Reshaping the Jaynes-Cummings ladder with Majorana bound states

We study the optical properties of a hybrid device composed by a quantum dot (QD) resonantly coupled to a photonic mode of an optical microcavity and a Majorana nanowire: a topological superconducting segment hosting Majorana bound states (MBSs) at the opposite ends. In the regime of strong light-matter coupling, it is demonstrated that the leakage of the Majorana mode into the QD opens new optical transitions between polaritonic states formed due to hybridization of material excitation with cavity photons, which leads to the reshaping of the Jaynes-Cummings ladder and can lead to the formation of a robust single peak at cavity eigenfrequency in the emission spectrum. Moreover, weak satellite peaks in the low- and high-frequency regions are revealed for the distinct cases of highly isolated MBSs, overlapped MBSs, and MBSs not well localized at the nanowire ends.

Figure 1

(a) The sketch of the proposed device: a quantum dot (QD) embedded in an optical microcavity and coupled to a Majorana nanowire hosting Majorana bound states (MBSs) at its opposite ends (light green circles). ${\lambda }_L$ and ${\lambda }_R$ represent the couplings between the QD excited level and the left (${\gamma }_L$) and right MBSs (${\gamma }_R$), respectively, while ${\varepsilon }_M$ stands for the overlap amplitude between the MBSs. (b) Optical transitions for a QD placed inside an optical microcavity. The dot interacts resonantly with a single-mode photonic field of the pumped cavity with frequency ${\omega }_c={\omega }_{eg}={\omega }_e-{\omega }_g$ brought in resonance with the energy of the optical transition, where ${\omega }_e$ and ${\omega }_g$ stand for the energies of the excited and ground states of the QD, respectively.

Figure 2

Left: Evolution of the first 20 energy levels of the system as a function of the parameters of QD-MBS couplings ${\lambda }_L$ and ${\lambda }_R$, as well as the overlap between the MBSs ${\varepsilon }_M$. Right: Optical transitions between the system eigenenergies for each corresponding case. The blue lines represent the initial and final states of each corresponding transition, which is represented by a red arrow. The specific parameters adopted in each of the right panels are indicated by the light green rectangle in the corresponding left ones. [(a), (b)] The case of well-isolated MBSs localized at the nanowire ends, with ${\lambda }_L=0.3$ in (b). [(c), (d)] The situation of overlapped MBSs well localized at the nanowire ends, for ${\lambda }_L=0.3$ and ${\varepsilon }_M=0.4$ in (d). [(e), (f)] The case wherein the MBSs barely overlap, but are not well localized at the ends of the nanowire, with ${\lambda }_L=0.6$ and ${\lambda }_R=0.3$ in (f).

Figure 3

Normalized emission spectrum [Eq. (10)] for the situation of isolated MBSs localized at opposite nanowire ends (${\lambda }_R={\varepsilon }_M=0$), with ${\omega }_c={\omega }_{eg}=1.0$, ${\mathrm{\Omega }}_R=0.1$, $P=0.015$, and ${\gamma }_{\text{ph}}=0.02$. [(a), (b)] Intensity of emission spectrum as a function of both emitted photon frequency $\omega$ and QD–left MBS coupling ${\lambda }_L$, in linear and logarithmic scales, respectively. [(c), (e), (g)] Linecuts indicated by geometric shape markers in (a); [(d), (f), (h)] the same linecuts, but in logarithmic scale.

Figure 4

Top: Normalized emission spectrum [Eq. (10)] as a function of the frequency of the emitted photon $\omega$, for the situation in which the MBSs are well localized at the nanowire ends and do not overlap with each other (${\lambda }_R={\varepsilon }_M=0.0$), with ${\omega }_c={\omega }_{eg}=1.0$, ${\mathrm{\Omega }}_R=0.1$, $P=0.015$, and ${\gamma }_{\text{ph}}=0.02$, considering distinct values of the QD–left MBS coupling ${\lambda }_L$. Bottom: Allowed optical transitions of the system which correspond to the peaks in the top panels, labeled by colored numbers.

Figure 5

Top: Normalized emission spectrum [Eq. (10)] as a function of the emitted photon frequency $\omega$, for the situation where the MBSs are well localized at the nanowire ends and do not overlap with each other (${\lambda }_R={\varepsilon }_M=0.0$), considering two values of Rabi frequency ${\mathrm{\Omega }}_R$, with ${\omega }_c={\omega }_{eg}=1.0$, $P=0.015$, ${\gamma }_{\text{ph}}=0.02$, and distinct values of QD–left MBS coupling ${\lambda }_L$. Middle: Same emission spectra of upper panels, but in logarithmic scale. In both top and middle panels, the plots are slightly shifted in the y axis for a better visualization. Bottom: Evolution of the normalized transition probabilities [Eq. (8)] at frequency $\omega \approx {\omega }_c$ as a function of the QD–left MBS coupling ${\lambda }_L$.

Figure 6

Normalized emission spectrum [Eq. (10)] for the situation of MBSs localized at opposite nanowire ends (${\lambda }_R=0.0$), but with a finite overlap ${\varepsilon }_M$ between each other, with ${\lambda }_L=0.3$, ${\omega }_c={\omega }_{eg}=1.0$, ${\mathrm{\Omega }}_R=0.1$, $P=0.015$, and ${\gamma }_{\text{ph}}=0.02$. [(a), (b)] Intensity of emission spectrum as a function of both emitted photon frequency $\omega$ and left-right MBS overlap ${\varepsilon }_M$, in linear and logarithmic scales, respectively. [(c), (e), (g)] Linecuts indicated by geometric shape markers in (a); [(d), (f), (h)] the same linecuts, but in logarithmic scale.

Figure 7

Normalized emission spectrum [Eq. (10)] for the situation where the MBSs are not well localized at the nanowire ends (${\lambda }_R\ne 0.0$) and have a negligible overlap between each other (${\varepsilon }_M\ll {\lambda }_R,{\lambda }_L$), with ${\lambda }_L=0.6$, ${\omega }_c={\omega }_{eg}=1.0$, ${\mathrm{\Omega }}_R=0.1$, $P=0.015$, and ${\gamma }_{\text{ph}}=0.02$. [(a), (b)] Intensity of emission spectrum as a function of both emitted photon frequency $\omega$ and QD–right MBS coupling ${\lambda }_R$, in linear and logarithmic scales, respectively. [(c), (e), (g)] Linecuts indicated by geometric shape markers in (a); [(d), (f), (h)] the same linecuts, but in logarithmic scale.