⚛️ quantum physics

Creating Qubit States with Degenerate Two-level Systems

This paper demonstrates that two distinct levels with multiply degenerate sub-states can function as practical qubits by showing that Rabi oscillations persist under continuous fields and by deriving the necessary two-atom interaction to construct a controlled-Z gate.

Original authors: Zhuoran Bao, Daniel F. V. James

Published 2026-03-19

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CC BY 4.0

Original authors: Zhuoran Bao, Daniel F. V. James

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a super-computer that uses the laws of quantum physics. To do this, you need tiny building blocks called qubits.

Traditionally, scientists have been very picky about what makes a good qubit. They usually insist on a system that has exactly two distinct states, like a light switch that is either strictly ON or strictly OFF. If an atom has more than two states (like a dimmer switch with many settings), scientists usually try to "break" the extra states so only two remain. They do this by applying a strong magnetic field to force the atom into a specific shape.

This paper asks a bold question: "Do we really need to break the extra states? Can we use the 'messy' atom with all its extra states as a qubit anyway?"

The authors, Zhuoran Bao and Daniel James, say: Yes, we can. Here is how they explain it, using some everyday analogies.

1. The "Degenerate" Atom: A Room with Identical Doors

Imagine an atom as a house with two floors: a Ground Floor (low energy) and an Excited Floor (high energy).

In a normal "two-state" atom, there is only one door on the ground floor and one door on the excited floor. You can only go in or out through that single door.
In a "degenerate" atom (the kind this paper studies), the floors are huge ballrooms. There are multiple identical doors on the ground floor and multiple identical doors on the excited floor.

Usually, scientists think this is a problem because the atom gets confused about which door to use. They try to lock all the doors except two.

The Paper's Idea: Instead of locking the doors, let's shine a specific kind of light (a laser) on the house. If the light is perfectly aligned (like a laser pointer hitting the center of the room), it turns out that every single door pair behaves exactly the same way.

2. The "Rabi Dance": Synchronized Swimmers

When you shine this light on the atom, the electrons start dancing between the ground floor and the excited floor. This is called Rabi Oscillation.

Think of the atom's multiple states as a team of synchronized swimmers.

In a traditional setup, you might try to get just one swimmer to perform a routine while the others sit on the bench.
In this paper's setup, the light makes all the swimmers perform the exact same routine at the exact same time.

Because they are all doing the same thing, the whole group acts like a single, perfect qubit. You don't need to lock the doors; you just need to make sure the music (the light) makes everyone dance in unison.

3. The "Hadamard Gate": The Magic Coin Flip

To build a computer, you need to perform operations, like flipping a coin to get a random result. In quantum terms, this is called a Hadamard Gate.

The authors showed that even with all those extra doors (degenerate states), you can still perform this "coin flip" perfectly.

The Analogy: Imagine you have a deck of cards with many identical pairs. You want to shuffle them so that every card has a 50/50 chance of being red or black.
The paper proves that if you use the right "shuffle" (the laser pulse), every single pair of cards gets shuffled perfectly at the same time. The result is a perfect quantum gate, even without removing the extra cards.

4. The "Weak Magnetic Field" Problem: A Slight Tilt

In the real world, it's hard to keep a lab perfectly free of magnetic fields (like the Earth's magnetic field). This is like trying to keep a spinning top perfectly upright while someone gently tilts the table.

The authors asked: "What happens if the table tilts slightly?"

They calculated that as long as the tilt is very small compared to the strength of the laser dance, the swimmers will still stay mostly in sync.
They proved that the "mistake" (error) introduced by the tilt is tiny. With current technology (using special magnetic shields), we can keep the tilt small enough that the computer still works perfectly.

5. The "Two-Atom Chat": The Controlled-Z Gate

A quantum computer needs two qubits to talk to each other to do complex math. This is called a Controlled-Z gate.

The authors figured out how to make two of these "multi-door atoms" talk to each other.

The Analogy: Imagine two ballrooms next to each other. If the people in Room A start dancing, it creates a vibration that makes the people in Room B change their dance steps.
They showed that if the atoms are identical and the light is aligned correctly, this "vibration" creates a perfect logic gate. It's like two synchronized swimmers high-fiving each other without ever touching, just by moving in a specific pattern.

Why Does This Matter? (The Big Picture)

Currently, building quantum computers is like trying to build a skyscraper with a hammer that only fits one specific nail. It's hard, expensive, and slow.

This paper suggests we can use a universal hammer.

No need for perfect isolation: We don't need to spend millions of dollars creating perfect magnetic vacuums to "lock" the extra states.
Simpler setup: We can use the natural, messy, multi-state atoms that are easier to find and control.
More flexibility: It opens the door to using different types of atoms and encoding information in new ways.

In short: The authors found that nature's "messy" atoms can be tamed not by forcing them to be simple, but by dancing with them in a way that makes their complexity work for us, rather than against us. It's a more relaxed, flexible, and potentially easier way to build the quantum computers of the future.

1. Problem Statement

Current quantum computing architectures (trapped ions, superconducting qubits, etc.) typically require the removal of degeneracy in energy levels to define a qubit as a strict two-state system. This is often achieved using external magnetic fields (Zeeman effect) or hyperfine splitting. However, as quantum computers scale toward millions of qubits, maintaining stable, uniform external magnetic fields becomes experimentally challenging; fluctuations in these fields induce dephasing.

The authors pose a fundamental question: Is degeneracy lifting mandatory for optical qubits? They investigate whether two distinct energy levels, each containing multiple degenerate sub-states (e.g., Zeeman sub-levels), can function collectively as a practical quantum bit without the need for an external static magnetic field to split them.

2. Methodology

The authors employ a theoretical framework based on quantum optics and atomic physics, focusing on a model system of two fine-structure levels: the ground state $2S_{1/2}$ and the excited state $2P_{1/2}$ .

Hamiltonian Formulation: They define the system Hamiltonian including the atomic energy levels ( $\hat{H}_0$ ), the dipole interaction with an oscillating optical field ( $\hat{d} \cdot \mathbf{E} \cos(\omega t)$ ), and, in later sections, a weak static magnetic field ( $\hat{H}_B$ ).
Symmetry Analysis: Utilizing the Wigner-Eckart theorem and 3j-Symbols, they analyze the selection rules for dipole transitions. They propose defining the quantization axis along the polarization direction of the linearly polarized driving field rather than an external magnetic field.
Decoupling Strategy: They demonstrate that under linear polarization, the equations of motion for the probability amplitudes of the degenerate sub-states decouple into independent pairs. Each pair behaves as an identical two-level system.
Perturbation Theory: To assess robustness, they introduce a weak static magnetic field as a perturbation. They expand the time-evolution operator as a power series in the ratio of the magnetic coupling strength to the Rabi frequency ( $\mu_B B_0 / \hbar\Omega$ ).
Gate Construction:
- Single Qubit: They construct a "degenerate Hadamard gate" by timing the Rabi oscillations.
- Two Qubits: They generalize the approach to two interacting atoms to derive the conditions necessary for a Controlled-Z (CZ) gate, analyzing the interaction Hamiltonian $\hat{V}_{AB}$ .

3. Key Contributions

A. Theoretical Proof of Degenerate Qubits

The paper establishes that a system with spherical symmetry and degenerate levels can exhibit qubit-like behavior. By aligning the quantization axis with the electric field polarization, the Hilbert space decomposes into a superposition of identical rank-2 subspaces. The transition operations (Rabi oscillations) apply identically to each subspace, effectively creating multiple parallel qubits within a single atom's degenerate manifold.

B. Degenerate Hadamard Gate

The authors explicitly derive the matrix form of a Hadamard gate for a degenerate system (Eq. 8). They show that a sequence of free evolution and driven evolution can transform the state such that every degenerate pair undergoes the same unitary transformation, effectively acting as a Hadamard gate on the logical qubit space.

C. Fidelity Analysis under Magnetic Perturbation

A critical contribution is the calculation of the average fidelity of the degenerate Hadamard gate in the presence of a weak, uncontrolled static magnetic field.

They derive a power series expansion for the time-evolution operator.
They calculate the average fidelity up to the second order (Eq. 18).
Result: The error scales with $(\mu_B B_0 / \hbar\Omega)^2$ . For typical Rabi frequencies ( $10^3$ to $10^6$ Hz), the magnetic field must be smaller than $10^{-8}$ T to $10^{-5}$ T. The authors argue this is achievable with standard $\mu$ -metal shielding, suggesting that perfect field cancellation is not strictly necessary if the Rabi frequency is sufficiently high.

D. Two-Qubit Controlled-Z Gate

The authors propose a mechanism for a two-qubit CZ gate between two degenerate atoms.

They assume the atoms are identical and interact via a specific Hamiltonian where only linearly polarized light mediates coupling.
They show that under these conditions, the time evolution operator can be manipulated (via local unitaries) to produce a CZ gate.
Crucially, the gate acts as four simultaneous CZ gates on the different pairs of degenerate sub-states, provided the interaction strength is uniform across the manifold.

4. Results

Rabi Oscillations: The system undergoes Rabi oscillations even with degeneracy, provided the driving field is linearly polarized and the quantization axis is defined by the field.
Gate Performance: The degenerate Hadamard gate is theoretically perfect in the absence of magnetic fields. With weak magnetic fields, the fidelity remains high ( $>99\%$ ) if the magnetic field is sufficiently shielded relative to the Rabi frequency.
Scalability Implication: The approach allows for "multi-level encoding" where sub-state populations can be used for encoding, offering flexibility in quantum computing codes without the overhead of complex magnetic field stabilization.
Limitations: The authors clarify that while the system has multiple degrees of freedom, it is not "hyperentangled" in the traditional sense because the total angular momentum ( $J$ ) and its projection ( $M_J$ ) cannot be operated on independently without ancillary states. Furthermore, degeneracy does not inherently provide error correction capabilities.

5. Significance

This work challenges the conventional dogma that qubits must be strictly non-degenerate two-level systems.

Experimental Simplification: It suggests that trapped-ion quantum computers could potentially operate without the stringent requirement of stabilizing external magnetic fields to lift degeneracy, reducing experimental complexity and cost.
Resource Efficiency: It opens the door to using the full degeneracy of fine-structure levels as a resource for encoding information, potentially increasing the density of qubits per ion.
New Architectures: By demonstrating that spherical symmetry can be broken by the driving field itself rather than an external magnet, the paper proposes a new class of optical qubits that are more robust against magnetic noise, provided the driving field is strong enough.

In conclusion, the paper provides a rigorous theoretical foundation for using degenerate atomic levels as functional qubits, demonstrating that with appropriate field polarization and shielding, high-fidelity quantum gates are achievable without the need for degeneracy lifting.