Rabi splitting and optical Kerr nonlinearity of quantum dot mediated by Majorana fermions

Majorana fermions (MFs), due to their significance in fundamental physics and potential applications in topological quantum computation based on solid-state devices, have drawn widespread attention. Here, we design a hybrid semiconductor/superconductor ring (S/SR) device and put forward an optical method for detecting a pair of MFs in the S/SR device with a quantum dot (QD) with the popular optical pump-probe scheme, which is very different from the present electrical method only considering one nearby MF coupled to the QD. The symmetric and unsymmetric Rabi splitting and optical Kerr nonlinear effect of QD mediated by MFs are investigated under uncoupled and coupled majorana modes, respectively. The coherent optical spectra indicate that a pair of MFs coupled to the QD induced remarkable splitting under different parameters regime and the physical origin of these phenomena are elaborated detailedly. Due to QD-MFs coupling, the probe absorption spectra present the phenomenon of Majorana modes induced transparency (MMIT) which will induce remarkable phenomena of slow light. The coherent optical spectra afford a potential supplement for probing MFs and support Majorana fermions-based topological quantum computation.

Majorana fermions (MFs) are exotic particles whose antiparticle is itself γ γ = † obeying non-Abelian statistics. Therefore MFs can realize subsequent potential applications in topological quantum computation and quantum information processing 1 . Despite proposed initially to investigate neutrinos, one has discovered the analogous Majorana zero modes in condensed matter systems 2 . Currently, Majorana signature have been observed experimentally in various hybrid systems including hybrid semiconducting nanowire (atomic chains)/superconductor structure 3-7 , iron-based superconductor 8 , topological structure 9,10 , and quantum anomalous Hall insulatorsuperconductor structure 11 . For detecting MFs, several typical means have also been presented experimentally, such as zero-bias peaks (ZBPs) in tunneling spectroscopy [3][4][5][6][7] , fractional a.c. Josephson effect 12 , Coulomb blockade spectroscopy experiment 9 , and spin-polarized scanning tunneling microscopy 13 . We notice that most of the recent theories and experiments for researching and detecting MFs proposed and carried out focus on electrical scheme, and other effective and alternative methods, for example, all-optical means for probing Majorana bound states have received less attention.
On the other hand, nanostructures such as quantum dots (QDs) have been obtained remarkable progress in modern nano-science and nano-technology in recent years 14 . QD, consiered as a simple stationary "artificial atom" with well optical property 15 , lays the foundation for numerous potential applications 16 , where the significant applications including entanglement concentration, logic gate construction and entangled state preparation with optical Kerr nonlinear effect have drawn widespread attention 14,17,18 . The optical Kerr nonlinear effect manifested by the third-order optical Kerr nonlinearity of QD, which is essential for light-controlled phase and refractive index modulation in various fields such as optical telecommunications, optical data storage, and information processing 14,19 . If coupling the QD to metallic nanoparticle (MNP) 20,21 , the plasmon resonance on MNP will amplify the local field near the QD and thereby intensify the mechanically induced optical Kerr nonlinearity, which will realize several orders of magnitude in the optical Kerr effect larger than in intrinsic Si and other hybrid systems 22 . Further, the detection of MFs with the QD have been proposed theoretically [23][24][25][26][27] , and most recently Majorana bound states have been investigated at the end of epitaxial hybrid Sm-S nanowires using tunneling spectroscopy through QDs 28 . However, in these schemes, QD is consider as only one resonant level with single spin state [23][24][25][26][27] , and we will consider the QD as a two-level system (TLS) (as shown in Fig. 1 In this work, rather than previous schemes for probing MFs in electrical domain, firstly, we design a hybrid semiconductor/superconductor ring (S/SR) device as shown in Fig. 1(a), and the QD is driven by the optical pump-probe technology 29 for detecting a couple of MFs emerging in the ends of the semiconducting nanowire ring under in a magnetic field. Due to the QD has no direct contact with the S/SR device, the scheme is immune to the disturbance of other signals in detecting of MFs with the optical means. Once a couple of MFs appear at the end of the nanowire ring and have interaction with the QD, the signatures of MFs will be brought out with the coherent optical response. Secondly, we further investigate the Rabi splitting and optical Kerr nonlinear effect of QD mediated by MFs. The probe absorption spectra present symmetrical and asymmetrical Rabi splitting under uncoupled MFs modes (ϵ MF = 0) and coupled MFs modes (ϵ MF ≠ 0), respectively. When a pair of MFs couple to the QD, the Rabi splitting is enhanced, which depends on the coupling strength between a pair of MFs and the QD. With adjusting the exciton resonant detuning and Majorana resonant detuning, the probe absorption intensity and the peak width of splitting are enhanced significantly. The optical Kerr nonlinear phenomena are also investigated under different parameter regions, and the peak width of splitting in the optical Kerr spectrum depends on the coupling strength and resonant detuning.

Results
The Model and Hamiltonian. Figure 1 shows a QD coupled to a pair of MFs emerging in the hybrid S/SR device. We consider the QD as a TLS which include the ground state g and the single exciton state e 30,31 , and its Hamiltonian can be described as H QD = ℏω e S z , where ω e is the exciton frequency. We introduce the pseudo-spin operators S z and S ± to describe the QD, which has the commutation relations [S z , S ± ] = ±S ± and [S + , S − ] = 2S z . In order to describe MFs, we also introduce an operator γ M with relation γ γ due to MFs are their own antiparticle. Then the coupling Hamiltonian of QD and a pair of MFs γ M1 and γ M2 is [23][24][25][26][27] .
, where † f M (f M ) is the fermion creation (annihilation) operator. Using rotating wave approximation 32 , Eq. (1) can be rewritten as We have neglected the non-conservation term of energy incluing β − , and numerical results indicate that the influence of them are too weak to be considered in the theoretical treatment.
The Hamiltonian of s indicates a strong pump laser and a weak probe laser simultaneously irradiating to the QD 33 and has a interaction with the QD, in which μ is the electric dipole moment of the QD, ω c and ω s with light intensity E c and E s are the frequencies of the pump laser and probe laser fields, respectively. We use the rotating frame of the pump laser frequency ω c , and then obtain the full Hamiltonian of our hybrid system 34 where Δ c = ω e − ω c is the exciton-pump field detuning, Δ MF = ω MF − ω c is the Majorana-pump field detuning, and δ = ω s − ω c is the probe-pump detuning. Ω c = μE c /ℏ indicates the Rabi frequency of the pump field.

The Quantum Langevin Equations (QLEs). We use the Heisenberg equation of motion iℏ∂ t ρ = [ρ, H]
(ρ = S z , S − , f M ) and introduce corresponding damping and noise operators, we obtain the QLEs as 35 : where Γ 1 (Γ 2 ) is the relaxation rate (dephasing rate) of the exciton, and κ M is MFs decay rate. τ in (t) describe the Langevin noise follows the relations of τ τ and 〈τ in (t)〉 = 0. MFs are also influenced by stochastic force process with the following correlation function where k B is Boltzmann constant, T indicates the reservoir temperature. In Eq. (7), both the Brownian processes and non-Markovian processes 36 will influence Majorana modes, and the quantum signatures of MFs modes can be observed only in the case of resolved sideband regime, i.e., ω MF /κ M >> 1 under low temperature T. In the weak coupling regime, the effect of Brownian noise is modeled as Markovian processes. Further, the interaction between the QD and MFs is stronger than the interation with the reservoir, considering second order approximation 36 , then the reservoir that affects Majorana modes can be obtained as shown in Eq. (7).

The Coherent Optical Response of QD.
When the QD is driven by a strong pump laser, the Heisenberg operator can be divided into two parts, i.e., steady-state mean value ρ 0 , and small fluctuation δρ with zero mean value 〈δρ〉 = 0. To solve the steady-state values, we obtain the the steady-state population inversion (θ = S 2 z 0 0 ) as follows ( We here make the ansatz to solve the fluctuation equation 33

solving the fluctuation equation set, the linear optical susceptibility is
, and χ (1) (ω s ) is given by where The real and imaginary parts of χ (1) (ω s ) show the dispersion and absorption, respectively. The nonlinear optical susceptibility can be calculated as ,  , and χ (3) (ω s ) is given by The real and imaginary parts of χ (3) (ω s ) characterize the Kerr coefficient and nonlinear absorption, respectively.
According to several recent experiment reports 3-7 , we expect that the decay rate κ M = 0.1 × 10 6 Hz. For the coupling strength β of the QD and MFs, we use the previous published work 38 , and the numerical value is β = 0.05 × 10 9 Hz which is related to the distance between the QD and the hybrid S/SR device as shown in Fig. 1.
Firstly, we consider uncoupled majorana modes (i.e. ϵ MF = 0), that is to say the two MFs are not coupled to each other and probe field is resonant with the exciton frequency (i.e. Δ c = 0). In this situation, the Hamiltonian of the coupled QD-MFs can be reduced to , which is similar to the J-C Hamiltonian in quantum optics. Figure 2(a) displays the probe absorption spectra of the probe laser (i.e., the imaginary part of the dimensionless susceptibility Im χ (1) ) vs. the detuning Δ s (Δ s = ω s − ω e ) under several different QD-MFs coupling strengths. The black solid curve in Fig. 2(a) shows the result of no QD-MFs coupling, which present Lorentz line shape under weak pump laser driven. When a pair of MFs emerge in the end of the hybrid S/SR device and interact with the two-level QD, the absorption spectrum presents a symmetric Rabi splitting when β 1 = 0.05 × 10 9 Hz and β 2 = 0 as shown the red curve in Fig. 2(a). Thus, the symmetric Rabi splitting in the probe absorption reveals a true signature of MFs emerge in the hybrid S/SR device. To determine the symmetric Rabi splitting of the QD is induced by MFs modes rather than other regimes induced the Majorana-like signatures, we consider several conditions. Due to Kondo effect can also leads to analogous Majorana signatures, it is necessary to distinguish the true Majorana signatures and Kondo effect. Kondo effect, in the detection of MFs with electrical methods, is generally related to the strong coupling to two normal leads in he hybrid S/SR structure, and when the Kondo temperature T k is smaller than the gap, a superconducting gap can suppress the effect. Recently, to control Kondo effect, Nadj-Perge et al. have reported an experimental project of the hybrid Fe-atoms-chain/Pb-superconductor (Fe/Pb) substrate 6 , and the present another evidence in the scheme is related to superconductivity and rather than with other phenomena. Therefore, in a way, the Fe/Pb substrate can replace the hybrid S/SR structure. On the other hand, due to there are several normal electrons in semiconductor nanowire ring, so it is necessary to determine whether the signals in the probe absorption spectra are true QD-MFs coupling, rather than the interation between the QD and the electrons in the nanowire induced the same Majorana signals. For this question, we once have used the tight-binding Hamiltonian 34 to elucidate the electrons in nanowire, and the numerical results indicate MFs signatures in the coherent optical spectra are the true MFs signals 34 .
When a pair of MFs couple to the QD (β 1 = 0.05 GHz, β 2 = 0.05 GHz), the peak width of Rabi splitting induced by MFs is enhanced significantly (the blue curve in Fig. 2(a)), and the bigger QD-MFs coupling induced larger peak width of splitting as green curve in Fig. 2(a). In Fig. 2(b), we also show three different QD-MFs coupling strength, i.e. (β 1 = 0.05 GHz, β 2 = 0.05 GHz), (β 1 = 0.07 GHz, β 2 = 0.03 GHz), and (β 1 = 0.09 GHz, β 2 = 0.01 GHz). The results indicate that the peak width of Rabi splitting depends on one of the pair of MFs coupling to the QD. The physical origin of such results are due to the coherent interaction between the QD and MFs, and we introduce the dressed-state theory to describe the interation between the QD and MFs for interpreting this physical  Fig. 1(c). Then, in Fig. 2, the left sharp peaks of Rabi splitting shows the transition from 0 to n 1, MF , and the right one gives the transition between 0 and + n 1, 1 MF . Secondly, we consider coupled Majorana modes (ϵ MF ≠ 0) and Δ c = 0 as shown in Fig. 3. In this condition, the interaction between the two MFs should be taken into consideration. As in uncoupled Majorana modes, if there is no QD-MFs coupling, the spectrum (the black curve in Fig. 3(a)) shows the same Lorentz line shape. When a pair of MFs coupled to the QD, the spectra display unsymmetric Rabi splitting under the detuning Δ MF = −0.5 × 10 9 Hz. Figure 3(b) is the detail parts of the left peaks in Fig. 3(a), and sharp peak locates at Δ s = −0.5 GHz which is different from in uncoupled majorana modes. In Fig. 3(c), we further discuss several different QD-MFs coupling strength, the results in probe absorption spectra manifest that when the QD is close to one MF (such as γ M1 , and then β 1 > β 2 ), the width of unsymmetric Rabi splitting is larger than the case of β 1 = β 2 , and Fig. 3(d) is the detail parts of the left peaks in Fig. 3(c). The QD-MFs coupling strength can be controlled with controlling the distance of the QD and the hybrid S/SR device. The results in Fig. 2 and Fig. 3 manifest that the signals are the ture MFs signature, and our all-optical detection means can work at the coupled-and uncoupled-Majorana edge states 38 .
In addition, when making a comparison between ϵ MF ≠ 0 in Fig. 3 and ϵ MF = 0 in Fig. 2, we find that the probe absorption spectra show the analogous phenomenon of electromagnetically induced transparency (EIT) 39 in the two situations. The absorption dip approaches zero at Δ s = 0 and Δ s = −0.5 × 10 9 Hz under ϵ MF = 0 and ϵ MF ≠ 0, respectively, which indicates the probe field is transmitted with experiencing nothing absorption. The result is due to the destructive quantum interference effect between the beat frequency δ (δ = ω s − ω c ) of the two optical fields (i.e. pump laser field and probe laser field) via the QD and Majorana modes 38 . When the beat frequency δ approximates to the MFs resonance frequency ω MF , then Majorana modes start oscillating coherently resulting in Stokes-like (Δ SM = ω c − ω MF ) and anti-Stokes-like (Δ ASM = ω c + ω MF ) light scattering from the QD. In highly off-resonant, the process of Δ SM is strongly suppressed, and only leaving the process of Δ ASM interferes with the probe field modifying the absorption spectra. We therefore define the phenomenon as Majorana modes induced transparency (MMIT) 38 , which will induce remarkable phenomena of slow light.
Thirdly, we further consider coupled majorana modes (ϵ MF ≠ 0) but adjust the detuning Δ c from Δ c = 0 to Δ c = 0.5 × 10 9 Hz. In this case, the location of the left peaks of unsymmetric Rabi splitting coincides with frequency shift induced by the exciton-pump field detuning, and then the resonant makes the coherent interaction between the QD and a couple of MFs more stronger. Figure 4(a) gives the absorption spectra with several different QD-MFs coupling strength under two detuning, i.e. Δ MF = −0.5 × 10 9 Hz, Δ c = 0.5 × 10 9 Hz, which shows more remarkable unsymmetric splitting and the intensity is also enhanced simultaneously. Figure 4(b) is the detail parts of the left peaks in Fig. 4(a). Moreover, we make a comparison between the two situations, i.e. (Δ c = 0, Δ MF = −0.5 × 10 9 Hz) and (Δ c = 0.5 × 10 9 Hz, Δ MF = −0.5 × 10 9 Hz) at a pair of coupling strength β 1 = 0.05 × 10 9 Hz and β 2 = 0.05 × 10 9 Hz. The probe absorption spectrum shows that the bandwidth is about 8 MHz and the intensity is about 4.5 with the left sharp peaks exactly locating at Δ s = −0.5 × 10 9 Hz under the condition of Δ c = 0 and Δ MF = −0.5 × 10 9 Hz as shown in Fig. 4(c). However, when adjusting the detuning Δ c from Δ c = 0 to Δ c = 0.5 × 10 9 Hz, one can obtain the narrower bandwidth (about 1 MHz) and the stronger intensity (about 0.9) locating at Δ s = −1.0 × 10 9 Hz as shown in Fig. 4(d). In this case, the sideband peak induced by coupled majorana modes coincides with sharp peaks induced by pump off-resonant Δ c = 0.5 × 10 9 Hz, which makes the coherent interaction of QD-MF more strong. Moreover, we slao investigate the optical Kerr nonlinear effect of QD mediated by MFs. Figure 5(a) gives the optical Kerr coefficient (Re(χ (3) )) as a function of Δ s under ϵ MF ≠ 0 with the resonance detuning Δ c = 0. In Fig. 5(a), the black curve displays the nonlinear optical Kerr spectrum without the QD-MFs coupling, the red curve gives the QD couples to one MFs with coupling strength β 1 = 0.05 GHz, the blue one is that the QD couple to a pair of MFs with coupling strength β 1 = 0.05 GHz and β 2 = 0.05 GHz under Δ c = 0 and Δ MF = −0.5 GHz. Figure 5(b,c) are the detail parts of the right and left peaks in Fig. 5(a). It is obvious that when the MFs emerge in the hybrid S/SR device and couple to the QD, two sharp sideband peaks will emerge in the optical Kerr spectra. The physical origin of the results are due to the coherent interaction between the QD and MFs, which makes the resonant enhancement of the optical Kerr effect in the QD. Figure 5(d) shows optical Kerr coefficient at uncoupled Majorana modes with no QD-MFs coupling, one QD-MFs coupling, and a pair of QD-MFs coupling, optical Kerr spectra also present splitting when the QD couple to MFs. The results signify that the sharp peaks in the nonlinear optical spectra may be the signature of real MFs which gives the nonlinear optical means to detect MFs.
Switching Δ c from Δ c = 0 to Δ c = 0.5 GHz, Fig. 6(a) presents the optical Kerr coefficient vs. the detuning Δ s at coupled Majorana modes under several different QD-MFs coupling strengths, and the details are shown in Fig. 6(b,c). It is obvious that the width of splitting induced by a pair of MFs couple to the QD is larger than in the case of one MF couple to the QD. Furthermore, the width of splitting in the optical Kerr spectrum is determined by the two detuning and QD-MFs coupling strength β 2 and β 2 as shown in Fig. 6(d).

Discussion
We have designed a hybrid S/SR device and introduced an optical pump-probe technology to probe MFs with QDs which is very different from the current electrical method. In the previous theoretical schemes for detecting MFs with QD, QD is considered as only one resonant level with single spin state and they only consider one adjacent MF interact with the QD. In this work, we investigate a pair of MFs coupled to the QD which induces remarkable splitting in the probe absorption spectrum and nolinear optical Kerr spectrum. The symmetric and unsymmetric Rabi splitting and optical Kerr nonlinear effect of QD mediated by MFs are studied under uncoupled and coupled majorana modes, respectively. Due to QD-MFs coupling, the probe absorption spectra present the phenomenon of Majorana modes induced transparency which will induce remarkable phenomena of slow light. With adjust the exciton resonant detuning and Majorana resonant detuning, the probe absorption intensity and the peak width of splitting are enhanced significantly, and the peak width of splitting in the optical Kerr spectrum depends on the coupling strength and resonant detuning. Compared with electrical measurement, the optical scheme for the detection of MFs have several advantage, the first one is that there is no contact between the QD and the hybrid S/SR device, thus the introduce of noises can be avoided effectively, and finally the sensitivity of the measurement will enhance observably. The second one is that nanostructures such as QD have obtained remarkable progress in modern nanoscience, which paves a way to detect MFs experimentally. Therefore, the optical detection of MFs with QD is scientific and feasible based on recent experiment. The scheme proposed here may provide potential applications in quantum computing based on MFs with optically controlled in solid-state systems.

Methods
The Heisenberg operator (ρ = S z , S − , f M ) can be divided into two parts including steady-state mean value ρ 0 and small fluctuation δρ, i.e.