⚛️ quantum physics

Temporal modes of quantum states of light scattered by a two-level system

This paper presents an analytic framework for characterizing light scattered by a two-level system and demonstrates that scattering two-photon inputs can deterministically generate highly non-Gaussian, two-mode entangled NOON states with enhanced Wigner negativity.

Original authors: Yann Bouchereau, Lucas Weitzel, Valerian Thiel, Valentina Parigi, Mattia Walschaers, Nicolas Treps

Published 2026-06-30

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CC BY 4.0

Original authors: Yann Bouchereau, Lucas Weitzel, Valerian Thiel, Valentina Parigi, Mattia Walschaers, Nicolas Treps

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

The Big Picture: Turning Light into a "Quantum Switch"

Imagine you have a stream of light (photons) flowing down a one-way street (a waveguide). In the middle of this street, there is a tiny, solitary traffic light (a two-level system, or an atom). The goal of this research is to understand exactly what happens to the stream of light after it hits this traffic light.

Specifically, the scientists wanted to know: Can we use this simple setup to turn a simple, single-file line of light into a complex, entangled quantum state?

In the world of quantum computing, "simple" light states (called Gaussian states) are easy to make but not very powerful. "Complex" light states (called non-Gaussian states) are the superpowers needed for advanced quantum computers, but they are notoriously difficult to create reliably. This paper proposes a way to create them deterministically (meaning it happens every time, not just by luck).

The Setup: A One-Way Street and a Bouncer

Think of the experiment like this:

The Light: A group of photons (particles of light) traveling in a specific pattern, like a marching band playing a specific song.
The Atom: A single "bouncer" standing in the path.
The Interaction: When the light hits the bouncer, the bouncer can absorb a photon and then immediately spit it back out.

The tricky part is that light is made of waves. If two photons arrive at the bouncer at the exact same time, they might interfere with each other. The researchers wanted to map out exactly how the "song" of the light changes after passing the bouncer.

The Discovery: A New Mathematical "Recipe"

The authors developed a new mathematical recipe (an analytic formula) to predict exactly what the light looks like after it leaves the bouncer.

Old Way: Previous methods were like trying to solve a massive, messy puzzle by brute force. You had to run heavy computer simulations that took a long time and were hard to interpret.
New Way: The authors found a shortcut. They realized that the output light is just a specific combination of the input light and the "echo" left behind by the atom. It's like knowing that if you shout in a canyon, the echo is just your voice delayed and slightly distorted. Their formula lets you calculate the result instantly without needing a supercomputer.

The Main Experiment: The "Two-Photon" Magic Trick

To test their recipe, they focused on a specific scenario: sending exactly two photons into the system at the same time, both following the exact same pattern (the same "temporal mode").

They asked: Can we tune the shape of the incoming light so that the two photons come out as a perfectly entangled pair?

The Result:
Yes! They found that if you tune the width of the light pulse just right (like tuning a radio to a specific frequency), the two photons that go in as a single unit come out as a NOON state.

What is a NOON state? (The Analogy)
Imagine you have two coins.

Before: You put both coins in your left pocket.
After: The magic happens, and now the coins are in a "superposition." They are in a state where they are both in the left pocket AND both in the right pocket at the same time, but they are perfectly linked. If you check one, you instantly know where the other is.

In this experiment, the "left pocket" and "right pocket" are two different "modes" (patterns) of light. The two photons, which started in one pattern, end up split between two different patterns in a highly entangled way.

Why This Matters (According to the Paper)

Deterministic Generation: Usually, creating these special quantum states is like flipping a coin; you might get the result once in a million tries. This method is "deterministic," meaning if you set up the light correctly, the transformation happens every single time.
Wigner Negativity: The paper mentions that the resulting state has more "Wigner negativity." In simple terms, this is a mathematical way of saying the state is "more quantum" and less like a classical wave. This is a key ingredient for building powerful quantum computers.
The "Tuning" Works for Different Shapes: They tested this with two different shapes of light pulses:
- Lorentzian: A bell-shaped curve that falls off slowly.
- Gaussian: A standard bell curve.
  They found that while both worked, the Gaussian shape was slightly better at creating the perfect entangled state.

Summary

The paper presents a new, easy-to-use mathematical tool to predict how light behaves when it bounces off a single atom. Using this tool, they showed that by carefully tuning the shape of a two-photon light pulse, you can reliably transform it into a highly complex, entangled quantum state. This offers a promising, reliable path toward creating the "fuel" needed for future quantum computers.

Technical Summary: Temporal Modes of Quantum States of Light Scattered by a Two-Level System

Problem Statement
Non-Gaussian quantum states of light are essential for quantum computing, yet their deterministic generation remains a significant challenge due to the difficulty of controlling nonlinearities in physical systems. Existing methods, such as photon addition/subtraction and catalysis, rely on probabilistic measurements, creating a bottleneck for large-scale implementations. While deterministic approaches using two-level systems (TLS) coupled to bosonic modes show promise, the theoretical description of these interactions becomes increasingly complex when moving beyond monochromatic light to multimode and multiphoton scenarios. Specifically, there is a need for an analytical framework that can efficiently describe the scattering of multimode, multiphoton light by a TLS without relying solely on computationally intensive numerical techniques.

Methodology
The authors present a fully analytical treatment of the unidirectional scattering of multimode, multiphoton light by a single TLS in a chiral waveguide. The approach combines tools from multimode quantum optics with scattering theory formalism.

Theoretical Framework: The system is modeled using a Hamiltonian describing a photonic field, a TLS, and their interaction. The authors utilize "time-localized" photon operators and define input states as superpositions of creation operators acting on temporal modes $f_j(t)$ .
Scattering Matrix Derivation: By employing the input-output formalism and the known analytical expression for the scattering matrix ( $S$ -matrix) coefficients in the time domain, the authors derive an explicit expression for the output wavefunction. They express the $n$ -photon output state solely in terms of the input temporal modes.
Modal Decomposition: To analyze the structure of the scattered state, the authors employ a modal decomposition technique. For the specific case of two photons ( $n=2$ ), the output wavefunction is treated as a complex symmetric matrix and decomposed using Takagi decomposition. This yields a set of "principal modes" and associated eigenvalues, allowing for the calculation of the effective number of modes ( $M$ ) required to describe the output state.

Key Contributions

Analytic Output Wavefunction: The paper provides a compact, explicit, and computationally efficient analytical formula (Eq. 24) for the output wavefunction of $n$ photons scattered by a TLS. Unlike previous numerical methods or recursive collisional models, this expression allows for systematic studies over arbitrary input modes.
Physical Interpretation: The derived formula reveals that the output wavefunction is a sum over all possible subsets of photons that interact with the TLS. Crucially, it demonstrates that the interaction is strictly sequential; there are no terms corresponding to the simultaneous interaction of two or more photons with the TLS, a manifestation of photon anti-bunching.
Mapping to Entangled States: In the specific case of two-photon scattering, the authors demonstrate that by tuning the spectral width of the input mode relative to the atomic decay rate, the scattering process can deterministically map a single-mode two-photon input state into a two-mode entangled output state.

Results
The authors applied their method to two-photon inputs in both Lorentzian and Gaussian temporal modes:

Effective Mode Number: The effective number of modes ( $M$ ) was calculated as a function of the ratio between the input mode width ( $\gamma$ or $\sigma$ ) and the TLS decay rate ( $\Gamma$ ). $M$ exhibits two local maxima.
NOON State Generation: At specific parameter values (e.g., $\gamma \approx 5.52 \Gamma$ for Lorentzian modes), the effective number of modes approaches 2 with high precision (deviation $< 1\%$ ). At these points, the output state is well-approximated by a superposition of two orthogonal principal modes.
State Mapping: The analysis shows that a single-mode two-photon input $|2f\rangle$ is mapped to an entangled two-mode state of the form $\frac{1}{\sqrt{2}}(|2\psi_1, 0\rangle - |0, 2\psi_2\rangle)$ , which corresponds to a two-photon NOON state.
Mode Preservation: In the optimal configuration, one of the output principal modes ( $\phi_2$ ) is found to be nearly identical to the original input mode $f$ , while the other ( $\phi_1$ ) is an orthogonal mode.
Profile Comparison: While both Lorentzian and Gaussian inputs yield similar results, the Gaussian profile provides a slightly better approximation to the ideal NOON state (error $\approx 0.35\%$ vs. $0.6\%$ for Lorentzian).

Significance and Claims
The paper claims that this analytical framework offers a straightforward physical interpretation and superior computational efficiency compared to existing numerical methods, particularly for $n > 2$ photons. The primary significance lies in the demonstration that a simple, deterministic scattering setup can generate non-Gaussian states (specifically, two-photon NOON states) with high fidelity. The authors note that these output states possess greater Wigner negativity than the input states, suggesting a potential application for the deterministic generation of non-Gaussian resources for quantum computing. The work also opens the door for "mode optimization" protocols, where input modes could be tailored to maximize the fidelity of the resulting entangled states. The authors conclude that their method provides a foundation for investigating the modal content of more complex scattered states, such as those involving squeezed states.