Topology and energy dependence of Majorana bound states in a photonic cavity

TL;DR

This work explores how a photonic cavity modifies Majorana bound states in a one-dimensional topological superconductor. By combining a fully quantum treatment with multi-photon sectors and a spectral localizer, the study reveals that MBS persist in a cavity, acquire a photon-number dependent energy shift (pseudo-dispersion), and exhibit enhanced stability against oscillations despite a reduced topological gap. The authors show that high-frequency photon regimes allow clean, energy-resolved topology via the spectral localizer, while low-frequency regimes require an engineered energy term to prevent cross-sector pollution. Overall, the work offers a new route to tune and stabilize MBS with light and provides a practical framework for cavity-modified topologies using an adaptable spectral localizer.

Abstract

Light-matter interaction plays a crucial role in modifying the properties of quantum materials. In this work, we investigate the effect of cavity induced photon fields on a topological superconductor hosting Majorana bound states (MBS). We model the system using a Peierls substitution of the photonic operator in the kinetic and spin-orbit terms, and utilize an exact diagonalization of Hamiltonian for a finite number of photons to investigate the coupled system. We find that the MBS persist even in the presence of a cavity field and notably appear at finite and tunable energy, in contrast to a usual 1D topological superconductor. The MBS energy is shifted by two processes: the cavity photon energy adds a constant energy shift, while the light-matter interaction induces additional parameter dependencies, such that the MBS experience a pseudo-dispersion as a function of both light-matter interaction and magnetic field. Additionally, we find that the MBS energy oscillations are suppressed with increasing light-matter interaction and that disorder stability is not impacted by the light-matter interaction. Combined, these offer additional tunability and stability of the MBS. As a second result, we establish a modified spectral localizer formalism as an essential tool for topological characterization of quantum matter in a cavity. The spectral localizer allows characterization at arbitrary energies, which is needed for probing different photon sectors. However, hybridization between different photon sectors in the low-frequency regime limits a straightforward application of a standard spectral localizer. We fully resolve this issue by judiciously applying an energy shift to the spectral localizer. Our work thus introduces a new avenue for controlling MBS via light-matter coupling and provides a framework for exploring cavity-modified topologies.

Figures (14)

• Figure 1: (a) Schematic view of a 1DTSC with light (dark) blue balls denoting the superconducting substrate (Rashba nanowire) exhibiting MBS at its ends, interacting with a single photon (red waveform) present between two photonic cavity mirrors. (b) Schematic representation of the bands of a 1DTSC before (left) and after (right) placing it in a cavity in the high-frequency regime, where zero energy of one sector does not overlap with conduction or valence bands from another. Here, the 0th photon sector has nontrivial topology, while the 1st photon sector is trivial. Parameters: $\mu=-2t, \ \Delta=\alpha=0.2t, \ B=0.3t$ and for the cavity (right) $\omega=6t$, $\gamma=0.4t$.
• Figure 2: (a) Eigenvalues of $\mathcal{H}_{0}$ [Eq. \ref{['Eq:InfiniteHam']}] as a function of magnetic field $B$ for uncoupled ($\gamma=0.0$, black) and coupled ($\gamma=0.5t$, red) system in the 0th photon sector for $N_{ph}=0$. (b) Eigenvalues of $\mathcal{H}_{1}$ as a function of $B$ in the 0th (green) and 1st (violet) photon sector for $N_{ph}=1$. Bottom (top) panel shows eigenvalues close to $\omega/2$ ($3\omega/2$), corresponding to the bare photon energy of the $0$th ($1$st) photon sector. Parameters are $\mu=-2t, \ \Delta=\alpha=0.2t$, $\omega=10t$, and $\gamma=0.5t$ for (b).
• Figure 3: Envelope plot of the MBS energy oscillations for increasing chemical potential $\mu=-2t \rightarrow \mu=-1.5t$ in the 0th photon sector for the bare 1DTSC for $\gamma = 0$ (blue lines) and increasing light-matter coupling $\gamma=0t \rightarrow \gamma=0.5t$ of the 1st photon sector for $\mu = 2t$ (red lines). Same colored dots mark the value of the critical magnetic field $B_{c_1}$ for each value of the varied parameter. Energy spectrum $E_{\rm MBS}$ (grey lines) shows the oscillations of a single MBS level enveloped by the line depicting the largest shift from the initial value ($\mu=-2t$ and $\gamma = 0.5t$). Parameters are $B=0.4t, \ \Delta=\alpha =0.2t$, $\omega = 10t$, $N_{ph}=1$.
• Figure 4: MBS energies of $\mathcal{H}_{0-5}$ for the lowest 6 photon sectors solved for up to $N_{ph} = 5$ photons. Apart from the case of $N_{ph}=0$, where the MBS do not exhibit an energy pseudo-dispersion (PD), the MBS energy is each photon sector is either increased (highest photon sector for given $N_{ph}$), decreased (photon sectors $N_{ph}-1$ and $N_{ph}-2$) for finite $\gamma$. Parameters are $\mu=-2t, \ \Delta=\alpha=0.2t, \ N_{ph}=1, \ B=0.3t$ and $\omega=10t$.
• Figure 5: (a) Eigenvector distribution of $\mathcal{H}_1$ showing the MBS localization at the ends of the nanowire for both photon sectors. (b) Localizer gap $\sigma(x,E)$ and (c) topological invariant $\nu(x,E)$ for both photon sectors showing the existence of a topological phase in the system (i.e. $\nu(x,E)=1$), with borders coinciding with the position of MBS in the nanowire. Parameters are $\mu=-2t, \ \Delta=\alpha=\gamma=0.2t, \ N_{ph}=1, \ B=0.3t$ and (b,c) $\omega=10t, E=\omega/2$ ($0$th photon sector) and $E=3\omega/2$ ($1$st photon sector).