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Experimental Realization and Phase-Space Winding of a Topological Defect State in the Quantum Rabi Model

This paper reports the experimental realization of a topological defect state in a single trapped $^{171}\mathrm{Yb}^{+}$ ion, demonstrating its tunable phase-space winding numbers and dynamical properties through characteristic-function tomography and carrier-amplitude modulation within the quantum Rabi model.

Original authors: Kyungmin Lee, Jiyong Kang, Sunkyu Yu, Jaehun You, Wonhyeong Choi, Taehyun Kim

Published 2026-09-29

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CC BY 4.0

Original authors: Kyungmin Lee, Jiyong Kang, Sunkyu Yu, Jaehun You, Wonhyeong Choi, Taehyun 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 microscopic world of quantum physics, particles do not always behave like tiny, solid billiard balls. Sometimes, they act more like waves, spreading out and interfering with one another in ways that create strange, stable patterns. One of the most fascinating of these patterns is the topological state. Imagine a knot tied in a piece of string; you can twist and turn the string, stretch it, or even shake it, but as long as you do not cut the string or untie the knot, the knot remains. In quantum systems, certain states of matter are like these knots. They are protected by the overall shape of the system's energy landscape, making them robust against small disturbances. Scientists have long sought to create and study these states because their stability could one day lead to more reliable quantum computers. To explore this, researchers often build "synthetic lattices," which are not made of physical atoms arranged in a grid, but are instead created by using light and magnetic fields to force particles to behave as if they are hopping between the rungs of a ladder.

A team of researchers at Seoul National University has now successfully created one of these protected states, known as a topological defect, inside a single trapped ion. They used a specific model of physics called the quantum Rabi model, which describes how a simple two-level system, like a spinning particle, interacts with a vibrating field. By carefully tuning laser beams, the scientists forced a single ion of the element ytterbium to act as if it were sitting on a special kind of ladder where the rules for moving up or down changed depending on how far the ion had already traveled. This setup created a boundary, or interface, within the system. At this boundary, a unique state formed that was trapped in place, unable to escape because of the topological protection. The researchers did not just predict this state; they built it, measured its shape in a high-dimensional space, and confirmed that it possessed a specific geometric property called a winding number, which acts like a fingerprint for this type of quantum knot.

To understand how this works, one must first picture the ion not as a stationary dot, but as a particle that can vibrate. In this experiment, the vibration is treated as a series of distinct energy levels, similar to the rungs of a ladder. The researchers used three different laser tones to control the ion. One laser acted as a steady bridge connecting two internal states of the ion, while the other two lasers allowed the ion to jump between the vibration rungs. Crucially, the strength of these jumps depended on which rung the ion was currently on. This created a situation where the "hopping" rules were not uniform; they changed as the ion moved along the ladder. This variation created a synthetic interface, a dividing line in the energy landscape where the rules shifted. On one side of this line, the ion preferred to stay in one internal state, and on the other side, it preferred the opposite state. At the interface itself, a special state emerged that was localized, meaning the ion's vibration was concentrated in a specific region rather than spreading out.

The team prepared this state by first cooling the ion to its lowest possible energy and then slowly turning on the laser controls. They ramped up the laser strengths over a period of half a millisecond, guiding the ion into the desired configuration. Once the state was prepared, they needed to see what it looked like. Because the state exists in a complex space involving both the ion's spin and its motion, they could not simply take a photograph. Instead, they used a technique called characteristic-function tomography. This method allowed them to reconstruct the full distribution of the ion's motion, effectively mapping out where the ion was likely to be found in its phase space, a mathematical representation that combines position and momentum. They repeated this process for many different settings of the laser phases, tracing out the path the center of the ion's motion took as they changed the conditions.

The results revealed a clear geometric signature. When the researchers adjusted the lasers to create one type of coupling, the path traced by the center of the ion's motion wound around a central point exactly once as they cycled through the laser phases. This is known as a winding number of one. When they switched the lasers to create the opposite type of coupling, the path did not wind around the center at all; it simply moved back and forth, resulting in a winding number of zero. This difference in winding confirmed that the state they had created was indeed the topological defect predicted by theory. The experiment showed that this winding property is robust; even when the researchers changed the strength of the main laser bridge, the size of the path changed, but the way it wound around the center remained exactly the same. This demonstrated that the topological nature of the state was preserved despite changes in the system's parameters.

To further test the stability and nature of this state, the researchers applied a rhythmic modulation to the strength of the main laser bridge. They observed how the ion's internal spin responded to this shaking. The response they measured matched the predictions for a system with specific energy gaps between its states. The data showed that the defect state was connected to only a few other energy levels, making it relatively isolated and stable. This isolation helps explain why the state remains intact even when the system is slightly disturbed. The researchers also noted that while their setup included some unavoidable shifts in energy caused by the lasers, these shifts did not destroy the fundamental shape of the state or its topological protection. The experiment successfully linked the abstract mathematical concept of a topological invariant to a measurable, physical geometry in a real quantum system.

This work represents a significant step in understanding how topological states can be engineered in hybrid systems where light and matter interact. By demonstrating that a single trapped ion can host a defect state with a measurable winding number, the researchers have provided a concrete example of how synthetic lattices can be used to explore complex quantum phenomena. The ability to tune the interface and observe the resulting geometric properties offers a powerful tool for future studies. The findings confirm that the competition between uniform driving forces and position-dependent interactions can create stable, protected states that are resilient to change. This level of control and measurement brings the theoretical world of topological defects closer to practical reality, opening new avenues for exploring the fundamental geometry of quantum matter.

Technical Summary: Experimental Realization and Phase-Space Winding of a Topological Defect State in the Quantum Rabi Model

Problem and Context
Topological phases, characterized by global invariants and boundary states, have been explored across various quantum platforms. In hybrid oscillator–qubit systems, specifically within a Fock-state lattice (FSL), oscillator number states act as lattice sites with interactions providing site-dependent hopping. Theoretical studies have proposed that number-dependent hopping in the resonantly driven quantum Rabi model (QRM) can support topological defect states. In this regime, uniform carrier coupling competes with the oscillator's intrinsic $\sqrt{n}$ sideband couplings, creating a synthetic, tunable SSH-like interface in the FSL. A key theoretical prediction is that these defect states occupy a single spin sublattice, and their topological structure can be identified through the winding of the bosonic centroid in phase space under a Hamiltonian phase cycle. However, the nonuniform hopping in these systems hinders a direct description via ordinary Bloch bands, necessitating experimental verification of these phase-space invariants and the defect state's stability.

Methodology
The authors experimentally realize this system using a single trapped $^{171}\text{Yb}^+$ ion. The spin is encoded in hyperfine states ( $|F=0, m_F=0\rangle$ and $|F=1, m_F=0\rangle$ ), and a radial motional mode provides the Fock basis.

Hamiltonian Implementation: The system is driven by three phase-coherent Raman tones: a carrier tone and two sideband tones (red and blue). These tones independently control the carrier amplitude ( $w$ ) and the sideband couplings ( $g_1 e^{i\phi_1}$ and $g_2 e^{i\phi_2}$ ), effectively implementing the driven anisotropic QRM Hamiltonian with vanishing on-site energies.
State Preparation: The target defect state, $|\psi_0\rangle = \hat{U}(\theta, \alpha, r)|0\rangle|A\rangle$ , is prepared adiabatically. The process involves initializing the ion, applying a carrier $\pi$ pulse for specific regimes, and ramping the carrier and weaker sideband amplitudes while keeping the stronger sideband active (to which the state is dark). This prepares the motional component as a rotated, displaced, and squeezed vacuum on the appropriate spin sublattice ( $|A\rangle$ ).
Characterization:
1. Phase-Space Tomography: The motional state is reconstructed using characteristic-function tomography. The measured characteristic functions are Fourier-transformed to obtain Wigner functions, from which the centroid trajectories are extracted.
2. Spin Response: The stability and excitation gaps of the defect state are probed via carrier-amplitude modulation ( $w(t) = w[1 + \delta_w^a \sin(2\pi f_N t)]$ ). The response is quantified by the RMS spin deviation from the unmodulated signal.

Key Contributions and Results

Experimental Realization of the Defect State: The authors successfully prepared the topological defect state in two distinct coupling regimes ( $g_1 > g_2$ and $g_1 < g_2$ ). The conditional motional-state fidelity, inferred from the overlap between the experimental characteristic function and the ideal theoretical state, was measured to be $0.915 \pm 0.013$ .
Observation of Phase-Space Winding: By varying the sideband phase $\phi_1$ $ϕ_{1}$ while keeping $\phi_2 = 0$ $ϕ_{2} = 0$ , the authors mapped the Wigner-centroid trajectories.
- In the regime where $g_1 > g_2$ , the trajectory winds once around the origin (winding number $W=1$ ).
- In the regime where $g_1 < g_2$ , the trajectory does not enclose the origin (winding number $W=0$ ).
- Changing the carrier amplitude $w$ rescaled the trajectories without altering their winding numbers, confirming the robustness of the topological invariant against parameter rescaling.
Dynamical Probe via Modulation: Carrier-amplitude modulation revealed a spin response consistent with the theoretical excitation spectrum. The measured RMS spin deviation showed frequency-dependent features corresponding to the energy gaps between the defect state and the first excited doublet ( $|\psi_{1,\pm}\rangle$ ). The results indicated that the defect state has the fewest directly coupled final eigenstates at first order, contributing to its stability away from resonant frequencies.
Handling Symmetry Breaking: The authors addressed the presence of a differential AC Stark term ( $\hat{H}_S \propto \hat{\sigma}_z$ ) arising from the Raman drives. While this term breaks the chiral symmetry of the ideal model and shifts the defect energy, it does not alter the ideal spin-polarized wavefunction or its centroid geometry, allowing the topological features to remain observable.

Significance
The paper demonstrates the experimental realization of a topological defect state in a nonuniform synthetic lattice formed by a hybrid spin–boson system. The work connects the theoretical concept of phase-space invariants to measurable geometric properties (Wigner-centroid winding) in a system where ordinary Bloch-band descriptions are insufficient. By independently preparing the state in different coupling regimes and verifying the distinct winding numbers (one and zero), the experiment validates the theoretical prediction that number-dependent hopping can support topological defect states. Furthermore, the complementary use of carrier-amplitude modulation provides a dynamical probe that links the measured spin response to the underlying excitation gaps and transition channels, offering a comprehensive characterization of the defect state's topological and energetic structure.

Technical Summary: Experimental Realization and Phase-Space Winding of a Topological Defect State in the Quantum Rabi Model

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