⚛️ quantum physics

Quantum dynamics of few-photon pulsed waveguide-QED with a single artificial atom: frequency-dependent scattering theory and time-dependent matrix product states

This paper presents a comparative quantum dynamical study of pulsed few-photon scattering from a single artificial atom in a waveguide QED system, demonstrating excellent agreement between frequency-dependent scattering matrix and time-dependent matrix product state (MPS) methods while highlighting the latter's superior capability to simulate higher-order nonlinear dynamics up to eight-photon excitations.

Original authors: Sofia Arranz Regidor, Matthew Kozma, Stephen Hughes

Published 2026-03-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sofia Arranz Regidor, Matthew Kozma, Stephen Hughes

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: A Quantum Traffic Jam

Imagine a very narrow, one-lane highway (the waveguide) where cars are not made of metal, but are tiny packets of light called photons. On the side of this road, there is a single, very picky toll booth (the artificial atom or qubit).

When a car (photon) drives by, it might stop at the toll booth, get excited, and then either drive past it or bounce back. The scientists in this paper wanted to understand exactly what happens when one car, two cars, or even eight cars drive by this toll booth at the same time.

They didn't just guess; they used two different "super-computers" (simulation methods) to predict the outcome and checked if they agreed with each other.

The Two "Super-Computers"

The paper compares two different ways to solve this traffic puzzle. Think of them as two different strategies for predicting a traffic jam:

1. The "Frequency Map" Method (Scattering Matrix Theory)

Imagine you are a traffic analyst who only looks at the speed of the cars, not their exact position on the road.

How it works: You take the pulse of light (the cars) and break it down into a spectrum of speeds (frequencies). You ask: "If a car comes in at this speed, how likely is it to pass through? How likely is it to bounce back?"
The Analogy: It's like looking at a weather map. You know the wind patterns (frequencies) and can predict the storm's path. It's great for understanding the "shape" of the interaction and separating the "linear" effects (cars just passing through) from the "non-linear" effects (cars crashing into each other and changing direction).
The Limit: It gets very messy if you have too many cars (photons) at once. The math becomes a tangled knot that is hard to untangle.

2. The "Time-Lapse Puzzle" Method (Matrix Product States - MPS)

Imagine you are a security guard watching a time-lapse video of the highway, broken into tiny frames (time bins).

How it works: Instead of looking at speeds, you look at the road second-by-second. You build a giant digital puzzle where each piece represents a tiny slice of time. You connect these pieces together to see how the cars move, stop, and interact with the toll booth in real-time.
The Analogy: It's like solving a Sudoku puzzle where the numbers represent photons. As you fill in the grid, you can see exactly when the toll booth gets excited and when the cars leave.
The Superpower: This method is incredibly robust. It can handle 8 cars (photons) at once without the math exploding. It's like a super-efficient video game engine that can render complex scenes without crashing.

What They Discovered

1. The "Bird" Shape (Two-Photon Interaction)

When they sent two photons through the system, they looked at how likely it was to find a photon at time $T$ and another at time $T + \text{delay}$ .

The Result: They saw a pattern that looked like a bird (specifically, a "bird-like" shape on a graph).
The Meaning: This "bird" is a signature of quantum weirdness. It shows that the two photons aren't just passing by independently; they are "talking" to each other through the toll booth. One photon changes the path of the other. This shape has been seen in real experiments, and the scientists' simulations matched it perfectly.

2. The "Rabi Dance" (Eight Photons)

This is where the "Time-Lapse Puzzle" (MPS) really shined. They simulated sending 8 photons at once.

The Result: The toll booth didn't just sit there; it started dancing. The population of the excited atom oscillated up and down rapidly, like a pendulum.
The Analogy: In classical physics, if you push a swing hard, it goes higher. But here, the "push" (the light pulse) has no average force (it's a quantum field with zero average electric field), yet the toll booth still starts swinging wildly. It's like a ghost pushing a swing, and the swing going crazy. This is a purely quantum effect called Rabi oscillations, and seeing it happen with 8 photons is a big deal.

3. The "Mirror" Effect (Symmetric vs. Chiral)

They tested two types of toll booths:

Symmetric: The booth has two doors (left and right). Cars can bounce back or go forward.
Chiral (One-way): The booth has a one-way sign. Cars can only go forward.
The Discovery: In the one-way (chiral) case, the toll booth gets much more excited. It's harder for the light to escape, so it gets trapped and interacts more strongly. In the symmetric case, the light splits, and the interaction is weaker.

Why Does This Matter?

Think of this research as building the blueprint for a quantum internet.

Current Internet: Uses light to send data, but the light beams don't really interact with each other. They just pass through.
Future Quantum Internet: We need light beams to interact (like cars crashing or merging) to process information. We need to know exactly how to control these interactions.

This paper proves that we have two powerful tools to design these systems:

Scattering Theory is great for understanding the "rules of the road" (frequencies and shapes).
MPS is the ultimate tool for simulating complex traffic jams with many cars (high photon numbers).

By showing that both methods agree perfectly on simple cases (1 and 2 photons), the scientists have validated their tools. Now, they can use the "Time-Lapse Puzzle" (MPS) to design complex quantum devices with many photons, paving the way for new technologies in quantum computing and secure communication.

The Takeaway

The scientists successfully mapped out how light behaves when it hits a single quantum object, using two different mathematical lenses. They confirmed that their models are accurate and showed that with enough photons, light can make a quantum atom "dance" in ways that look like classical physics but are actually deeply quantum. It's a step toward building machines that use light to think.

1. Problem Statement

The paper addresses the challenge of simulating the quantum dynamics of few-photon pulses interacting with a single artificial atom (a two-level system or TLS/qubit) coupled to a one-dimensional waveguide. While waveguide Quantum Electrodynamics (waveguide-QED) is a promising platform for quantum nonlinearities, existing theoretical methods face limitations:

Green-function/Master Equation approaches: Often rely on weak excitation approximations (WEA) or trace out waveguide modes, failing to capture strong interactions, finite populations, or photon-matter entanglement.
Scattering Matrix (S-matrix) Theory: Powerful for few-photon processes but traditionally assumes continuous-wave (CW) inputs or long-time asymptotic limits. It struggles to directly calculate time-dependent population dynamics during finite pulse interactions and becomes computationally intractable for high photon numbers ( $N > 2$ ).
Matrix Product States (MPS): A robust method for handling time-dependent dynamics and retardation effects, but its application to specific few-photon scattering problems with arbitrary pulse envelopes requires careful formulation and comparison with established analytical methods.

The authors aim to bridge these gaps by directly comparing and contrasting frequency-dependent S-matrix theory and time-dependent MPS for pulsed few-photon excitations, specifically focusing on extracting time-dependent observables and extending simulations to higher photon numbers.

2. Methodology

The study employs two distinct theoretical frameworks to model a TLS coupled to an open waveguide (symmetrically or chirally):

A. Scattering Matrix Theory (Frequency Domain)

Derivation: The authors derive S-matrices for single-photon and two-photon Fock state inputs with arbitrary temporal envelopes.
Time-Dependent Extraction: They demonstrate how to convert frequency-domain scattering coefficients into time-dependent observables.
- Output States: Constructed as superpositions of transmitted and reflected Fock states using frequency integrals of the pulse spectrum and scattering coefficients.
- Correlation Functions: First-order ( $G^{(1)}$ ) and second-order ( $G^{(2)}$ ) correlation functions are calculated by integrating over the pulse spectrum.
- Population Dynamics: A novel contribution is the derivation of the TLS population ( $n_{TLS}$ ) using photon conservation laws. By calculating the integrated photon flux of the input pulse and subtracting the transmitted and reflected fluxes, the excited state population is extracted: $n_{TLS}(t) = N_{total} - N_{wg}(t)$ .
Linear vs. Nonlinear: The method explicitly separates linear and nonlinear contributions in the two-photon subspace, allowing for the analysis of pulse envelope effects on interference.

B. Matrix Product States (Time Domain)

Discretization: The waveguide is discretized into time bins, converting the continuous field into a tensor network.
Input State: A general $N$ -photon Fock state pulse is encoded as a Matrix Product State (MPS) using quantum noise operators acting on time bins.
Evolution: The system evolves via a time-evolution operator (MPO) applied to the initial MPS.
Observables: Correlation functions and populations are extracted directly from the evolved tensor network. The authors utilized the open-source Python package QwaveMPS for these calculations.

3. Key Contributions

Direct Method Comparison: The paper provides a rigorous, side-by-side comparison of S-matrix theory and MPS for pulsed inputs. It proves that both methods yield numerically exact and identical results for 1-photon and 2-photon pulses in both symmetric and chiral coupling regimes.
Time-Dependent Population Extraction from S-Matrix: The authors present a specific formalism to extract the time-dependent population of the TLS using only photon flux data derived from scattering theory, a capability often considered difficult or impossible with standard S-matrix approaches.
Pulse Envelope Analysis: The study highlights the critical role of pulse duration (bandwidth) in nonlinear interactions. It demonstrates that for finite pulses, the interaction is a convolution of the pulse shape and scattering coefficients, rather than a simple multiplication, leading to complex interference patterns in the frequency domain.
Scalability to High Photon Numbers: The paper showcases the superior scalability of MPS. While S-matrix theory becomes intractable for $N > 2$ , the authors successfully simulate interactions with up to 8-photon Fock-state pulses, revealing complex nonlinear dynamics.

4. Key Results

Validation: For 1-photon and 2-photon Gaussian pulses, the first-order correlation functions ( $G^{(1)}$ ) and second-order correlation functions ( $G^{(2)}$ ) calculated via S-matrix and MPS are in perfect agreement.
Two-Photon Correlation Shape: The simulations reproduce the characteristic "bird-like" shape in the two-time second-order correlation function ( $G^{(2)}$ ), a feature recently observed in experiments, confirming the models' accuracy.
Symmetric vs. Chiral Coupling:
- Symmetric: Shows both transmission and reflection. The reflected flux is found to be exactly half the TLS population ( $n_L = n_{TLS}/2$ ) due to the vacuum input on the left channel.
- Chiral: Shows only transmission. The TLS population reaches higher maximums, and the dynamics are faster.
Pulse Duration Effects:
- Short pulses (broad bandwidth) show strong frequency-domain interference.
- Long pulses (narrow bandwidth) result in weaker nonlinear interactions and distinct interference patterns in the real and imaginary parts of the nonlinear scattering terms.
High-Photon Dynamics ( $N=1$ to $8$):
- Symmetric Case: As photon number increases, the TLS population increases, and stimulated emission accelerates the decay.
- Chiral Case: The TLS population maximum actually decreases slightly as $N$ increases due to rapid stimulated emission. A distinct peak splitting in the population dynamics emerges for $N \geq 4$ , resembling classical Rabi oscillations but occurring with quantum fields that have a vanishing electric field expectation value.

5. Significance

This work is significant for several reasons:

Methodological Benchmark: It establishes a gold standard for simulating waveguide-QED by validating two powerful, complementary methods. It clarifies the limitations of S-matrix theory (difficulty with time-dependence and high $N$ ) and the strengths of MPS (scalability and direct time-domain access).
Experimental Relevance: The ability to model finite-duration pulses and extract time-dependent populations is crucial for interpreting recent experiments with superconducting qubits and semiconductor quantum dots, where pulse shaping is a key control parameter.
Nonlinear Quantum Optics: The demonstration of "Rabi-like" oscillations with $N$ -photon Fock states (where $\langle E \rangle = 0$ ) deepens the understanding of quantum nonlinearities. It shows that strong interactions can be driven purely by photon number statistics rather than classical field amplitude.
Tool Development: The use and validation of the QwaveMPS package provide the community with a robust, open-source tool for simulating non-Markovian waveguide-QED systems with retardation and high photon numbers.

In conclusion, the paper successfully unifies frequency-domain scattering theory with time-domain tensor network methods, providing a comprehensive toolkit for analyzing and designing quantum photonic devices operating in the few-photon regime.