Effect of interatomic repulsion and quasi-degenerate states on a Kitaev-transmon qubit based on double quantum dots

Technical Summary: Effect of Interatomic Repulsion and Quasi-Degenerate States on a Kitaev-Transmon Qubit

Problem Statement
The paper investigates the robustness and spectral properties of a Kitaev-transmon qubit, a system proposed by Pino et al. consisting of a Josephson junction connecting two double quantum dots (DQDs) modeled by the spinless Kitaev Hamiltonian. While previous studies established the existence of "poor man's Majorana" (PMM) states at specific parameter "sweet spots," those analyses often neglected interatomic Coulomb repulsion ($V$) and certain quasi-degenerate states. The authors aim to determine how the inclusion of $V$ affects the stability of PMMs in an isolated DQD and how both $V$ and previously disregarded states influence the full transmon qubit's eigenstates and microwave spectrum.

Methodology
The study employs a theoretical framework based on exact diagonalization of the system's Hamiltonian.

1. Isolated DQD Model: The authors first analyze a single DQD described by a Hamiltonian including chemical potentials ($\mu_i$), crossed Andreev reflection (CAR, $\Delta$), single-electron elastic cotunneling (ECT, $t$), and interdot Coulomb repulsion ($V$). They derive analytical expressions for the ground states in even and odd parity subspaces.
2. Full System Model: The model is extended to two DQDs coupled by a Josephson junction ($t_J$) and embedded in a superconducting circuit with a charging energy ($E_C$) and a second Josephson junction ($E_J$). The phase difference across the junction is treated as an operator conjugate to the number of Cooper pairs.
3. Numerical Simulation: The total Hamiltonian is numerically diagonalized by truncating the basis of the charge operator $\hat{n}$ to $|n| < 16$. The authors vary parameters such as the offset charge ($m_g$) and the energy difference between even and odd parity states ($E_{ee}$ vs. $E_{oo}$) to map the energy spectrum and calculate the microwave transition rates in linear response.

Key Contributions and Results

• Persistence of PMM Sweet Spots with Repulsion:
  The authors demonstrate that a "sweet spot" hosting well-defined PMMs persists in the presence of interatomic Coulomb repulsion $V$, provided system parameters are tuned. Specifically, the conditions for the sweet spot shift from the $V=0$ case to:

  • $\mu_1 = \mu_2 = V/2$
  • $\Delta = |t| + V/2$
    Under these conditions, the four criteria for PMMs (degeneracy of ground states, zero charge change per site, perfect localization, and a finite excitation gap) are satisfied. The authors note that while theoretically achievable, the requirement to tune $\Delta$ or $t$ by $V/2$ may be experimentally challenging for large $V$.

• Double Degeneracy at Sweet Spots:
  For the full Kitaev-transmon system, the authors prove that when both DQDs are at their respective sweet spots ($E_{ee} = E_{oo}$) and the external flux is zero ($\phi_{ext} = 0$), all eigenstates of the system are doubly degenerate. This degeneracy arises from a specific symmetry of the Hamiltonian: the existence of an operator $\hat{O}$ that maps between two decoupled Hilbert subspaces (one containing states $|ee, n\rangle$ with even $n$ and $|oo, n\rangle$ with odd $n$, and the other with the parity of $n$ reversed). The authors show that $\hat{O}$ commutes with the Hamiltonian and satisfies $\hat{O}^2 = -1$, necessitating double degeneracy.

• Impact of Disregarded States and Level Crossings:
  Away from the sweet spots, the system's behavior becomes sensitive to the initial state. The authors identify that states previously disregarded in similar models—specifically $|ee, n\rangle$ with even $n$ and $|oo, n\rangle$ with odd $n$—play a crucial role. As the offset charge $m_g$ is varied, level crossings occur between the ground states of these different subspaces.

  • When $|E_{ee}| < |E_{oo}|$, the ground state is dominated by $|oo, n\rangle$ states.
  • When $|E_{ee}| > |E_{oo}|$, the ground state is dominated by $|ee, n\rangle$ states.
    These crossings result in jumps in the intensity of the microwave spectrum.

• Microwave Spectrum:
  The calculated microwave spectrum ($S(\omega)$) shows that while the transition energies remain identical for the two subspaces due to the underlying symmetry, the spectral intensities depend on the ground state configuration. The authors note that quasiparticle poisoning could induce switching between these subspaces at crossing points, potentially leading to a superposition of spectra in experimental observations.

Significance and Claims
The paper claims to provide a necessary extension to the theoretical understanding of Kitaev-transmon qubits by explicitly incorporating interatomic Coulomb repulsion. The primary significance lies in:

1. Parameter Renormalization: Showing that the effects of $V$ can be accounted for by simple parameter renormalization (shifting chemical potentials and adjusting CAR/ECT amplitudes), thereby validating the robustness of the PMM concept against this specific interaction.
2. Completeness of the Spectrum: Highlighting the existence of a second, decoupled Hilbert subspace that was previously overlooked. The authors demonstrate that outside the fine-tuned sweet spots, the system's spectrum is richer and dependent on the specific subspace occupied, which is relevant for systems initially in thermal equilibrium at low temperatures.
3. Experimental Detection: The authors suggest that the calculated microwave spectrum, particularly the intensity jumps associated with ground state crossings, could serve as a tool to experimentally detect the sweet spots and the influence of interaction parameters.

The authors maintain a modest tone, acknowledging that while the sweet spots are theoretically robust, large values of $V$ may make experimental realization difficult. They do not propose new experimental setups but rather refine the theoretical model of the existing Pino et al. proposal.