Researchers Reveal Novel Discrete States in Josephson Qubits at Odd Parity

Researchers have long explored the quantum mechanics underpinning the Josephson effect as a cornerstone of quantum technologies, but a recent theoretical prediction suggests a fundamentally different behaviour when operating in the odd parity sector with a quasiparticle trapped in an Andreev bound state. Manuel Houzet, Julia S. Meyer, and Yuli V. Nazarov, from institutions including Univ. Grenoble Alpes and Delft University of Technology, now investigate this phenomenon in a simplified system consisting of a single Josephson junction coupled to a capacitance. Their work reveals a novel structure for low-lying discrete states in the odd parity sector, distinct from those found in the even sector, spanning the range from Coulomb-dominated to Josephson-dominated regimes. This prediction offers a testable pathway for experiments utilising superconductor/semiconductor/superconductor junctions, increasingly prominent in nanowire and two-dimensional electron gas research, and could significantly advance the development of superconducting qubits.

This breakthrough, detailed in recent work, centres on the quantum mechanics of the Josephson effect, a cornerstone of quantum technologies employing superconducting circuits.

Researchers focused on a single Josephson junction coupled to a capacitive electromagnetic environment, a distinct scenario from previous investigations involving more complex bosonic environments. The study establishes that the low-lying discrete states exhibit a unique arrangement, particularly when transitioning between the Coulomb-dominated and Josephson-dominated regimes of the junction.
The team achieved a comprehensive analysis of the bound-state spectrum, ranging from configurations where the charging energy dominates, known as the Cooper pair box, to those where the Josephson energy prevails, characteristic of a transmon. This work builds upon prior predictions suggesting fundamentally different Josephson quantum mechanics in the odd parity sector when a quasiparticle is trapped in an Andreev bound state.

By deriving a low-energy effective Hamiltonian, the researchers were able to simplify the study of the Andreev spectrum, effectively integrating out fermionic degrees of freedom and focusing on the electromagnetic environment. Experiments show that multiple bound states can emerge within a single channel when the Josephson energy becomes comparable to or exceeds the charging energy.

This structure of bound states is strikingly different from that predicted by the Mathieu equation applicable to the even sector, potentially revealing itself through microwave spectroscopy. The research establishes a clear distinction between the odd and even parity sectors, a differentiation that would not be possible using methods reliant on detecting excess quasiparticles on the islands.

This prediction could be tested in forthcoming experiments utilising superconductor/semiconductor/superconductor junctions, which have received considerable attention in recent years, employing both nanowires and two-dimensional electron gases. The work opens avenues for exploring the implications of Andreev bound states in superconducting qubits, potentially leading to improved qubit coherence and performance in future quantum technologies. The derived eigenvalue problem simplifies the study of the Andreev spectrum, focusing on an electromagnetic degree of freedom and highlighting differences from previous studies with galvanic coupling to the electromagnetic environment.

Eigenproblem formulation for quasiparticle bound states in capacitively shunted Josephson junctions reveals interesting energy level structures

Scientists investigated the quantum mechanics of Josephson junctions, focusing on the odd parity sector where a quasiparticle is trapped in an Andreev bound state. The study pioneered an analysis of a Josephson junction coupled to a capacitive environment, differing from previous work examining ohmic environments.

Researchers determined a novel structure for low-lying discrete states in the odd sector, contrasting with the conventional structure found in the even sector, and explored this spectrum across the Coulomb-dominated and Josephson-dominated regimes. To define the bound-state spectrum, the team employed a scalar eigenproblem, q Ω+ H − p 2EJ sin φ 2 ψ(φ) = 0, derived from the Josephson Hamiltonian.

This approach assigned a specific sign to the chirality index, differing from alternative formulations where a quasiparticle “poisons” the Coulomb island. The method was extended to systems with two islands flanking the junction, where φ represents the superconducting phase difference between them and the charging energy depends on the charge difference.

In the Cooper-pair box regime, where EJ ≤E∗ J ≪EC, the lowest eigenstates |0⟩ and |1⟩ of H yielded energies E0,1 = EC[(N −1/2)2 + 1/4] ± EC(N −1/2). Lifting the degeneracy at N = 1/2, researchers found a single bound state with binding energy Ω= −EC|N −1/2| + q E2 C(N −1/2)2 + E2 J/4, valid for |N −1/2| ≪1.

Conversely, in the transmon regime, E∗ J ≫EC, the Hamiltonian was approximated as a quantum harmonic oscillator with resonance frequency ħω0 = p 8E∗ JEC. Linearizing the sine function as sin φ/2 ≈φ/2 = (EC/8E∗ J)1/4(b+b†), the team perturbatively calculated the binding energy Ω= ħω0/16N 2 ch. As Nch decreased, the study formulated the eigenproblem in dimensionless units, resulting in Heff(E) = q X2 + P 2 −E − 1 √Nch X (28), with ladder operators X and P, and dimensionless energy E.

At 0 Numerical solutions interpolated between these equations, accurately describing the spectrum. Finally, at Nch = 1, the problem was mapped onto a particle in a Rosen-Morse potential, defined by the Hamiltonian H = EC N 2 + 1 4 tan2( φ/2), revealing the energy landscape in this regime.

Odd-parity Andreev bound states exhibit unique energy level structure and spatial profiles

Scientists have uncovered a novel structure for low-lying discrete states within the odd-parity sector of Josephson junctions, differing significantly from the conventional even sector. The research focused on a Josephson junction coupled to a capacitive electromagnetic environment, extending previous work on galvanic coupling.

Experiments revealed that multiple bound states can emerge in a given channel when the Josephson energy, E∗J, becomes comparable to or exceeds the charging energy, EC. The team measured the binding energy of these Andreev bound states across various regimes, both analytically and numerically. Results demonstrate a striking difference in the structure of these bound states compared to those predicted by the Mathieu equation applicable to the even sector.

Specifically, the analysis applies to superconducting qubits with an arbitrary ratio of E∗J/EC and any number of channels, revealing a complex interplay between these parameters. Data shows that the charge dispersion of these bound states is periodic with the gate charge, presenting a cusp at charge degeneracy points.

This periodicity is crucial, as it distinguishes these states from those affected by excess quasiparticles on the islands, which would obscure the clear e-periodic signature. The study derives a low-energy effective eigenvalue problem that simplifies the analysis of the Andreev spectrum in the odd-parity sector, integrating out fermionic degrees of freedom.

Measurements confirm that the environment Hamiltonian depends on the location of additional quasiparticles, highlighting the importance of capacitive coupling in this system. The breakthrough delivers a pathway for revealing these multiple bound states through microwave spectroscopy, potentially advancing the development of more robust quantum technologies. This work predicts that the observed structure could be tested in superconductor/semiconductor/superconductor junctions, particularly those utilising nanowires or two-dimensional electron gases.

Odd-parity bound states emerge via capacitive coupling in a simplified Josephson junction array

Researchers have identified a novel structure for low-lying discrete states within the odd parity sector of a Josephson junction, differing significantly from the conventional even sector. This work centres on a single Josephson junction coupled to a capacitive environment, contrasting with previous studies focusing on more complex electromagnetic environments.

The investigation spans the range from Coulomb-dominated to Josephson-dominated regimes, examining the bound-state spectrum of the system. The findings demonstrate that the Andreev spectrum in the odd parity sector exhibits unique characteristics due to the capacitive coupling to the electromagnetic environment.

This contrasts with earlier research where the coupling was galvanic, highlighting the importance of the environmental nature in determining the system’s quantum behaviour. By deriving an effective Hamiltonian, the authors simplified the study of bound states in a superconducting qubit, integrating out fermionic degrees of freedom and focusing on the electromagnetic aspects.

The authors acknowledge that distinguishing between the odd and even sectors experimentally may be challenging due to the e-periodic dispersion with gate charge and the potential for quasiparticle poisoning. Future research could explore the predicted effects in superconductor/semiconductor/superconductor junctions, particularly those utilising nanowires or two-dimensional electron gases, potentially through microwave spectroscopy. These results contribute to a deeper understanding of Josephson quantum mechanics and could inform the development of advanced quantum technologies, although further investigation is needed to fully realise their potential.