2010). [CrossRef] [PubMed]

**18. **Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and Majorana bound states in quantum wires,” Phys. Rev. Lett. **105**, 177002 (2010). [CrossRef]

**19. **T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majorana fermions in semiconductor nanowires,” Phys. Rev. B **84**, 144522 (2011). [CrossRef]

**20. **X. H. Wang, T. T. Liu, Y. Xiong, and P. Q. Tong, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian Kitaev and extended Kitaev models,” Phys. Rev. A **92**, 012116 (2015). [CrossRef]

**21. **C. Yuce, “Majorana edge modes with gain and loss,” Phys. Rev. A **93**, 062130 (2016). [CrossRef]

**22. **Q. B. Zeng, B. G. Zhu, S. Chen, L. You, and R. Lü, “Non-Hermitian Kitaev chain with complex on-site potentials,” Phys. Rev. A **94**, 022119 (2016). [CrossRef]

**23. **Q. B. Zeng, S. Chen, and R. Lü, “Anderson localization in the non-Hermitian Aubry-André-Harper model with physical gain and loss,” Phys. Rev. A **95**, 062118 (2017). [CrossRef]

**24. **M. Klett, H. Cartarius, D. Dast, J. Main, and G. Wunner, “Relation between 𝒫𝒯-symmetry breaking and topologically nontrivial phases in the Su-Schrieffer-Heeger and Kitaev models,” Phys. Rev. A **95**, 053626 (2017). [CrossRef]

**25. **R. Wakatsuki, M. Ezawa, Y. Tanaka, and N. Nagaosa, “Fermion fractionalization to Majorana fermions in a dimerized Kitaev superconductor,” Phys. Rev. B **90**, 014505 (2014). [CrossRef]

**26. **Y. Xiong and P. Q. Tong, “A NOT operation on Majorana qubits with mobilizable solitons in an extended Su-Schrieffer-Heeger model,” New. J. Phys. **17**, 013017 (2015). [CrossRef]

**27. **Y. C. Wang, J. J. Miao, H. K. Jin, and S. Chen, “Characterization of topological phases of dimerized Kitaev chain via edge correlation functions,” Phys. Rev. B **96**, 205428 (2017). [CrossRef]

**28. **L. J. Lang and S. Chen, “Majorana fermions in density-modulated p-wave superconducting wires,” Phys. Rev. B **86**, 205135 (2012). [CrossRef]

**29. **D. P. Liu, “Topological phase boundary in a generalized Kitaev model,” Chin. Phys. B **25**, 057101 (2016). [CrossRef]

**30. **Q. B. Zeng, S. Chen, and R. Lü, “Generalized Aubry-André-Harper model with p-wave superconducting pairing,” Phys. Rev. B **94**, 125408 (2016). [CrossRef]

**31. **M. N. Chen, F. Mei, W Su, H. Q. Wang, S. L. Zhu, L. Sheng, and D. Y. Xing, “Topological phases of the kicked Harper–Kitaev model with ultracold atoms,” J. Phys. Condens. Matter **29**, 035601 (2017). [CrossRef]

**32. **H. Q. Wang, M. N. Chen, R. W. Bomantara, J. B. Gong, and D. Y. Xing, “Line nodes and surface Majorana flat bands in static and kicked p-wave superconducting Harper model,” Phys. Rev. B **95**, 075136 (2017). [CrossRef]

**33. **L. Fidkowski and A. Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B **83**, 075103 (2011). [CrossRef]

**34. **A. M. Turner, F. Pollmann, and E. Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B **83**, 075102 (2011). [CrossRef]

**35. **G. Goldstein and C. Chamon, “Exact zero modes in closed systems of interacting fermions,” Phys. Rev. B **86**, 115122 (2012). [CrossRef]

**36. **G. Kells, “Many-body Majorana operators and the equivalence of parity sectors,” Phys. Rev. B **92**, 081401 (2015). [CrossRef]

**37. **J. J. Miao, H. K. Jin, F. C. Zhang, and Y. Zhou, “Exact solution for the interacting Kitaev chain at the symmetric point,” Phys. Rev. Lett. **118**, 267701 (2017). [CrossRef] [PubMed]

**38. **T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science **321**, 1172–1176 (2008). [CrossRef] [PubMed]

**39. **F. Marquardt and S. M. Girvin, “Trend: Optomechanics,” Physics **2**, 40 (2009). [CrossRef]

**40. **I. Favero and K. Karrai, “Optomechanics of deformable optical cavities,” Nat. Photonics **3**, 201–205 (2009). [CrossRef]

**41. **C. Cao, X. Chen, Y. W. Duan, L. Fan, R. Zhang, T. J. Wang, and C. Wang, “Concentrating partially entangled W-class states on nonlocal atoms using low-Q optical cavity and linear optical elements,” Sci. China-Phys. Mech. Astron. **59**, 100315 (2016). [CrossRef]

**42. **Y. P. Gao, C. Cao, T. J. Wang, Y. Zhang, and C. Wang, “Cavity-mediated coupling of phonons and magnons,” Phys. Rev. A **96**, 023826 (2017). [CrossRef]

**43. **D. Kleckner and D. Bouwmeester, “Sub-kelvin optical cooling of a micromechanical resonator,” Nature (London) **444**, 75–78 (2006). [CrossRef]

**44. **I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. **99**, 093901 (2007). [CrossRef] [PubMed]

**45. **F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. **99**, 093902 (2007). [CrossRef] [PubMed]

**46. **C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A **77**, 033804 (2008). [CrossRef]

**47. **Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Physics **5**, 489–493 (2009). [CrossRef]

**48. **Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A **90**, 053841 (2014). [CrossRef]

**49. **Y. M. Liu, C. H. Bai, D. Y. Wang, T. Wang, M. H. Zheng, H. F. Wang, A. D. Zhu, and S. Zhang, “Ground-state cooling of rotating mirror in double-laguerre-gaussian-cavity with atomic ensemble,” Opt. Express **26**, 6143–6157 (2018). [CrossRef] [PubMed]

**50. **J. Q. Liao and C. K. Law, “Parametric generation of quadrature squeezing of mirrors in cavity optomechanics,” Phys. Rev. A **83**, 033820 (2011). [CrossRef]

**51. **H. Shi and M. Bhattacharya, “Quantum mechanical study of a generic quadratically coupled optomechanical system,” Phys. Rev. A **87**, 043829 (2013). [CrossRef]

**52. **G. S. Agarwal and S. M. Huang, “Strong mechanical squeezing and its detection,” Phys. Rev. A **93**, 043844 (2016). [CrossRef]

**53. **D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity,” Sci. Rep. **6**, 24421 (2016). [CrossRef] [PubMed]

**54. **D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Steady-state mechanical squeezing in a double-cavity optomechanical system,” Sci. Rep. **6**, 38559 (2016). [CrossRef] [PubMed]

**55. **D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. **98**, 030405 (2007). [CrossRef] [PubMed]

**56. **Q. Wu, Y. Xiao, and Z. M. Zhang, “Entanglements in a coupled cavity-array with one oscillating end-mirror,” Chin. Phys. B **24**, 104208 (2015). [CrossRef]

**57. **C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Robust entanglement between a movable mirror and atomic ensemble and entanglement transfer in coupled optomechanical system,” Sci. Rep. **6**, 33404 (2016). [CrossRef] [PubMed]

**58. **P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A **90**, 043825 (2014). [CrossRef]

**59. **C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A **95**, 033820 (2017). [CrossRef]

**60. **C. H. Bai, D. Y. Wang, H. F. Wang, A. D. Zhu, and S. Zhang, “Classical-to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction,” Sci. Rep. **7**, 2545 (2017). [CrossRef] [PubMed]

**61. **L. Qi, Y. Xing, H. F. Wang, A. D. Zhu, and S. Zhang, “Simulating Z_{2} topological insulators via a one-dimensional cavity optomechanical cells array,” Opt. Express **25**, 17948–17959 (2017). [CrossRef] [PubMed]

**62. **L. L. Wan, X. Y. Lü, J. H. Gao, and Y. Wu, “Controllable photon and phonon localization in optomechanical Lieb lattices,” Opt. Express **25**, 17364–17374 (2017). [CrossRef] [PubMed]

**63. **J. Cao, Y. Xing, L. Qi, D. Y. Wang, C. H. Bai, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and studying the topological properties of generalized commensurate Aubry-André-Harper model with microresonator array,” Laser Phys. Lett. **15**, 015211 (2018). [CrossRef]

**64. **Y. Xing, L. Qi, J. Cao, D. Y. Wang, C. H. Bai, H. F. Wang, A. D. Zhu, and S. Zhang, “Spontaneous 𝒫𝒯-symmetry breaking in non-Hermitian coupled-cavity array,” Phys. Rev. A **96**, 043810 (2017). [CrossRef]

**65. **L. Qi, Y. Xing, J. Cao, X. X. Jiang, C. S. An, A. D. Zhu, S. Zhang, and H. F. Wang, “Simulating and detecting the topological properties of modulated Rice-Mele model in one-dimensional circuit-QED lattice,” Sci. China-Phys. Mech. Astron. **61**, 080313 (2018). [CrossRef]

**66. **S. Ganeshan, K. Sun, and S. D. Sarma, “Topological zero-energy modes in gapless commensurate Aubry-André-Harper models,” Phys. Rev. Lett. **110**, 180403 (2013). [CrossRef]

**67. **F. Mei, S. L. Zhu, Z. M. Zhang, C. H. Oh, and N. Goldman, “Simulating Z_{2} topological insulators with cold atoms in a one-dimensional optical lattice,” Phys. Rev. A **85**, 013638 (2012). [CrossRef]