Efficient imaging of quantum emitters using compressive sensing

This paper demonstrates a compressive sensing-based imaging technique that utilizes random binary excitation patterns and GPSR-BB reconstruction to efficiently image sparse quantum emitters, such as NV centers in diamond, achieving high-fidelity intensity and $g^{(2)}(0)$ maps with only 20% of the measurements required by conventional raster scanning.

The Gist

Imagine you are trying to take a picture of a dark room where only a few fireflies are blinking.

The Old Way (Raster Scanning):
Traditionally, to see where these fireflies are, you would use a very narrow flashlight. You would shine it on one tiny spot, wait to see if a firefly is there, then move the flashlight to the next spot, and the next, and the next. You have to check every single inch of the room, even the empty corners where no fireflies exist.

• The Problem: This takes forever. If the room is huge or the fireflies are very dim (hard to see), you might run out of battery (photons) or time before you finish the picture. It's like reading a book by shining a laser pointer on one letter at a time.

The New Way (Compressive Sensing):
The researchers in this paper came up with a clever shortcut. Instead of a narrow flashlight, they use a "smart projector" (called a Digital Micromirror Device, or DMD) that can flash complex, random patterns of light onto the whole room at once.

Think of it like this:

1. The Pattern: Instead of checking one spot, you flash a pattern of light that looks like a random maze or a checkerboard. Some parts of the room are lit up, others are dark.
2. The "One-Pixel" Eye: You don't use a camera with millions of pixels. Instead, you use a single, super-sensitive "ear" (a detector) that just listens to the total amount of light coming back from the whole room.
3. The Puzzle: You do this many times, but with different random patterns each time.
   • Pattern A: Lights up the left side. The detector hears a "beep" (light).
   • Pattern B: Lights up the right side. The detector hears silence.
   • Pattern C: Lights up the middle. The detector hears a loud "beep."

Even though you never looked at the fireflies directly, the computer can now solve a giant puzzle. Because it knows the fireflies are sparse (there are only a few of them in a huge room), it can mathematically figure out exactly where they are based on which patterns made the detector "sing."

The Magic Result:
The paper shows that you don't need to check every single spot. By using this "random pattern" method, they were able to create a perfect picture of the fireflies using only 20% of the data that the old method would have required.

• Analogy: It's like guessing the layout of a sparse forest by throwing a net over it a few times and seeing where the trees get caught, rather than walking every single step to count every tree.

The "Super" Upgrade (Finding Single Fireflies):
The researchers didn't just stop at finding where the fireflies are; they also wanted to know what kind of fireflies they were. Some fireflies blink one by one (single photons), while others blink in groups.

• The Challenge: Usually, to tell them apart, you have to measure how their blinks correlate with each other, which is even slower than just taking a picture.
• The Solution: They applied the same "random pattern" trick to this correlation measurement. They proved that even with very little data, they could still identify which fireflies were "single blippers" (single-photon emitters) by looking for a specific "anti-bunching" signature (a way of blinking that proves there is only one source).

Why This Matters:

• Speed: You get the picture 5 times faster.
• Efficiency: You don't waste energy checking empty spaces.
• Versatility: This works for any "sparse" system, not just diamond defects. It's a new way to see the quantum world that is faster, cheaper, and less exhausting on the equipment.

In a Nutshell:
Instead of scanning a room one brick at a time, this method flashes random patterns of light and uses a smart computer to reconstruct the image from the shadows and highlights, saving 80% of the time and effort while still seeing everything clearly.

Technical Summary

1. Problem Statement

Conventional optical imaging of quantum emitters (e.g., nitrogen-vacancy centers in diamond) typically relies on confocal microscopy with point-by-point raster scanning. While this method offers high spatial resolution, it suffers from two critical inefficiencies:

• Time-Consuming: Sequential excitation of every pixel results in long acquisition times, especially for large fields of view.
• Photon Inefficiency: In sparse systems (where emitters occupy only a small fraction of the field of view), the majority of measurement time and photon budget is wasted on empty regions.
• Limitations: These inefficiencies restrict the ability to image weak emitters, large areas, or photon-limited samples, which are common in quantum photonics and nanoscale sensing.

2. Methodology

The authors propose a Compressive Sensing (CS) framework that replaces raster scanning with spatially structured wide-field excitation.

• Hardware Setup:
  • Excitation: A 532 nm continuous-wave laser illuminates a Digital Micromirror Device (DMD). The DMD projects a sequence of random binary masks (patterns of "bright" and "dark" pixels) onto the sample.
  • Optics: A $4f$ imaging system relays the DMD pattern to the sample plane with a demagnification factor of ~100x, creating an illumination area of ~60 $\mu$m.
  • Detection: Fluorescence is collected via a high-NA objective, coupled into a multimode fiber, and detected by a Single-Photon Avalanche Diode (SPAD). For correlation measurements, the path splits into two arms with two SPADs and a time-tagger.
• Mathematical Framework:
  • Sparsity Assumption: The image $X$ is assumed to be sparse in the canonical pixel basis (i.e., most pixels are zero).
  • Measurement Model: Instead of measuring individual pixels, the system acquires $M$ linear measurements ($y = \Phi x + n$), where each measurement is the total integrated fluorescence corresponding to a specific DMD mask.
  • Reconstruction Algorithm: The sparse coefficient vector is recovered by solving an $\ell_1$-regularized least-squares problem:
    $$\hat{s} = \arg \min_s \frac{1}{2} \|y - As\|_2^2 + \tau \|s\|_1$$
    This is solved using the GPSR-BB (Gradient Projection for Sparse Reconstruction with Barzilai–Borwein step size) algorithm.
• Extension to Correlation Imaging ($g^{(2)}(0)$):
  • The authors extend the framework to reconstruct spatial maps of the second-order correlation function, $g^{(2)}(0)$, which identifies single-photon emitters (antibunching).
  • Since $g^{(2)}(0)$ is non-linear, they derive a modified linear forward model: $y_k = [1 - g^{(2)}_{tot, k}(0)] (I_{tot, k})^2 = \sum A_{k,i} [1 - g^{(2)}_i(0)] I_i^2$.
  • This allows the reconstruction of the $g^{(2)}(0)$ map in a two-step process: first reconstructing the intensity map, then using it to recover the correlation map from compressive correlation data.

3. Key Contributions

1. Demonstration of CS for Quantum Emitters: The first experimental demonstration of compressive sensing imaging specifically for Nitrogen-Vacancy (NV) centers in diamond, replacing raster scanning with random binary mask illumination.
2. Significant Reduction in Data Acquisition: Achieving high-fidelity image reconstruction using only ~20% of the measurements required for conventional raster scanning.
3. Correlation-Based Imaging: Theoretical and numerical extension of the CS framework to reconstruct spatial maps of $g^{(2)}(0)$, enabling the identification of single-photon sources with reduced data overhead.
4. Generalizability: The method relies only on spatial sparsity, making it applicable to a broad class of quantum emitters beyond NV centers.

4. Results

• Simulation: Numerical simulations of sparse NV center distributions showed that the CS algorithm could perfectly reconstruct the ground-truth image (100% correlation) using only 20% of the data, even with realistic noise modeling (shot noise, detector jitter, dead time).
• Experimental Validation (Intensity):
  • Using a sample of nanodiamonds with NV centers, the team reconstructed images using 20% of the measurement time.
  • Metrics: The reconstructed images achieved a Peak Signal-to-Noise Ratio (PSNR) of 35 dB and a Pearson correlation coefficient of 0.927 compared to the full raster-scanned reference.
  • Efficiency: This represents a 5-fold reduction in acquisition time.
  • Sampling Threshold: Performance metrics saturated rapidly, showing reliable recovery even at sampling ratios as low as 20-30%.
• Correlation Imaging (Numerical):
  • Simulations demonstrated that the modified CS framework could accurately reconstruct $g^{(2)}(0)$ maps.
  • The algorithm correctly identified single-photon emitter locations (defined as $g^{(2)}(0) < 0.5$) despite the strong undersampling, validating the potential for correlation-based quantum imaging.

5. Significance

This work addresses a critical bottleneck in quantum imaging: the trade-off between acquisition speed, photon budget, and spatial resolution.

• Photon Efficiency: By multiplexing measurements, the method maximizes the information gained per photon, which is crucial for delicate or photon-limited samples.
• Speed: The 5x reduction in acquisition time enables faster characterization of quantum systems and potentially real-time monitoring.
• Scalability: The approach is scalable to larger fields of view where raster scanning becomes prohibitively slow.
• Future Applications: The ability to reconstruct correlation functions ($g^{(2)}(0)$) from compressive data opens new avenues for characterizing quantum light sources and entanglement without the time penalty of traditional scanning correlation measurements. The framework is also adaptable to adaptive optics and machine learning-enhanced illumination patterns for further optimization.