Phase-Resolved Electron Guiding in Optimized Chip-Based Microwave Potentials

Surface-electrode chips are a versatile and well-established tool for trapping and guiding charged particles. The technique, usually applied to ions, has recently been adapted for electrons using a two-dimensional quadrupole guide at microwave driving frequencies. During injection into the guiding potential, the electron trajectories show a strong dependence on the phase and amplitude of fringing electric fields at the coupling entrance of the guide. Here we study the corresponding electron dynamics using a pulsed electron source with pulse durations of several 100 ps to temporally resolve fringing electric fields oscillating at the microwave drive frequency. By synchronizing the timing of the electron pulses to a certain microwave phase, we can increase the electron guiding efficiency by 50%. Furthermore, we present numerically optimized electrode structures for which the amplitude of fringing electric fields at the coupling entrance of the guide is drastically reduced. Particle-tracing simulations suggest that the optimized electrode layout allows the direct injection of electrons into the lowest-lying motional quantum states of the transverse guiding potential.

Figure 1

Guiding potential in the pseudopotential approximation. (a) Pseudopotential $\mathrm{\Psi }$ in the $x\text{−}z$ plane with a cross-sectional view of the electrodes. The height of the electrodes is exaggerated for better visibility. (b) Pseudopotential along the $x$ direction (blue) with harmonic fit (green). (c) Pseudopotential along the $z$ direction (blue) with harmonic fit (green). The confinement of electrons is weakest along the $z$ direction with a potential depth of 30 meV.

Figure 2

Electron guiding experiment. (a) Photograph of the electron source (top left), microwave guiding chip (center), and the imaging electron microchannel-plate detector (bottom). Here the mu-metal shield and a gold-plated copper housing enclosing the experiment are removed. (b) Electron guiding signal with the guided electrons recorded at a position $x=0\mathrm{mm}$. When the microwave drive is turned off, the unguided electron spot is detected at $x=21\mathrm{mm}$ (not shown). The color scale depicts the electron count rate. (c) Simulated guiding signal. Here the color scale corresponds to the phase of the microwave driving signal at the instant in time when the electron is injected into the guide.

Figure 3

Simulation of electric fields close to the substrate edge where electrons are injected into the guiding potential. The color scale displays the absolute value of the electric field $|\mathbf{E}|$ at a snapshot in time where it is at its maximum. The last aperture of the electron gun is shown schematically at $y=0\mathrm{mm}$ (drawn in black). The microwave substrate at $z=0\mathrm{mm}$ is indicated in gray starting from $y=0.5\mathrm{mm}$. Fringing fields near the substrate edge lead to significant local deviations of the electric field from the ideal quadrupolar shape. As a consequence, an electron impinging at a height $R_0=500\mu \mathrm{m}$ experiences an electric-field amplitude $E$ and becomes deflected during injection.

Figure 4

Electron trajectory simulations resolving the phase of the microwave drive. Electrons enter the guiding potential at $y=0\mathrm{mm}$ through the exit aperture of the electron gun (drawn in black) at a height of $z=500\mu \mathrm{m}$ above the substrate. The initial electron spot size in the simulations is $60\mu \mathrm{m}$ in diameter. After propagation through the guiding potential, electrons are released into free space at $y=37\mathrm{mm}$ and detected in the detector plane at $y=45\mathrm{mm}$ (not shown). (a),(b) show a side view of ray-tracing simulations in the $y\text{−}z$ plane along the guiding electrodes. The color of the electron trajectories is chosen for better visibility and does not contain any further information here. (a) For a microwave phase of $\varphi =0^{\circ }$, all electrons are confined in the guiding potential and oscillate with the trap frequency $\omega$ corresponding to a spatial period of approximately 9.5 mm. In (b), a significant portion of electrons is lost for a microwave phase of $\varphi =180^{\circ }$. Here electrons are lost from the guiding potential in the vertical $z$ direction starting from $y=15\mathrm{mm}$. (c),(d) show the simulated guiding signals for the corresponding microwave phases. In (c), for $\varphi =0^{\circ }$, all trajectories are guided along the curve and hit the detector around $x=0$, whereas in (d), a large fraction of the trajectories is lost for $\varphi =180^{\circ }$.

Figure 5

Conductor layout, vertical pseudopotential gradient along the guide axis, and electron-trajectory simulations for optimized coupling structures. In (a),(d),(g), the signal electrodes are drawn in blue, whereas white corresponds to grounded areas (gapless-plane approximation). In the numerical optimization of the electrode shape, the width of each electrode is constrained to be larger than $20\mu \mathrm{m}$. In (b),(e),(h), the dimensionless, vertical pseudopotential gradient $F_z$ is shown. The last column (c),(f),(i) shows simulated trajectories (white curves) together with a color plot of the pseudopotential. For better visibility, trajectories are shown for two selected microwave phases where the electron performs the most- and the least-excited oscillation within the guiding potential. Note the different scaling of the $z$ axis as well as the color scale going from Figs. 5 to 5 and 5. Clearly, the reduction in the vertical pseudopotential gradient is reflected in a smaller oscillation amplitude of the electron after injection.

Figure 6

Pulsed electron source. (a) Schematic of the deflection element together with results of a particle-tracing simulation (blue trajectories). The trajectories are simulated with a static deflection voltage of 0.5 V applied to the upper (green) and $-0.5\mathrm{V}$ to the lower (red) deflection electrode. The electron beam enters through an aperture at $y=-7\mathrm{mm}$ with a diameter of $150\mu \mathrm{m}$ and becomes deflected onto the exit aperture with a diameter of $50\mu \mathrm{m}$ at $y=11\mathrm{mm}$. Only electrons that pass through the exit aperture, which is just visible at $z=0\mathrm{mm}$, are later injected into the guide. (b) Simulated transit time as a function of the deflection voltage $V_{\text{defl}}$. A minimum transit time of 51 ps can be reached by applying $V_{\text{defl}}=1.3\mathrm{V}$. (c) Simulated (black) and measured (red) electron count rate after the exit pinhole as a function of $V_{\text{defl}}$.

Figure 7

Schematic of the microwave phase-resolved measurement setup. A stable phase lock is established via the 10-MHz reference signal between two analog signal generators in order to separately drive the microwave guiding chip and the pulsed source. We insert a variable phase shifter in the signal path of the guiding chip to shift the microwave phase of the guide relative to the pulsed source. The $-180^{\circ }$ phase shifter in the signal path of the pulsed electron source yields an opposite sign of the deflection voltage $V_{\text{defl}}$ on the upper and lower part of the deflection element.

Figure 8

Microwave phase-resolved measurements. (a) Electron signals are shown for $\mathrm{\Delta }\varphi =3^{\circ }$ and $\mathrm{\Delta }\varphi =185^{\circ }$ where the number of guided electrons is at its maximum and minimum with respect to the microwave phase $\mathrm{\Delta }\varphi$. (b) Relative change in count rate of guided (red triangles) and lost (blue dots) electrons as a function of the relative microwave phase for synchronous operation of the electron gun. When the gun is operated in asynchronous mode, with ${\mathrm{\Omega }}_{\text{defl}}=\mathrm{\Omega }/(2n)+2\pi \times 1\mathrm{mHz}$, we observe no change in the guided and unguided signal (black squares). Every data point is an average over 240 images with 1-s exposure time which corresponds to $4.78\times 10^{10}$ electron pulses. The lines correspond to sinusoidal fits of the measured data.