Majorana Quantum Oscillations in Topological Systems

• Majorana quantum oscillations are interference phenomena resulting from the coherent overlap and hybridization of spatially separated Majorana quasiparticles in topological nanowires and hybrid structures.
• They manifest as energy splitting, parity crossings, and conductance oscillations, providing key diagnostics to differentiate true Majorana modes from trivial bound states.
• Advanced protocols such as Rabi, Ramsey, and LZS oscillations enable coherent qubit manipulation, paving the way for robust topological quantum computing architectures.

Majorana quantum oscillations are a collection of interference, spectral, and transport phenomena arising from quantum coherence, hybridization, and topological properties of systems hosting Majorana modes. These oscillations are observed in diverse contexts including semiconductor-superconductor nanowires, quantum dot–Majorana hybrids, chiral Majorana platforms, Kitaev spin liquids, and even certain Kondo insulators. The phenomena share a unifying physical origin: the coherent superposition, controlled hybridization, or collective response of spatially separated or topologically protected Majorana quasiparticles, manifesting as oscillatory energy splitting, Rabi or LZS qubit dynamics, parity-dependent observables, and magnetic or strain-induced quantum oscillations.

1. Overlap-induced Majorana Oscillations in Nanowires and Hybrid Structures

Majorana bound states (MBSs) emerging at the ends of proximitized, spin-orbit-coupled nanowires in the topological regime hybridize over finite wire length, leading to oscillatory energy splitting characterized by

$\delta E(L) = A e^{-L/\xi} \cos(k_F L + \phi)$

where $\xi$ is the Majorana coherence length and $k_F$ is an effective Fermi momentum determined by the chemical potential and Zeeman field (Avila et al., 2020, Aksenov, 2022, Cao et al., 2018). These oscillations arise from the Friedel-like overlap of spatially separated MBS wavefunctions. As $B$ or $\mu$ is tuned, the splitting $\delta E$ exhibits periodic zero crossings, leading to repeated "fermion parity crossings" in the ground state—an essential feature for identifying Majorana physics versus trivial Andreev bound states.

In transmon qubits based on such nanowire junctions, these oscillations translate to periodic "spectral holes" in the microwave absorption spectrum, shifts in gate-charge periodicity, and abrupt redistribution of oscillator strength among transitions, all tracking the underlying Majorana parity crossings (Avila et al., 2020). Conductance measurements in Aharonov-Bohm geometries involving nanowires or quantum-dot-coupled MBSs also reveal oscillatory signatures (e.g., period-doubling of AB peaks, fractional Fano resonances), where the shift and amplitude are direct diagnostics of Majorana overlap and nonlocality (Aksenov, 2022, Ricco et al., 2018).

2. Quantum Oscillations from Interferometric and Parity-dependent Protocols

Interference between Majorana modes, either across engineered Josephson junctions or in ring and interferometer settings, produces quantum oscillations in a range of physical quantities. In Josephson junctions between topological superconductors, the parity-qubit energy splitting as a function of superconducting phase difference $\varphi$ is described by a $4\pi$-periodic term $E_M \cos(\varphi/2)$, identifiable via Landau-Zener–Stückelberg (LZS) interferometry. Periodic current or phase drives induce coherent oscillations between qubit states, with an effective mixing angle determined by the accumulated Stückelberg phase, and the resulting population oscillations are detectable in the microwave spectrum emitted from the junction (Wang et al., 2016).

Capacitance-based measurements in nanowire-dot loops exploit parity-dependent quantum oscillations: the charge susceptibility (quantum capacitance) of the dot exhibits $\Phi_0=h/2e$-periodic oscillations in loop flux, with parity branches shifted by half a period—a robust parity readout mechanism (Sau et al., 2024, Stanescu et al., 29 May 2025). Remarkably, such flux-induced oscillations are not exclusive to topologically protected modes; partially separated non-topological states and ps-ABSs can generate similar patterns, highlighting the necessity for complementary topological invariants or dynamic tests (Stanescu et al., 29 May 2025).

3. Rabi, Ramsey, and LZS Oscillations: Majorana Qubit Manipulation

Majorana quantum qubit experiments leverage the coherence and controlled hybridization between end modes or engineered MBSs in quantum dot–superconductor arrays and nanowire devices. The minimal qubit realization employs two spatially separated MBSs forming an effective two-level system. Time-dependent tunnel couplings, AC gate drives, or controlled fluxes enable Rabi and Ramsey oscillations, in which the qubit coherently cycles between its logical states (Pan et al., 2024, Sau et al., 2024, Wang et al., 2014). The Rabi frequency typically scales linearly with the effective tunnel amplitude and displays parity and phase (flux) dependence.

High-coherence Majorana qubits are characterized by enhanced dephasing times and low leakage outside the computational manifold, provided the induced gaps are large. Quantum capacitance readout acting as a high-contrast, noninvasive probe can simultaneously monitor qubit parity and poisoning events, opening paths to advanced error detection and correction schemes (Pan et al., 2024). LZS and Floquet-driven dynamics permit the implementation of arbitrary single-qubit rotations, with gate operations mapped to the Stückelberg phase control (Wang et al., 2016, Wang et al., 2014).

4. Topological and Spatial Quantum Oscillations in Extended Systems

Majorana quantum oscillations also arise in more complex and extended systems, where topology plays a direct role in both real- and momentum-space oscillatory behavior. In 1D $p$-wave chains with quasiperiodic potential, the spatial profile of edge Majorana modes acquires a damped oscillatory envelope whose period is fixed by a Chern number associated with an underlying 2D Harper model—establishing a direct link between quantum oscillations and bulk topological invariants (Satija, 2017). In $d$-wave proximitized 2DEGs, the oscillatory properties of induced Majorana edge modes remain robust and nearly indistinguishable from the $s$-wave case, with the oscillation wavevector and decay rate determined by the induced $f$-wave pairing parameters (Ortiz et al., 2017).

Quantum walks and Floquet-engineered systems with symmetry-protected zero modes exhibit field-induced coherent oscillations—Rabi-type evolution between boundary-localized Majorana states—providing potential schemes for topologically protected qubit manipulation and robust memory (Yu et al., 2016, Yu et al., 2018).

5. Majorana Quantum Oscillations in Chiral and Correlated Systems

In quantum anomalous Hall (QAH) platforms proximitized by superconductors, chiral Majorana edge modes generate conductance oscillations via Fabry-Pérot interference, with characteristic dependence on junction length, phase, and transverse dimensions (Osca et al., 2018, Lian et al., 2015, Li et al., 2018). The oscillation period directly encodes the group velocity of the chiral mode and provides "smoking-gun" evidence of Majorana transport through the oscillatory modulation rather than the static $e^2/2h$ plateau. In interferometric geometries with dislocations or strain fields, topological defects alter the parity and period of the quantum oscillations in magneto-conductance, further emphasizing the nontrivial topology of the underlying quasiparticle transport (Mesaros et al., 2010).

Strongly correlated systems, such as Kondo insulators and Kitaev spin liquids, produce bulk quantum oscillations of entirely neutral Majorana Fermi surfaces. In SmB$_6$, oscillatory bulk magnetization and specific heat—the "Majorana de Haas–van Alphen effect"—are theoretically explained by a scalar Majorana Fermi sea that, while electrically insulating, responds to magnetic fields identically to a charged Fermi liquid (Baskaran, 2015). In Kitaev spin liquids, triaxial strain acts as an effective gauge field inducing pseudo-Landau levels, resulting in quantum oscillations of the density of states and specific heat akin to those in metallic systems—directly probing the presence and topology of Majorana Fermi surfaces (Yokoyama et al., 14 Dec 2025).

6. Limitations, Non-Ideality, and Experimental Considerations

In realistic devices, several sources modify or obscure the expected Majorana oscillatory signatures. Steplike or nonuniform spin-orbit coupling, disorder, and partial non-topological overlap lead to non-universal or decaying oscillation amplitudes, altered periods, and the washing out of characteristic phase shifts, potentially mimicking or masking true Majorana physics (Cao et al., 2018, Stanescu et al., 29 May 2025). Consequently, observation of quantum or parity oscillations must be accompanied by corroborating evidence—such as the exponential length scaling of energy splitting, bulk gap closure, or evidence for non-Abelian statistics—to conclusively establish the topological Majorana character. In many cases, the same oscillatory phenomena can be generated by trivial Andreev bound states, ps-ABSs, or non-topological overlapping modes, indicating the fundamental challenge of unambiguously identifying Majorana zero modes via quantum oscillations alone.

7. Summary Table of Majorana Quantum Oscillation Contexts

System/Class	Oscillation Origin	Physical Observable(s)
Nanowire MBS (finite L)	Overlap splitting	Energy splitting, parity crossing, conductance, capacitance (Avila et al., 2020, Aksenov, 2022, Cao et al., 2018, Stanescu et al., 29 May 2025)
Qubit arrays, dots, drives	Rabi/LZS/Floquet interference	Population, capacitance, microwave emission (Wang et al., 2016, Pan et al., 2024, Wang et al., 2014, Sau et al., 2024)
Chiral/edge Majoranas (QAH)	Interference, phase accumulation	Conductance oscillations, shot noise (Osca et al., 2018, Lian et al., 2015, Li et al., 2018, Mesaros et al., 2010)
Quasiperiodic/lattice chains	Topological (Chern/invariant)	Spatial oscillation period, edge-state profile (Satija, 2017, Ortiz et al., 2017)
Quantum-walk/ring/ion-trap	Floquet/topological, field-induced	Two-level (Rabi) oscillations, qubit manipulation (Yu et al., 2016, Yu et al., 2018)
Kondo insulators/Spin Liquids	Landau quantization/strain-induced	dHvA-type oscillations, specific heat, magnetization (Baskaran, 2015, Yokoyama et al., 14 Dec 2025)

Majorana quantum oscillations represent a diverse and technically rich set of interference and hybridization effects in topological and non-topological systems. Their identification, manipulation, and application rely on precise experimental control and comprehensive multi-modal validation of the underlying Majorana character.