Andreev bound states in a superconducting qubit at odd parity

This paper predicts a novel structure for low-lying discrete states in the odd-parity sector of a capacitively shunted superconducting qubit containing a trapped quasiparticle, revealing a spectrum fundamentally distinct from the conventional even-parity case across both Coulomb-dominated and Josephson-dominated regimes.

The Gist

Imagine a superconducting qubit (a tiny quantum computer chip) as a small, isolated island floating in a sea of electricity. Usually, this island is perfectly balanced, with an even number of electrons dancing around. This is the "even" state, and it's the standard way these quantum bits work.

However, sometimes a single, uninvited guest—a "quasiparticle" (a rogue electron-like particle)—gets stuck on the island. This puts the system in an "odd" state. In the past, scientists thought this was just a glitch or a nuisance that ruined the computer's memory.

This paper explores what happens when that rogue guest gets trapped in a specific "room" on the island called an Andreev bound state. The authors discovered that when this happens, the rules of the game change completely.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of Islands

The paper looks at two different ways to build this quantum island:

• The "Cooper-pair Box" (The Sensitive Scale): This is a very sensitive setup where the island is small and the electricity is tightly controlled. It's like a delicate scale that reacts strongly to the slightest change in charge.
• The "Transmon" (The Heavyweight): This is a more robust setup where the island is "heavier" and less sensitive to outside noise. This is the type used in most modern quantum computers today.

2. The Rogue Guest and the New Rules

When a single quasiparticle gets trapped in the Andreev state (the "odd" sector), the authors found that the energy levels of the system don't behave like they do in the normal "even" state.

• The Old Way (Even Sector): Think of the energy levels like rungs on a ladder. In the standard setup, the rungs are spaced out in a predictable, smooth pattern.
• The New Way (Odd Sector): When the rogue guest is trapped, the "ladder" changes shape entirely.
  • In the sensitive setup, the guest creates a single, deep trap where it can hide.
  • In the robust setup (Transmon), something surprising happens: instead of just one or two rungs, the system can suddenly support multiple distinct energy levels (multiple rungs) for that single trapped guest.

3. The "Channel" Analogy

Imagine the junction (the bridge between the two parts of the island) has several "lanes" or channels for traffic.

• If there is only one lane, the trapped guest creates a specific pattern of energy levels.
• If the "Josephson energy" (the strength of the bridge) becomes very strong compared to the "charging energy" (the cost of adding charge), the system acts like a radial oscillator.
• The Metaphor: Imagine a marble rolling in a bowl. In the standard case, the marble rolls in a simple circle. In this new "odd" case with a strong bridge, the marble can settle into multiple distinct orbits inside the bowl, depending on how strong the bridge is. The paper predicts that as you tune the strength of the bridge, you can see these multiple orbits appear one by one.

4. Why This Matters (According to the Paper)

The authors predict that if you shine microwaves (like a radio signal) at these devices, you will see a unique "fingerprint" in the sound.

• In the past, scientists thought the trapped guest just made the system messy.
• This paper says: No, the trapped guest creates a whole new, structured spectrum of energy levels.
• These levels repeat every time you add a specific amount of charge (an "e-periodic" pattern), which is different from the usual patterns.

5. The Bottom Line

The paper claims that by studying superconducting qubits made of specific materials (like nanowires or 2D electron gases), scientists can experimentally see these new, multiple energy levels. They are essentially saying: "We found a hidden structure in the quantum mechanics of a trapped particle that looks nothing like what we see in normal operation. It's a new kind of quantum 'music' that only plays when the system is 'poisoned' by a single quasiparticle."

What the paper does NOT claim:

• It does not say this will immediately fix quantum computers.
• It does not claim this will be used for medical devices.
• It does not say we can use this to build a better computer today.
• It strictly focuses on the theoretical prediction of these energy levels and suggests they can be tested in upcoming lab experiments using microwave signals.

Technical Summary

Technical Summary: Andreev Bound States in a Superconducting Qubit at Odd Parity

Problem Statement
The quantum mechanics of the Josephson effect is central to superconducting circuit technologies. While the even-parity sector (where the number of quasiparticles is even) is well-understood, the odd-parity sector—where a single quasiparticle is trapped in an Andreev bound state (ABS)—presents distinct physics. Previous studies have addressed the odd-parity sector in specific contexts: either for a Cooper-pair box with a single-channel junction coupled to a gate [16], or for a junction galvanically coupled to an electromagnetic environment modeled as a collection of harmonic oscillators (including ohmic environments) [17].

However, a gap remains in understanding the odd-parity spectrum for a superconducting qubit where the electromagnetic environment is reduced to a simple capacitance (a capacitive coupling). This configuration is relevant for standard superconducting qubits ranging from the charge-sensitive Cooper-pair box ($E_J^* \ll E_C$) to the charge-insensitive transmon ($E_J^* \gg E_C$). The authors aim to determine the structure of low-lying discrete states in this capacitive regime and compare them to the conventional even-sector results and previous odd-parity studies.

Methodology
The authors derive a low-energy effective eigenvalue problem to describe the Andreev spectrum in the odd-parity sector of a single Josephson junction qubit.

1. Model Setup: The system consists of a superconducting island with total capacitance $C$ (including gate capacitance $C_g$ and junction capacitance $C_J$) connected to a superconducting lead via a tunnel junction. The island is subject to a gate charge $Q_g$.
2. Hamiltonian Derivation: Starting from the microscopic Hamiltonian, the authors integrate out fermionic degrees of freedom, retaining only the electromagnetic phase degree of freedom $\phi$. The total Hamiltonian includes:
   • Charging energy ($\hat{H}_C$) dependent on the location of the quasiparticle (island vs. lead).
   • Josephson coupling ($\hat{H}_J$) arising from second-order quasiparticle tunneling.
   • Effective quasiparticle tunneling terms ($\hat{H}_{eff}^T$) projected onto the low-energy subspace.
3. Effective Eigenproblem: By projecting the total Hamiltonian onto a spinor wavefunction in the island/lead space, they derive a scalar eigenvalue equation (Eq. 24) for the binding energy $\Omega$:
   $$\left( \sqrt{\Omega + \hat{H}} - \sqrt{2E_J} \sin\frac{\hat{\phi}}{2} \right) \psi(\phi) = 0$$
   Here, $\hat{H} = \hat{H}_J + E_C(-2i\partial_\phi - N)^2 - E_{min}(N)$, where $N$ is the dimensionless gate charge. Crucially, the boundary conditions for the wavefunction are modified to $|\Phi(2\pi)\rangle = -\tau_z |\Phi(0)\rangle$, reflecting the odd-parity nature and the specific capacitive coupling, distinguishing it from the galvanic coupling case in Ref. [17].

Key Contributions and Results
The study analyzes the binding energy spectrum across different regimes defined by the ratio of Josephson energy to charging energy ($E_J^*/E_C$) and the effective number of channels ($N_{ch} = E_J^*/E_J$).

• Cooper-Pair Box Regime ($E_J^* \ll E_C$):
  In the limit of small Josephson energy, the system accommodates a single, spin-degenerate bound state. The binding energy is derived analytically as:
  $$\Omega = -E_C|N - 1/2| + \sqrt{E_C^2(N - 1/2)^2 + E_J^2/4}$$
  This result aligns with previous findings for single-channel junctions [16], confirming the $e$-periodicity of the dispersion with respect to the gate charge.

• Transmon Regime ($E_J^* \gg E_C$):
  In the transmon limit, the spectrum is analyzed using a harmonic oscillator approximation.

  • Large $N_{ch}$ ($N_{ch} \gg 1$): The binding energy is small, scaling as $\Omega \propto \hbar\omega_0 / N_{ch}^2$.
  • Intermediate $N_{ch}$ ($0 < N_{ch} - 1 \ll 1$): As $N_{ch}$ decreases towards unity, the authors find a novel structure where multiple bound states appear within a single channel. The eigenproblem maps to a radial harmonic oscillator in three dimensions, yielding a discrete spectrum of bound states below the continuum. The number of these states scales approximately as $1/\sqrt{N_{ch}-1}$.
  • $N_{ch} = 1$: When the effective channel number is unity, the states localized near $\phi=0$ and $\phi=2\pi$ hybridize. The effective Hamiltonian describes a particle in a Rosen-Morse potential, yielding a distinct set of eigenenergies that differ significantly from the even-sector Mathieu equation results.

• General Findings:
  The authors demonstrate that the structure of odd-parity bound states is fundamentally different from the even sector. Specifically, in the transmon regime with $N_{ch}$ comparable to or slightly larger than 1, multiple discrete bound states can exist, which is not predicted by the standard even-parity theory.

Significance and Claims
The paper claims to generalize previous theories of odd-parity Josephson quantum mechanics to the case of capacitive coupling, which is the standard environment for superconducting qubits. The primary significance lies in the prediction of a novel structure for low-lying discrete states in the odd sector, characterized by the potential existence of multiple bound states in a single channel when $E_J^*$ is comparable to or larger than $E_C$.

The authors assert that these predictions could be tested in forthcoming experiments using superconductor/semiconductor/superconductor junctions (utilizing nanowires or two-dimensional electron gases) via microwave spectroscopy. They note that while quasiparticle poisoning typically obscures parity sectors in charge dispersion, the unique $e$-periodic dispersion of these bound states offers a potential signature, though distinguishing sectors via poisoning effects alone may be challenging. The work suggests that the interaction between the Josephson junction and its electromagnetic environment profoundly modifies the bound state spectrum, opening new perspectives for understanding quasiparticle dynamics in qubits.