⚛️ quantum physics

Matrix Product State Theory of Few-Photon Squeezed Pulses Interacting with a Two-Level Emitter in a Waveguide

This paper presents a numerically exact matrix product state (MPS) framework to model the interaction of few-photon squeezed pulses with a two-level emitter in a waveguide, revealing unique nonlinear population dynamics and multi-photon correlations without relying on Markov or Born approximations.

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

Published 2026-08-10

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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

Imagine the world of light not just as a beam you can see, but as a bustling city of tiny, invisible messengers called photons. Usually, we think of these messengers as behaving like a calm, predictable crowd, where everyone keeps a steady distance from their neighbors. But in the strange and wonderful realm of quantum physics, light can get "squeezed." Think of it like a crowd of people in a hallway: if you squeeze them together on the left side, they get packed tight, but on the right side, they spread out wildly. This is "squeezed light," a special state where the uncertainty of one property (like how many photons are there) is reduced, while another property (like their timing) gets a bit more chaotic. Scientists love this stuff because it's a superpower for future technologies, from building unbreakable codes for computers to creating sensors so sensitive they could feel a gravitational wave ripple through the universe.

Now, usually, when scientists study this squeezed light, they look at it as a continuous, never-ending stream, like a river flowing forever. But what happens when you take just a tiny, finite splash of this squeezed light—just a few photons—and send it zooming down a microscopic highway (a waveguide) to crash into a single, tiny atom? This is the tricky part. When you only have a few photons, the usual rules of "average" behavior break down, and the wild quantum fluctuations take over. It's like trying to predict the path of a single raindrop in a storm versus predicting the flow of the whole ocean. Understanding how these few-photon squeezed pulses interact with matter is crucial for building the next generation of quantum devices, but it's been incredibly hard to calculate because the math gets messy and complex very quickly.

This paper steps in with a clever new way to solve that puzzle. The authors, a team from Queen's University, developed a powerful computer simulation method called "Matrix Product States" (MPS) to model exactly what happens when a few-photon squeezed pulse hits a single two-level atom (a simple quantum system) inside a waveguide. Instead of making simplifying guesses that often fail for these tiny, short bursts of light, their method treats the light and the atom with perfect mathematical precision, step-by-step, in time.

What they found is a mix of the expected and the surprising. First, they confirmed that for very weak squeezed pulses (where the average number of photons is tiny, around 0.01), the interaction looks a lot like what happens when you send a pure "two-photon" packet at the atom. The atom gets excited, and the light scatters in a way that matches our existing theories for two-photon collisions. However, the real magic happens when they looked at the "hidden" correlations that only squeezed light possesses. They discovered that squeezed pulses create unique quantum fingerprints—specific patterns in how the photons relate to each other over time—that simply cannot exist in normal light or even in pure two-photon packets. For instance, they measured a specific type of correlation that showed a "bird-like" pattern after the light bounced off the atom, a signature that is unique to the squeezed nature of the input.

Furthermore, they explored how the "color" or spectrum of the squeezed light changes after the crash. They found that the atom doesn't just scatter the light; it actually reshapes the squeezed spectrum, flipping its signs and sharpening its peaks depending on how long the pulse was. This proves that the interaction is deeply nonlinear and depends heavily on the duration of the pulse. The authors emphasize that their results come from these highly accurate simulations, which allow them to explore regimes that were previously too difficult to calculate. By showing that squeezed pulses carry rich, unique nonlinear interactions even with very few photons, this work opens the door to designing better quantum sensors and computers that can harness these subtle, squeezed states without needing massive, continuous beams of light.

Technical Summary: Matrix Product State Theory of Few-Photon Squeezed Pulses Interacting with a Two-Level Emitter in a Waveguide

Problem Statement
While squeezed light is a fundamental non-classical state with established applications in quantum sensing and metrology, its role as a fundamental quantum input for nonlinear light-matter interactions in the few-photon regime remains underexplored. Existing theoretical research largely focuses on the continuous wave (CW) limit or large photon numbers where mean-field approximations suffice. In contrast, the few-photon regime is dominated by quantum fluctuations, requiring an exact Fock state basis to capture photon-photon correlations. Furthermore, treating finite-duration squeezed pulses interacting with a two-level system (TLS) exactly is computationally complex, particularly when avoiding Markov and Born approximations. Current scattering matrix approaches struggle to extend beyond simple two-photon Fock states or higher quanta, and few works have modeled squeezed few-photon pulses using tensor network techniques.

Methodology
The authors develop a numerically exact Matrix Product State (MPS) framework to model finite-duration squeezed vacuum pulses interacting with a chiral two-level emitter in a one-dimensional waveguide.

Hamiltonian and Discretization: The system is defined by a waveguide Hamiltonian, a TLS Hamiltonian, and an interaction term under the rotating wave approximation. The electromagnetic field is discretized into time bins, transforming frequency-dependent operators into time-bin noise operators. This allows the Hilbert space growth to be controlled, enabling numerically tractable solutions.
State Construction: The initial squeezed state is constructed directly within the discretized waveguide Hilbert space as a coherent superposition of Fock states. The squeezing operator is applied to the vacuum, and the resulting state is expanded in the Fock basis ( $|2n\rangle$ ). The authors explicitly handle the entanglement of time bins required to represent photon pairs created at different parts of the pulse, encoding squeezing information into edge tensors ( $\alpha$ and $\Omega$ ) of the MPS.
Observables: The framework calculates local observables (TLS population, transmitted flux) and, crucially, multi-time quantum correlation functions. This includes first-order correlations ( $\langle b^\dagger(t)b(t+t')\rangle$ , $\langle b(t)b(t+t')\rangle$ ), second-order correlations ( $\langle b^\dagger b^\dagger b b \rangle$ , $\langle b b b^\dagger b^\dagger \rangle$ ), and the squeezing spectrum $S(\omega, \phi)$ .
Validation: The MPS results for population dynamics are benchmarked against an alternative master equation solution developed for squeezed wave packets, showing consistency across different squeezing parameters.

Key Results
The study simulates a weak squeezed Gaussian pulse ( $\zeta = 0.1$ , mean photon number $\bar{n} \approx 0.01$ ) interacting with a chiral TLS.

Population Dynamics: The TLS population and transmitted flux for the weak squeezed pulse are dominated by the two-photon subspace component. When normalized, the dynamics closely resemble those of a pure two-photon Fock state pulse, though the squeezed pulse exhibits nonlinear interactions driven by multi-photon contributions.
Unique Correlation Signatures:
- The first-order correlation $\langle b^\dagger(t)b(t+t')\rangle$ behaves similarly to the 2-photon Fock case.
- However, the correlation $\langle b(t)b(t+t')\rangle$ , which is identically zero for Fock and coherent states, yields finite values for the squeezed pulse. Upon interaction with the TLS, this correlation develops a "bird-like" pattern, a signature previously associated with second-order correlations of 2-photon Fock scattering, but here appearing in a first-order correlation function unique to squeezing.
- Similarly, the four-photon correlation $\langle b(t)b(t+t')b(t+t')b(t)\rangle$ yields finite values for the squeezed pulse, revealing a unique quantum feature of the 4-photon component even in weak pulses.
Squeezing Spectra: The study analyzes the squeezing spectra before and after interaction. The interaction with the TLS causes a sign flip in the spectral values and narrows the spectrum. The temporal length of the pulse dictates the spectral width; longer pulses result in narrower spectra where the sign flip becomes total and the central peak sharpens.

Significance and Claims
The paper claims to provide the first numerically exact MPS framework for modeling squeezed few-photon Fock pulses interacting with a TLS in a waveguide, extending beyond the CW limit.

Theoretical Advancement: The method allows for the modeling of arbitrary bandwidth squeezing without Markov or Born approximations, providing a complete temporal envelope description of the scattering process.
Physical Insight: The work demonstrates that even weak squeezed pulses contain rich nonlinear light-matter interactions captured by specific correlation functions ( $\langle bb \rangle$ and $\langle bbbb \rangle$ ) that are inaccessible to standard Fock-state descriptions.
Scalability: The authors assert that the MPS approach is easily extensible to waveguide systems with multiple emitters and time-delayed feedback, offering a route to study non-Markovian regimes in nonlinear quantum optics that are currently inaccessible to analytical techniques.
Experimental Relevance: The parameters used are consistent with existing experimental platforms, including semiconductor waveguides and circuit-QED systems, suggesting the predictions are accessible with current quantum technologies.

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