🔬 optics

Photon pairs, squeezed light and the quantum wave mixing effect in a cascaded qubit system

This paper theoretically demonstrates that in a cascaded two-qubit waveguide-QED system, the resonance fluorescence of a driven source qubit can effectively act as broadband squeezed light on a probe qubit, leading to a quantum wave mixing spectrum with suppressed odd-photon sidebands that serves as a probe for the incident field's nonclassical photon statistics.

Original authors: R. D. Ivanovskikh, W. V. Pogosov, A. A. Elistratov, S. V. Remizov, A. Yu. Dmitriev, T. R. Sabirov, A. V. Vasenin, S. A. Gunin, O. V. Astafiev

Published 2026-04-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: R. D. Ivanovskikh, W. V. Pogosov, A. A. Elistratov, S. V. Remizov, A. Yu. Dmitriev, T. R. Sabirov, A. V. Vasenin, S. A. Gunin, O. V. Astafiev

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 Dance Floor

Imagine a tiny, microscopic dance floor where two "dancers" (superconducting qubits) live. These aren't human dancers, but artificial atoms that can exist in quantum states.

The Source (Dancer A): This dancer is being spun around wildly by a strong, rhythmic music track (a strong laser/microwave drive). Because they are spinning so fast, they aren't just moving randomly; they are throwing out "confetti" (photons) in a very specific, synchronized pattern.
The Probe (Dancer B): This dancer is standing nearby, listening to a gentle, steady beat from a different speaker (a weak coherent tone). However, they are also trying to dance to the confetti flying off Dancer A.

The paper is about what happens when Dancer B tries to mix their own steady beat with the chaotic, yet strangely organized, confetti coming from Dancer A.

The Problem: The "Fluorescence Triplet"

When Dancer A spins fast, they emit light that looks like a three-part sandwich (scientists call this a "fluorescence triplet"):

The Center: A bright, steady beam of light (like a laser).
The Sides: Two fuzzy, blurry bands of light on the left and right.

Usually, if you mix a steady laser with a fuzzy light, you get a messy mix of frequencies. But the researchers discovered something weird happens when Dancer A spins really fast.

The Magic Trick: The "Invisible Filter"

The researchers found that if Dancer A spins fast enough, the Center of the sandwich (the steady beam) disappears. It gets suppressed.

What's left? Just the fuzzy sidebands. But here is the kicker: The confetti isn't flying out randomly anymore. It is flying out in pairs.

Think of it like this:

Normal Light: Imagine a machine gun firing bullets one by one at random times.
This Special Light: Imagine a machine gun that only fires two bullets at exactly the same time, every single time. You never get one bullet alone; you always get a pair.

In physics terms, this is called Squeezed Light or Correlated Photon Pairs. The two photons are "entangled" in their timing and energy.

The Discovery: The "Odd-Even" Rule

When Dancer B (the Probe) tries to mix this "pair-only" light with their own steady beat, a strange rule kicks in.

Imagine you are trying to build a tower using blocks.

Normal Light: You can add 1 block, 2 blocks, 3 blocks, or 4 blocks. You can build any height.
This Special "Pair" Light: Because the light only comes in pairs, you can only add 2 blocks at a time. You can build a tower of 2, 4, 6, or 8 blocks. But you cannot build a tower of 1, 3, 5, or 7 blocks.

The paper shows that the "spectrum" (the list of frequencies Dancer B emits) follows this rule perfectly.

The peaks that represent "odd" numbers of photons (1, 3, 5) vanish completely.
The peaks that represent "even" numbers of photons (2, 4, 6) stay strong.

This is called a Selection Rule. It's like a bouncer at a club who only lets in people wearing matching pairs of shoes. If you show up with just one shoe (an odd number), you don't get in.

Why Does This Matter?

The authors didn't just guess this; they built a mathematical model (a "Master Equation") to prove it. They showed that the messy, complex light coming from the spinning Dancer A acts exactly like a theoretical "Squeezed Light" source, even though it was created by a simple spinning atom.

The Takeaway:
By looking at the "spectrum" (the list of frequencies) of the second dancer, we can tell exactly how the first dancer is behaving.

If we see "odd" peaks, the light is normal.
If the "odd" peaks disappear and only "even" peaks remain, we know the light is made of correlated pairs.

The Real-World Application

This is a big deal for quantum computing and sensing.

Detecting the Invisible: This setup acts like a super-sensitive detector. You don't need to catch the photons to know they are paired; you just need to listen to the "music" (the spectrum) of the second qubit.
New Light Sources: It suggests we can use a simple, driven qubit as a factory to produce "squeezed light" (a special type of quantum light used for ultra-precise measurements) without needing complex, bulky equipment.

In Summary:
The paper describes a quantum experiment where a fast-spinning atom acts like a machine that only shoots light in pairs. When a second atom tries to mix this light with its own signal, it creates a pattern where "odd" frequencies disappear. This proves that the light is special (squeezed/correlated) and gives scientists a new, easy way to detect these quantum properties just by looking at the sound of the light.

1. Problem Statement

The paper addresses the challenge of characterizing nonclassical light fields, specifically those generated by a strongly driven superconducting qubit, using a second "probe" qubit in a cascaded waveguide-QED setup.

Context: In Quantum Wave Mixing (QWM), a qubit driven by two fields (a coherent tone and a source field) scatters radiation into sidebands. The amplitudes of these sidebands encode the photon statistics of the incident fields.
Gap: While QWM with coherent and squeezed light (generated by parametric amplifiers) has been studied, the specific case where the nonclassical field is the resonance fluorescence of a strongly driven source qubit requires a deeper theoretical understanding.
Specific Question: How does the QWM spectrum of a probe qubit change when the incident field is the resonance fluorescence of a source qubit driven so strongly that its coherent (Rayleigh) component is suppressed? The authors aim to explain the observed selection rules (suppression of specific sidebands) and link them to the underlying photon correlations (pairing) in the source emission.

2. Methodology

The authors employ a combination of analytical derivation and numerical simulation:

Analytical Framework:
- Source Characterization: They start with the Lindblad master equation for a strongly driven source qubit. Using the dressed-state basis, they derive the normal and anomalous two-time correlation functions of the source fluorescence.
- Identification of Squeezing: They demonstrate that in the strong-drive limit ( $\Omega_s \gg \Delta$ ), the coherent Rayleigh component vanishes, and the field exhibits non-vanishing anomalous correlators ( $\langle a a \rangle \neq 0$ ). This indicates the field behaves as broadband squeezed light (or a source of correlated photon pairs) with total energy conservation $2\omega_s$ .
- Effective Master Equation: For the probe qubit, they derive an effective master equation. By treating the source field as a reservoir with specific correlation properties (characterized by parameters $N$ and $M$ ), they show the probe dynamics are equivalent to a qubit driven by a coherent tone and squeezed white noise.
- Selection Rules: They solve the equations of motion for the probe averages ( $\langle \sigma_- \rangle$ and $\langle \sigma_z \rangle$ ) to predict the QWM spectrum.
Numerical Simulation:
- They simulate the full cascaded two-qubit system using a set of coupled differential equations for the density matrix elements and correlation functions (including cross-correlations between source and probe).
- They vary the ratio of radiative decay rates ( $\gamma_s / \gamma_{pr}$ ) to test the validity of the analytical approximations (specifically the Born approximation and the narrow-filter limit).

3. Key Contributions

Theoretical Equivalence: The paper establishes a rigorous theoretical equivalence between a probe qubit driven by the resonance fluorescence of a strongly driven source qubit and a probe qubit driven by a coherent tone plus broadband squeezed light.
Origin of Selection Rules: The authors identify that the suppression of the coherent Rayleigh peak in the source spectrum leads to a regime where the incident field is dominated by correlated photon pairs. This results in a strict selection rule for the QWM spectrum: sidebands associated with processes involving an odd number of photons from the source field are strongly suppressed.
Analytical Derivation of Correlators: They explicitly calculate the normal and anomalous correlators for the fluorescence triplet in the dressed-state picture, showing how the anomalous correlator $M$ becomes non-zero and approaches unity in the strong-drive, narrow-filter limit.
Validation of Regimes: The work clarifies the conditions under which the probe acts as a narrow-band filter for the source, effectively detecting the squeezed nature of the radiation, versus the wide-filter limit where correlations vanish.

4. Results

Spectral Asymmetry: Numerical simulations confirm that as the source drive strength increases ( $\Omega_s/\gamma_s \gg 1$ $Ω_{s} / γ_{s} ≫ 1$ ), the QWM spectrum exhibits a pronounced even/odd asymmetry.
- Suppressed Peaks: Sidebands at frequencies corresponding to odd photon numbers from the source (e.g., $-5\delta\omega, -1\delta\omega, +3\delta\omega$ ) are suppressed.
- Surviving Peaks: Peaks corresponding to even photon numbers from the source remain and become independent of the source drive strength.
Agreement with Theory: The analytical solution derived from the effective master equation (assuming squeezed light input) matches the numerical simulation of the full cascaded system perfectly in the regime where $\gamma_{pr} \ll \gamma_s$ (narrow filter limit).
Photon Statistics: The results confirm that the probe qubit effectively detects photon bunching and pair correlations. In the narrow-filter limit, the intensity correlation function $g^{(2)}$ reaches a maximum of 2, confirming the emission of correlated pairs.
Decay Rate Dependence: The study shows that the selection rule is most distinct when the probe decay rate is much smaller than the source decay rate ( $\gamma_{pr} \ll \gamma_s$ ). In the opposite limit ( $\gamma_{pr} \gg \gamma_s$ ), the filter is too broad to resolve the correlations, and the selection rule disappears.

5. Significance

Nonclassical Field Probing: The research demonstrates that QWM spectroscopy can be used as a powerful tool to probe the quantum statistics of incident nonclassical fields without needing direct photon counting. The amplitude of specific sidebands serves as a signature for the presence of squeezed light or correlated photon pairs.
On-Chip Quantum Sources: It validates the concept of using a strongly driven superconducting qubit as an on-chip source of correlated photon pairs, a resource essential for quantum information processing and continuous-variable quantum optics.
Fundamental Physics: The work bridges the gap between resonance fluorescence theory and squeezed light theory, showing that the fluorescence of a single artificial atom, under specific conditions, mimics the statistical properties of a degenerate parametric amplifier.
Experimental Guidance: The paper provides clear guidelines for experimentalists on how to tune drive strengths and decay rates to observe these quantum interference effects, facilitating the design of future waveguide-QED experiments.