In trapped ion experiments, RF voltages of several hundreds of volts are commonly applied to the trap, which has a simple capacitor as its electrical circuit equivalent. Thus, we need to design a resonator that maximizes the RF voltage at this capacitor for a given input power. In this section, we discuss general considerations in resonator design which are not limited to a specific resonator type.
Choosing the trap drive frequency
At first, we consider the losses in an RLC resonator and its scaling with frequency. From the solution of the equation of motion of a trapped ion in a Paul trap (Mathieu equation), one obtains the trap voltages with the stability parameter q [18], which scales as
$$q = \frac{2 e V}{M r_{0}^{2} {\varOmega }^2} \propto \frac{V}{{\varOmega }^{2}}$$
(1)
where V is the amplitude of the RF voltage at the trap and \({\varOmega }\) is the trap drive (angular) frequency. Hence, for constant q, the loss power scales as
$$P \propto V^{2} \propto {\varOmega }^{4}.$$
(2)
A low trap drive frequency will reduce the losses. On the other hand, high secular motion frequencies of the ion in the trap are desirable for operation. Higher secular frequencies require a higher \({\varOmega }\), and thus, the desired secular frequencies set a lower limit for \({\varOmega }\). In this study, we aim for a q around 0.25, an axial secular frequency of 1 MHz, and both radial frequencies to be 3.7 MHz. This will require a trap drive frequency of \({\varOmega }= 2 \pi\) 42.6 MHz. Simulations of our trap show that we need a drive voltage of about \(170\,{V_{\mathrm {rms}}}\) to reach the desired trap frequencies. In order to limit the thermal load onto the cryostat from dissipated power in the resonator, this voltage should be reached with less than 100 mW of RF input power. This power corresponds to a consumption of about 1/7 l of liquid helium per hour when operating a wet cryostat at 4.2 K [19].
Voltage gain of an RLC resonator
Figure 1 shows an RLC series resonator driven through a matching network. In the following, we will assume perfect impedance matching with a loss-free matching network and that the losses in the connecting cable do not influence the resonator circuit. Thus, we can set the input power equal to the loss power in the resonator \(P_{\mathrm {input}} = P_{\mathrm {loss,resonator}}\), where \(P_{\mathrm {input}} = \frac{V_{\mathrm {in}}^2}{R_{\mathrm {wave}}}\) with \(V_{\mathrm {in}}\) being the input voltage supplied to the circuit and \(R_{\mathrm {wave}}\) the wave impedance of the connecting cable, commonly \(50\,\Omega\). We can further write
$$\begin{aligned} P_{\mathrm {loss,resonator}} = \left| I\right| ^2 \cdot R_{\mathrm {eff}} = \left| \frac{V_{\mathrm {c}}}{\frac{1}{i {\varOmega } C}}\right| ^2 \cdot R_{\mathrm {eff}}, \end{aligned}$$
(3)
where I is the current in the resonator, \(R_{\mathrm {eff}}\) its effective loss resistance, \({\varOmega }\) its frequency, \(V_{\mathrm {c}}\) the voltage at the capacitor, and C the capacitor representing the trap. Thus, we obtain
$$\begin{aligned} \frac{V_{\mathrm {in}}^2}{R_{\mathrm {wave}}} = \left| {\varOmega } C V_{\mathrm {in}} G_{\mathrm {V}}\right| ^2 \cdot R_{\mathrm {eff}} \end{aligned}$$
(4)
where \(G_{\mathrm {V}}\) is the voltage gain of the circuit, defined as \(G_{\mathrm {V}}=\frac{V_{\mathrm {c}}}{V_{\mathrm {in}}}\).
The quality factor Q of a resonator is defined as the resonance frequency \(f_{\mathrm {0}}\) divided by the bandwidth \({\Delta } f\)
$$\begin{aligned} Q = \frac{f_{\mathrm {0}}}{{\Delta } f}. \end{aligned}$$
(5)
In trapped ion experiments, we are usually not interested in a small bandwidth around \({\varOmega }\) but rather in a large voltage gain at the trap drive frequency. It should be noted that the voltage gain and bandwidth are qualitatively but not necessary quantitatively the same for different types of resonators. The quantity that is of direct interest for our application is the voltage gain, which can be derived from (4) as
$$\begin{aligned} G_{\mathrm {V}} = \frac{1}{{\varOmega } \cdot C} \cdot \frac{1}{\sqrt{R_{\mathrm {wave}} \cdot R_{\mathrm {eff}}}} = \sqrt{\frac{Q}{R_{\mathrm {wave}} \cdot {\varOmega } \cdot C}} \end{aligned}$$
(6)
with the quality factor of an ideal RLC resonator \(Q=\frac{1}{R_{\mathrm {eff}} \cdot {\varOmega } \cdot C}\). This indicates that both, the capacitive load and the effective resistance of the resonator. should be minimized.
Pickup
Voltage pickups are not required for the operation of the trap, but are useful to measure the voltage on the trap and are required to actively stabilize the RF voltage on the trap. In this section, we discuss inductive and capacitive pickups, as shown in Fig. 2.
A capacitive pickup consists of a voltage divider parallel to the trap, where \(C_{\mathrm {pickup2}} \gg C_{\mathrm {pickup1}}\), which leads to the pickup voltage
$$\begin{aligned} V_{\mathrm {pickup,C}} = \frac{C_{\mathrm {pickup1}}}{C_{\mathrm {pickup1}}+C_{\mathrm {pickup2}}} \cdot V_{\mathrm {c}}. \end{aligned}$$
(7)
\(C_{\mathrm {pickup1}}\) has to be small compared to the trap capacitance C, or the capacitive load of the RLC resonator will increase significantly, reducing \(G_{\mathrm {V}}\) as shown in (6). Typical values for \(C_{\mathrm {pickup2}}\) are several hundred times the value of \(C_{\mathrm {pickup1}}\).
In an ideal RLC series resonator, the voltage on the coil is the same as the voltage at the capacitor but with a \(180^{\circ }\) phase shift. We can monitor the voltage at the coil as a signal proportional to the voltage at the trap since we are only interested in the amplitude of the trap voltage. Here, L and \(L_{\mathrm {pickup}}\) are coupled, and the pickup voltage can be estimated with the derivations from ref. [13].
In order to maintain low losses, the pickup should not add significant losses in the resonator. If one uses a coaxial cable with a wave impedance \(R_{\mathrm {wave}}\) to monitor the pickup signal, a resistor \(R_{\mathrm {wave}}\) is used as a termination to avoid reflections. The losses are then \(V_{\mathrm {pickup}}^2/R_{\mathrm {wave}}\) which need to be small compared to the losses in the resonator P.
In general, we recommend using an inductive pickup, because it does not increase the capacitive load, which would reduce the voltage gain. However, for experiments, which frequently test different traps, a capacitive pickup may be preferable since the more accurate ratio between pickup and applied voltage facilitates estimating trapping parameters.