A Radio-Frequency Emitter Design for the Low-Frequency Regime in Atomic Experiments

TL;DR

This work tackles delivering high-current, broadband RF drive in the low-frequency regime for cold-atom experiments by introducing a capacitive transformer network (CTN) that treats the RF coil as an intrinsic inductor and includes a virtual load for flexible impedance matching. It presents both broadband and narrowband CTN configurations, enabling stable current across a wide frequency range and high Rabi frequencies at low power. The broadband CTN enables efficient evaporative cooling of a $^{87}$Rb-$^{40}$K Bose-Fermi mixture, dramatically reducing input power while maintaining coil current, whereas the narrowband CTN supports rapid Zeeman-state manipulation with an estimated $\Omega_r$ near $9\ \text{kHz}$ at $0.1$ dBW. Collectively, the CTN approach offers compact, robust RF delivery with broad applicability to complex experimental environments, including potential use in space-based quantum gas platforms.

Abstract

Radio-frequency (RF) control is a key technique in cold atom experiments. We present a compact and efficient RF circuit based on a capacitive transformer network, where a low-frequency coil operating up to 30MHz serves as both an intrinsic inductor and a power-sharing element. The design enables high current delivery and flexible impedance matching across a wide frequency range. We integrate both broadband and narrowband RF networks into a unified configuration that overcomes the geometric constraints imposed by the metallic chamber. In evaporative cooling, the broadband network allows a reduction of the applied RF input power from 14.7dBW to -3.5dBW, owing to its non-zero coil current even at ultra-low frequencies. This feature enables the Bose-Fermi mixture to be cooled below 10μK. In a Landau-Zener protocol, the coil driven by the narrowband network transfers 80% of rubidium atoms from |F = 2,mF = 2> to |2,-2> in 1 millisecond, achieving a Rabi frequency of approximately 9 kHz at an input power of 0.1dBW.

Figures (6)

• Figure 1: Schematic of the CTN for impedance matching of a low-frequency RF coil, with $C_{\rm op}$ as an optional capacitor that is replaced by a wire unless otherwise specified. (a) The RF coil serves as the built-in inductor, canceling the reactance introduced by the capacitors. (b) Conventional designs for low-frequency RF networks, featuring a series-to-load scheme and a dual-capacitor LTN. (c) Fully numerically calculated $Q_L$ for the CTN (solid green line) is lower than that of the LTN (dashed line), indicating the CTN's potential for broadband impedance matching. The analytical result (green and gray line) agrees well with the numerical simulation. (d) Comparison of delivered power ratio from the amplifier, defined as $1 - |S_{11}|^2$, and coil current for the CTN and LTN at the resonant frequency of $12\,\mathrm{MHz}$ (indicated by the star in (c)). The LTN behaves as an open circuit at low frequencies due to its capacitive nature, while the CTN retains its inductive response, allowing continued power delivery in this regime. Parameters for (c)(d) are $L_{\rm C} = \unitqty{100}{\nano\henry}, R_{\rm C}=\unitqty{1}{\ohm}$, and $R_{\rm S} = R_{\rm L} =\unitqty{50}{\ohm}$, with $C_{1,2} = (0.67, 1.6)\,{\rm nF}$ for LTN and $C_{1,2}=(2.7, 4.7)\,{\rm nF}$ for CTN. The coil current in (d) is evaluated at a nominal input power of 0 dBW.
• Figure 2: Effective current distributions for different inductances $L_{\rm C}$ and resistances $R_{\rm C}$ of the RF coil under resonant conditions at 12 (top row) and 20.2 (bottom row). The red dashed line denotes the boundary of feasible impedance matching (see text), where the coil receives the maximum power deliverable from the amplifier. The white region indicates the parameter range where at least 70% of this maximum power is delivered to the coil. All impedance matching networks use the CTN configuration from Fig. \ref{['fig1']}(a), unless otherwise specified. (a) Comparison of four coil materials with different resistance and inductance values. (b) Adding an optional capacitor $C_{\rm op}$ shifts the current distribution, the resonance solution region, and the impedance matching region along the inductance axis. This adjustment enables the enameled copper wire to achieve higher current while maintaining good impedance matching. (c) $\abs{S_{11}}$ before and after the inclusion of $C_{\rm op}$, with comparison to our previous LTN configuration using the same enameled copper wire. (d)-(f) Corresponding measurements and simulations at approximately 20.2. Capacitor values used in the CTN are: For (a): $C_1 = \unitqty{5.6}{\nano\farad}$, $C_2 = \unitqty{3.3}{\nano\farad}$; For (b): $C_1 = \unitqty{3.3}{\nano\farad}$, $C_2 = \unitqty{10}{\nano\farad}$, $C_{\rm op} = \unitqty{5.6}{\nano\farad}$; For (d) and (e): $C_1 = \unitqty{15}{\nano\farad}$, $C_2 = \unitqty{0.36}{\nano\farad}$ and $C_1 = \unitqty{3.3}{\nano\farad}$, $C_2 = \unitqty{1.8}{\nano\farad}$, $C_{\rm op} = \unitqty{0.56}{\nano\farad}$, respectively. All data points are obtained by fitting measured $\abs{S_{11}}$ curves and serve as approximate estimates of the parameters. $R_{\rm C}$ denotes the total resistance in the coil branch. The coils are circular single-turn loops with a diameter of 4.1 in (a)-(c) and 6 in the other panels. The current distributions are calculated under an assumed 0 dBW input power.
• Figure 3: Configuration of the two coils (made from RG316 and $d$=1.0 enameled copper wire) used for evaporative cooling and sublevel manipulation, with impedance matching measurements and simulations corresponding to the former (right) coil. (a) Photograph of the two half-annular coils mounted on a 3D-printed resin holder. The holder is installed just above the upper viewport of the vacuum chamber. $d_{\rm mount}$ denotes the inner diameter of the holder and the outer diameter of the coil. (b) Engineering schematic of the coil holder. (c) Simulated horizontal RF magnetic field at the atomic position for an ideal conductor in vacuum. The dot indicates the relative location of the atomic cloud. (d) Simulated current and power through the evaporation coil as a function of frequency, based on the experimentally applied input power of 14.7 dBW, with transmission losses taken into account. Impedance mismatch at ultra-low frequencies is not a concern, as the input RF power has already decreased by two orders of magnitude at these frequencies (see Fig. \ref{['fig4']}(b)). The electronic components used are $R_{\rm C} \approx \unitqty{0.7}{\ohm}$, $L_{\rm C} \approx \unitqty{175}{\nano\henry}$, and dual capacitors $C_1 = \unitqty{4.7}{\nano\farad}$, $C_2 = \unitqty{0.3}{\nano\farad}$ for evaporation, and $R_{\rm C} \approx \unitqty{0.35}{\ohm}$, $L_{\rm C} \approx \unitqty{275}{\nano\henry}$, $C_1 = \unitqty{10}{\nano\farad}$, $C_2 = \unitqty{15}{\nano\farad}$ for sublevel manipulation.
• Figure 4: RF evaporative cooling of a Bose-Fermi mixture in a quadrupole magnetic trap. (a) The variation in $^\text{87}\text{Rb}$ atom number and temperature over four stages. The atom number decreases from $1.2 \times 10^7$ to $3.8 \times 10^5$, while the temperature drops from 260(12) to 9(2). Shaded bands represent the standard deviation obtained from fitting. Individual error bars are smaller than the marker size. (b) Radio frequency is decreased almost linearly from 28 to 0.6 over 10 seconds. The power was reduced by more than 18 dB (from 14.7 dBW to -3.5 dBW) during the final two stages to minimize atom loss, since the coil maintains substantial current at ultra-low frequencies. (c) In situ absorption images of $^\text{40}\text{K}$ atoms at each stage reveal a significant increase in density. The plug beam has a waist of about 40μ, a power of 800, and a wavelength of 760. The magnetic field gradient is held constant at 150 G/cm throughout the evaporation sequence.
• Figure 5: Measurement of the reflection coefficient and Landau-Zener tunneling using a narrowband emitter based on a CTN design. (a) The reflection coefficient $\abs{S_{11}}$ reaches a minimum near 4 MHz, confirming resonance. Low-pass behavior persists below this frequency. (b)(c) Population transfer from $\ket{2,2}$ (red circles) to $\ket{2,-2}$ (blue circles), measured using the Stern-Gerlach method. Data are normalized to the initial atom number. Error bars represent the standard deviation. The gray line shows the full numerical simulation, while solid lines represent smoothed sigmoid-like fits $\tilde{S}(t)$, used to extract the smoothed population dynamics. A vertical bias field (aligned with gravity) is linearly ramped down over 2.9, while the radio frequency is fixed at 3.91 MHz and the input power is set to 0.1 dBW. The magnetic gradient is adjusted to balance the gravitational force on the $\ket{2,2}$ state. Horizontal confinement is provided by a weak crossed optical dipole trap.