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Quantum Simulation of Two-Dimensional Free Dirac Hamiltonian in Multimode Circuit QED

This paper reports the experimental realization of a programmable two-dimensional massive Dirac Hamiltonian in a multimode circuit QED platform, enabling the observation of rotational Zitterbewegung and establishing a compact system for exploring relativistic quantum dynamics in gapped Dirac systems.

Original authors: Jiwon Kang, Jiuk Lee, Eliya Blumenthal, Shay Hacohen-Gourgy, Eunseong Kim

Published 2026-09-16

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CC BY 4.0

Original authors: Jiwon Kang, Jiuk Lee, Eliya Blumenthal, Shay Hacohen-Gourgy, Eunseong Kim

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

In the subatomic world, particles do not always behave like tiny, solid marbles rolling along a track. Instead, they follow rules that blend the mechanics of the very small with the laws of motion that govern light and speed. One of the most famous equations in physics describes how a particle with a specific kind of spin moves when it is subject to these relativistic rules. A striking prediction of this equation is that such a particle should not move in a straight line, even when no forces are pushing or pulling it. Instead, it should jitter and wobble in a rapid, oscillating motion. This phenomenon, known by a German name meaning "trembling motion," arises because the particle exists in a superposition of two different energy states at once, causing it to interfere with itself. While this behavior is a fundamental part of the theory, it happens so fast and on such a tiny scale for real particles that it has never been directly observed in a free particle. To study it, scientists have turned to quantum simulations, where they build artificial systems that mimic the math of these particles, allowing them to watch the trembling motion unfold on a human timescale.

A team of researchers has now taken this concept into two dimensions, creating a programmable simulation of a massive particle moving in a flat plane. In their experiment, they used a superconducting circuit containing a single artificial atom, known as a transmon qubit, coupled to a special microwave cavity that can hold two distinct modes of electromagnetic vibration. By carefully driving the qubit with microwave signals, they engineered a situation where the qubit acts as the particle's internal spin, while the vibrations in the cavity represent the particle's position and momentum in two perpendicular directions. The researchers tuned the system so that the qubit and the cavity vibrations interacted in a way that perfectly matched the mathematical description of a free particle with mass. This setup allowed them to program the particle's mass and watch how it moved, effectively creating a miniature, controllable universe where the laws of relativistic quantum mechanics could be tested directly.

The team observed that when they prepared the system in a specific state, the simulated particle did not travel in a straight line. Instead, it traced out a spiraling path, a two-dimensional version of the trembling motion. As they increased the effective mass of the particle by adjusting the frequencies of their control signals, the nature of this motion changed in a precise way. The particle began to spin faster, completing its loops more quickly, but the size of the loops shrank. This behavior matched the theoretical prediction that heavier particles should exhibit a tighter, more rapid trembling motion. The researchers found that the rotational frequency of the path increased while the amplitude of the oscillation decreased as the mass grew, a relationship that holds true for massive Dirac particles in the real world.

To ensure these observations were not just artifacts of their equipment, the researchers compared their experimental data with detailed computer simulations that included the real-world imperfections of their device. These imperfections included the natural loss of energy in the circuit, small deviations in how the microwave signals interacted with the qubit, and the fact that the artificial atom is not a perfect two-level system. The computer models, which accounted for these specific flaws, reproduced the experimental results with high fidelity. This close agreement confirmed that the observed spiraling motion was indeed the signature of the simulated Dirac particle and not a glitch in the machinery. The study explicitly ruled out the idea that the motion was a simple combination of two independent one-dimensional wobbles; instead, the movement in the two directions was deeply linked because they shared the same qubit, creating a complex, correlated dance that could not be separated into simpler parts.

This work establishes a new platform for exploring the dynamics of particles in gapped systems, where a finite mass creates an energy barrier. The ability to independently control the spin-momentum couplings and the mass of the simulated particle makes this circuit a versatile tool for future investigations. The researchers suggest that this programmable setup could be used to simulate how particles behave in external magnetic fields or to explore topological phenomena, which are properties of matter that remain stable even when the material is deformed. By demonstrating that a single qubit coupled to multiple cavity modes can faithfully reproduce the behavior of a two-dimensional massive Dirac particle, the team has provided a robust foundation for studying complex quantum dynamics that were previously out of reach. The experiment confirms that the rotational trembling motion is a real, observable feature of massive Dirac systems, offering a clear window into the relativistic quantum world.

Technical Summary: Quantum Simulation of Two-Dimensional Free Dirac Hamiltonian in Multimode Circuit QED

Problem Statement
While quantum simulations have successfully demonstrated one-dimensional Dirac dynamics and two-dimensional massless Weyl dynamics, the programmable simulation of massive two-dimensional Dirac dynamics has remained experimentally unexplored. In two dimensions, a finite Dirac mass opens an energy gap at the Dirac point and introduces a rotational component to the Zitterbewegung (oscillatory motion arising from interference between positive- and negative-energy states), characterized by a well-defined handedness linked to the system's Berry curvature. Previous approaches, such as photonic honeycomb lattices or superconducting-qubit experiments, were limited by a lack of independent control over constituent terms or the treatment of momentum as fixed parameters rather than dynamical quantum degrees of freedom.

Methodology
The authors realize a programmable two-dimensional free Dirac Hamiltonian using a multimode circuit QED architecture. The experimental setup employs a single transmon qubit coupled to two distinct modes of a flute cavity (TE101 and TE102), which serve as the two spatial dimensions.

Hamiltonian Engineering: The target Hamiltonian is $H_D = c \hat{P}_1\sigma_x + c \hat{P}_2\sigma_y + mc^2\sigma_z$ $H_{D} = c \hat{P}_{1} σ_{x} + c \hat{P}_{2} σ_{y} + m c^{2} σ_{z}$ .
- Spinor Degree of Freedom: A Rabi-driven qubit defines a dressed basis ( $|\tilde{g}\rangle, |\tilde{e}\rangle$ ) acting as the Dirac spinor.
- Momentum Operators: The conjugate quadratures of the two cavity modes represent the position ( $\hat{X}_j$ ) and momentum ( $\hat{P}_j$ ) operators.
- Coupling and Mass: Double-sideband drives applied to the cavity modes generate controlled spinor-momentum couplings ( $\sigma_x$ and $\sigma_y$ ). The effective mass term ( $mc^2\sigma_z$ ) is engineered via the detuning ( $\Delta\Omega$ ) between the Rabi drive frequency and the sideband drive frequencies.
Experimental Protocol: The simulation follows a four-stage pulse sequence:
1. Preparation: Double-sideband drives prepare the initial cavity state and set the effective phase.
2. Evolution: Simultaneous Rabi and sideband drives evolve the system under the effective Dirac Hamiltonian.
3. Projection & Ringdown: The entangled qubit-cavity state is projected onto a selected qubit state using the DRACHMA readout protocol to minimize backaction, followed by a ringdown period to ensure photon decay.
4. Measurement: The position expectation values $\langle \hat{X}_j \rangle$ are extracted from qubit-state-dependent characteristic functions.

Key Contributions

First Realization of 2D Massive Dirac Dynamics: The work presents the first experimental realization of a programmable two-dimensional free massive Dirac Hamiltonian where both momentum components are represented by quantum operators and independently controlled.
Observation of Rotational Zitterbewegung: The authors observe the rotational Zitterbewegung of a massive Dirac particle, demonstrating how the trajectory depends on the effective mass.
Comprehensive Modeling: The experimental dynamics are reproduced using time-dependent master-equation simulations that incorporate measured decoherence, corrections beyond the rotating-wave approximation (RWA), transmon anharmonicity, and control-parameter fluctuations.

Results

1D Benchmarking: In one dimension, the system successfully simulated Zitterbewegung for various effective masses ( $\Delta\Omega/2\pi = 0$ to $0.5$ MHz). As the effective mass increased, the oscillation frequency increased while the amplitude decreased, consistent with the approach to the non-relativistic limit.
2D Dynamics: In two dimensions, the system exhibited a rotational trajectory. Increasing the effective mass resulted in faster rotational motion with a smaller trajectory radius.
Decoherence and Limitations: The experimental data showed good agreement with full master-equation simulations. The primary limitation identified was spin-locking decoherence ( $T_{2\rho}$ ) of the Rabi-driven qubit. Additional deviations were attributed to RWA breakdown, finite transmon anharmonicity, and control-parameter fluctuations. Despite these limitations, the experimental results for larger effective masses remained remarkably close to the ideal Dirac Hamiltonian predictions.

Significance
The paper establishes multimode circuit QED as a compact and programmable platform for simulating higher-dimensional relativistic quantum dynamics. By demonstrating that the two-dimensional dynamics are not merely a superposition of independent one-dimensional motions but arise from correlated multimode dynamics via a shared qubit degree of freedom, the work provides a foundation for exploring dynamical and topological phenomena in gapped Dirac systems. The authors suggest this programmability opens avenues for future simulations involving external electromagnetic fields (e.g., Klein tunneling) and topological Dirac physics, though these are presented as potential extensions rather than immediate results of the current study.

Technical Summary: Quantum Simulation of Two-Dimensional Free Dirac Hamiltonian in Multimode Circuit QED

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