⚛️ quantum physics

Resonance fluorescence of an artificial atom with a time-delayed coherent feedback

This paper presents the first experimental demonstration of time-delayed coherent feedback in a superconducting transmon qubit, showing how non-Markovian effects arising from significant delay times fundamentally modify the resonance fluorescence spectrum, including the generation of non-Markovian Mollow triplets.

Original authors: Ching-Yeh Chen, Gavin Crowder, Zheng-Qi Niu, Ping Yi Wen, Yen-Hsiang Lin, Jeng-Chung Chen, Zhi-Rong Lin, Franco Nori, Stephen Hughes, Io-Chun Hoi

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ching-Yeh Chen, Gavin Crowder, Zheng-Qi Niu, Ping Yi Wen, Yen-Hsiang Lin, Jeng-Chung Chen, Zhi-Rong Lin, Franco Nori, Stephen Hughes, Io-Chun Hoi

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 shouting in a large, empty canyon. Normally, your voice travels out, hits the far wall, bounces back, and you hear an echo a split second later. If you shout again before the echo returns, your new shout and the old echo mix together. Sometimes they cancel each other out (silence), and sometimes they make the sound louder.

This paper is about doing something very similar, but instead of a canyon and a human voice, the scientists used a tiny artificial atom (a superconducting circuit) and microwaves (invisible radio waves).

Here is the story of what they did, broken down into simple concepts:

1. The "Memoryless" Rule vs. The "Echo" Reality

In most physics textbooks, there's a rule called the Markovian approximation. It's like saying, "What happens next only depends on what is happening right now." It assumes the system has no memory of the past.

However, in the real world, things often have retardation effects (delays). If you shout in a canyon, the echo comes back later. When that echo returns, it interacts with your current state. This creates a non-Markovian system—a system with "memory."

The scientists wanted to see what happens when you force an artificial atom to interact with its own "echo" after a specific delay.

2. The Setup: The Artificial Atom and the Mirror

The Artificial Atom: They built a tiny circuit called a "transmon qubit." Think of this as a super-fast, super-small lightbulb that can only be "on" or "off" (a two-level system).
The Waveguide: This is a "highway" for microwaves. The atom is connected to this highway.
The Mirror: At the end of the highway, they put a mirror. When the atom emits a microwave photon, it travels down the highway, hits the mirror, and bounces back.
The Delay: The highway is long enough that the "echo" takes about 3.6 nanoseconds to return. This is a very short time, but it's roughly the same amount of time it takes for the atom to "relax" or calm down after being excited. This timing is crucial.

3. The Experiment: Singing and Listening

The scientists shined a steady microwave beam (the "pump") at the atom to make it excited. They then listened to what the atom shouted back.

Part A: The Linear Regime (Whispering)

First, they used a very weak signal, like whispering.

The Result: They found that depending on the distance to the mirror (the delay), the atom would either reflect the signal perfectly or become completely invisible to it.
The Analogy: Imagine standing in a room with a mirror. If you stand in the right spot, your reflection cancels out your voice, making you silent. If you stand in the wrong spot, your voice bounces back and makes you sound louder. They mapped out exactly where these "silent spots" and "loud spots" were.

Part B: The Non-Linear Regime (Singing Loudly)

This is the big discovery. They turned up the volume (the pump power) so the atom was screaming.

The Expected Result (The Mollow Triplet): Usually, when you shout at an atom, it sings back in a specific pattern called the Mollow Triplet. Imagine a musical chord with three notes: a central note and two side notes (one higher, one lower). This is a famous phenomenon in quantum physics.
The Surprise: Because of the time-delayed echo, the Mollow Triplet didn't look normal.
- The "Nodes": At certain frequencies, the side notes completely disappeared! It was as if the echo came back exactly out of sync with the atom's voice, canceling those specific notes out.
- New Notes: In other places, the echo interfered in a way that created new notes that shouldn't exist in a normal system.

4. Why This Matters

This isn't just about making weird sounds. It proves that we can control quantum systems using time delays rather than just changing the strength of the signal.

Memory Matters: It shows that quantum systems aren't "memoryless." If you time the feedback just right, you can trap energy, cancel out noise, or create new types of light.
Future Tech: This could help build better quantum computers and quantum networks. For example, if you want to send a single photon (a packet of light) perfectly without losing it, you can use this "echo" technique to guide it. It's like using the canyon's echo to aim your voice perfectly at a friend on the other side.

Summary

The scientists took a tiny artificial atom, made it shout into a microwave "canyon," and listened to the echo. They discovered that by timing the echo perfectly, they could make the atom's song change completely—hiding some notes and inventing new ones. This proves that time delays are a powerful tool for controlling the future of quantum technology.

1. Problem Statement

The standard model of light-matter interaction in quantum electrodynamics (QED) typically relies on the Markovian approximation, which assumes the system is "memoryless" (future evolution depends only on the current state). This approximation fails when retardation effects (time delays) are significant, such as when a system interacts with its own emitted field after a finite propagation time.

While non-Markovian effects are theoretically predicted to enable new quantum phenomena (e.g., in quantum networks and information processing), experimental realization in the nonlinear regime (strong driving) has been challenging. Specifically, observing how time-delayed coherent feedback modifies the well-known Mollow triplet (the resonance fluorescence spectrum of a driven two-level system) in a non-Markovian regime had not been previously demonstrated experimentally.

2. Methodology

Experimental Platform

System: A superconducting transmon qubit (treated as a two-level artificial atom) embedded in a 1D open transmission line waveguide.
Feedback Loop: The waveguide is terminated by a mirror (a shorted superconducting coaxial cable), creating a feedback loop.
Time Delay: The round-trip length ( $L_0$ ) of the feedback loop is approximately 0.3 meters, resulting in a delay time ( $\tau$ ) of 3.63 ns. This delay is comparable to the qubit's radiative lifetime ( $1/\Gamma$ ), satisfying the condition $\Gamma\tau > 0.1$ required to enter the non-Markovian regime.
Tunability: The qubit frequency ( $\omega_0$ ) is tuned via an external magnetic flux, allowing researchers to control the round-trip phase change ( $\phi$ ) relative to the waveguide mode, effectively moving the qubit between nodes and antinodes of the standing wave.

Theoretical Framework

Model: The authors utilize the Quantum Trajectory Discretized Waveguide (QTDW) method.
Advantage: Unlike standard master equations that fail with time delays, QTDW discretizes the waveguide into bins. This allows the non-Markovian system to be described using Markovian equations by treating the feedback as a system-level interaction rather than an instantaneous coupling.
Regimes: The study covers both:
- Linear Regime: Weak pumping ( $\Omega_L \ll \gamma_d$ ) to characterize elastic scattering and reflection coefficients.
- Nonlinear Regime: Strong resonant pumping ( $\Omega_{eff} \gg \gamma_d$ ) to observe resonance fluorescence and the Mollow triplet.

3. Key Contributions

First Experimental Observation of Non-Markovian Mollow Triplets: The paper presents the first experimental report of Mollow triplets in a system dominated by time-delayed coherent feedback, demonstrating how the spectrum deviates from standard Markovian predictions.
Demonstration of Non-Markovian Signatures in Linear Regime: The authors identified and characterized non-Markovian signatures in the linear reflection coefficient, specifically the tilt of the response circle in the complex plane, caused by feedback-induced frequency shifts and phase differences.
Control of Spectral Nodes and Antinodes: The study demonstrates that the time delay creates specific frequency conditions (nodes) where destructive interference suppresses the Mollow sidebands, and conditions (antinodes) where constructive interference creates new resonances (Fabry-Pérot modes) beyond the standard triplet.
Validation of QTDW Theory: The experimental results show excellent qualitative and quantitative agreement with the QTDW simulations, validating this theoretical approach for modeling non-Markovian waveguide QED.

4. Key Results

Linear Regime (Elastic Scattering)

Reflection Coefficient ( $r$ ): By sweeping the pump frequency and qubit frequency, the team mapped the reflection coefficient.
Non-Markovian Tilt: In the complex plane, the reflection coefficient trace forms a circle. In the non-Markovian regime, this circle is tilted. The tilt angle is determined by $(\tilde{\omega}_0 - \omega_0)\tau$ , where $\tilde{\omega}_0$ is the feedback-shifted resonance frequency. This asymmetry is a direct signature of the time delay.
Coupling Control: At the antinode ( $\phi=0$ ), the effective relaxation rate is doubled ( $\tilde{\Gamma} \approx 2\Gamma$ ), leading to perfect reflection. At the node ( $\phi=\pm\pi$ ), the qubit becomes transparent, and the mirror reflects perfectly.

Nonlinear Regime (Resonance Fluorescence)

Mollow Triplet Modification: Under strong driving, the standard Mollow triplet (central peak + two sidebands at $\pm \Omega_{eff}$ ) is observed but significantly modified.
Suppression of Sidebands (Nodes): Unlike the continuous sidebands in a Markovian system, the time delay introduces nodes where the sidebands disappear due to destructive interference. These nodes occur when the round-trip phase satisfies $\phi_{node} = (2k-1)\pi$ .
Emergence of New Resonances (Antinodes): When the delay creates constructive interference (Fabry-Pérot modes), new resonances appear in the spectrum between the standard Mollow peaks.
Asymmetry: For non-zero phase shifts (e.g., $\phi = -0.632\pi$ ), the suppression of the red and blue sidebands becomes asymmetric, and the central peak narrows due to suppressed effective relaxation.

5. Significance

Fundamental Physics: This work bridges the gap between theoretical predictions of non-Markovian dynamics and experimental reality in the nonlinear regime. It proves that time-delayed feedback can fundamentally alter the quantum optical response of an atom, creating "dressed" states that are a hybrid of the atom and the cavity modes formed by the delay loop.
Quantum Technology Applications:
- Single Photon Sources: The ability to strategically choose time delays to interfere constructively or destructively can improve the coherence and indistinguishability of on-demand single-photon sources.
- Quantum Networks: The system serves as a prototype for engineering non-Markovian reservoirs, which are essential for long-distance quantum communication and entanglement distribution.
- Cluster State Generation: The ability to manipulate emission between sequential excitations via time delays offers a pathway for generating multi-dimensional photonic cluster states.
Methodological Impact: The successful application of the QTDW method provides a robust tool for simulating and designing future quantum devices involving finite-time delays, moving beyond the limitations of the Born-Markov approximation.

In conclusion, this paper establishes a superconducting qubit-mirror system as a versatile platform for exploring non-Markovian quantum optics, demonstrating that time-delayed coherent feedback can be used to engineer exotic quantum states and modify fundamental light-matter interaction spectra.