Stimulated emission of signal photons from dark matter waves

This paper demonstrates a quantum-enhanced dark matter search technique using a superconducting qubit to prepare a microwave cavity in a non-classical Fock state, which stimulates signal photon emission to increase the scan rate by a factor of 2.78 and excludes kinetic mixing angles $\epsilon \geq 4.35 \times 10^{-13}$ for dark photons around 5.965 GHz.

The Gist

The Big Picture: Listening for a Ghost

Imagine the universe is filled with a mysterious, invisible substance called Dark Matter. We know it's there because it holds galaxies together with its gravity, but we've never seen it or touched it. Scientists think it might be made of tiny, invisible particles (like "dark photons") that act like a giant, invisible ocean wave washing over the Earth.

The problem? This wave is incredibly weak. Trying to detect it is like trying to hear a single whisper in a hurricane.

The Old Way: The Empty Room

Traditionally, scientists use a special metal box (a cavity) to catch these whispers. They tune the box to a specific pitch (frequency). If a dark matter particle hits the box, it should turn into a tiny flash of light (a photon) that the box can catch.

However, the box is usually empty (a vacuum). When the dark matter wave hits an empty box, it's like trying to start a fire with a single spark on a cold, damp log. It's hard to get a reaction, and the "noise" of the universe often drowns out the signal.

The New Trick: The "Crowded" Room

This paper introduces a clever new trick. Instead of leaving the box empty, the scientists fill it with a specific number of photons first (specifically, 4 photons). They call this a Fock State.

Think of it like this:

• The Old Way: You are in a silent library (the empty box). Someone whispers a secret (dark matter). You might not hear it over the sound of your own breathing (quantum noise).
• The New Way: You are in a room where 4 people are already clapping in perfect rhythm (the Fock state). When the whisper comes, it doesn't just get heard; it triggers the clappers to clap one more time in sync.

This is called Stimulated Emission. By having a "crowd" of photons already there, the dark matter wave can easily push them to create one more photon. It's like a snowball effect: a tiny push on a large snowball creates a bigger avalanche than a push on a tiny pebble.

The Magic Number: 2.78x Faster

The scientists found that by preparing the box with 4 photons instead of 0, they didn't just get a little more signal; they got a 2.78 times stronger signal.

Why is this a big deal?

• Dark matter searches are like tuning a radio. You have to scan through millions of frequencies to find the one where the dark matter is hiding.
• Because their signal is now nearly 3 times stronger, they can scan through the frequencies 3 times faster.
• This turns a search that might take 10 years into one that takes 3 or 4.

How They Did It: The Quantum Jukebox

To make this work, they used a Superconducting Qubit (a tiny, artificial atom made of superconducting circuits).

1. The Setup: They connected a high-quality metal box (the cavity) to this artificial atom.
2. The Preparation: Using precise microwave pulses (like a very sophisticated DJ), they forced the box to hold exactly 4 photons. This is hard to do because photons usually like to be messy or random; forcing them to be exactly 4 is like organizing a chaotic crowd into a perfect square dance formation.
3. The Detection: They waited to see if the dark matter wave would push that formation to become 5 photons. They used a special counting method (photon counting) to see if the number jumped from 4 to 5.

The Results: Catching the Ghost

They tested this in a specific frequency range (around 6 GHz).

• The Outcome: They didn't find dark matter this time (which is actually good news for physics, as it rules out some theories).
• The Exclusion: They proved that if dark matter did exist at that frequency, it would have to be weaker than a certain limit. They effectively drew a line on a map saying, "Dark matter is not here."
• The Significance: They set a new record for sensitivity in this specific area, excluding a range of possibilities that no one had ruled out before.

The "Catch" (Why it's not perfect yet)

There is a trade-off. When you pack 4 photons into a box, they are a bit more unstable than 0 photons. They want to leak out faster.

• Analogy: Imagine a bucket with a hole. If you fill it with 4 cups of water, it leaks faster than if it's empty.
• The Fix: The scientists had to be very fast. They prepared the state, waited a tiny fraction of a second for the dark matter to do its work, and then checked the result before the photons leaked away.

The Bottom Line

This paper is a breakthrough in Quantum Metrology (using quantum mechanics to measure things). It proves that by using the weird rules of quantum physics—specifically, by preparing a system in a specific, non-classical state—we can amplify the faintest whispers of the universe.

It's like upgrading from a paper cup to a high-tech megaphone to listen for a ghost. Even though they didn't find the ghost this time, they proved the megaphone works, and now they can search the whole universe much, much faster.

Technical Summary

1. Problem Statement

The search for dark matter (DM), particularly axions and dark photons, relies on detecting the conversion of these particles into microwave photons within a resonant cavity.

• The Challenge: As the search moves to higher frequencies (microwave range), the signal power drops significantly due to smaller cavity volumes and increased thermal/quantum noise.
• The Limitation: Current searches use near-quantum-limited amplifiers that operate at the Standard Quantum Limit (SQL), adding at least 0.5 photons of noise per mode. While photon counting techniques can evade the SQL by achieving sub-SQL background rates, they typically rely on the cavity starting in the vacuum state ($|n=0\rangle$). In this regime, the signal builds up via a process akin to spontaneous emission, which is inherently slow and weak.
• The Goal: To accelerate the frequency scan rate ($d\nu/dt$) by enhancing the signal photon rate ($R_s$) without proportionally increasing the background noise ($R_b$) or compromising the coherence time of the probe.

2. Methodology

The authors propose and demonstrate a technique using stimulated emission to amplify the dark matter signal. Instead of starting from the vacuum, they prepare the microwave cavity in a non-classical Fock state (a state with a definite number of photons, $|n\rangle$) before the dark matter interaction occurs.

• Physical Principle:

  • When a dark matter wave interacts with a cavity prepared in a Fock state $|n\rangle$, it induces a transition to $|n+1\rangle$ (stimulated emission) or $|n-1\rangle$ (stimulated absorption).
  • Quantum mechanically, the probability of stimulated emission into the $|n+1\rangle$ state is enhanced by a factor of $(n+1)$ compared to spontaneous emission from the vacuum.
  • Crucially, the authors argue that because the limiting coherence time is that of the dark matter wave (not the probe state), the signal rate retains this $(n+1)$ enhancement factor, unlike typical quantum metrology where probe decoherence cancels out the gain.

• Experimental Setup:

  • Device: A 3D multimode superconducting cavity (Storage mode at 5.965 GHz) dispersively coupled to a superconducting transmon qubit.
  • State Preparation: The transmon qubit is used to prepare specific Fock states ($|n\rangle$) in the storage cavity using GRadient Ascent Pulse Engineering (GRAPE) based optimal control pulses. This allows for the creation of states like $|n=4\rangle$ with high fidelity.
  • Detection: The system employs number-resolved photon counting. The qubit acts as a detector; by driving the qubit at a frequency shifted by the photon number ($\omega_q + n\chi$), the team can perform Quantum Non-Demolition (QND) measurements to determine if the cavity contains $n$ or $n+1$ photons.
  • Protocol:
    1. Initialize cavity in $|n\rangle$.
    2. Wait for a "dwell time" to allow the dark matter field to drive the transition.
    3. Perform repeated QND measurements to check if the state has flipped to $|n+1\rangle$.
    4. Use a Hidden Markov Model (HMM) to analyze the sequence of measurement outcomes, distinguishing true signal events from detector errors and background noise.

3. Key Contributions

• Demonstration of Fock-Enhanced Stimulated Emission: The first experimental realization of using a pre-prepared Fock state to stimulate the emission of a photon from a dark matter wave (simulated via a coherent drive).
• Quantitative Signal Enhancement: The team demonstrated a signal rate enhancement of 2.78 times when initializing the cavity in the $|n=4\rangle$ state compared to the vacuum state ($|n=0\rangle$). This matches the theoretical prediction of $\eta(n+1)$, where $\eta$ is the detection efficiency.
• Integration with Photon Counting: The work successfully combines Fock state preparation with the previously demonstrated sub-SQL background rates of photon counting, proving that the two techniques are compatible.
• New Dark Photon Exclusion Limits: The technique was applied to a real dark photon search, setting new exclusion limits in an unexplored parameter space.

4. Results

• Signal Enhancement:
  • By initializing the cavity in $|n=4\rangle$, the measured signal rate increased by a factor of 2.78 relative to $|n=0\rangle$.
  • The efficiency ($\eta$) for $|n=4\rangle$ was measured at 0.45, slightly lower than for lower $n$ states due to the increased decay rate of higher Fock states ($T_1 \propto 1/n$) and higher "demolition probability" (measurement-induced loss).
• Dark Photon Search:
  • The experiment searched for dark photons in a band around 5.965 GHz (24.67 $\mu$eV).
  • Exclusion Limit: The study excluded kinetic mixing angles $\epsilon \geq 4.35 \times 10^{-13}$ at the 90% confidence level.
  • This result sets a new constraint on dark photon models, particularly those arising from high-scale cosmic inflation.
• Systematic Effects:
  • The team identified that higher Fock states suffer from faster decay and potential leakage to nearby cavity modes (observed as anomalous behavior in the $|n=3\rangle$ data).
  • The "demolition probability" (non-QND nature of the measurement) increases with $n$, limiting the maximum achievable enhancement in current hardware.

5. Significance and Outlook

• Accelerating the Search: This technique offers a viable path to accelerate the scan rate of dark matter searches by a factor of $\sim 2.8$ (and potentially higher with optimized hardware) without requiring a fundamental change in the detector architecture.
• Beyond Dark Matter: The method is not limited to dark matter. It represents a general quantum-enhanced metrology technique for sensing ultra-weak forces or signals where the signal accumulation time is limited by the signal's coherence time rather than the probe's.
• Future Improvements: The authors note that future improvements in cavity quality factors ($Q$), reduced measurement demolition probabilities, and better state preparation techniques (e.g., Echoed Conditional Displacement) could push the enhancement factor closer to the theoretical limit of $(n+1)$ and enable searches at even higher frequencies.

In summary, this paper provides a proof-of-concept that quantum state engineering (specifically Fock states) can be used to leverage stimulated emission to significantly boost the sensitivity of dark matter detectors, effectively bypassing some of the limitations imposed by the Standard Quantum Limit.