Radiated Power in Radiation Emission Spectroscopy Experiments

A. Ashtari Esfahani,1, ∗ V. Bansal,2 S. B¨oser,3 N. Buzinsky,4 R. Cervantes,1 C. Claessens,3 L. de Viveiros,5 P. J. Doe,1 M. Fertl,1 J. A. Formaggio,4 L. Gladstone,6 M. Guigue,2, † K. M. Heeger,7 J. Johnston,4 A. M. Jones,2 K. Kazkaz,8 B. H. LaRoque,2 M. Leber,9 A. Lindman,3 E. Machado,1 B. Monreal,6 E. C. Morrison,2 J. A. Nikkel,7 E. Novitski,1 N. S. Oblath,2 W. Pettus,1 R. G. H. Robertson,1 G. Rybka,1, ‡ L. Salda˜na,7 V. Sibille,4 M. Schram,2 P. L. Slocum,7 Y-H. Sun,6 J. R. Tedeschi,2 T. Th¨ummler,10 B. A. VanDevender,2 M. Wachtendonk,1 M. Walter,10 T. E. Weiss,4 T. Wendler,5 and E. Zayas4 (Project 8 Collaboration) 1Center for Experimental Nuclear Physics and Astrophysics and Department of Physics, University of Washington, Seattle, WA 98195, USA 2Pacific Northwest National Laboratory, Richland, WA 99354, USA 3Institut f¨urPhysik, Johannes-Gutenberg Universit¨atMainz, 55128 Mainz, Germany 4Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 5Department of Physics, Pennsylvania State University, State College, PA 16801, USA 6Department of Physics, Case Western Reserve University, Cleveland, OH 44106, USA 7Wright Laboratory, Department of Physics, Yale University, New Haven, CT 06520, USA 8Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 9Department of Physics, University of California Santa Barbara, CA 93106, USA 10Institut f¨urKernphysik, Karlsruher Institut f¨urTechnologie, 76021 Karlsruhe, Germany (Dated: January 10, 2019) 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.

PACS numbers: 29.40.-n, 23.40.-s, 52.50.Qt

I. INTRODUCTION TO CYCLOTRON purely magnetic trap, eliminating the possibility of a full RADIATION EMISSION SPECTROSCOPY Penning trap configuration.

Following the invention of the Penning trap [1], low- Current implementations of CRES [6,7] consist of elec- energy bound with electric and magnetic fields trons produced and trapped inside a waveguide. The have been used to make some of the most precise mea- waveguide propagates the cyclotron radiation emitted by surements of fundamental physics values (e.g., the g−2 of the electrons to a receiver with minimal losses. The back- the electron [2,3]). The success of these measurements ground magnetic field within the waveguide consists of was contingent on a well-developed theory relating the two contributions: a strong, uniform, background field, signal from the axial motion of the electron in the trap which is parallel to the axis of the waveguide, and a mag- to the electron’s kinematic parameters [4]. netic distortion which forms the magnetic bottle. Recently it has been proposed [5] that the tech- nique called Cyclotron Radiation Emission Spectroscopy (CRES) be used to make precise measurements of the Here we develop a mathematical description that re- energies of electrons trapped in a magnetic bottle. The lates the characteristics of the apparatus, the motion of arXiv:1901.02844v1 [physics.ins-det] 9 Jan 2019 cyclotron radiation from these particles gives direct in- the electron, and the measured signal. In sectionII, we formation about their total energy. Since an electric field investigate the variation of the cyclotron frequency due would introduce a position-dependent component to the to the electron’s motion. In sectionIII, we derive the particle energy, studies of radioactive decay require a electron’s radiation spectrum into the waveguide. In sec- tionIV, we study the effects of signal reflections on the measured radiation spectrum. In sectionV, we apply the formulae from sectionsII,III, andIV to two examples of ∗ [email protected] † Now at Sorbonne Universit´e and Laboratoire de Physique magnetic bottle configurations. Finally, in sectionsVI Nucl´eaireet des Hautes Energies,´ CNRS/IN2P3, 75005 Paris, andVII, we demonstrate that there is sufficient informa- France; [email protected] tion in the signal to reconstruct the kinematic parameters ‡ [email protected] of the electron. 2

II. MOTION AND CYCLOTRON FREQUENCY OF AN ELECTRON IN A MAGNETIC BOTTLE

A. The need for a magnetic trap

The angular cyclotron frequency, Ωc, of an electron with kinetic energy Ke and mass me, in a magnetic field B, is given by eB eB Ωc = = 2 , (1) γme me + Ke/c FIG. 1. Axial motion of an electron in a magnetic bottle. where e is the elementary charge, c is the speed of , The magnetic trap has depth ∆B and maximum value Bmax. and γ the electron’s Lorentz factor. For a known mag- The electron’s pitch angle is defined as the angle between the netic field, a measurement of the frequency of an elec- electron’s momentum vector and the direction of the local tron’s cyclotron radiation is also a determination of its magnetic field. If the electron’s pitch angle at the bottom of kinetic energy [6]. The frequency resolution of the mea- the trap satisfies Eq. (4), the electron undergoes an oscilla- surement, and therefore the energy resolution, improves tory axial motion inside the trap. The turning point for the with increasing observation time. Therefore a no-work electron corresponds to the position when the pitch angle is trap is necessary for an electron to be observed for a suf- 90 degrees. ficiently long time. trap. For every electron, we define the pitch angle at the bottom of the trap to be θbot. Due to conservation of en- B. Magnetic bottle and pitch angle definition ergy, the condition on pitch angle for a trapped electron is A magnetic bottle consists of a local minimum in the r ! magnitude of background magnetic field. The behavior −1 ∆B θbot ≥ sin 1 − , (4) of a charged particle in a magnetic bottle has been well- Bmax described [4,8], so here we highlight only the elements important for our results. If we define an electron’s in- where Bmax is the maximum value of the magnetic field stantaneous pitch angle, θ(t), as the angle between the and ∆B is the trap depth. local magnetic field and the electron’s momentum, then Existing CRES experiments operate at a background the kinetic energy for an electron undergoing cyclotron field of 1 T, which we use for all examples throughout. motion can be decomposed to its parallel and perpendic- A 4 mT trap depth on this background field can trap ular components as electrons with pitch angles greater than 86 degrees. As a consequence, electrons trapped in the magnetic bottle

Ke = Kek + Ke⊥ with a pitch angle other than 90 degrees at the bottom of the trap will undergo periodic axial motion as depicted 1 p2 (2) = 0 cos2 θ(t) + µ(t)B(t), in Fig.1. 2 me where p is the magnitude of the electron’s initial mo- 0 C. Time-varying cyclotron frequency of an electron mentum and µ is the equivalent magnetic moment of the electron, given by The magnetic field experienced by an electron varies 2 2 1 p0 sin θ(t) with time due to its axial motion, resulting in a time- µ(t) = . (3) varying cyclotron frequency, given by 2 me B(t) eB(t) In the adiabatic regime, where the change in the mag- Ωc(t) = 2 . (5) netic field direction is slow compared with the cyclotron me + Ke/c frequency, an electron’s equivalent magnetic moment is Additionally, the electron’s cyclotron motion causes it a constant of motion. For the remainder of this deriva- to radiate, reducing its kinetic energy and therefore in- tion µ is treated as time-independent and the term µB(t) creasing its cyclotron frequency. This energy loss can be behaves as a magnetic potential energy. Electrons with expressed as pitch angles of 90 + δθ or 90 − δθ degrees will have the dK (t) same motion; therefore we will only consider electrons e = −P (t), (6) with pitch angles between 0 and 90 degrees. The pitch dt angle approaches 90 degrees for an electron exploring re- where P , the power radiated by the electron, can be as- gions of increasing magnetic field, whereas the pitch angle sumed to be constant over short times. The energy ra- decreases for an electron approaching the bottom of the diated is much smaller than the electrons initial total 3 energy. Therefore, the instantaneous frequency of radia- tion emitted by the electron can be derived from Eq. (1) as 1.2 eB(t)  P t  1 Power [a.u.] Ωc(t) ' 2 1 + 2 , (7) me + K0/c mec + K0 0.8 where K0 is the initial kinetic energy of the electron. 0.6 Because the cyclotron radiation is observed for a finite amount of time, the frequency is shifting by the electron 0.4 P t power loss 2 . mec +K0 Existing CRES experiments operate with midly rela- 0.2 tivistic electrons, we take a 30 keV electron for examples 00 10 20 30 40 50 60 70 throughout. Such an electron in a 1 T background field Frequency [a.u.] radiates 1 fW of power. Over 10 µs this results in a cy- clotron frequency shift of 3 kHz, which is equivalent to FIG. 2. The comb structure of the frequency spectrum an energy shift of 60 meV. This effect can be ignored in of cyclotron power from a trapped 30 keV electron in a 1 T the following calculation of CRES power spectral density. background field. The central peak is located at the average We will consider it again when we introduce the slope of cyclotron frequency, and the axial frequency, which defines tracks in Sec.VI. the separation between the peaks, is 15 MHz.

D. Axial motion and Doppler shift magnetic field being the smaller of the two effects. Fre- quency modulated signals have been studied extensively As a trapped electron oscillates axially in a magnetic as a form of encoding information in radio frequency sig- bottle, the frequency of radiation collected by the receiver nals [9]. The expected signal at the receiver consists of on the same axis, Ωr, is shifted by the Doppler effect and a frequency comb structure, where the main carrier is can be expressed as at the average cyclotron frequency and is surrounded by sidebands which are evenly spaced by the frequency of  −1 vz(tret) axial motion as shown in Fig.2. Ωr(t) = Ωc(tret) × 1 − , (8) vp The relative magnitude of the sidebands can be char- acterized by the modulation index, h = ∆ω , where ∆ω ωa where tret is the retarded time, vz is the electron axial is the maximum frequency change due to the Doppler velocity, and vp is the phase velocity of the wave inside shift and magnetic field, and ωa is the axial frequency. the waveguide. For mildly relativistic electrons with the The magnitude of the nth sideband is given by the Bessel vz (tret) function J (h). For values of h greater than 0.5, a signif- large pitch angles required for trapping, the term v n p icant fraction of power is present in the sidebands. For is small compared to 1. Substituting Ωc from Eq. (7) into Eq. (5) results in h ' 2.41, all of the power is radiated in sidebands, and no power is radiated into the carrier, as shown in Fig.5.   eB(tret) vz(tret) As a simple case, we can calculate the sideband struc- Ωr(t) ' 2 1 + , (9) me + K0/c vp ture from only the Doppler shift for an electron moving axially in simple harmonic motion, with axial frequency in which the second-order contributions in vz/vp have ωa and maximum travel zmax. From Eq. (9) we note that been neglected. Eq. (9) introduces two systematic ef- the maximum frequency change is ∆ω = Ωcωazmax/vp. fects that must be accounted for to understand the rela- The modulation index is then h = Ωczmax . The thresh- vp tionship between the electron’s energy and the observed old for significant received signal power in the sidebands, signal. h ∼ 0.5, is therefore equivalent to an axial travel for elec- First, the average value of B(t) is greater than the trons greater than a half-wavelength of light in waveg- value of B at the center of the trap and depends on the uide. magnitude of the electron’s axial motion. This causes the average measured frequency to be dependent on the elec- tron’s motion in the trap. This feature has been briefly discussed in [6], for electrons in a harmonic trap, and will E. Grad-B and Curvature Drifts be discussed in detail in SectionV. Second, the terms B(t) and vz(t) vary periodically at The electron undergoes a cyclotron motion, an axial harmonics of the frequency of the electron’s axial mo- motion, and two drift motions induced by non-uniformity tion. This imposes frequency modulation on the cy- in the magnetic field. The first force is prompted by the clotron signal, both by the varying magnetic field and magnetic gradient in the trap. These local magnetic field by the Doppler shift, with the modulation due to the gradients exert a force on the electron that gives rise to 4 a drift velocity perpendicular to both the magnetic field a sum over all modes in the ± z-directions as and its gradient, which we call grad-B motion, given by Z ∞ ± X ± [10] E (r, t) = Aλ (ω)(Etλ(x, y) ± Ezλ(x, y)ˆz) λ −∞ µ × e±ikλze−iωtdω vgrad−B = B × ∇B. (10) Z ∞  1  mΩcB ± X ± H (r, t) = Aλ (ω) ± ˆz × Etλ(x, y) + Hzλ(x, y)ˆz Zλ λ −∞ This slow grad-B motion is analogous in its effect to × e±ikλze−iωtdω, magnetron motion in a Penning trap; it pins the guiding (12) center of the electrons cyclotron motion to a larger circle. where kλ is the wave number and Zλ the mode The radius of this larger circle is set by the electrons impedance. The amplitude, Aλ(ω), of each mode is found radial position in the trap at the moment it is created. via Poynting’s theorem and is given by The grad-B velocity for a 30 keV electron with pitch Z Z ± λ ∓ikλz 3 angle of 86 degrees in a 1 T magnetic field with a Aλ (ω) = − J(ω)·(Etλ(x, y) ∓ Ezλ(x, y)ˆz) e d r, 2 V 10 mT/m field gradient is smaller than 300 m/s. This ve- (13) locity corresponds to a frequency of 5 kHz for an electron with V being the waveguide volume and the current in- orbit with a 1 cm radius. For power spectral densities side the waveguide, J(ω), defined as calculated for a finite time length smaller than grad-B motion’s period, this effect can be ignored and the elec- 1 Z ∞ J(ω) = J(r, t)eiωtdt. (14) tron’s guiding center can be assumed fixed. 2π −∞ The curvature in the field lines introduces another drift motion, which we call curvature drift, given by [10] These mode amplitudes fully determine the signal in the waveguide. The transverse electric field modes, Etλ(x, y), are normalized over the waveguide cross-section A such 2 2 v0 cos (θ(t)) that vcurv = 3 B × (B · ∇) B. (11) ΩcB Z Etλ · Etµda = δλµ. (15) For the conditions described below Eq.(10), the curvature A drift is smaller than 3 m/s and therefore negligible. For The longitudinal electric field modes, Ezλ(x, y), are nor- a detailed study of these two effects look at [11] and [12]. malized for TM modes such that Z 2 γλ Ezλ · Ezµda = − 2 δλµ, (16) A kλ

III. RADIATION OF A TRAPPED ELECTRON with γλ being the mode eigenvalues, which are zero for INTO A WAVEGUIDE MODE TE modes. The power transmitted in the ± z-direction is a spatial We now derive generic expressions for the spectral dis- integral of the normal component of the Poynting vector, tribution of the cyclotron radiation of a trapped electron. taken over the waveguide’s cross section, A. It can be We first expand the radiation from a generic current in- written as side the waveguide volume in terms of waveguide modes, Z ± ± ± X 1  ± 2 and derive the power that propagates through the waveg- P (t) = E (r, t)×H (r, t)·(±ˆz) da = Bλ (t) , Zλ uide. We then discuss the specific case of an electron A λ (17) coupling to a waveguide and the associated approxima- ± tions (as done with more detail in [13]). This allows us in which the mode excitation, Bλ (t), not to be confused to show that the mode excitation can be written in term with the B-field, is defined as of harmonics, corresponding to the axial modes, which Z ∞ ± ± ±ikλz −iωt demonstrates the comb structure of the measured cy- Bλ (t) = Aλ (ω)e e dω. (18) clotron power. Finally, we discuss the implications of our −∞ results in two examples, rectangular and circular waveg- uide. B. Power spectral density

Power spectral density is the quantity which we ulti- mately aim to calculate. To that end, we define the power A. Waveguide modes and transmitted power spectral density of the mode excitation as 2π 1 2 Generalizing the notation in Jackson, [12], the electric ˜± X ˜± P (ω) = Bλ (ω) , (19) T Zλ and magnetic fields inside a waveguide can be written as λ 5 with where (e1, e2) is an orthonormal basis in the plane trans- verse to the z direction, v0 is the electron’s initial veloc- 1 Z +∞ ˜± ± iωt ± ±ikλz ity, and Φc(t) is the phase of the electron in its cyclotron Bλ (ω) = Bλ (t)e dt = Aλ (ω)e , (20) 2π −∞ orbit, defined as Z t and T being the total time of observation. Eq. (19) can 0 0 Φc(t) = Ωc(t )dt . (27) be interpreted as the sum of the power in each waveguide 0 mode, This phase can also be written as a combination of con- X P˜±(ω) = P˜±(ω), (21) stant phase progression at the average cyclotron fre- λ quency, Ω , and a periodic perturbation at the electron’s λ 0 axial frequency. where The v·E term in Eq.(25), at the position (x0(t), y0(t)), can then be written as 2π 1 2 ˜± ˜± Pλ (ω) = Bλ (ω) . (22) T Zλ v(t) · (Etλ ∓ Ezλˆz) = v sin θ(t)(E cos(Φ (t)) + E sin(Φ (t))) ∓ cos θ(t)E ), In the case of a single electron, moving on the trajec- 0 1λ c 2λ c zλ (28) tory r = r0(t) with the velocity v(t), the current density is where E1λ and E2λ are the components of the transverse electric field for the mode λ. J(r, t) = −ev(t)δ3(r − r (t)). (23) 0 The radius of the cyclotron motion, rc, and the wave- length of the cyclotron radiation, λ , are related via From this and Eq. (13), the mode amplitudes can be c found to be v rc = λc. (29) Z Z ∞ 2πc ± Zλ A (ω) = − ev(t) · (Etλ(x, y) ∓ Ezλ(x, y)ˆz) As a result, the radius of cyclotron motion is small com- λ 4π V −∞ pared to the wavelength of cyclotron radiation and there- 3 iωt ∓ikλz 3 × δ (r − r0(t))e e dtd r. fore the waveguide dimensions. The variation in coupling (24) due to the cyclotron motion can be neglected, and one can replace the actual position of the electron by its gyro- By changing the order of integrals and taking the spatial center, defined to be the center of the electron’s cyclotron integral, we find that motion. In this work, we will further assume the trans- Z ∞ verse position of the electron’s gyrocenter (xc, yc) does ± eZλ Aλ (ω) = − v(t) · [Etλ(x0(t), y0(t)) not change with time. This may not be true in experi- 4π −∞ (25) ments with significant drift motion. iωt ∓ikλz0(t) ∓Ezλ(x0(t), y0(t))ˆz] e e dt, The z component of the v · E term in Eq. (25) is equal to zero for Transverse Electric (TE) modes and small in where the field is evaluated at the electron’s position, Transverse Magnetic (TM) modes for electrons with large r0(t) = (x0(t), y0(t), z0(t)). Using the mode amplitudes, pitch angles. The phase oscillation induced by sin θ(t) in the procedure from the preceding section is used to find Eq. (28) is thus small compared with the cyclotron phase ˜± Bλ (ω), from which the energy losses and signal power Φc and can be neglected. follow. Using the above approximations, Eq. (28) can be rewritten as

C. Field amplitudes for a CRES electron v · E = v0 [E1λ(x, y) cos(Φc(t)) + E2λ(x, y) sin(Φc(t))] v0 h i = (E − iE )eiΦc(t) + (E + iE )e−iΦc(t) . 2 1λ 2λ 1λ 2λ Eq. (25) describes the coupling of an electron inside (30) a waveguide, without any assumptions about its motion. A number of reasonable approximations can be used in Replacing the above expression for v · E in Eq. (25) we the case of an electron in a CRES experiment. get The electron’s periodic motion can be decomposed into eZ v  Z ∞ a cyclotron motion, an axial motion and a drift mo- ± λ 0 iΦc(t) ∓ikλz0(t) iωt Aλ (ω) = − (E1λ − iE2λ) e e e dt tion. Following the discussion of sectionIIE we assume 8π −∞ this last motion is slow compared with the first two, so Z ∞  −iΦc(t) ∓ikλz0(t) iωt the electron’s transverse and longitudinal velocity com- +(E1λ + iE2λ) e e e dt , −∞ ponents can be written as (31)

vt(t) = v0 sin θ(t) (cos Φc(t)e1 + sin Φc(t)e2) (26) where the electric fields are being evaluated at the elec- vz(t) = v0 cos θ(t), tron’s gyrocenter (xc, yc). 6

D. Mode expansion of motion and phase frequency, the power spectral density for the waveguide mode λ, Eq. (22) is

Because z0(t) and Φc(t) − Ω0t are periodic at the elec- ∞   2 tron’s axial motion frequency Ωa, these terms can be ex- ˜± X Ω0 + nΩa P (ω) = P0,λ an ± panded in a Fourier series as λ v n=−∞ p,λ (38)

∞ × [δ(ω − (Ω0 + nΩa)) + δ(ω + Ω0 + nΩa)] , iΦc(t)−iΩ0t X imΩat e = αme (32) m=−∞ where P0,λ is defined as and 2 2 e v0Zλ  2 2  P0,λ = E + E , (39) ∞ 8 1λ 2λ ikλz(t) X imΩat e = βm(kλ)e . (33) m=−∞ and vp,λ is the phase velocity in the waveguide for the mode λ. Note that there are possible cross terms between As a result, the exponential term in Eq. (31) can be the nth positive and the mth negative frequencies when 2Ω0 n + m = − . Because of the small values of an for written as Ωa large n, these terms can be neglected. ∞ iΦc(t)+ikλz0(t) X i(Ω0+nΩa)t The measured power spectrum thus exhibits a comb e = an(kλ)e , (34) structure in the frequency domain as shown in Fig.2. n=−∞ For an electron with no axial motion, all the power will in which be radiated with a frequency Ω0. An electron with pitch angle other than 90 degrees at the bottom of the trap, ∞ will undergo axial motion, and as a result some power X an(kλ) = αm(kλ)βn−m(kλ). (35) will be radiated at the harmonic frequencies which are m=−∞ nΩa away from the main peak. Eq. (38) indicates that the power in the nth harmonic is These coefficients, an, can be computed from a decompo- sition of the axial motion and the cyclotron phase evolu-   2 Ω0 + nΩa tion into harmonics of the axial frequency. This greatly Pn = P0,λ an ± . (40) v simplifies the study of the radiated power spectral den- p,λ sity. Based on Eq. (35), we get the following: F. Power in particular waveguide geometries ∞ X eiΦc(t)−ikλz0(t) = a (−k )ei(Ω0+nΩa)t, n λ The simplest experimental design choice is a waveguide n=−∞ geometry in which the radiation from the electron will ∞ −iΦc(t)−ikλz0(t) X ∗ −i(Ω0+nΩa)t only couple significantly to a single propagating mode. e = an(kλ)e , (36) Detailed calculations of P0,λ for two interesting exam- n=−∞ ∞ ples are included in Appx.B. For the TE10 mode in a −iΦc(t)+ikλz0(t) X ∗ −i(Ω0+nΩa)t rectangular waveguide we get e = an(−kλ)e . n=−∞ 2 2 Z10e v0 2 πxc  P0,T E = cos , (41) Expanding the exponential terms in Eq. (31) using the 10 4wh w above Fourier series results in in which Z10 is the TE10 mode impedance, v0 is the elec- " ∞ eZλv0 X tron velocity, w and h are the waveguide’s width and A±(ω) = − (E − iE ) a (∓k )δ(ω + Ω + nΩ ) λ 2 1λ 2λ n λ 0 a height, defined to be along x and y directions respectively, n=−∞ ∞ # and xc is the x position of the electron’s gyrocenter. X ∗ +(E1λ + iE2λ) an(±kλ)δ(ω − Ω0 − nΩa) . For the TE11 mode in a circular waveguide we get n=−∞ (37) Z e2v2  1  11 0 02 2 (42) P0,T E11 = J1 (kcρc) + 2 2 J1 (kcρc) , 8πα kc ρc E. Frequency comb structure of cyclotron power in which Z11 is the TE11 mode impedance, ρc defines the radial position of the gyrocenter of the electron in cylin- 2 Utilizing conventional techniques of handling δ func- drical coordinates, kc is the wavenumber for the cutoff tions and the relationship between the wave-number and frequency of the mode, and α is defined in Eq. (B8). 7

density then follows by using Eq. (19),

∞   2 X Ω0 + nΩa Pλ(ω) = 4P0,λ an v n=−∞ p.λ   2 Ω0 + nΩa cos (zt − zs) vp.λ

[δ(ω − (Ω0 + nΩa)) + δ(ω + Ω0 + nΩa)] . (45) FIG. 3. Schematic of an experiment with an electron un- dergoing cyclotron motion in a waveguide with a conductive Here we have assumed that the trap is symmetric, in short. The relevant parameters include: the origin, O, the po- which case an(−k) can be written in terms of an(k) as in sition of the electron, z0, the position of the waveguide short, Eq. (A8) (see Appx.A). zs, on the left side of the waveguide, and the position of the This power spectrum still has a comb structure, similar receiver, zr, on the right side of the waveguide. The magnetic to the one in the absence of a reflector at the end of the field, B, is parallel to the waveguide axis. waveguide. However, the amplitude of each peak is now modulated with an extra cos2 factor, which depends on the distance between the reflector and trap center, zt−zs. IV. EFFECTS OF WAVEGUIDE REFLECTION Therefore, while the introduction of a reflector increases the total power collected by the receiver, it also intro- duces a frequency-dependent amplitude for each peak in In our discussion of waveguides we have assumed in- the power spectrum. finite length, whereas any experimental realization of a CRES experiment must be finite in length. Allowing that one end of the waveguide must have a receiver, we are left V. TRAPPING GEOMETRIES with several options for the treatment of signals at the other end. In Sec.III, we built the foundation for calculating the One option is to add a second receiver. The signal CRES signal’s spectral features. From the obtained equa- observed by each receiver is then available for analysis, tions, it is clear that it is impossible to extract a simple at the cost of supporting two receiver systems. Another analytical solution that is valid and usable for every trap option is to install a terminator on one end of the waveg- configuration. Therefore, in this section we describe a uide. The receiver will detect only half of the electron’s step-by-step procedure to obtain the spectral properties radiated power and the signal will be the same as the of a CRES signal. We will then apply this procedure to case of the infinite waveguide. The final option, shown two simple and useful trap geometries, enabling us to de- in Fig.3, is to install a conductive short to the end rive numerical solutions for more complicated geometries of the waveguide, reflecting signals back to the first re- following these steps: ceiver. The first two options have been already analyzed. In this section we calculate the effects of the reflector on • An appropriate field approximation B(z) must be the power spectral density of the CRES signal. found. In some cases, where the expression of the The total mode excitation at the receiver, B˜λ(ω), is exact magnetic field is complex, one can consider ˜+ using a piecewise approximation of the field. a superposition of the direct wave, Bλ (ω), and the re- ˜− flected wave, Bλ (ω). The reflection induces a phase shift • With the assumed field profile, the electron’s equa- of 180 degrees. As a consequence, the total mode excita- tion of axial motion, Eq. (2), can be solved. Since tion at the receiver can be written as the effective potential in this equation depends only on the axial position of the electron, we can find a ˜ ˜+ − iπ Bλ(ω) = B (ω) + B (ω)e general solution, λ λ (43) = B+(ω) − B−(ω). λ λ Z z0(t) dz0 t = q . (46) z (0) 2 0 ˜± 0 (Ke − µB(z )) Using the definition of Bλ (ω) given by Eq. (20), we then m have • Once the axial motion of the electron is calculated, ˜ + ikλzr − ikλ(2|zs−zt|+zr ) the axial frequency follows. For the special case of Bλ(ω) = Aλ (ω)e − Aλ (ω)e , (44) a symmetric trap, we find where the expression is being evaluated at the receiver’s Z zmax −1 2 dz position, zr, and zs and zt are the positions of the reflec- Ωa = q . (47) π 0 2 tor and the trap center respectively. The power spectral m (E0 − µB(z)) 8

• Once the axial position of the electron is found at any given time, the value of magnetic field experi- 0 enced by the electron at that time, B(t), follows. −0.5 Finally, the cyclotron phase, Eq. (27), is found to −1 be −1.5 Z t eB(t0) −2 Φ (t) = dt0. (48) c −2.5 0 γme Total field - main [mT] −3 • To find the power in each peak, the Fourier coeffi- −3.5 cients introduced in Eq. (34) should be determined −4 by −6 −4 −2 0 2 4 6 z [cm] 1 Z Ta i(Φc(t)+kλz(t)) −i(Ω0+nΩa)t an = e e dt, (49) Ta 0 FIG. 4. The on-axis magnetic field profile of a “harmonic” trap (black line), generated by a single coil, and the corre- in which Ω0 is the average cyclotron frequency sponding approximation given by Eq. (52) with L0 = 20 cm given by (red line).

Φc(Ta) Ω0 = . (50) Ta and the maximum displacement for the electron is zmax = L0 cot θbot. • The power in each peak of the spectrum can be The magnetic field seen by the electron as a function determined, using Eq. (40), to be of time is 2 2 2  z z  Pn = P0,λ|an| . (51) max max Bz(t) = B0 1 + 2 − 2 cos(2Ωat) . (55) 2L0 2L0 • Finally, the total power radiated by the electron The cyclotron frequency Eq. (1) of a trapped electron, can be calculated by summing over the power of all peaks. This power will define the slopes of tracks  2 2  eB0 zmax zmax in Sec.VI. Ωc(t) = 1 + 2 − 2 cos(2Ωat) , (56) γme 2L0 2L0 follows. The last term describes the modulation in fre- A. Power spectrum in a “harmonic trap” quency and the first two terms determine the average cyclotron frequency The simplest magnetic bottle is realized with a single trapping coil producing a field anti-parallel to a back- eB  z2  Ω = 0 1 + max . (57) ground field. This geometry can be approximated as a 0 2 γme 2L0 purely axial field with parabolic z dependence as repre- sented by Fig.4. It can be described by The cyclotron phase, which can then be found by inte- grating over the cyclotron frequency, is  z2  B (z) = B 1 + , (52) z 0 2 Φ (t) = Ω t + q sin(2Ω t), (58) L0 c 0 a in which L0 is the characteristic length of the trap. Note in which the magnitude of the modulation is that this approximation is accurate for trapped electrons 2 with large pitch angle values that cannot travel to high eB0 zmax q = − 2 . (59) field regions. γme 4L0Ωa For the harmonic field approximation, electrons un- dergo simple harmonic motion in the axial direction, To find the power spectrum of the electron’s radiation, Fourier coefficients in Eq. (35) are needed, and can be

z(t) = zmax sin(Ωat), (53) calculated using the Jacobi-Anger expansion given by

iΦ (t)+ik z (t) i(Ω t+q sin(2Ω t)+k z sin(Ω t)) in which the axial frequency is determined by the axial e c λ 0 = e 0 a λ max a velocity at the trap minimum, ∞ X i(Ω0+(2m+p)Ωa)t = Jm(q)Jp(kλzmax)e , v0 sin θbot m,p=−∞ Ωa = , (54) L0 (60) 9

Modulation Index 5 0 1 2 3 4 5 6 7 4.5 1 Main Peak 4 First Order Sideband 3.5 0.8 Second Order Sideband 3 Relative Power 0.6 2.5 2 Total field - main [mT] 0.4 1.5 1 0.2 0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 −6 −4 −2 0 2 4 6 zmax [cm] z [cm]

FIG. 5. Relative magnitudes of the sidebands in a harmonic FIG. 6. Magnetic field profile generated by two coils sepa- trap as a function of the maximum axial travel, zmax, of a rated by 5 cm, forming a “bathtub” shape (black line), and trapped 30 keV electron. Here we consider an ideal harmonic the corresponding approximation given by Eq. (63) with L0 trap as described in Eq. (52), with a background field of 1 T = 35 cm and L1 = 0.5 cm(red line). and an L0 of 20 cm. No reflection effect is taken into account.

th B. Power spectrum in a “bathtub trap” where Jn is the n Bessel function of the first kind. Therefore, the power for each harmonic can be found from Eq. (51) by squaring The harmonic trap described previously has a limited trapping volume. A “bathtub trap,” generated using two ∞ coils, includes a wide flat region to extend the trapping X an(kλ) = Jm(q)Jn−2m(kλzmax) (61) volume. This field geometry is depicted in Fig.6. In this m=−∞ case, we approximate the field as a region of constant magnetic field between two half parabolas given piecewise by and using the appropriate P0,λ as found in sectionIIIF. Let us note that these coefficients, an, corresponds to the coefficients αm and βm defined by Eq. (32) and Eq. (33).   2  (z+L1/2) L1 This result matches well with our original intuition be-  B0 1 + L2 z < − 2  0 cause the modulation is harmonic, with a modulation in- L1 L1 Bz(z) = B0 − 2 < z < 2 , (63)  2  dex of q for the magnetic field induced modulation, and a  (z−L1/2) L1  B0 1 + L2 2 < z modulation index of kλzmax for the Doppler shift induced 0 modulation. The relative magnitude of the main peak and sideband powers for typical parameters are shown in in which L is a measure of field gradient in the curved Fig.5. 0 region and L1 is the width of the flat region. From Eq. (4), a 4 mT deep trap in a 1 T background magnetic field can trap electrons with pitch angles as For convenience, here we define t0 to be the time when small as 86 degrees. In this case, the magnitude of the the electron first enters the flat region from the curved modulation of the magnetic field experienced by the elec- region with negative z and we define t2 to be the time tron, q, will be smaller than 0.6, while the Doppler effect’s one half-period later when it enters the flat region from the opposite direction. This field configuration results in modulation, kλzmax, can be as large as 10.5. Therefore, constant velocity motion when the electron is in the flat Jm(q) can be approximated with δm0. In this case, the L1 power spectrum will be simplified to region, from t0 = 0 to t1 = v cos θ , and half harmonic 0 bot π motion at the two ends, for t2 − t1 = , in which the ωa angular frequency ωa of the half harmonic motion is de- v0 sin θbot fined by ωa = . The period of axial motion is L0 ∞ 2π ± X 2 then T = 2t2 = , which means that the frequency of P˜ (ω) = P J (k z ) Ωa λ 0,λ n λ max the axial motion is n=−∞ (62)

× [δ(ω − (Ω0 + nΩa)) + δ(ω + Ω0 + nΩa)] .  −1 2π L1 Ωa = 2L 2πL = ωa 1 + tan θbot . 1 + 0 πL0 This approximation works well for shallow traps in which v0 cos θbot v0 sin θbot ∆B B < 0.002. (64) 10

The equation of axial motion for an electron is thus 65  v t − L1 0 < t < t  z0 2 1 60  z sin[ω (t − t )] + L1 0 < t < t < t z(t) = max a 1 2 1 2 , 55 −v (t − t ) + L1 t < t < t ≡ t + t  z0 2 2 2 3 1 2 50  L1  −z sin[ω (t − t )] − t < t < T Frequency [a.u.] max a 3 2 3 45 (65) 40 with zmax = L0 cot θbot being the maximum displacement for the electron into the harmonic potential. 35 Using this equation and magnetic field configuration 30 from Eq. (63), the magnetic field seen by the electron as 25 a function of time is 20 0 1 2 3 4 5  B0 0 < t < t1 Time [a.u.]  z2 z2  max max  B0(1 + 2L2 − 2L2 cos[2ωa(t − t1)]) t1 < t < t2 Bz(t) = 0 0 . FIG. 7. Schematic of the power (represented by line width), B0 t2 < t < t3  2 2 as a function of time and frequency in the absence of a waveg-  zmax zmax  B0(1 + 2 − 2 cos[2ωa(t − t3)]) t3 < t < T 2L0 2L0 uide reflector. The main track and first order sidebands are (66) shown. Sudden losses of energy (and thus increases of fre- The cyclotron frequency of the electron is therefore quency), induced by collisions with background gas particles, happen at 1.5 ms and 3.5 ms.  1 0 < t < t1  z2 z2  max max  1 + 2 − 2 cos[2ωa(t − t1)] t1 < t < t2 eB0 2L 2L Ωc(t) = 0 0 . γme 1 t < t < t can be written as  2 3  z2 z2  max max 1 + 2L2 − 2L2 cos[2ωa(t − t3)] t3 < t < T 0 0 Ωc (67) S = 2 P. (69) mec + K0 and the average frequency of cyclotron radiation is Electrons can scatter off a molecule of the residual gas in the waveguide, causing abrupt energy losses, changes of 2  −1! eB0 zmax L1 pitch angle, and breaks in the observed tracks. Ω0 = 1 + 2 1 + tan θbot . (68) γme 2L0 πL0 The tracks parallel to the main tracks we call side- bands. These tracks are located at multiples of the axial Ωa The detailed calculation of the coefficients an, which frequency, fa = 2π , away from the main track; the order are used to calculate the power, can be found in Appx. of a sideband corresponds to this multiplicity. As long C. as we only consider time intervals short enough that the power radiated does not significantly change the axial fre- quency, sidebands will appear parallel to the main track. VI. SPECTRAL FEATURES IN CYCLOTRON Eq. (54) and Eq. (64) show how the axial frequency, RADIATION EMISSION SPECTROSCOPY measured from the frequency-distance to sidebands, can be used to relate the pitch angle and kinetic energy of an In this section we identify the features required for re- electron in a harmonic or bathtub traps. constructing the kinematics of an electron in a CRES The distribution of power between a main track and its experiment, based on the relationships in previous sec- sidebands depends on the electron’s energy and pitch an- tions. We also develop a common terminology for these gle. In the presence of a reflector on one end of the waveg- features. uide, as described in SectionIV, the distance between the The power spectrum of the signal generated by an elec- trap and the reflector will also impact the power distri- tron possesses a comb structure given by Eq. (38). If bution as shown in Fig.8. In realistic experiments, this we represent the power spectrum as a function of time is further complicated as tracks with suppressed power in a spectrogram, the excess of power forms connected will be indistinguishable from noise. structures that we call tracks. Fig.7 represents the tracks coming from the comb structure of the spectrum. The track at the average cyclotron frequency, given by VII. EXTRACTION OF KINEMATIC Eq. (7), is called the main track. As the electron ra- PARAMETERS FROM A MEASURED diates energy, the cyclotron frequency increases, caus- SPECTRUM ing the tracks to have a positive slope. For any given trap configuration, the track’s slope, S, is proportional The previous sections show that the primary observ- to the power radiated into both propagating and non- able parameters of a CRES signal are the frequency of propagating modes. According to Eq. (7) this relation the main track, the frequency separating the sidebands, 11

in Appx.C. Therefore we have 65 2 2 Z10e v0 2 πxc  60 P (ω) = cos λ πwh w 55 ∞ 50 X 2 2

Frequency [a.u.] |an (kλ)| cos [(zt − zs)kλ] 45 n=−∞ 40 [δ(ω − (Ω + nΩ )) + δ(ω + Ω + nΩ )] . 35 0 a 0 a (70) 30 Of the observables, the start frequency of the main 25 track is the most strongly related to the electron’s kinetic 20 energy. The cyclotron frequency of a 30 keV electron, 0 1 2 3 4 5 Time [a.u.] with a pitch angle of 90 degrees at the center of the trap, is 26.44 GHz. However, this frequency is increased by FIG. 8. Schematic of the power (represented by line width) the electron’s pitch angle as described in Sec.IID. The as a function of time and frequency in the presence of a waveg- distortion of the distribution of main track frequencies uide reflector. The main track appears and disappears as the by pitch angle is shown in Fig. 10. kinematic parameters (such as the pitch angle) change as a The other signal parameters can be used to correct result of collisions with background gas particles. for the pitch angle effect and recover the true kinetic energy of the electron. Decreasing the pitch angle will simultaneously increase the start frequency and effects the other parameters discussed above. In principle, only a subset of the parameters are needed to find the pitch angle and recover the correct energy. the power in both the main track and sidebands, and the A sufficiently precise measurement of the axial fre- slope of the main track. quency alone can be used to correct the main track fre- quency, yielding the cyclotron frequency at the center of For a given configuration of trapping field and waveg- the trap. However, extraction of the axial frequency is uide, these parameters are completely determined by the possible only for pitch angles for which there are at least electron’s kinetic energy and pitch angle. However, the two visible tracks above the noise level. converse is not in general true. The axial frequency in In other cases, other parameters must be used. Track a real magnetic trap is double valued with respect to power carries valuable information, though typically the pitch angle whenever the floor of the trap is flatter power measurements in CRES experiments are less pre- than harmonic. This is because the axial frequency is cise than frequency measurements, and may not be pos- relatively low both for small amplitudes and for ampli- sible if the noise level is high. Furthermore, the power is tudes that almost eject the electron over the trap-field double-valued for non-shallow trap geometries that can maxima, and it reaches a broad maximum for intermedi- trap electrons with smaller pitch angles. Determining ate amplitudes. Other ambiguities arise when resonant a track’s slope is a frequency measurement, measurable structures such as those described in Sec.IV cause the even if the main track or sidebands are absent, and there- slope to have multiple values. These ambiguities can be fore is a most reliable parameter for correction. However, mitigated at the design stage and by making use of all the slope is double-valued for this example. The precise the available information in the signal. We will now give algorithm for combining the parameters to achieve a high a concrete example of predicting the observable parame- resolution energy measurement will, therefore, depend on ters from a particular trapping field, and then speculate the particular geometry and signal-to-noise of the CRES on the observations needed to reconstruct the electron’s experiment. initial kinetic energy.

For our example in Fig.9, we will use a bathtub trap VIII. CONCLUSION with an L0 of 35 cm and an L1 of 0.5 cm in a 1.07 cm wide rectangular waveguide. We consider a short on one We have found that electrons in a CRES experiment end of the waveguide, a distance 0.6 cm away from the undergo nontrivial but predictable motion within a mag- trap center, and a 1 T background magnetic field. We netic bottle, and this motion affects the detected cy- will examine predicted signals from electrons with 30 keV clotron signal. We identified the carrier and sideband of kinetic energy and with different pitch angles. We structure of the signal, and have shown that these fea- find the power in the nth harmonic for this situation us- tures encode the entirety of the electron’s kinematic pa- ing Eq. (45), which includes the short, using the power rameters. Following the results derived here, a suffi- from Eq. (41), which is for the rectangular waveguide. ciently precise measurement of these features should al- The Fourier coefficients for the bathtub trap are found low complete reconstruction of the electron’s kinetic en- 12

40

35 Counts [a.u.]

30

Axial Frequency [MHz] 25

20

15

40 42 44 46 48 50 52 54 39 40 41 42 43 44 Start Frequency - 26.4 GHz [MHz] Measured Start Frequency - 26.4 GHz [MHz]

0.9 Main Peak FIG. 10. Simulation of the energy spectrum of electrons sampled from a 60 eV wide Lorentzian, centered at 30 keV, 0.8 First Sideband in a 1 T background magnetic field. In blue, the spectrum 0.7 Second Sideband of the extracted start frequencies for an idealized case where 0.6 the magnetic field is flat and all electrons have a 90 degrees Track Power [fW] pitch angle. In red, the actual lineshape when the electrons 0.5 have an isotropic momentum distribution and are confined in 0.4 a 4 mT deep ideal harmonic trap as given in Eq. (52). The 0.3 blue histogram is scaled down, so it can be compared with the red one. 0.2 0.1

0 40 42 44 46 48 50 52 54 and may be able to calibrate some of the detector config- Start Frequency - 26.4 GHz [MHz] uration as well. Notably, we point out that for configu- rations where the electron undergoes axial motion larger 0.5 than a half wavelength of cyclotron radiation, the mod- ulation is such that detection and interpretation of side- 0.45 bands is necessary to detect all trapped electrons. The practicalities of signal detection and reconstruc- 0.4 tion will depend on the particular apparatus design and

Track Slope [MHz/ms] detection scheme, in particular the signal-to-noise ratio 0.35 of the sidebands, and we leave a discussion of the pre- cise reconstruction algorithm and ultimate resolution to 0.3 future work.

0.25

40 42 44 46 48 50 52 54 IX. ACKNOWLEDGMENTS Start Frequency - 26.4 GHz [MHz] This material is based upon work supported by the FIG. 9. Spectral features of the CRES signal for 30 keV following sources: the U.S. Department of Energy Of- electrons with different values of pitch angle and single radial position at ρ = 0 cm, trapped in an ideal baththub trap de- fice of Science, Office of Nuclear Physics, under Award No. DE SC0014130 to UCSB, under Award No. de- scribed by Eq.(63) with L0 = 35 cm and L1 = 0.5 cm, in a 1 T background field including the effect of a short. An elec- sc0011091 to MIT, under the Early Career Research Pro- tron with a pitch angle of 90 degrees has a start frequency of gram to Pacific Northwest National Laboratory (PNNL), 26.44 GHz while electrons with lower pitch angles are subject a multiprogram national laboratory operated by Bat- to pitch angle effects which increase their start frequencies. telle for the U.S. Department of Energy under Con- This shift can systematically affect energy measurements in tract No. DE-AC05-76RL01830, under Award No. DE- CRES experiments. The above plots illustrate how different FG02-97ER41020 to the University of Washington, and measurable quantities in a CRES experiment can be used to under Award No. de-sc0012654 to Yale University; the correct for this frequency shift. National Science Foundation under Award Nos. 1205100 and 1505678 to MIT; Lab-Directed Research and Devel- opment at LLNL (18-ERD-028), Prepared by LLNL un- ergy, which is necessary for proposed CRES experiments der Contract DE-AC52-07NA27344; the Massachusetts to achieve their desired sensitivity. In fact, the mea- Institute of Technology (MIT) Wade Fellowship; the Lab- surable features over-constrain the kinematic parameters oratory Directed Research and Development Program at 13

PNNL; the University of Washington Royalty Research By equating the coefficients with those of the first ex- Foundation. A portion of the research was performed us- pression in Eq. (36), we arrive at the form, ing Research Computing at Pacific Northwest National Laboratory. We further acknowledge support from Yale n −2ikλzt University, the PRISMA Cluster of Excellence at the Uni- an(−kλ) = (−1) e an(kλ), (A7) versity of Mainz, and the KIT Center Elementary Parti- cle and Astroparticle Physics (KCETA) at the Karlsruhe which is consistent with our expectation of equal power Institute of Technology. propagating in both directions, since

Appendix A: Property of the Fourier coefficients in 2 2 |an(−kλ)| = |an(kλ)| . (A8) a symmetric trap

In a trap where the magnetic field distortion is sym- Appendix B: P0,λ calculation for two specific metric with respect to the center of the trap, we expect waveguide geometries the same amplitude of radiation to propagate in both di- rections in the waveguide. This means that we need to show that The power amplitude, P0,λ, was introduced in Eq. (39) as a measurement of an electron’s coupling to a waveg- 2 2 |an(−kλ)| = |an(kλ)| . (A1) uide mode. The calculation details for two particularly relevant cases are shown here. Two useful expressions in symmetric traps will assist us in deriving this relation. The first relates to the peri- odicity of the electron’s position, z , in a symmetric trap, 0 1. Rectangular waveguide TE10 mode given by

 T  The first example is the fundamental mode of a rect- −(z (t) − z ) = z t + a − z , (A2) 0 t 0 2 t angular waveguide. For such a waveguide, with w being its longer dimension (defined to be along the x axis) and in which zt is the axial position of the center of the trap. h the smaller one (along the y axis), the electric field has Furthermore, a symmetric trap forces the cyclotron fre- the form quency to be periodic, with period equal to half of the πx axial motion’s period. Therefore the cyclotron phase sat- Ey(x) = K cos y.ˆ (B1) isfies w   Eq. (15) can now be used to find the normalization fac- Ta Ta Φc t + = Ω0 + Φc(t). (A3) tor, giving 2 2 r Utilizing Eq. (A2) for Φ (t) and Eq. (A3) to rewrite Z πx 2 c K2 cos2 dxdy = 1 ⇒ K = . (B2) kλz0(t), we can write A w wh  T  T Φ (t) − k z (t) = Φ t + a − Ω a With the normalized field, the expression for P0,T E10 c λ 0 c 2 0 2 follows from the definition in Eq. (39) and is found to be   (A4) Ta + kλz0 t + − 2kλzt. 2 !2 Z e2v2 r 2 πx  P = 10 0 cos c Therefore we have 0,T E10 8 wh w (B3) iΦ (t)−ik z (t) −iΩ Ta −2ik z iΦ (t+ Ta )+ik z (t+ Ta ) 2 2 e c λ 0 = e 0 2 λ t e c 2 λ 0 2 . Z10e v πxc  = 0 cos2 . (A5) 4wh w

Using Eq. (34), we expand the second exponent to get 2. Circular waveguide TE mode ∞ 11 Ta Ta iΦc(t)−ikλz0(t) −2ikλzt−iΩ0 X i(Ω0+nΩa)(t+ ) e = e 2 an(kλ)e 2 n=−∞ The second example to consider is that of a circular ∞ waveguide with radius R. The TE11 mode has the lowest −2ikλzt X n i(Ω0+nΩa)t = e (−1) an(kλ)e . cutoff frequency in a circular waveguide and the associ- 1.841 n=−∞ ated wavenumber is kc = R . This mode consists of (A6) two degenerate modes for which the electric field can be 14 found in [14], The perturbation to the average cyclotron phase can be written as −iωµ E1ρ(ρ, φ) = K 2 cos(φ)J1(kcρ), kc ρ Φc(t) − Ω0t = iωµ 0 (B4) E (ρ, φ) = K sin(φ)J (k ρ),  −∆Ωt 0 < t < t 1φ 1 c  1 kc  ωa ∆Ω  ∆Ω (t − t1) − sin[2ωa(t − t1)] − ∆Ωt t1 < t < t2 Ωa 2Ωa . −∆Ω(t − t ) t < t < t E1z(ρ, φ) = 0  2 2 3  ωa ∆Ω  ∆Ω (t − t3) − sin[2ωa(t − t3)] − ∆Ω(t − t2) t3 < t < T Ωa 2Ωa and (C2) −iωµ 0 The coefficients, αn, can then be found to be E2ρ(ρ, φ) = K 2 sin(φ)J1(kcρ), kc ρ 0 iωµ 0 (B5) E2φ(ρ, φ) = K cos(φ)J1(kcρ), 1 Z T kc α = eiΦc(t)−iΩ0te−inΩatdt. (C3) n T E2z(ρ, φ) = 0. 0 The same technique is used to find the normalized fields, The integral can be computed by splitting it into four Z pieces as  2 2  1 = E1ρ(ρ, φ) + E1φ(ρ, φ) ρdρdφ A (B6) 2 2 Z R  2  1 2 ω µ J1 (kcρ) 02 αn = (An + Bn + Cn + Dn) , (C4) = −K π 2 2 2 + J1 (kcρ) ρdρ. T 2kc 0 kc ρ Hence the normalization factor can be found to be in which ik K = K0 = √c (B7) ωµ πα Z t1 iΦc(t)−iΩ0t −inΩat in which An = e e dt 0 R 2   Z   t1 t J1 (kcρ) 02 −i(∆Ω+nΩa) 1 = t1e 2 sinc (∆Ω + nΩa) , (C5) α = 2 2 + J1 (kcρ) ρdρ. (B8) 0 kc ρ 2 Z t2 iΦc(t)−iΩ0t −inΩat The calculation of the coefficients P0,T E11 follows the Bn = e e dt rectangular waveguide calculation with one difference. t1 That is, to find the power in the waveguide, the two de- ∞   π ∆Ω π −i(∆Ω+nΩa)t1/2 X −in generate modes’ powers should be added together. This = e Jm e 2 ω 2Ω gives us a m=−∞ a   2 2 t1 nπ Ωa Z11e v0  2 2 2 2  sinc ∆Ω − + mπ , (C6) P0,T E11 = E1φ + E1ρ + E2φ + E2ρ 2 2 ωa 8 t 2 2   (B9) Z 3 Z e v 1 iΦc(t)−iΩ0t −inΩat 11 0 02 2 Cn = e e dt = J (kcρc) + J (kcρc) . 1 2 2 1 t 8πα kc ρc 2 n = (−1) An, (C7) Z T iΦc(t)−iΩ0t −inΩat Appendix C: Bathtub Trap Calculation Dn = e e dt t3 n The “bathtub” trapping geometry was introduced in = (−1) Bn. (C8) Sec.V. Here we present detailed calculations of both the phase and the axial motion Fourier expansion coefficients, defined by Eq. (32) and Eq. (33), respectively. Note that for odd values of n the coefficient αn is zero. First, we define the frequency difference between the The determination of βn follows in a similar manner. average cyclotron and the frequency at the bottom of the The electron’s equation of motion (Eq. (65)) gives trap using Eq. (64),

2 −1 eB Ω z  L  T ∆Ω ≡ Ω − 0 = c max 1 + 1 tan θ 1 Z 0 2 β = eikλz(t)e−inΩatdt γme 2 L0 πL0 n (C1) T 0 (C9) Ω z2 Ω = c max a . 1 2 = (En + Fn + Gn + Hn) , 2 L0 ωa T 15 where

Z t1 Z t3 ikλz(t) −inΩat ikλz(t) −inΩat En = e e dt Gn = e e dt 0 t2   t1 t t   −inΩa 1 n −inΩ 1 t1 = t1e 2 sinc (kλvz0 − nΩa) , (C10) = (−1) t e a 2 sinc (k v + nΩ ) , (C12) 2 1 λ z0 a 2 Z t2 Z T ikλz(t) −inΩat ikλz(t) −inΩat Fn = e e dt Hn = e e dt t1 t3 ik L /2 π −inΩ t1/2 π = e λ 1 e a = (−1)ne−ikλL1/2 e−inΩat1/2 ωa ωa ∞ mπ nπ Ω  ∞   X m−n a X −m−n mπ nπ Ωa Jm(kλzmax)i sinc − , Jm(kλzmax)i sinc − . 2 2 ωa 2 2 ω m=−∞ m=−∞ a (C11) (C13)

The coefficients αn and βn can be used to find an as defined in Eq. (35). These an coefficients are a mea- sure of the relative power in the nth peak of the power spectrum, according to Eq. (38).

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