⚛️ quantum physics

Nonlinear Optics Mediated by Chiral Waveguide QED: Generation of Momentum-anticorrelated Photon Pairs

This paper demonstrates that chirally coupled transmons in a waveguide can generate momentum-anticorrelated photon pairs through nonlinear two-photon processes, where the interplay between waveguide and local driving allows for precise engineering of the transmitted field's quantum statistics across anti-bunched, coherent, and bunched regimes.

Original authors: Wenqi Tong, Spencer J Walsh, H. Alaeian, F. Robicheaux

Published 2026-07-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Wenqi Tong, Spencer J Walsh, H. Alaeian, F. Robicheaux

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 as a smooth, continuous beam like a laser pointer, but as a stream of tiny, discrete particles called photons. In the realm of quantum physics, scientists are constantly trying to figure out how to make these photons talk to each other, dance together, or even change their personalities. Usually, light is very polite and predictable; if you shine a steady beam through a window, the light that comes out is just the same steady beam. But what if you could make the light "messy" or "bunchy," where photons clump together like a crowd at a concert, or "anti-bunchy," where they keep their distance like shy introverts? This is the playground of nonlinear optics, a field where light interacts with matter so strongly that it changes its own behavior.

To do this, researchers often use "artificial atoms"—tiny, man-made quantum systems that act like the atoms in a star but are built on a computer chip. A special trick called chirality (from the Greek word for "hand") is used here. Think of it like a one-way street for light. In a normal world, light can travel left or right and bounce back and forth. In a chiral system, the light is forced to only move in one direction, like a train on a single-track loop that can never go backward. This one-way rule creates a unique environment where light and matter can have very strange, one-sided conversations. Scientists care about this because if we can control how photons group together, we might build better quantum computers or super-secure communication networks that use light instead of wires.

Now, let's look at what a team of researchers from Purdue University discovered in this specific playground. They set up a simulation involving a chain of these artificial atoms (called transmons) sitting next to a one-way highway for light (a waveguide). They wanted to see what happens when they shine a steady, calm beam of light into this system.

Here is the magic trick they found: When a single artificial atom sits on this one-way highway, it acts like a chaotic DJ. It takes the calm, orderly stream of incoming light and scrambles it. The light that comes out the other side loses its smooth, steady rhythm. Instead, the photons start "bunching" together, clumping up in groups. The researchers call this a "loss of coherence." It's as if the orderly line of people walking into a room suddenly turns into a chaotic mosh pit as they exit.

But here is the plot twist: If they add a second artificial atom right behind the first one, something incredible happens. The second atom acts like a "undo" button. It takes that chaotic, clumpy mess of light and magically restores it to a calm, orderly stream again. The light leaves the system looking just as perfect as it did when it entered, as if the chaos never happened. This happens regardless of how bright the light is, which is a very surprising result.

To understand why this happens, the team ran deep, full-quantum simulations (essentially solving the math for every single particle involved). They discovered that the "chaos" created by the first atom isn't just random noise. It comes from a specific two-step dance. When two independent photons from the incoming light hit the first atom, the atom grabs them and turns them into a special pair. These two new photons are "momentum-anticorrelated," which is a fancy way of saying they are linked in a way that makes them move in opposite directions relative to the original flow. Because this new pair is so different from the original light, it can't mix back in with the calm stream, creating the "incoherent" mess.

However, when the second atom is added, it recognizes this special pair and cancels it out, turning the light back into its original, calm state. The researchers also found that the light in this "bunched" state is truly weird—it behaves in ways that classical physics can't explain, showing "non-classical" traits like negative probabilities in its mathematical description (a sign of true quantum weirdness).

Finally, the team showed that they could control this whole process like a volume knob. By adding a second, local light source that they could tune independently (changing its strength and timing), they could make the output light do anything they wanted. They could make the photons:

Anti-bunch: Keep their distance (perfectly spaced out).
Be coherent: Stay calm and orderly.
Bunch: Clump together in groups.
Super-bunch: Clump together so intensely that the probability of finding two photons at once becomes astronomically high (reaching values like $2.5 \times 10^7$ in their simulations).

In short, this paper suggests that by using a simple one-way street for light and a couple of artificial atoms, we can turn a boring, steady beam of light into a dynamic, shape-shifting stream of particles. We can make them dance, separate, or clump together at will, and even reverse the chaos if we add just one more atom. While these results are currently based on computer simulations and theoretical models, they offer a promising blueprint for engineering the future of quantum light.

Technical Summary: Nonlinear Optics Mediated by Chiral Waveguide QED

Problem Statement
The paper investigates the interaction between classical and quantum light and a chain of superconducting transmons chirally coupled to a one-dimensional (1D) waveguide. While chiral quantum optics has been explored for deterministic state transfer and non-reciprocal scattering, the specific effect of chirality on the statistical properties of transmitted light—particularly the generation of incoherent components and photon correlations—remains less understood. The authors aim to characterize the transition of light from a coherent state (Poissonian statistics, long-range coherence) to an incoherent state (bunched statistics, spatially localized coherence) after passing through chiral emitters, and to understand the mechanism by which a second chiral emitter can restore the original coherence.

Methodology
The study employs a dual-methodological approach combining semi-classical and full-quantum simulations:

System Setup: The system consists of $N$ transmons (modeled as two-level systems) coupled to a 1D waveguide. The coupling is chiral, meaning the transmons interact only with right-propagating photons. The system is driven by a continuous-wave field through the waveguide ( $\Omega_{wg}$ ) and, in later sections, by local side-port drives ( $\Omega_d$ ) with tunable amplitude and phase.
Semi-Classical Simulations: Using the Born-Markov approximation, the photon field is traced out, and the dynamics are governed by a master equation for the reduced density operator of the transmons. The transmitted field is decomposed into coherent ( $T_{coh}$ ) and incoherent ( $T_{inc}$ ) components using input-output theory. The coherent component interferes with the incident field, while the incoherent component does not.
Full-Quantum Calculations: To uncover the microscopic origin of the incoherent light, the authors perform wavefunction-based calculations in the weak-driving limit. The system state includes the transmons and a two-photon wavepacket propagating in the waveguide. The dynamics are solved via the Schrödinger equation. The incoherent component is isolated by orthogonalizing the final wavefunction against the free-propagating input wavepacket. This allows for the direct calculation of two-photon probability densities, power spectral densities, and second-order correlation functions ( $g^{(2)}$ ).

Key Contributions and Results

Loss and Revival of Coherence:
- Single Transmon: A single chiral transmon converts incident coherent light (with long-range coherence and Poissonian statistics) into an incoherent field characterized by spatially localized coherence and photon bunching ( $g^{(2)}(0) > 1$ ). At a specific Rabi frequency ( $\Omega = \frac{\sqrt{2}}{2}\Gamma$ ), the coherent transmission vanishes completely ( $T_{coh}=0$ ), resulting in $T_{inc}=1$ .
- Two Transmons: Remarkably, a second chiral transmon placed downstream completely reverses this effect. Regardless of the incident power, the two-transmon system restores the light to a coherent state ( $T_{inc} \to 0$ ). This is attributed to the formation of a pure "dark" steady state (a dimer) where the emitters are effectively decoupled from the waveguide field.
Mechanism of Incoherent Generation (Two-Photon Process):
- Full-quantum calculations in the weak-driving limit reveal that the incoherent light generated by a single transmon arises primarily from a two-photon process.
- The transmon converts two independent incoming photons into a photon pair with opposite momentum shifts ( $k - k_0 = -(k' - k_0)$ ).
- The wavefunction of this generated photon pair is orthogonal to the incoming wavepacket, explaining the loss of interference (coherence).
- The second transmon suppresses the population of this specific entangled photon pair, thereby restoring the coherent transmission.
- The generated field exhibits non-classicality, evidenced by the negativity in the one-photon Wigner function, which the authors attribute mainly to the two-photon process in the weak-driving regime.
Engineering Quantum Statistics via Local Driving:
- By introducing a local drive ( $\Omega_d$ ) with independently tunable amplitude and phase alongside the waveguide drive ( $\Omega_{wg}$ ), the authors demonstrate control over the photon statistics of the transmitted field.
- By adjusting the ratio and phase between $\Omega_{wg}$ $Ω_{w g}$ and $\Omega_d$ $Ω_{d}$ while keeping the total Rabi frequency fixed, the system can be tuned to produce:
  - Anti-bunched light ( $g^{(2)}(0) = 0$ ) when driven solely by the local port.
  - Coherent light ( $g^{(2)}(0) = 1$ ) when the drives cancel out at the transmon.
  - Bunched light ( $g^{(2)}(0) \approx 9$ ) with standard waveguide driving.
  - Super-bunched light ( $g^{(2)}(0) \to \infty$ ) when the drives are tuned to suppress coherent transmission entirely while maintaining a non-zero total drive.

Significance
The paper claims that these results provide a deeper understanding of nonlinear quantum optics mediated by chiral waveguide QED. Specifically, it identifies the microscopic two-photon mechanism responsible for the "loss of coherence" in chiral systems and explains the "revival of coherence" via the suppression of entangled photon pairs by a second emitter. Furthermore, the work demonstrates a practical method for engineering the quantum statistics of transmitted light (spanning anti-bunched to super-bunched regimes) through the interplay of waveguide and local driving fields, offering a new degree of freedom for manipulating quantum light in superconducting circuits. The authors note that while the current full-quantum analysis is limited to the weak-driving regime, the findings suggest that the reversal mechanism holds even in strongly driven scenarios, though a full description of higher-order photon processes in that regime requires further methodological development.

Technical Summary: Nonlinear Optics Mediated by Chiral Waveguide QED

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