document: majorana fermions ( mfs ) are real solutions of the dirac equation and which are their own antiparticles @xmath0 @xcite . 
   although proposed originally as a model for neutrinos , mfs have recently been predicted to occur as quasi - particle bound states in engineered condensed matter systems @xcite . 
   this exotic particle obeys non - abelian statistics , which is one of important factors to realize subsequent potential applications in decoherence - free quantum computation @xcite and quantum information processing @xcite . over the recent few years , the possibility for hosting mfs in exotic solid state systems focused on topological superconductors @xcite . 
   currently , various realistic platforms including topological insulators @xcite , semiconductor nanowires ( snws ) @xcite , and atomic chains @xcite have been proposed to support majorana states based on the superconducting proximity effect . although various schemes have been presented , observing the unique majorana signatures 
   experimentally is still a challenging task to conquer . 
   mfs are their own antiparticles , and they are predicted to appear in tunneling spectroscopy experiments , in which majoranas manifest themselves as characteristic zero - bias peaks ( zbps ) @xcite . 
   the theoretical predictions of zbps have been observed experimentally in snws which are interpreted as the signatures of mfs mourikv , dasa , dengmt , churchillhoh , finckadk . 
   remarkably , nadj - perge et al . 
   @xcite recently designed a chain of magnetic fe atoms deposited on the surface of an s - wave superconducting pb with strong spin - orbit interactions , and reported the striking observation of a zbp at the end of the atomic chains  with stm , which provides evidence for majorana zero modes . however , these above experimental results can not serve as definitive evidences to prove the existence of mfs in condensed matter systems , and it is also a major challenge in these experiments to uniquely distinguish majoranas from conventional fermionic subgap states . 
   the first reason is that the zero - bias conductance peaks can also appear in terms of the other mechanisms @xcite , such as the zero - bias anomaly due to kondo resonance @xcite and the disorder or band bending in the snw @xcite . 
   the second one is that andreev bound states in a magnetic field can also exhibit a zero - energy crossing as a function of exchange interaction or zeeman energy @xcite , and therefore give rise to similar conductance features . 
   as far as we know , most of the experimental evidences for majorana bound states largely relies on measurements of the tunneling conductance at present , and the observation of majorana signature based on electrical methods still remains a subject of debate . 
   identifying mfs only through tunnel spectroscopy is somewhat problematic . 
   therefore , to obtain definitive signatures of mfs , alternative setups or proposals for detecting mfs are necessary . here , we will propose an alternative all - optical scheme to detect mfs . 
   benefitting from recent advances in nanotechnology and nanofabrication , nanostructures such as quantum dots ( qds ) and nanomechanical resonators ( nrs ) have been obtained significant progress in modern nanoscience and nanotechnology . 
   qd , as a simple stationary atom with well optical property @xcite , lays the foundation for numerous possible applications urbaszekb . 
   due to high natural frequencies and large quality factors of nrs @xcite , if qds coupled to nrs @xcite to form hybrid systems , the coherent optical properties will be enhanced remarkably , which will be an alternative ultrasensitive detection means . 
   although probing mfs with qds @xcite have been proposed , we notice that all the schemes are still based on electrical means . in the present work , we propose an optical measurement scheme to detect the existence of mfs in iron chains on the superconducting pb surface @xcite via a coupled hybrid qd - nr system with optical pump - probe scheme @xcite .    compared with electrical detection means where the qds are coupled to mfs via the tunneling @xcite , in our optical scheme , there is no direct contact between mfs and the hybrid qd - nr system , which can effectively avoid introducing  other signals disturbing the detecting of mfs . 
   the interaction between mfs in iron chains and qd in hybrid qd - nr system is mainly due to the dipole - dipole interaction , and the distance between the two systems can be adjusted by several tens of nanometers , therefore the tunneling between the qd and mfs can be neglected safely . in addition , 
   the qd is considered as a two - level system rather than a single resonant level with spin - singlet state , and once mfs appear in  the end of iron chains and couple to the qd , the majorana signature will be carried out via the coherent optical spectrum of the qd . 
   the change in the coherent optical spectrum as a possible signature for mfs is another potential evidence in the iron chains . 
   this optical scheme will provide another method for the detection of mfs , which is very different from the zero - bias peak in the tunneling experiments mourikv , dasa , dengmt , churchillhoh , finckadk , nadj - perges2 . 
   furthermore , in order to investigate the role of the nr in the hybrid system , we further introduce the exciton resonance spectrum to detect mfs . 
   the results shows that the vibration of the nr acting as a phonon cavity will enhance the exciton resonance spectrum significantly and make mfs more sensitive to be detectable . 
   the technique proposed here provide a new platform for applications ranging from robust manipulation of mfs and mfs based quantum information processing . 
   figure 1(b ) shows the schematic setup that will be studied in this work , where iron ( fe ) chains on the superconducting pb(110 ) surface nadj - perges2 , and we employ a two - level qd with optical pump - probe technology to detect mfs . 
   the fe chain is ferromagnetically ordered nadj - perges2 with a large magnetic moment , which takes the role of the magnetic field in the nanowire experiments @xcite . 
   different from the proposal of mourik et al . 
   @xcite , this `` magnetic field '' is mostly localized on the fe chain , with small leakage outside , and superconductivity is not destroyed along the chain . in this setup , the energy scale of the exchange coupling of the fe atoms is typically much larger than that of the rashba spin - orbit coupling and the superconducting pairing . 
   figure 1(c ) displays that a qd is implanted in the nr to form a coupled hybrid qd - nr system . 
   the whole system includes two kinds of couplings which are qd - mf coupling and qd - nr coupling as shown in fig . 
   1(a ) . in the following 
   , we will discuss the two kinds of coupling in detail , respectively .    in the hybrid qd - nr system , 
   the qd is modeled as a two - level system consisting of the ground state @xmath1 and the single exciton state @xmath2 at low temperatures zrennera , stuflers , and the hamiltonian of the qd can be described as @xmath3 with the exciton frequency @xmath4 , where @xmath5 and @xmath6 are the pseudospin operator describing the two - level exciton with the commutation relation @xmath7 = \pm s^{\pm } $ ] and @xmath8 = 2s^{z}$ ] . for the nr , 
   the thickness of the beam is much smaller than its width , the lowest - energy resonance corresponds to the fundamental flexural mode that will constitute the resonator mode @xcite which can be characterized by a quantum harmonic oscillator with hamiltonian @xmath9 , where @xmath10 is the resonator frequency and @xmath11 is the annihilation operator of the resonator mode . 
   since the flexion induces extensions and compressions in the structure @xcite , this longitudinal strain will modify the energy of the electronic states of qd through deformation potential coupling . 
   then the coupling between the resonator mode and the qd is described by @xmath12 , where @xmath13 is the coupling strength between the resonator mode and qd wilson - raei . 
   thus we obtain the hamiltonian of the coupled hybrid qd - nr system@xmath14    for the qd - mf coupling , as each mf is its own antiparticle , we introduce an operator @xmath15 with @xmath16 and @xmath17 to describe mfs . 
   supposed that the qd couples to the nearby mf @xmath18 in the end of iron chains , then the hamiltonian is written by liude , flensbergk , leijnsem , caoys , lij @xmath19to detect mfs , it is helpful to switch the majorana representation to the regular fermion one via the exact transformation @xmath20 and @xmath21 @xmath22 , where @xmath23 and @xmath24 are the fermion annihilation and creation operators obeying the anti - commutative relation @xmath25 . 
   accordingly , in the rotating wave approximation @xcite , the above hamiltonian can be rewritten as@xmath26where the first term gives the energy of mf with frequency @xmath27 and @xmath28 with the iron chains length ( @xmath29 ) and the pb superconducting coherent length ( @xmath30 ) . 
   if the iron chains length ( @xmath29 ) is large enough , @xmath31 will approach zero . in the following 
   , we will discuss the two conditions of @xmath32 and @xmath33 , and define the two conditions as coupled mfs ( @xmath34 ) and uncoupled mfs ( @xmath33 ) , respectively . 
   the second term describes the coupling between the nearby mf and the qd with the coupling strength @xmath35 , where the coupling strength is related to the distance between the hybrid qd - nr system and the iron chains . 
   it should be also noted that the term of non - conservation for energy , i.e. @xmath36 , is generally neglected . 
   we have made the numerical calculations ( not shown in the following figures ) and shown that the effect of this term is too small to be considered in our theoretical treatment . 
   currently , the optical pump - probe technique has become a popular topic , which affords an effective way to investigate the light - matter interaction . 
   the optical pump - probe technology includes a strong pump laser and a weak probe laser @xcite . in the optical pump - probe technology , 
   the strong pump laser is used to stimulate the system to generate coherent optical effect , while the weak laser plays the role of probe laser . 
   therefore , the linear and nonlinear optical effects can be observed via the probe absorption spectrum based on the optical pump - probe scheme . 
   xu et al . have obtained coherent optical spectroscopy of semiconductor qd when driven simultaneously by two optical fields @xcite . 
   their results open the way for the demonstration of numerous quantum level - based applications , such as qd lasers , optical modulators , and quantum logic devices . in terms of this scheme , we apply the pump - probe scheme to the qd of the hybrid qd - nr system simultaneously . 
   when the optical pump - probe technology is applied on the qd , the majorana signature will be carried out via the coherent optical spectrum . 
   the hamiltonian of the exciton of the qd coupled to the two fields is given by @xcite @xmath37 , where @xmath38 is the dipole moment of the exciton , and @xmath39 is the slowly varying envelope of the field . 
   therefore , we obtain the whole hamiltonian of the hybrid system as @xmath40 . in a rotating frame at the pump field frequency @xmath41 , we obtain the total hamiltonian of the system as @xmath42where @xmath43 is the detuning of the exciton frequency and the pump frequency , @xmath44 is the rabi frequency of the pump field , and @xmath45 is the detuning of the probe field and the pump field . 
   @xmath46 is the detuning of the mf frequency and the pump frequency . 
   actually , we have neglected the regular fermion like normal electrons in the nanowire that interact with the qd in the above discussion . to describe the interaction between the normal electrons and the exciton in qd 
   , a tight binding hamiltonian of the whole iron chains is introduced chenhj .    according to the heisenberg equation of motion and introducing the corresponding damping and noise terms , the quantum langevin equations of the whole system 
   are derived as@xmath47@xmath48s^{-}+2(\beta f - i\omega _ { pu})s^{z}-\frac{2i\mu e_{pr}}{\hbar } e^{-i\delta t}s^{z}+\hat{% \tau}(t)\text{,}\]]@xmath49@xmath50where @xmath51 ( @xmath52 ) is the exciton spontaneous emission rate ( dephasing rate ) , @xmath53 is the position operator , @xmath54 is the decay rate of the nr , and @xmath55 is the decay rate of the mf . 
   @xmath56 is the @xmath57-correlated langevin noise operator , which has zero mean @xmath58 and obeys the correlation function @xmath59 . 
   the resonator mode is affected by a brownian stochastic force with zero mean value , and @xmath60 has the correlation function@xmath61,\]]where @xmath62 and @xmath63 are the boltzmann constant and the temperature of the reservoir of the coupled system . 
   mfs have the same correlation relation as the resonator mode as @xmath64.\]]in eq.(9 ) and eq.(10 ) , both the nr and majorana mode will be affected by a thermal bath of brownian and non - markovian processes @xcite . in the low temperature , the quantum effects of both the majorana and nr mode 
   are only observed in the case of @xmath65 and @xmath66 . 
   due to the weak coupling to the thermal bath , the brownian noise operator can be modeled as markovian processes . 
   in addition , both the qd - mfs coupling and qd - nr mode coupling in the hybrid system are stronger than the coupling to the reservoir that influences the two kinds coupling . in this case , owing to the second order approximation gardinercw , we can obtain the form of the reservoir that affects both the nr mode and majorana mode as eq.(9 ) and eq.(10 ) .    to go beyond weak coupling , 
   the heisenberg operator can be rewritten as the sum of its steady - state mean value and a small fluctuation with zero mean value@xmath67since the driving fields are weak , but classical coherent fields , we will identify all operators with their expectation values , and drop the quantum and thermal noise terms . 
   simultaneously , inserting these operators into the langevin equations eqs.(5)-(8 ) and neglecting the nonlinear term , we can obtain two equation sets about the steady - state mean value and the small fluctuation . 
   the steady - state equation set consisting of @xmath68 , @xmath69 and @xmath70  is related to the population inversion ( @xmath71 ) of the exciton which is determined by@xmath72 + 4w_{0}\gamma _ { 2}\omega _ { pu}^{2}(\delta _ { m}^{2}+\kappa _ { m}^{2}/4)=0.\end{gathered}\]]for the equation set of small fluctuation , we make the ansatz @xcite @xmath73 ( @xmath74 ) . solving the equation set and working to the lowest order in @xmath75 but to all orders in @xmath76 , we can obtain the linear susceptibility as @xmath77 , where @xmath78 is given by@xmath79\gamma _ { 2}}{\pi { % _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]@xmath68 , @xmath70 and @xmath69 can be derived from the steady - state equations , and @xmath80 , @xmath81 , @xmath82 , @xmath83/(\gamma _ { 1}-i\delta ) $ ] , @xmath84/(\gamma _ 
   { 1}-i\delta ) $ ] , @xmath85 , @xmath86 , @xmath87 , @xmath88 , @xmath89 ( @xmath90 indicates the conjugate of @xmath91 ) . 
   the imaginary and real parts of @xmath92 indicate absorption and dissipation , respectively . 
   in addition , the average population of the exciton states can be obtained as@xmath93}{\pi { _ { 2}\pi _ { 4}^{\ast } -\lambda _ { 1}\lambda _ { 2}\pi _ { 1}\pi _ { 3}^{\ast } } } , \]]which is benefited for readout the exciton states of qd . 
   for illustration of the numerical results , we choose the realistic hybrid systems of the coupled qd - nr system @xcite and the iron chains on the superconducting pb surface @xcite . for an inas qd in the coupled qd - nr system , 
   we use parameters @xcite : the exciton relaxation rate @xmath94 ghz , the exciton dephasing rate @xmath95 ghz . 
   the physical parameters of gaas nr are @xmath96 , @xmath97 , @xmath98 ghz , @xmath99 kg , @xmath100 , where @xmath101 , @xmath97 , and @xmath102 are the resonator frequency , the effective mass , and quality factor of the nr , respectively . 
   the decay rate of the nr is @xmath103 @xmath104 khz , and the coupling strength between the qd and nr is @xmath105 . for mfs , there are no experimental values for the lifetime of the mfs and the coupling strength between the exciton and mfs in the recent literature . 
   however , according to a few recent experimental reports @xcite , it is reasonable to assume that the lifetime of the mfs is @xmath106 mhz . 
   since the coupling strength between the qd and nearby mfs is dependent on their distance , we also expect the coupling strength @xmath107 ghz via adjusting the distance between the hybrid qd - nr system and the iron chains . 
   figure 2(a ) shows the coherent optical properties of the qd as functions of probe - exciton detuning @xmath108 at the detuning of the exciton frequency and the pump frequency @xmath109 , i.e. , the absorption ( @xmath110 ) and dissipation ( @xmath111 ) properties of the qd without considering any coupling ( @xmath112 ) , which indicates the normal absorption and dissipation of the qd , respectively . turning on the qd - nr coupling ( @xmath105 ) and without considering the qd - mf coupling ( @xmath113 ) 
   , two sharp peaks will appear in both the absorption and dissipation spectra as shown in fig . 
   2(b ) . from the curves , we find that the two sharp peaks at both sides of the spectra just correspond to the vibrational frequency of the nr . 
   the physical origin of this result is due to mechanically induced coherent population oscillation , which makes quantum interference between the resonator and the beat of the two optical fields via the qd when the probe - pump detuning is equal to the nr frequency @xcite . 
   this reveals that if fixing the pump field on - resonance with the exciton and scan through the frequency spectrum , the two sharp peaks can obtain immediately in the coherent optical spectra , which also indicates a scheme to measure the frequency of the nr . 
   this phenomenon stems from the quantum interference between the vibration nr and the beat of the two optical fields via the exciton when probe - pump detuning @xmath114 is adjusted equal to the frequency of the nr . 
   therefore , the qd - nr coupling play a key role in the hybrid system , and if we ignore the coupling ( @xmath115 ) , the above phenomenon will disappear completely as shown in fig . 
   compared with fig.2(b ) , in fig.2(c ) , we consider the qd coupled with the nearby mf @xmath18 without taking the qd - nr coupling into account , i.e. the condition of @xmath115 and @xmath107 ghz . as 
   the mfs appear in the ends of iron chains and coupled to the qd , both the probe absorption ( the blue curve ) and dissipation ( the green curve ) spectra will present an remarkable signature of mfs under @xmath116 ghz . 
   the physical origin of this result is due to the qd - mf coherent interaction and we can interpret this physical phenomenon with dressed state between the exciton and mfs . 
   the qd coupled to the nearby mf will induce the upper level of the state @xmath117 to split into @xmath118 and @xmath119 ( @xmath120 denotes the number states of the mfs ) . 
   the left peak in the coherent optical spectra signifies the transition from @xmath1 to @xmath121 while the right peak is due to the transition of @xmath122 to @xmath119 chenhj . to determine this signature is the true mfs rather than the normal electrons that couple with the qd 
   , we have used a tight binding hamiltonian to describe the electrons in whole iron chains , the numerical results indicate the signals in the absorption and dissipation spectra are the true mfs signature @xcite . if we consider both the two kinds coupling , i.e. the qd - nr coupling ( @xmath105 ) and qd - mfs coupling ( @xmath107 ghz ) as shown in fig . 
   2(d ) , not only the two sharp peaks locate at the nr frequency induced by its vibration , i.e. two peaks are at @xmath123 ghz ( @xmath124ghz ) , there is also mfs signal appear at @xmath125 ghz ( @xmath116 ghz ) induced by the qd - mf coupling .    in fig . 
   2(c ) , we only consider the situation of @xmath34 . 
   in fact , if the iron chains length @xmath29 is much larger than the pb superconducting coherent length @xmath30 , @xmath31 will approach zero . 
   therefore , it is necessary to consider the conditions of @xmath126 and @xmath33 , and we define them as coupled mfs mode ( @xmath32 ) and uncoupled mfs mode ( @xmath33 ) , respectively . 
   figure 3(a ) and figure 3(b ) show the absorption and dissipation spectra as a function of detuning @xmath127 with qd - mf coupling constants @xmath128 ghz under @xmath34 and @xmath33 , respectively . compared with the coupled mfs mode 
   , the uncoupled qd - mf hamiltonian will reduce to @xmath129 which is analogous j - c hamiltonian of standard model under @xmath33 , and the probe absorption spectrum ( the blue curve ) shows a symmetric splitting as the qd - mf coupling strength @xmath107 ghz which is different from of coupled mfs mode presenting unsymmetric splitting due to a detuning @xmath130 ghz . 
   therefore , our results reveal that the signals in the coherent optical spectra is a real signature of mf , and the optical detection scheme can work at both the coupled majorana edge states and the uncoupled majorana edge states .    in fig . 
   3(c ) , we further make a comparison of the probe absorption spectrum under the coupled mfs mode ( @xmath34 ) and uncoupled mfs mode ( @xmath131 ) . 
   it is obvious that the probe absorption spectrum display the analogous phenomenon of electromagnetically induced transparency ( eit ) @xcite under both the two conditions . 
   the dip in the probe absorption spectrum goes to zero at @xmath132 and @xmath133 ghz with @xmath33 and @xmath34 , respectively , which means the input probe field is transmitted to the coupled system without absorption . 
   such a phenomenon is attributed to the destructive quantum interference effect between the majorana modes and the beat of the two optical fields via the qd . if the beat frequency of two lasers @xmath57 is close to the resonance frequency of mfs , the majorana mode starts to oscillate coherently , which results in stokes - like ( @xmath134 ) and anti - stokes - like ( @xmath135 ) scattering of light from the qd . 
   the stokes - like scattering is strongly suppressed because it is highly off - resonant with the exciton frequency . 
   however , the anti - stokes - like field can interfere with the near - resonant probe field and thus modify the probe field spectrum . here 
   the majorana modes play a vital role in this coupled system , and we can refer the above phenomenon as majorana modes induced transparency , which is analogous with eit in atomic systems @xcite . 
   on the other hand , we can propose a means to determine the qd - mf coupling strength @xmath35 via measuring the distance of the two peaks with increasing the qd - mf coupling strength in the probe absorption spectrum . 
   figure 3(d ) indicates the peak - splitting width as a function of the qd - mf coupling strength @xmath35 under the condition of the coupled mfs mode ( @xmath136 ) and the uncoupled mfs mode ( @xmath33 ) which follows a nearly linear relationship . 
   it is obvious that the two lines ( the uncoupled mfs and the coupled mfs mode ) have a slight deviation . 
   however , the deviation becomes slighter with increasing coupling strength . 
   therefore , it is essential to enhance the coupling strength for a clear peak splitting via adjusting the distance between the qd and the nearby mfs . in this case 
   the coupling strength can obtain immediately by directly measuring the distance of the two peaks in the probe absorption spectrum .    as shown in fig . 
   2(d ) , there are not only two sharp peaks locate at the nr frequency induced by its vibration but also the mfs signal appear at @xmath137 induced by the qd - mf coupling in the probe absorption spectrum ( the blue curve ) under the two kinds coupling . in fig . 
   4(a ) , we further consider switching the detuning @xmath116 ghz to @xmath138 ghz at small exciton - pump detuning @xmath139 ghz . 
   it is obvious that the resonance amplification process ( 1 ) and the resonance absorption process ( 2 ) in the probe absorption spectrum without considering the qd - mf coupling ( the blue curve , @xmath113 ) will accordingly transform into the the resonance absorption process ( 3 ) and the resonance amplification process ( 4 ) due to the qd - mf coupling ( the green curve , @xmath140 ghz ) . 
   return to fig . 
   1(a ) , there are two kinds of coupling which are qd - nr coupling and qd - mf coupling in the hybrid system . for the qd - nr system 
   , the two sharp peaks in the probe absorption corresponding to the resonance amplification ( 1 ) and absorption process ( 2 )  can be elaborated with dressed states @xmath141 , @xmath142 , @xmath143 , @xmath144 ( @xmath145 denotes the number state of the resonance mode ) , and the two sharp peaks indicate the transition between the dressed states @xcite . 
   however , once mfs appear in the ends of iron chains and coupled to the qd , the ground state @xmath146 and the exciton state @xmath2 of the qd will also modify by the number states of the mfs @xmath120 and induce the majorana dressed states @xmath147 , @xmath148 , @xmath118 , @xmath149 . with increasing the qd - mf coupling , the majorana dressed states will affect the amplification ( 1 ) and absorption process ( 2 ) significantly , and even realize the inversion between the absorption ( 3 ) and amplification ( 4 ) process due to the qd - mf coherent interaction ( the green curve ) .    to illustrate the advantage of the nr in the hybrid system 
   , we introduce the exciton resonance spectrum to investigate the role of nr in the coupled qd - nr , which is benefited for readout the exciton states of qd . in fig . 
   4(b ) , we adjust the detuning @xmath116 ghz to @xmath150 ghz , therefore , the location of the two sideband peaks induced by the qd - mf coupling coincides with the two sharp peaks induced by the vibration of nr , thus the nr is resonant with the coupled qd - mf system and makes the coherent interaction of qd - mf more strong . 
   figure 4(b ) shows the exciton resonance spectrum of the probe field as a function of the probe detuning @xmath151 with the detuning @xmath139 ghz under the coupled mfs mode @xmath34 . 
   the black and red curves correspond to @xmath115 and @xmath152 for the qd - mf coupling @xmath153 ghz , respectively . 
   it is obvious that the role of nr is to narrow and to increase the exciton resonance spectrum . in this case 
   , the nr behaves as a phonon cavity will enhance the sensitivity for detecting mfs . 
   we have proposed an all - optical means to detect the existence of mfs in iron chains on the superconducting pb surface with a hybrid qd - nr system . 
   the signals in the coherent optical spectra indicate the possible majorana signature , which provides another supplement for detecting mfs . due to the vibration of nr , 
   the exciton resonance spectrum becomes much more significant and then enhances the detection sensitivity of mfs . 
   in addition , the qd - mf coupling in our system is a little feeble , while ref . 
   [ 35 ] presents a strong qd - mf coupling and the coupling strength can reach kilohertz , which is beneficial for mfs detection . on the other hand , if we consider embedding a metal nanoparticle - quantum dot ( mnp - qd ) complex chenhj , lijj in the nr , the surface plasmon induced by the mnp will enhance the coherent optical property of qd , which may be robust for probing mfs . 
   the concept proposed here , based on the quantum interference between the nr and the beat of the two optical fields , is the first all - optical means to probe mfs . 
   this coupled system will provide a platform for applications in all - optically controlled topological quantum computing based on mfs . 
   the authors gratefully acknowledge support from the national natural science foundation of china ( no.11574206 , no.10974133 , no.11274230 , no.61272153 , no.61572035 , no.51502005 , and no.11404005 ) , the key foundation for young talents in college of anhui province ( no . 
   2013sqrl026zd ) , and the foundation for phd in anhui university of science and technology . 
   i. yeo , p. l. de assis , a. gloppe , e. dupont - ferrier , p. verlot , n. s. malik , e. dupuy , j. claudon , j. m. grard , a. auffves , g. nogues , s. seidelin , j. p. poizat , o. arcizet , and m. richard , nat . nanotechnol . 
   * 9 * , 106 ( 2014 ) . 
   fig.1 sketch of the proposed setup for optically detecting majorana fermions ( mfs ) . 
   ( a ) the energy - level diagram of a qd coupled to mfs and nr , which includes two kinds coupling , i.e. the qd - mf coupling ( the dotted frame ) and the qd - nr coupling ( the dashed frame ) . 
   ( b ) the iron chains on the superconducting pb surface , and a pair of mfs appear in the ends of the iron chains . 
   the nearby mf is coupled to ( c ) a qd embedded in a nanomechanical resonator ( nr ) with optical pump - probe technology . 
   fig.2 the absorption ( the blue curve ) and dispersion ( the green curve ) spectra of probe field as a function of the probe detuning @xmath127 under different conditions . 
   ( a ) without considering any coupling , i.e. , @xmath115 and @xmath113 . 
   ( b ) the qd - nr coupling strength is @xmath105 and @xmath113 . 
   ( c ) the qd - mf coupling strength is @xmath107 ghz and @xmath115 . 
   ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath154 ghz . 
   the parameters used are @xmath94 ghz , @xmath95 ghz , @xmath155 khz , @xmath124 ghz , @xmath156 mhz , @xmath157(ghz)@xmath158 , @xmath116 ghz , and @xmath109 . 
   fig.3 ( a ) and ( b ) show the probe absorption ( the blue curve ) and dispersion ( the green curve ) spectra with qd - mf coupling strengths @xmath107 ghz under @xmath34 and @xmath33 , respectively . 
   ( c ) the probe absorption spectrum under @xmath34 ( the green curve ) and @xmath131 ( the blue curve ) , respectively . 
   ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath159 . 
   the other parameters used are the same as in fig.2 . 
   fig.4 ( a ) the probe absorption spectrum as a function of the probe detuning @xmath160 with considering ( the blue curve , @xmath153 ghz ) and without considering ( the green curve , @xmath113 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . 
   ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath105 at the qd - mf coupling strength @xmath153 ghz . 
   @xmath150 ghz , @xmath139 ghz , @xmath161(ghz)@xmath158 , the other parameters used are the same as fig.2 .         under different conditions . 
   ( a ) without considering any coupling , i.e. , @xmath115 and @xmath162 . 
   ( b ) the qd - nr coupling strength is @xmath105 and @xmath163 . 
   ( c ) the qd - mf coupling strength is @xmath164 ghz and @xmath115 . 
   ( d ) considering both the qd - nr coupling and qd - mf coupling , i.e. , @xmath105 and @xmath164 ghz . 
   the parameters used are @xmath165 ghz , @xmath166 ghz , @xmath167 khz , @xmath168 ghz , @xmath169 mhz , @xmath170(ghz)@xmath158 , @xmath116 ghz , and @xmath109.,width=453 ]     ghz under @xmath171 and @xmath172 , respectively . 
   ( c ) the probe absorption spectrum under @xmath173 ( the green curve ) and @xmath174 ( the blue curve ) , respectively . 
   ( d ) the linear relationship between the distance of peak splitting and the coupling strength of qd - mf @xmath175 . 
   the other parameters used are the same as in fig.2.,width=453 ]     with considering ( the blue curve , @xmath176 ghz ) and without considering ( the green curve , @xmath162 ) the qd - mf coupling under the qd - nr coupling strength @xmath105 . 
   ( b ) the exciton resonance spectrum as a function of @xmath127 with @xmath115 and @xmath152 at the qd - mf coupling strength @xmath177 ghz . 
   @xmath138 ghz , @xmath139 ghz , @xmath178(ghz)@xmath158 , the other parameters used are the same as fig.2.,width=453 ]
summary: motivated by a recent experiment [ nadj - perge et al . , science 346 , 
   602 ( 2014 ) ] providing evidence for majorana zero modes in iron chains on the superconducting pb surface , in the present work , we theoretically propose an all - optical scheme to detect majorana fermions , which is very different from the current tunneling measurement based on electrical means . 
   the optical detection proposal consists of a quantum dot embedded in a nanomechanical resonator with optical pump - probe technology . with the optical means , 
   the signal in the coherent optical spectrum presents a distinct signature for the existence of majorana fermions in the end of iron chains . 
   further , the vibration of the nanomechanical resonator behaving as a phonon cavity will enhance the exciton resonance spectrum , which makes the majorana fermions more sensitive to be detectable . 
   this optical scheme affords a potential supplement for detection of majorana fermions and supports to use majorana fermions in fe chains as qubits for potential applications in quantum computing devices .