# A Feedback Control Method to Maintain the Amplitude of the RF Signal Applied to Ion Traps
## 1 Introduction  
The development of high-fidelity quantum gates with trapped ions is very important for quantum computing [1; 2; 3; 4; 5]. In the realization of ion trap-based quantum computing systems, the motional mode of the trapped ions is applied for quantum computation [6; 7; 8; 9]. The frequency of the motional mode must be fixed as the quantum gate fidelity decreases when the frequency drifts. Since the frequency of the motional mode and secular frequency of the trapped ion are directly related, the drift of the secular frequency affects the reliability of the internal state manipulation. The noisy secular frequency excites the motional state and produces heating [10]. In addition, the noisy frequency changes the parameters such as the trap depth and position in the ion shuttling experiment [11; 12]. Therefore, the secular frequency must be accurately controlled [13; 14]. If the mass of the trapped ion, and thedistance between the trapped ion and the electrodes of the ion trap do not change, the secular frequency \(\omega\) can be expressed as follows [13; 15; 16; 17; 18]:  
\[\omega\propto\frac{V}{\Omega}, \tag{1}\]  
where \(V\) and \(\Omega\) denote the amplitude and frequency of the radio frequency (RF) signal applied to the ion trap, respectively. Since the frequency of the RF signal from a signal generator does not drift, the secular frequency of the trapped ion changes only by the amplitude of the RF signal. The amplitude is increased by an RF amplifier and a helical resonator [18], which is vulnerable to the changes in the ambient temperature of the helical resonator and vibration of the optical table. These factors cause a drift in the resonant frequency and eventually change the amplitude of the RF signal. Therefore, the amplitude of the RF signal increased by the helical resonator needs to be maintained at a constant value. The change in the ambient temperature and the vibration of the optical table are less than a few kHz. Resistors, capacitors, and other electrical components may also be slightly affected by these factors. However, those effects are ignored because the bandwidth of the field-programmable gate array (FPGA)-based feedback controller developed in this paper is 133 kHz.  
A method for reducing the fluctuation in the amplitude of the RF signal applied to the ion trap, using a commercial universal proportional-plus-integral (PI) controller (LB1005), has been developed [13]. With this method, the control method cannot be changed to suit different experimental conditions, including derivative control or lead-lag compensation. It is also well known that a PI controller can have a long-term drift error. In addition, the feedback configuration cannot be changed to increase the resolution of the controller. The developed method is also unable to increase the bandwidth of the PI controller to more than 1 MHz (LB1005 limited to 1 MHz). We implement the use of a feedback controller that is similar to that developed in the previous work and define it as the baseline method. Subsequently, we compare this method with the field-programmable gate array (FPGA)-based feedback control method under the same conditions. When using the baseline method, the fluctuation in the amplitude of the RF signal is 1.806%. The amplitude is measured by an analog-to-digital converter (ADC). In this paper, we develop a method for reducing the fluctuation in the amplitude of the RF signal using FPGA-based feedback control [19; 20]. With this method, the control method can be freely selected, and various feedback configuration for increasing the resolution can be implemented. Furthermore, we design and utilize a PI controller and a lag compensator using Verilog. With the FPGA-based PI controller and lag compensator, the fluctuations in the amplitude of the RF signal are 0.099% and 0.102%, respectively. These values are 94.510% and 94.359% lower, respectively, than that obtained with the baseline method.