Nonlinear optomechanical detection for Majorana fermions via a hybrid nanomechanical system
© Chen and Zhu; licensee Springer. 2014
Received: 8 January 2014
Accepted: 29 March 2014
Published: 5 April 2014
The pursuit for detecting the existence of Majorana fermions is a challenging task in condensed matter physics at present. In this work, we theoretically propose a novel nonlinear optical method for probing Majorana fermions in the hybrid semiconductor/superconductor heterostructure. Our proposal is based on a hybrid system constituted by a quantum dot embedded in a nanomechanical resonator. With this method, the nonlinear optical Kerr effect presents a distinct signature for the existence of Majorana fermions. Further, the vibration of the nanomechanical resonator will enhance the nonlinear optical effect, which makes the Majorana fermions more sensitive to be detected. This proposed method may provide a potential supplement for the detection of Majorana fermions.
The search for Majorana fermions (MFs) in hybrid nanostructures of condensed matter systems has become an important topic in quantum information processing. Unlike the usual Dirac particles, MFs obey non-Abelian statistics, which will open the potential applications in topological quantum computation [1–3]. In recent years, a number of systems that might host MFs in solid-state scenarios have been proposed. Several typical proposals include atoms trapped in optical lattices [4, 5], heterostructures of topological insulators and superconductor [6, 7], carbon-based materials , p-wave superconductors [9–11], and graphene or graphene-like materials . Beyond these proposals, one promising scheme is to use semiconducting nanowires (such as InAs and InSb nanowires) with strong spin-orbit coupling placed in proximity with a superconductor and biased with an external magnetic field [13, 14]. After the prediction that Majorana bound states (MBSs) can be observed in the hybrid semiconductor/superconductor heterostructure, various experiments have indeed reported signatures of MFs in such systems recently [15–20].
Since MFs are their own antiparticles, they are predicted to appear in tunneling spectroscopy experiments as zero-bias peaks [21–23]. Such peaks have been observed in several experiments and have been interpreted as the signatures of MFs [15–19]. Unfortunately, a zero-bias anomaly might also occur under similar conditions due to a Kondo resonance once the magnetic field has suppressed the superconducting gap enough to permit the screening of a localized spin [18, 24], and these experiments are not spatially resolved to detect the position of the MFs. Additionally, in many instances, the presence of disorder can also result in spurious zero-bias anomalies even when the system is not topological [25–27]. Except zero-bias conductance peak, the Josephson effect is another signature which can demonstrate Majorana particles in the hybrid semiconductor-superconductor junction [20, 28, 29]. However, most of the recent experiments proposed and carried out have focused on electrical scheme, and the observation of Majorana signature based on electrical methods still remains a subject of debate. Meanwhile, other effective methods, such as optical technique [30, 31], for detecting MFs in the hybrid semiconductor/superconductor heterostructure have received less attention until now.
In recent years, 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 , lays the foundation for numerous possible applications . On the other hand, NRs are applied to ultrasensitive detection of mechanical signal , mass [35, 36], mechanical displacements , and spin  due to their high natural frequencies and large quality factors . Further, the hybrid system where a QD is coupled to the NR also attracts much interest [40–42]. Based on the advantages of QD or NR, several groups propose a scheme for detecting MFs via the QD [43–48] or the NR  coupled to the nearby MFs. Here, we will propose an optical scheme to detect the existence of MFs in such a hybrid semiconductor/superconductor heterostructure via a hybrid QD-NR system.
Model and theory
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 with the exciton frequency ωQD, 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 . We use a Hamiltonian of quantum harmonic oscillator 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 , 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 , where η is the coupling strength between the resonator mode and QD . Therefore, the Hamiltonian of the hybrid QD-NR system is .
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 γMF such that and to describe MFs. Supposing the QD couples to γMF1, the Hamiltonian of the hybrid system [43–46] is , 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 and . f M and are the fermion annihilation and creation operators obeying the anti-commutative relation . Accordingly, in the rotating wave approximation , H M can be rewritten as , where the first term gives the energy of MF at frequency ωMF, and 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. , 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 , 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.  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 . 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 , where µ is the dipole moment of the exciton, ωpu (ωpr) is the frequency of the pump (probe) laser, and Epu (Epr) is the slowly varying envelope of the pump (probe) laser. Therefore, one can obtain the total Hamiltonian of the hybrid system as H=HQD-NR+HMBS+HQD-L.
where N=b++b. Γ1 (Γ2) is the exciton relaxation rate (dephasing rate), κMF (γ m ) is the decay rate of the MF (nanomechanical resonator). Δpu=ωQD-ωpu is the detuning of the exciton frequency and the pump frequency, is the Rabi frequency of the pump field, and δ=ωpr-ωpu is the probe-pump detuning. ΔMF=ωMF-ωpu is the detuning of the MF frequency and the pump frequency. is the δ-correlated Langevin noise operator, which has zero mean and obeys the correlation function . The resonator mode is affected by a Brownian stochastic force with zero mean value, and has the correlation function , 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], where c k and are the regular fermion annihilation and creation operators with energy ω k and momentum obeying the anti-commutative relation 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 ).
where b1=g/[i(ΔMF-δ)+κMF/2], b2=g/[ i(ΔMF+δ)+κMF/2], , , , , , d2=i(Δpu-δ+ω m η N0)+Γ2-g b1w0-d1h2, , d4=i(Δpu+δ+ω m η N0)+Γ2-g b2w0-d3h5 (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.
Numerical results and discussions
For illustration of the numerical results, we choose the realistic hybrid systems of the coupled QD-NR system  and the hybrid semiconductor/superconductor heterostructure [15–17, 20]. For an InAs QD in the coupled QD-NR system, the exciton relaxation rate Γ1=0.3 GHz, the exciton dephasing rate Γ2=0.15 GHz. The physical parameters of GaAs nanomechanical resonator are (ω m , m, Q)=(1.2 GHz, 5.3×10-15 g, 3×104), where m and Q are the effective mass and quality factor of the NR, respectively. The decay rate of the NR is γ m = ω m /Q=4×10-5 GHz. The coupling strength between quantum dot and nanomechanical resonator is η=0.06. For MFs in the the hybrid semiconductor/superconductor heterostructure, 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 experimental reports [15–17], it is reasonable to assume that the lifetime of the MFs is κMF=0.1 MHz. Since the coupling strength between the QD and nearby MFs is dependent on their distance, we also expect the coupling strength g=0.03 GHz via adjusting the distance between the QD-NR hybrid structure and the nanowire.
We have proposed a nonlinear optical method to detect the existence of Majorana fermions in semiconductor nanowire/superconductor hybrid structure via a single quantum dot coupled to a nanomechanical resonator. The optical Kerr effect may provide another supplement for detecting Majorana fermions. Due to the nanomechanical resonator, the nonlinear optical effect becomes much more significant and then enhances the detectable sensitivity of Majorana fermions. Finally, we hope that our proposed scheme can be realized experimentally in the future.
The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 10974133 and No. 11274230).
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