Abstract
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.
Keywords:
Majorana fermions; Nanomechanical resonator; Coherent nonlinear optical spectroscopyBackground
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 nonAbelian statistics, which will open the potential applications in topological quantum computation [13]. In recent years, a number of systems that might host MFs in solidstate scenarios have been proposed. Several typical proposals include atoms trapped in optical lattices [4,5], heterostructures of topological insulators and superconductor [6,7], carbonbased materials [8], pwave superconductors [911], and graphene or graphenelike materials [12]. Beyond these proposals, one promising scheme is to use semiconducting nanowires (such as InAs and InSb nanowires) with strong spinorbit 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 [1520].
Since MFs are their own antiparticles, they are predicted to appear in tunneling spectroscopy experiments as zerobias peaks [2123]. Such peaks have been observed in several experiments and have been interpreted as the signatures of MFs [1519]. Unfortunately, a zerobias 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 zerobias anomalies even when the system is not topological [2527]. Except zerobias conductance peak, the Josephson effect is another signature which can demonstrate Majorana particles in the hybrid semiconductorsuperconductor 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 [32], lays the foundation for numerous possible applications [33]. On the other hand, NRs are applied to ultrasensitive detection of mechanical signal [34], mass [35,36], mechanical displacements [37], and spin [38] due to their high natural frequencies and large quality factors [39]. Further, the hybrid system where a QD is coupled to the NR also attracts much interest [4042]. Based on the advantages of QD or NR, several groups propose a scheme for detecting MFs via the QD [4348] or the NR [49] 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 QDNR system.
In the present article, we consider a scheme closed to that of the recent experiment by Mourik et al. [15]. Compared with zerobias peaks and the Josephson effect, we adopt an optical pumpprobe technique to detect MFs. The nonlinear optical Kerr effect, as a distinct signature for demonstrating the existence of MFs in the hybrid semiconductor/superconductor heterostructure, is the main result of this work. Further, in our system (see Figure 1), the NR as a phononic cavity will enhance the nonlinear optical effect significantly, which makes MFs more sensitive to be detected.
Figure 1. Sketch of the proposed setup for optically detecting MFs. An InSb semiconductor nanowire (SNW) with strong spinorbit interaction (SOI) in an external aligned parallel magnetic field B is placed on the surface of a bulk swave superconductor (SC). The two green stars at the ends of the nanowire represent a pair of MFs. The nearby MF is coupled to a semiconductor QD embedded in a nanomechanical resonator under a strong pump laser and a weak probe laser simultaneously. The inset is an energylevel diagram of a semiconductor QD coupled to MFs and NR.
Model and theory
Figure 1 presents the schematic setup that will be studied in this work. An InSb semiconductor nanowire with spinorbit coupling in an external aligned parallel magnetic field B is placed on the surface of a bulk swave 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 pumpprobe 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 [4042]. In such a hybrid system, the QD is modeled as a twolevel system consisting of the
ground state g〉 and the single exciton state ex〉 at low temperatures [50,51]. The Hamiltonian of the QD can be described as
Since several experiments [1520] 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
The optical pumpprobe technology includes a strong pump laser and a weak probe laser
[54], which provides an effective way to investigate the lightmatter interaction. Based
on the optical pumpprobe 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 pumpprobe 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]
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:
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,
To go beyond weak coupling, the Heisenberg operator can be rewritten as the sum of
its steadystate mean value and a small fluctuation with zero mean value:
where b_{1}=g/[i(Δ_{MF}δ)+κ_{MF}/2], b_{2}=g/[ i(Δ_{MF}+δ)+κ_{MF}/2],
Numerical results and discussions
For illustration of the numerical results, we choose the realistic hybrid systems of the coupled QDNR system [40] and the hybrid semiconductor/superconductor heterostructure [1517,20]. For an InAs QD in the coupled QDNR system, the exciton relaxation rate Γ_{1}=0.3 GHz, the exciton dephasing rate Γ_{2}=0.15 GHz. The physical parameters of GaAs nanomechanical resonator are (ω_{m}, m, Q)=(1.2 GHz, 5.3×10^{15} g, 3×10^{4}), 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 [1517], it is reasonable to assume that the lifetime of the MFs is κ_{MF}=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 QDNR hybrid structure and the nanowire.
Firstly, we consider the case that there is no coupling between the QD and NR (η=0), i.e. only a single QD is coupled to the nanowire. Figure 2 plots the optical Kerr coefficient Re(χ^{(3)}) as a function of the probe detuning Δ_{pr}. In Figure 2, the blue curve indicates the nonlinear optical spectrum without the QDMF coupling, and the red one shows the result with the QDMF coupling g=0.03 GHz. It is obvious that when the MFs are presented at the ends of the nanowire, the two sharp sideband peaks will appear in the optical Kerr spectrum of the QD. The physical origin of this result is due to the QDMF coherent interaction, which makes the resonant enhancement of the optical Kerr effect in the QD. This result also implies that the sharp peaks in the nonlinear optical spectrum may be the signature of MFs at the ends of the nanowire. Because there also includes normal electrons in the nanowire, in order to determine whether or not this signature (i.e. the sharp peaks) is the true MFs, we plot the inset of Figure 2, which uses the tight binding Hamiltonian to describe the normal electrons. In the figure, the parameters of normal electrons are chosen the same as MFs so that we can compare with the case of MFs. From the figure, we can observe that there is no sharp peak and only a nearly zero line in the spectrum (see the green line in the inset). This result demonstrates that the coupling between the QD and the normal electrons in the nanowire can be neglected in our theoretical treatment. In this case, one may utilize the optical Kerr effect in QD to detect the existence of MFs provided that the QD is close enough to the ends of the nanowire.
Figure 2. Optical Kerr coefficient as function of probe detuningΔ_{pr} with two different QDMF coupling strengths. The inset shows the result for the normal electrons in the nanowire that couple to
the QD at the coupling strength ζ=0.03 GHz. The parameters used are Γ_{1}=0.3 GHz, Γ_{2}=0.15 GHz, η=0, γ_{m}=4×10^{5} GHz, ω_{m}=1.2 GHz, κ_{MF}=0.1 MHz,
Secondly, we turn on the coupling to the NR (η≠0) and then plot the optical Kerr coefficient as a function of probe detuning Δ_{pr} for η=0.06 as shown in Figure 3. Taking the coupling between the QD and NR into consideration, the other two sharp peaks located at ±ω_{m} will also appear. The red and blue curves correspond to the optical Kerr coefficient with and without the QDMF coupling, respectively. Without the QDMF coupling, the two sharp peaks locate at the resonator frequency of nanomechanical resonator induced by its vibration, i.e. two peaks are at Δ_{pr}=±1.2 GHz as shown in Figure 3. The physical origin of this result is due to mechanically induced coherent population oscillation (MICPO), which makes quantum interference between the resonator and the beat of the two optical fields via the QD when the probepump detuning is equal to the resonator frequency [58]. Turning on the QDMF coupling, in addition to two sharp peaks located at ±1.2 GHz, the other two sideband peaks induced by the QDMF coupling appear at Δ_{pr}=±0.5 GHz simultaneously.
To illustrate the advantage of the NR in our system, we adjust the detuning Δ_{MF}=0.5 GHz to Δ_{MF}=1.2 GHz, in this case, the location of the two sideband peaks induced by the QDMF coupling coincides with the two sharp peaks induced by the vibration of NR, so the NR is resonant with the coupled QDMF system and makes the coherent interaction of QDMF more strong. Figure 4 gives the result of the optical Kerr coefficient as a function of probe detuning with or without the QDNR coupling for the QDMF coupling g=0.03 GHz. The blue and red curves correspond to η=0 and η=0.06, respectively. It is obvious that the role of NR is to narrow and to increase the optical Kerr effect. In this case, the NR as a phonon cavity will enhance the sensitivity for detecting MFs.
Conclusion
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HJC finished the main work of this paper, including deducing the formulas, plotting the figures, and drafting the manuscript. KDZ conceived of the idea, participated in the discussion, and provided some useful suggestion. Both authors are involved in revising the manuscript. Both authors read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge support from the National Natural Science Foundation of China (No. 10974133 and No. 11274230).
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