Electron radiated power in cyclotron radiation emission spectroscopy experiments

The recently developed technique of Cyclotron Radiation Emission Spectroscopy (CRES) uses frequency information from the cyclotron motion of an electron in a magnetic bottle to infer its kinetic energy. Here we derive the expected radio-frequency signal from an electron in a waveguide CRES apparatus from first principles. We demonstrate that the frequency-domain signal is rich in information about the electron's kinematic parameters and extract a set of measurables that in a suitably designed system are sufficient for disentangling the electron's kinetic energy from the rest of its kinematic features. This lays the groundwork for high-resolution energy measurements in future CRES experiments, such as the Project 8 neutrino mass measurement.

Figure 1

Axial motion of an electron in a magnetic bottle. The magnetic trap has depth $\mathrm{\Delta }B$ and maximum value $B_{\max }$. The electron's pitch angle is defined as the angle between the electron's momentum vector and the direction of the local magnetic field. If the electron's pitch angle at the bottom of the trap satisfies Eq. (4), then the electron undergoes an oscillatory axial motion inside the trap. The turning point for the electron corresponds to the position when the pitch angle is ${90}^{\circ }$.

Figure 2

The comb structure of the frequency spectrum of cyclotron power from a trapped 30-keV electron in a 1-T background field. The central peak is located at the average cyclotron frequency, and the axial frequency, which defines the separation between the peaks, is $15\mathrm{MHz}$.

Figure 3

Schematic of an experiment with an electron undergoing cyclotron motion in a waveguide with a conductive short. The relevant parameters include the origin, $O$; the position of the electron, $z_0$; the position of the waveguide short, $z_s$, on the left side of the waveguide; and the position of the receiver, $z_r$, on the right side of the waveguide. The magnetic field, $B$, is parallel to the waveguide axis.

Figure 4

The on-axis magnetic field profile of a “harmonic” trap (solid black line), generated by a single coil, and the corresponding approximation given by Eq. (52) with $L_0=20\mathrm{cm}$ (dashed red line).

Figure 5

Relative magnitudes of the sidebands in a harmonic trap as a function of the maximum axial travel, $z_{max}$, of a trapped 30-keV electron. Here we consider an ideal harmonic trap as described in Eq. (52), with a background field of 1 T and an $L_0$ of 20 cm. No reflection effect is taken into account.

Figure 6

Magnetic field profile generated by two coils separated by 5 cm, forming a “bathtub” shape (solid black line), and the corresponding approximation given by Eq. (63) with $L_0$ = 35 cm and $L_1=0.5$ cm (dashed red line).

Figure 7

Schematic of the power (represented by line width), as a function of time and frequency in the absence of a waveguide reflector. The main track and first-order sidebands are shown. Sudden losses of energy (and thus increases of frequency), induced by collisions with background gas particles, happen at $1.5\mathrm{ms}$ and $3.5\mathrm{ms}$.

Figure 8

Schematic of the power (represented by line width) as a function of time and frequency in the presence of a waveguide reflector. The main track appears and disappears as the kinematic parameters (such as the pitch angle) change as a result of collisions with background gas particles.

Figure 9

Spectral features of the CRES signal for 30-keV electrons with different values of pitch angle and single radial position at $\rho =$ 0 cm, trapped in an ideal baththub trap described by Eq. (63) with $L_0=$ 35 cm and $L_1=$ 0.5 cm, in a 1-T background field including the effect of a short. An electron with a pitch angle of ${90}^{\circ }$ has a start frequency of 26.44 GHz while electrons with lower pitch angles are subject to pitch-angle effects which increase their start frequencies. This shift can systematically affect energy measurements in CRES experiments. The above plots illustrate how different measurable quantities in a CRES experiment can be used to correct for this frequency shift.

Figure 10

Simulation of the energy spectrum of electrons sampled from a 60-eV wide Lorentzian, centered at $30\mathrm{keV}$, in a 1-T background magnetic field. In blue (light gray in black and white print), the spectrum of the extracted start frequencies for an idealized case where the magnetic field is flat and all electrons have a ${90}^{\circ }$ pitch angle. In red (dark gray in black and white print), the actual line shape when the electrons have an isotropic momentum distribution and are confined in a 4-mT deep ideal harmonic trap as given in Eq. (52). The blue histogram is scaled down, so it can be compared with the red one.