Figure 1 presents the schematic setup that will be studied in this work. An InSb semiconductor nanowire with spin-orbit coupling in an external aligned parallel magnetic field **B** is placed on the surface of a bulk s-wave superconductor (SC). A MF pair is expected to locate at the ends of nanowire. To detect MFs, we employ a hybrid system in which an InAs semiconductor QD is embedded in a GaAs NR. By applying a strong pump laser and a weak probe laser to the QD simultaneously, one could probe the MFs *via* optical pump-probe technique [30, 31].

Benefitting from recent progress in nanotechnology, the quantum nature of a mechanical resonator can be revealed and manipulated in the hybrid system where a single QD is coupled to a NR [40–42]. In such a hybrid system, the QD is modeled as a two-level system consisting of the ground state |*g*〉 and the single exciton state |*e* *x*〉 at low temperatures [50, 51]. The Hamiltonian of the QD can be described as ${H}_{\text{QD}}=\hslash {\omega}_{\text{QD}}{S}^{z}$ with the exciton frequency *ω*_{QD}, where *S*^{
z
} is the pseudospin operator. In a structure of the NR where 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 [40]. We use a Hamiltonian of quantum harmonic oscillator ${H}_{m}=\hslash {\omega}_{m}{b}^{+}b$ with the frequency *ω*_{
m
} and the annihilation operator *b* of the resonator mode to describe the eigenmode. Since the flexion induces extensions and compressions in the structure [52], 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 $\hslash {\omega}_{m}\eta {S}^{z}({b}^{+}+b)$, where *η* is the coupling strength between the resonator mode and QD [40]. Therefore, the Hamiltonian of the hybrid QD-NR system is ${H}_{\text{QD-NR}}=\hslash {\omega}_{\text{QD}}{S}^{z}+\hslash {\omega}_{m}{b}^{+}b+\hslash {\omega}_{m}\eta {S}^{z}({b}^{+}+b)$.

Since several experiments [15–20] have reported the distinct signatures of MFs in the hybrid semiconductor/superconductor heterostructure *via* electrical methods, we assure that the MFs may exist in these hybrid systems under some appropriate conditions. Based on these experimental results, in the present article, we will try to demonstrate the MFs by using nonlinear optical method. As each MF is its own antiparticle, one can introduce a MF operator *γ*_{MF} such that ${\gamma}_{\text{MF}}^{\u2021}={\gamma}_{\text{MF}}$ and ${\gamma}_{\text{MF}}^{2}=1$ to describe MFs. Supposing the QD couples to *γ*_{MF1}, the Hamiltonian of the hybrid system [43–46] is ${H}_{\text{MF}}=\mathrm{i\hslash}{\omega}_{\text{MF}}{\gamma}_{\text{MF1}}{\gamma}_{\text{MF2}}/2+\mathrm{i\hslash g}({S}^{-}-{S}^{+}){\gamma}_{\text{MF1}}$, where *S*^{±} are the pseudospin operators. To detect the existence of MFs, it is helpful to switch from the Majorana representation to the regular fermion one *via* the exact transformation ${\gamma}_{\text{MF1}}={f}_{M}^{+}+{f}_{M}$ and ${\gamma}_{\text{MF2}}=i({f}_{M}^{+}-{f}_{M})$. *f*_{
M
} and ${f}_{M}^{+}$ are the fermion annihilation and creation operators obeying the anti-commutative relation $\left\{{f}_{M},{f}_{M}^{+}\right\}=1$. Accordingly, in the rotating wave approximation [53], *H*_{
M
} can be rewritten as ${H}_{\text{MF}}=\hslash {\omega}_{\text{MF}}({f}_{M}^{+}{f}_{M}-1/2)+\mathrm{i\hslash g}({S}^{-}{f}_{M}^{+}-{S}^{+}{f}_{M})$, where the first term gives the energy of MF at frequency *ω*_{MF}, and $\hslash {\omega}_{\text{MF}}={\epsilon}_{\text{MF}}\sim {e}^{-l/\xi}$ with the wire length (*l*) and the superconducting coherent length (*ξ*). This term is small and can approach zero as the wire length is large enough. The second term describes the coupling between the right MF and the QD with coupling strength *g*, where the coupling strength depends on the distance between the hybrid QD-NR system and the hybrid semiconductor/superconductor heterostructure. Compared with electrical detection scheme which the QD is coupled to MF *via* the tunneling, here in our optical scheme, the exciton-MF coupling is mainly due to the dipole-dipole interaction. Since in current experiments the distance between QD and MF can be adjusted to locate the distance by about several tens of nanometers. In this case, the tunneling between the QD and MF can be neglected. It should be also noted that the term of non-conservation for energy, i.e. $\mathrm{i\hslash g}({S}^{-}{f}_{M}-{S}^{+}{f}_{M}^{+})$, 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, especially for calculating the nonlinear optical properties of the QD.

The optical pump-probe technology includes a strong pump laser and a weak probe laser [54], which provides an effective way to investigate the light-matter interaction. Based on the optical pump-probe scheme, the linear and nolinear optical effects can be observed *via* the probe absorption spectrum. Xu et al. [30] have obtained coherent optical spectroscopy of a strongly driven quantum dot without a nanomechanical resonator. Recently, this optical pump-probe scheme has also been demonstrated experimentally in a cavity optomechanical system [31]. In terms of this scheme, we apply a strong pump laser and a weak probe laser to the QD embedded in the NR simultaneously. The Hamiltonian of the QD coupled to the pump laser and probe laser is given by [54]${H}_{\text{QD-L}}=-\text{\xb5}{E}_{\text{pu}}({S}^{+}{e}^{-i{\omega}_{\text{pu}}t}+{S}^{-}{e}^{i{\omega}_{\text{pu}}t})-\text{\xb5}{E}_{\text{pr}}({S}^{+}{e}^{-i{\omega}_{\text{pr}}t}+{S}^{-}{e}^{i{\omega}_{\text{pr}}t})$, where µ is the dipole moment of the exciton, *ω*_{pu} (*ω*_{pr}) is the frequency of the pump (probe) laser, and *E*_{pu} (*E*_{pr}) is the slowly varying envelope of the pump (probe) laser. Therefore, one can obtain the total Hamiltonian of the hybrid system as *H*=*H*_{QD-NR}+*H*_{MBS}+*H*_{QD-L}.

According to the Heisenberg equation of motion and introducing the corresponding damping and noise terms, in a rotating frame at the pump laser frequency

*ω*_{pu}, we derive the quantum Langevin equations of the coupled system as follows:

$\begin{array}{c}\phantom{\rule{-15.0pt}{0ex}}{\stackrel{\u0307}{S}}^{z}=-\phantom{\rule{0.3em}{0ex}}{\Gamma}_{1}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\left({S}^{z}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}\frac{1}{2}\right)\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}i{\Omega}_{\text{pu}}\phantom{\rule{0.3em}{0ex}}\left({S}^{+}\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}{S}^{-}\right)\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}g\phantom{\rule{0.3em}{0ex}}\left({S}^{-}{f}_{M}^{+}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{S}^{+}{f}_{M}\right)\hfill \\ \phantom{\rule{2.5em}{0ex}}+\frac{i\text{\xb5}{E}_{\text{pr}}}{\hslash}({S}^{+}{e}^{-\mathrm{i\delta t}}-{S}^{-}{e}^{\mathrm{i\delta t}}),\hfill \end{array}$

(1)

$\begin{array}{c}\phantom{\rule{-15.0pt}{0ex}}{\stackrel{\u0307}{S}}^{-}=-\left[i\right({\Delta}_{\text{pu}}+{\omega}_{m}\mathrm{\eta N})+{\Gamma}_{2}]{S}^{-}-\frac{2i\text{\xb5}{E}_{\text{pr}}}{\hslash}{e}^{-\mathrm{i\delta t}}{S}^{z}\hfill \\ \phantom{\rule{2.8em}{0ex}}+2({\mathit{\text{gf}}}_{M}-i{\Omega}_{\text{pu}}){S}^{z}+{\widehat{F}}_{\text{in}}\left(t\right),\hfill \end{array}$

(2)

$\begin{array}{c}\phantom{\rule{-14.0pt}{0ex}}{\stackrel{\u0307}{f}}_{M}=-(i{\Delta}_{\text{MF}}+{\kappa}_{\text{MF}}/2){f}_{M}+g{S}^{-}+{\widehat{\xi}}_{\text{MF}}\left(t\right),\end{array}$

(3)

$\begin{array}{c}\phantom{\rule{-15.0pt}{0ex}}\stackrel{\u0308}{N}+{\gamma}_{m}\stackrel{\u0307}{N}+{\omega}_{m}^{2}N=-2{\omega}_{m}^{2}\eta {S}^{z}+\widehat{\xi}\left(t\right),\end{array}$

(4)

where *N*=*b*^{+}+*b*. *Γ*_{1} (*Γ*_{2}) is the exciton relaxation rate (dephasing rate), *κ*_{MF} (*γ*_{
m
}) is the decay rate of the MF (nanomechanical resonator). *Δ*_{pu}=*ω*_{QD}-*ω*_{pu} is the detuning of the exciton frequency and the pump frequency, ${\Omega}_{\text{pu}}=\text{\xb5}{E}_{\text{pu}}/\hslash $ is the Rabi frequency of the pump field, and *δ*=*ω*_{pr}-*ω*_{pu} is the probe-pump detuning. *Δ*_{MF}=*ω*_{MF}-*ω*_{pu} is the detuning of the MF frequency and the pump frequency. ${\widehat{F}}_{\text{in}}\left(t\right)$ is the *δ*-correlated Langevin noise operator, which has zero mean $\u3008{\widehat{F}}_{\text{in}}\left(t\right)\u3009=0$ and obeys the correlation function $\u3008{\widehat{F}}_{\text{in}}\left(t\right){\widehat{F}}_{\text{in}}^{\u2021}\left({t}^{\prime}\right)\u3009=\delta (t-{t}^{\prime})$. The resonator mode is affected by a Brownian stochastic force with zero mean value, and $\widehat{\xi}\left(t\right)$ has the correlation function $\u3008{\widehat{\xi}}^{+}\left(t\right)\widehat{\xi}\left({t}^{\prime}\right)\u3009=\frac{{\gamma}_{m}}{{\omega}_{m}}\int \frac{\mathrm{d\omega}}{2\pi}\omega {e}^{-\mathrm{i\omega}(t-{t}^{\prime})}[1+coth(\frac{\mathrm{\hslash \omega}}{2{\kappa}_{B}T}\left)\right]$, where *k*_{
B
} and *T* are the Boltzmann constant and the temperature of the reservoir, respectively. MFs have the same correlation relation as the resonator mode. Actually, we have neglected the regular fermions (i.e. normal electrons) in the nanowire that interact with the QD in the above discussion. To describe the interaction between the normal electrons and the QD, we use the tight-binding Hamiltonian of the whole wire as [55, 56]${H}_{\text{QD-e}}=\hslash {\omega}_{\text{QD}}{S}^{z}+\hslash \sum _{k}{\omega}_{k}{c}_{k}^{+}{c}_{k}+\mathrm{\hslash \zeta}\sum _{k}({c}_{k}^{+}{S}^{-}+{S}^{+}{c}_{k})$, where *c*_{
k
} and ${c}_{k}^{\u2021}$ are the regular fermion annihilation and creation operators with energy *ω*_{
k
} and momentum $\mathrm{\hslash k}$ obeying the anti-commutative relation $\left\{{c}_{k},{c}_{k}^{\u2021}\right\}=1$ and *ζ* is the coupling strength between the normal electrons and QD (here, for simplicity, we have neglected the *k*-dependence of *ζ* as in [57]).

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:

${S}^{z}={S}_{0}^{z}+\delta {S}^{z}$,

${S}^{-}={S}_{0}^{-}+\delta {S}^{-}$,

*f*_{
M
}=

*f*_{M 0}+

*δ* *f*_{
M
} and

*N*=

*N*_{0} +

*δ* *N*. Since 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 [

31]. Simultaneously, inserting these operators into the Langevin equations (Equations 1 to 4) 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

*f*_{M 0},

*N*_{0} and

${S}_{0}^{-}$ is related to the population inversion (

${w}_{0}=2{S}_{0}^{z}$) of the exciton which is determined by

${\Gamma}_{1}({w}_{0}+1)\left[\right({\Delta}_{\text{MF}}^{2}+{\kappa}_{\text{MF}}^{2}/4\left)\right({\Delta}_{\text{pu}}^{2}+{\Gamma}_{2}^{2}+{\omega}_{m}^{2}{\eta}^{4}{w}_{0}^{2}-2{\omega}_{m}{\Delta}_{\text{pu}}{\eta}^{2}{w}_{0})+{g}^{2}{w}_{0}^{2}({g}^{2}-2{\omega}_{m}{\Delta}_{\text{MF}}{\eta}^{2}+2{\Delta}_{\text{pu}}{\Delta}_{\text{MF}}-{\Gamma}_{2}{\kappa}_{\text{MF}}\left)\right]+4{\Omega}_{\text{pu}}^{2}{w}_{0}{\Gamma}_{2}({\Delta}_{\text{MF}}^{2}+{\kappa}_{\text{MF}}^{2}/4)=0$. For the equation set of small fluctuation, we make the ansatz [

54]

$\u3008\delta {S}^{z}\u3009={S}_{+}^{z}{e}^{-\mathrm{i\delta t}}+{S}_{-}^{z}{e}^{\mathrm{i\delta t}}$, 〈

*δ* *S*^{-}〉=

*S*_{+}*e*^{-i δ t}+

*S*_{-}*e*^{
i
δ
t
}, 〈

*δ* *f*_{
M
}〉=

*f*_{M+}*e*^{-i δ t}+

*f*_{M-}*e*^{
i
δ
t
}, and 〈

*δ* *N*〉=

*N*_{+}*e*^{-i δ t}+

*N*_{-}*e*^{
i
δ
t
}. Solving the equation set and working to the lowest order in

*E*_{pr} but to all orders in

*E*_{pu}, we can obtain the nonlinear optical susceptibility as

${\chi}_{\text{eff}}^{\left(3\right)}\left({\omega}_{\text{pr}}\right)=\text{\xb5}{S}_{-}\left({\omega}_{\text{pr}}\right)/\left(3{E}_{\text{pu}}^{2}{E}_{\text{pr}}\right)={\Sigma}_{3}{\chi}^{\left(3\right)}\left({\omega}_{\text{pr}}\right)$, where

${\Sigma}_{3}={\text{\xb5}}^{4}/\left(3{\hslash}^{3}{\Gamma}_{2}^{3}\right)$ and

*χ*^{(3)}(

*ω*_{pr}) is given by

${\chi}^{\left(3\right)}\left({\omega}_{\text{pr}}\right)=\frac{({d}_{2}^{\ast}+{h}_{4}{d}_{1}^{\ast}){d}_{3}{h}_{6}-{\mathit{\text{iw}}}_{0}{d}_{3}{h}_{4}}{({h}_{4}{h}_{5}{d}_{3}{d}_{1}^{\ast}-{d}_{4}{d}_{2}^{\ast}){\Omega}_{\text{pu}}^{2}}{\Gamma}_{2}^{3},$

(5)

where *b*_{1}=*g*/[*i*(*Δ*_{MF}-*δ*)+*κ*_{MF}/2], *b*_{2}=*g*/[ *i*(*Δ*_{MF}+*δ*)+*κ*_{MF}/2], ${b}_{3}=2{\omega}_{m}^{2}\eta /({\delta}^{2}+\mathrm{i\delta}{\gamma}_{m}-{\omega}_{m}^{2})$, ${h}_{4}=\left[\right(i{\Omega}_{\text{pu}}-{\mathit{\text{gf}}}_{M0})-{\mathit{\text{gS}}}_{0}^{-}{b}_{1}^{\ast}]/({\Gamma}_{1}+\mathrm{i\delta})$, ${h}_{5}=-\left[\phantom{\rule{0.3em}{0ex}}\right(i{\Omega}_{\text{pu}}+{\mathit{\text{gf}}}_{M0}^{\ast})+{\mathit{\text{gS}}}_{0}^{-\ast}{b}_{2}]/({\Gamma}_{1}+\mathrm{i\delta})$, ${h}_{6}={\mathit{\text{iS}}}_{0}^{-}/({\Gamma}_{1}+\mathrm{i\delta})$, ${d}_{1}=2({\mathit{\text{gf}}}_{M0}-i{\Omega}_{\text{pu}})-i{\omega}_{m}\eta {S}_{0}^{-}{b}_{3}$, *d*_{2}=*i*(*Δ*_{pu}-*δ*+*ω*_{
m
}*η* *N*_{0})+*Γ*_{2}-*g* *b*_{1}*w*_{0}-*d*_{1}*h*_{2}, ${d}_{3}=2({\mathit{\text{gf}}}_{M0}-i{\Omega}_{\text{pu}})-i{\omega}_{m}\eta {S}_{0}^{-}{b}_{3}^{\ast}$, *d*_{4}=*i*(*Δ*_{pu}+*δ*+*ω*_{
m
}*η* *N*_{0})+*Γ*_{2}-*g* *b*_{2}*w*_{0}-*d*_{3}*h*_{5} (where *O*^{∗} indicates the conjugate of *O*). The quantum Langevin equations of the normal electrons coupled to the QD have the same form as MFs; therefore, we omit its derivation and only give the numerical results in the following.