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Stimulated emission for a three-level artificial atom in waveguide quantum electrodynamics

This theoretical study demonstrates that in waveguide quantum electrodynamics, the presence of a third energy level in a three-level artificial atom with low anharmonicity (such as a transmon) significantly suppresses stimulated emission while enabling tunable photon bunching in the reflected field.

Original authors: O. A. Chuikin, Ya. S. Greenberg, V. V. Taravkova, O. V. Kibis

Published 2026-10-08

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CC BY 4.0

Original authors: O. A. Chuikin, Ya. S. Greenberg, V. V. Taravkova, O. V. Kibis

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, light and matter do not just pass each other by; they can become deeply entangled, trading energy and information in ways that defy everyday intuition. This interaction is the foundation of a field known as waveguide quantum electrodynamics, where scientists study how individual particles of light, called photons, behave when forced to travel along a narrow, one-dimensional path, much like a train on a single track. In this confined space, the rules of the game change. A single atom placed in this path can act as a mirror, a filter, or a switch, reflecting or absorbing light in ways that depend on its internal energy structure. While natural atoms have complex, rigid energy levels, engineers have created "artificial atoms" using superconducting circuits. These man-made structures mimic the behavior of natural atoms but offer a unique advantage: their energy levels can be tuned and adjusted with precision. One of the most popular types of these artificial atoms is called a transmon, which typically has three distinct energy steps. Understanding how these three-step systems interact with light is crucial for building future quantum computers and communication networks, yet the subtle influence of that third, often overlooked, energy step has remained a mystery.

Researchers set out to solve this puzzle by simulating a specific, fundamental scenario: a single pulse of light hitting an artificial atom that is already excited. In a simpler, two-step system, this interaction would typically trigger a process called stimulated emission, where the incoming light encourages the atom to release a second photon in the exact same direction, effectively cloning the light pulse. However, the team wanted to see what happens when the atom has a third energy level, a feature common in transmons but often ignored in theoretical models. By using a mathematical approach that tracks the position of the light and the atom in real space, they calculated exactly how the system would evolve over time. They focused on a pulse of light shaped like an exponential curve, a form that naturally occurs when atoms release energy, and watched how it scattered off the excited artificial atom.

The results revealed a surprising and significant twist. When the researchers modeled the artificial atom with the specific, low-energy spacing found in real-world transmons, they found that the third energy level acted as a brake on the stimulated emission process. Instead of the light pulse being amplified and cloned as expected, the presence of the third level largely suppressed this effect. In their simulations, the probability of finding two photons moving together in the same direction dropped significantly compared to what would happen in a simple two-level system. The team determined that for this suppression to occur, the spacing between the energy levels needed to be very small, specifically less than four percent difference. This finding suggests that for the most common type of artificial atom used today, the third level is not a passive bystander; it actively interferes with the generation of new light, potentially hindering applications that rely on efficient light amplification.

Yet, the story does not end with suppression. The study uncovered a different kind of behavior that emerged from this complex interaction: the photons began to cluster together in a phenomenon known as bunching. When the light was reflected back from the atom, the two photons were more likely to be found close to each other in space, rather than spread apart. This clustering effect was a direct result of the three-level structure and was not seen in the simpler two-level models. Perhaps most intriguingly, the researchers discovered that this behavior was not fixed. By simply adjusting the characteristics of the incoming light pulse, such as its frequency or its duration, they could effectively "switch off" the bunching effect. This means that the quantum behavior of the reflected light is not a permanent trait of the atom but can be controlled by the way the light is delivered.

These findings challenge the assumption that artificial atoms can be treated as simple two-level systems when designing quantum devices. The research demonstrates that even a small deviation in energy levels, typical of the transmons used in modern laboratories, can drastically alter how light is scattered and how photons correlate with one another. For scientists aiming to route information or amplify signals in quantum circuits, this implies that the third energy level must be accounted for, as it can either dampen the desired signal or introduce new, controllable patterns of light. The work provides a clearer map of the quantum landscape, showing that the path of a single photon is deeply influenced by the hidden architecture of the atom it encounters, and that by tuning the light itself, we can steer these quantum outcomes with surprising precision.

Technical Summary: Stimulated Emission for a Three-Level Artificial Atom in Waveguide Quantum Electrodynamics

Problem Statement
This work addresses the theoretical study of stimulated emission within the context of waveguide quantum electrodynamics (QED). Specifically, it investigates the scattering of a single-photon exponential pulse by an excited three-level ladder system (3LE) located in a one-dimensional open waveguide. While standard models often approximate artificial atoms (such as superconducting transmons) as two-level systems (qubits), the authors argue that the influence of the third energy level is significant, particularly for transmons which exhibit low anharmonicity. The study aims to rigorously determine how the presence of this third level modifies scattering dynamics, stimulated emission efficiency, and photon-photon correlations compared to a regular two-level system.

Methodology
The authors employ a real-space formalism under the rotating wave approximation (RWA) to describe the interaction between the ladder-type emitter and the continuous electromagnetic field.

Hamiltonian: The system is modeled using a Hamiltonian that includes the energy levels of the ground ( $|g\rangle$ ), first-excited ( $|e\rangle$ ), and second-excited ( $|f\rangle$ ) states, coupled to right-moving ( $a(x)$ ) and left-moving ( $b(x)$ ) photon fields via coupling constants $V_1$ and $V_2$ .
Wavefunction: The system is analyzed within a two-excitation subspace. The general wavefunction is constructed using probability amplitudes for the atom in state $|f\rangle$ (with zero photons), the atom in state $|e\rangle$ (with one photon), and the atom in state $|g\rangle$ (with two photons).
Analytical Derivation: The authors derive a set of coupled differential equations for these probability amplitudes. By solving these equations for an arbitrary incident pulse shape and then specializing to an exponential pulse (characteristic of spontaneous emission), they obtain analytical expressions for the wavefunction components.
Parameters: The study focuses on parameters relevant to superconducting transmons, characterized by a relative anharmonicity $\alpha_r = (\Omega_{ge} - \Omega_{ef})/\Omega_{ge}$ typically ranging from $-3\%$ to $-6\%$ .

Key Contributions and Results

Analytical Solution for Scattering Dynamics:
The paper provides explicit analytical expressions for the wavefunction amplitudes for an arbitrary single-photon pulse incident on an excited 3LE. This allows for a complete description of the scattering process, including the temporal evolution of single-photon and two-photon states.
Suppression of Stimulated Emission by the Third Level:
A central finding is that the third energy level significantly suppresses the stimulated emission effect in low-anharmonicity systems.
- In a standard two-level system (TLS), the probability of finding two co-propagating photons (moving in the same direction as the incident pulse) can exceed 0.5, indicating stimulated emission.
- For a three-level system with low anharmonicity (typical of transmons, $\alpha_r \approx -0.04$ ), this probability is reduced to approximately 0.5.
- The authors determine that a relative anharmonicity of $|\alpha_r| < 4\%$ is sufficient to largely suppress the stimulated emission effect. Consequently, superconducting transmons are identified as suboptimal candidates for applications relying specifically on stimulated emission, whereas more anharmonic systems (e.g., flux qubits, quantum dots) behave more like ideal two-level systems in this regard.
Photon-Photon Correlations and Bunching:
The study reveals distinct correlation effects in the reflected field that differ from the two-level case:
- Bunching in Reflected Fields: Unlike two-level systems, where the reflected field typically exhibits antibunching (photons avoiding each other), the three-level system with low anharmonicity exhibits photon bunching in the reflected field (two left-moving photons sharing the same spatial coordinate).
- Switching Mechanism: The authors demonstrate that this bunching behavior is not static; it can be "switched off" by tuning the incident photon parameters. By adjusting the central frequency and spectral width of the pulse, the system can transition from a bunching regime to a state resembling the antibunching behavior of a two-level system.
Dependence on Anharmonicity:
The results show a symmetrical dependence on the sign of the anharmonicity. As the magnitude of anharmonicity increases (moving away from equidistant levels), the behavior of the 3LE converges toward that of a regular two-level qubit. The maximum deviation from two-level behavior occurs when the levels are equidistant ( $\alpha_r = 0$ ).

Significance and Claims
The paper claims that its primary significance lies in demonstrating that the third energy level in artificial atoms cannot be neglected, even when only the two lowest levels are utilized for operation.

Theoretical Rigor: The work provides a more rigorous theoretical framework for waveguide QED involving artificial atoms by explicitly accounting for the ladder structure and low anharmonicity inherent to transmons.
Impact on Stimulated Emission: It establishes that the presence of the third level acts as a suppressor of stimulated emission in low-anharmonicity regimes, a factor that must be considered in the design of quantum amplifiers or routing devices.
Control of Correlations: The study highlights that photon-photon correlations, such as bunching in the reflected field, are intrinsic to the three-level structure but can be dynamically controlled via the incident pulse parameters.

The authors conclude that these findings provide essential insight into two-photon scattering processes in waveguide QED and emphasize the necessity of accounting for higher energy levels when modeling artificial atoms in continuous fields.

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