Cyclotron Radiation Emission Spectroscopy

• Cyclotron Radiation Emission Spectroscopy (CRES) is a high-precision, non-destructive technique that determines electron kinetic energy by measuring emitted microwave signals during cyclotron motion in a magnetic field.
• It employs a rigorous relativistic frequency–energy relation and Fourier-based spectrogram analysis to achieve sub-eV energy resolution with high signal-to-noise ratio.
• Demonstrated by Project 8, CRES offers scalability for direct neutrino mass measurements and broad applications in high-resolution beta decay and X-ray spectroscopy.

Cyclotron Radiation Emission Spectroscopy (CRES) is a precision technique for determining the kinetic energy of charged particles—principally electrons—by measuring the frequency of electromagnetic radiation emitted during their cyclotron motion in a magnetic field. Developed as a next-generation approach to direct neutrino mass measurement, CRES leverages the strict relativistic frequency–energy relation and the minute, non-destructive microwave signals generated by single electrons, offering both high-resolution spectroscopy and scalability to large detection volumes. The first demonstration was realized by the Project 8 collaboration for electrons from internal-conversion of $^{83m}$Kr, subsequently extending to tritium $\beta$ decay and other applications (Asner et al., 2014, Esfahani et al., 2017, Collaboration et al., 2022, Esfahani et al., 11 Mar 2025).

1. Theoretical Framework of CRES

CRES exploits the relativistic cyclotron frequency of an electron with rest mass $m_e$, charge $e$, and kinetic energy $K$ in a uniform magnetic field $B$. The fundamental relation is: $$f_{c} = \frac{1}{2\pi} \frac{e B}{\gamma m_e}, \quad \text{with} \quad \gamma = 1 + \frac{K}{m_e c^2},$$ making $f_{c}$ an explicit probe of the electron energy $K$. The energy can be reconstructed by inverting this relation,

$$K = m_e c^2 \left( \frac{eB}{2\pi m_e f_c} - 1 \right),$$

so that a frequency measurement at the $10^{-5}$-level corresponds directly to an eV- or sub-eV energy resolution at $K \sim 18$--30 keV (Asner et al., 2014, Esfahani et al., 2017).

The radiated power follows the Larmor formula (relativistically, the Liénard formula):

$$P = \frac{e^4 B^2}{6 \pi \epsilon_0 m_e^2 c^3} (\gamma^2-1) \sin^2\theta,$$

where $\theta$ is the pitch angle. For $K \simeq 18.6$ keV and $B=1$ T, $P \sim 1$ fW at $\theta=90^\circ$ (Asner et al., 2014).

CRES spectrograms display "tracks"—time-evolving frequency curves corresponding to individual electrons. The slow upward drift in $f_c$ is the radiative cooling of the electron; discrete jumps arise from inelastic scattering (Esfahani et al., 2017).

2. Experimental Principles and Key Hardware Elements

A CRES apparatus consists of the following essential components:

• Magnet: A superconducting solenoid furnishes a highly uniform $B_0 \simeq 0.95$--1 T field, with inhomogeneities controlled to $\ll 10^{-4}$ for energy resolution below a few eV (Esfahani et al., 2017, Esfahani et al., 11 Mar 2025).
• Magnetic Trap: Auxiliary coils ("pinch" or "bottle" configurations) create a shallow field minimum, axially confining electrons with large pitch angles by reflection. Depths of $\Delta B \sim$ 1--10 mT are typical (Esfahani et al., 2017).
• Detection Cell/Waveguide: The decay cell (e.g., WR42 rectangular, or circular guide) is filled with a source gas—$^{83m}$Kr, tritium, etc.—and forms a transmission line for the 26 GHz-range cyclotron emission, matching the fundamental TE mode to ensure efficient coupling (Asner et al., 2014, Esfahani et al., 11 Mar 2025).
• RF Front-End: A cryogenic InP or HEMT LNA (noise figures $T_{sys} \sim 50$–150 K) directly amplifies the microwave emission. Down-conversion stages bring signals to baseband ($\sim 0$–250 MHz) for digitization (Esfahani et al., 2017).
• Digitization & Trigger: The digitizer samples at hundreds of MS/s with $>8$-bit resolution. Real-time spectrum analyzers or continuous streaming provide spectrograms for subsequent event reconstruction (Esfahani et al., 2017).

Notable technical metrics from Project 8's Phase-1 detector include a SNR $>40$ dB per 32 $\mu$s window, frequency resolution $\delta f \sim 30$ kHz/bin, and background rates below 1% for spectroscopically resolved events (Esfahani et al., 2017).

3. Event Reconstruction and Data Analysis

CRES signal analysis follows a multi-step pipeline:

1. Spectrogram Formation: The digitized time series is Fourier-transformed (windowed blocks of 16,384 or 8,192 samples, Hann window, e.g., $\sim$30.5 kHz/bin for a 500 MS/s digitizer). The resulting $S(t,f)$ time-frequency map isolates spectral power in localized tracks (Asner et al., 2014, Esfahani et al., 2017).
2. Track Identification: Power thresholds (e.g., $5\sigma$ over noise) select candidate bins; a clustering algorithm groups adjacent bins, and Hough transforms fit near-linear "tracks" characterized by their initial frequency $f_0$ and slope (chirp rate) (Esfahani et al., 2017).
3. Sideband Association: Axial motion in a trap imprints sidebands at $f_0 \pm nf_z$ ($f_z \sim 40$ MHz in two-coil traps). Event building algorithms group mainband and sideband clusters with common start/stop and $f_z$-spaced structure (Esfahani et al., 2017).
4. Energy Extraction: The fitted track intercept $f_0$ yields $\gamma$ and hence $K$ as above. Statistical resolutions of $\sim$0.1 eV per event are achievable for high-SNR tracks (Esfahani et al., 2017).
5. Background Rejection: After clustering, spurious triggers from noise constitute $<1\%$ of the event pool; gas scattering causes premature track terminations in $<5\%$ of cases (Esfahani et al., 2017).

4. Performance Metrics and Systematics

Experimental results from Project 8's Phase-1 and follow-up detectors demonstrate:

Trap Configuration	Trap $\Delta B$ (mT)	Energy Resolution (keV)	FWHM (eV)
Single coil	$-$3.2	30	$\sim$140
Single coil (shallow)	$-$1.6	30	$\sim$15
Two-coil trap	$+$3.5 each	30	$<$4

Systematic effects are dominated by:

• Magnetic-Field Inhomogeneity: $\delta K_{sys} \lesssim 2$ eV broadening for the trap volume and coil material (Esfahani et al., 2017).
• B-field Calibrations: Uncertainties in absolute $B$ from NMR-probe calibration limit energy-scale accuracy to $\lesssim 1$ eV (Esfahani et al., 2017).
• Residual Gas Effects: At $\sim10^{-5}$ mbar $^{83m}$Kr, collisional broadening is negligible ($<$0.1 eV); gas-scattering losses are $<$5% (Esfahani et al., 2017).
• Sidebands and Magnetic Geometry: Axial frequency sidebands, readily visible ($\sim$40 MHz separation), provide handles for cross-checking trap uniformity and systematic broadening (Esfahani et al., 2017).

5. Scientific Impact and Applications

CRES was conceived to provide non-destructive, frequency-based measurement of the entire electron spectrum in nuclear and particle physics, specifically to enable direct searches for the absolute neutrino mass via tritium $\beta$ decay (Asner et al., 2014, Esfahani et al., 2017). Simultaneous, event-by-event energy reconstruction replaces the stepped-integration technique of MAC-E filter experiments. By detecting single electrons in situ, CRES enables the use of dense, gaseous sources, bypassing the limitations of electron transport, extraction, or absorber-induced loss (Esfahani et al., 2017).

The Project 8 roadmap sets out four phases, ultimately scaling CRES to atomic tritium sources and large detection volumes in pursuit of $\sim$40 meV neutrino-mass sensitivity—the regime of the inverted-mass hierarchy (Esfahani et al., 2017).

Beyond neutrino physics, CRES is extensible to high-resolution X-ray spectroscopy, broad-band $\beta$ decay studies in nuclear structure, and potentially, searches for beyond Standard Model physics through the paper of spectral endpoints and decay-forbidden regions (Kazkaz et al., 2019, Byron et al., 2022).

6. Outlook: Upgrades, Scalability, and Future Directions

Phase-2 Project 8 upgrades target tritium compatibility, event rate improvements, and systematic reduction:

• Cell Design: Transition to a cylindrical waveguide, tripling the fiducial volume; use of tritium-compatible metallurgy and ultra-high-vacuum components; robust getter cascades for safety and containment (Esfahani et al., 2017).
• Trap and Field Control: A five-coil solenoid permits finer manipulation of the trap, with in situ ESR magnetometers for $\lesssim$1 ppm $B$ calibration (Esfahani et al., 2017).
• RF and SNR Enhancements: Implementation of endcap reflectors and cryogenic circulators, halving the system noise temperature and boosting SNR by up to a factor of 3 (Esfahani et al., 2017).
• Data Chain: Digitization rates of up to 2 GS/s, advanced FFT-triggered readouts, and the development of Bayesian unfolding algorithms to reconstruct continuous spectra in real time (Esfahani et al., 2017).

Projected performance anticipates sub-eV energy resolutions, with event rates scalable by factors of 3--10 as the size increases, and backgrounds consistently below $10^{-3}$ triggered events per day in the endpoint window (Esfahani et al., 2017).

Next steps involve scaling to meter-scale detection cells, phased antenna arrays for signal collection, and the transition to atomic tritium, eliminating the irreducible $\sim$0.5 eV broadening observed with molecular sources (Esfahani et al., 2017).

In sum, CRES has been experimentally validated as a single-electron, frequency-based spectroscopic technique achieving sub-5 eV energy resolution, with demonstrated SNR, systematics, and scalability adequate for future direct neutrino mass measurement initiatives (Esfahani et al., 2017, Asner et al., 2014, Esfahani et al., 2017).