In 2016, we discovered that, to measure the spectrum of a stochastic signal on an optical interferometer (a task I shall call phase noise spectroscopy), photon counting is far superior to homodyne detection. We also found that photon counting is in fact the best measurement allowed by quantum mechanics for this task. Our paper:

Shilin Ng, Shan Zheng Ang, Trevor A. Wheatley, Hidehiro Yonezawa, Akira Furusawa, Elanor H. Huntington, and Mankei Tsang | Spectrum analysis with quantum dynamical systems | Physical Review A 93, 042121 (2016) |

It looks like Caltech scientists have picked up the idea recently and will be using it to detect signatures of quantum gravity; their technical paper by Lee McCuller is here: https://arxiv.org/abs/2211.04016, and some articles about their proposed experiment are here: https://www.caltech.edu/about/news/at-the-edge-of-physics and https://magazine.caltech.edu/post/quantum-gravity.

We also discovered recently that, with a squeezed input, the optimal measurement for phase noise spectroscopy and detection is unsqueezing + photon counting:

Mankei Tsang | Quantum noise spectroscopy as an incoherent imaging problem | Physical Review A 107, 012611 (2023) |

Researchers studying the use of microwave cavities to detect axion dark matter have reached similar conclusions; see Lamoreaux et al. (2003), Dixit et al. (2021), Gorecki et al. (2023), and Shi and Zhuang (2023).

Let me explain the history behind the idea.

## Thermal radiation measurement

Let’s start with the fact that, when it comes to the detection and spectroscopy of thermal optical radiation, it is well known that photon counting is far better than “linear” field detectors—which include heterodyne, homodyne, and linear amplifiers; see, for example, Siegman (1966), Kingston (1978), Charles Townes’ notes [Chapter 4], and Prasad (1994). To be specific, photon counting is far superior when the average signal photon number per mode, which I denote as ε, is much smaller than 1. ε << 1 for typical thermal sources at optical frequencies, and it is well known in optical astronomy that photon counting is the best while linear detectors are terrible.

The superiority of photon counting is fundamental, in the sense that we are assuming the same number of optical modes (i.e., same bandwidth, same integration time, and one spatial and polarization mode), 100% efficient detectors in both cases, and no excess noise other than that mandated by quantum mechanics.

A rough explanation is as follows. When ε << 1, the probability that you get one or more photons in each optical mode is very small, and most of the time there is no photon. With photon counting, you can tell very well when there is no photon and just ignore those null events, but linear detectors must suffer from the vacuum fluctuations all the time, even when there is no photon, so any signal would just be buried in the constant noise.

(This is just a heuristic explanation after the fact—we also have to account for the possibility of cleaning up the measurement with data processing. This is why any rigorous calculation must use proper statistics.)

In fact, photon counting is the best measurement of thermal radiation allowed by quantum mechanics; see Helstrom (1976) and Nair and Tsang (2015). Note, however, that linear detectors can become almost as good as photon counting when ε >> 1 (e.g., radio frequency or microwave), or when the radiation is coherent (e.g., laser source).

## Axion dark matter as thermal noise

For axion dark matter searches, people apply a DC magnetic field across a microwave cavity at a cryogenic temperature. Assuming that the cavity is initially empty, the magnetic field may convert dark matter to thermal microwave radiation through some hypothetical axionic interaction. Since the microwave output is thermal and the photon count per mode is low on average, our intuition about thermal radiation measurement would suggest that photon counting is far superior to linear detectors, and indeed people have confirmed that theoretically and experimentally; see Lamoreaux et al. (2003) and Dixit et al. (2021).

## Thermal noise versus phase noise

This conversion of dark matter to thermal microwave can be modeled as displacements of the microwave field in *both* quadratures by classical random processes. For an optical interferometer, the situation is quite different, as there is a coherent laser beam at the input, and the signal usually applies a very weak phase modulation to the light, effectively displacing only *one* quadrature of the optical field. When the signal is stochastic, we have a random displacement in one optical phase quadrature only, and the quantum state of the output optical field is a bit more complicated.

It was therefore a nontrivial open problem what the optimal measurement would be for phase noise spectroscopy. Does the random displacement make the problem similar to the case of thermal radiation, implying that photon counting would be far superior, or does the coherent input beam make the problem similar to the case of coherent radiation, implying that homodyne would be just as good?

Every study before our 2016 work suggested that, for many other tasks with interferometers, e.g., phase estimation, homodyne is optimal or close to optimal, and LIGO indeed uses homodyne detection. To our knowledge, no one expected that homodyne would be far from optimal for tasks such as phase noise spectroscopy and detection.

We certainly didn’t expect it, but that’s what we found in 2016 for phase noise spectroscopy with a coherent-state input. Inspired by our previous work on astronomical quantum optics, we discovered that photon counting in the spectral domain turns out to be far better than homodyne when the signal is small (analog of ε << 1). We extended that result for squeezed input and noise spectroscopy and detection in 2023.

## Summary

Of course, now that we’ve done the calculation, it is clear in hindsight that thermal radiation measurement and phase noise spectroscopy are similar problems. The essential difference is random displacements in both quadratures in the thermal case and random displacement in the phase quadrature in the interferometer case. In both cases, photon counting turns out to be far superior to linear detectors for low signals, while quantum estimation and detection theory shows that photon counting is in fact the best one can do for those tasks.

With nonclassical input light, the situation is a bit more complicated—see Tsang (2023), Gorecki et al. (2023), and Shi and Zhuang (2023) for details.

[Last update: 13 Oct 2023 (changed one word LIGO to Caltech)]