As a first step, the desired resonant trap frequency Ω needs to be specified. This follows for example from the required secular frequency of the ions. Then Ω follows from Ω = 2√2ωs/q, which for typical values ofq (0.1. . .0.2) is about 10-20 times the secular frequency. In general, high trap frequencies tend to be advantageous. Assuming (a, q) =const. high secular frequen- cies∝Ω imply well separated motional sidebands, further, adverse effects due to micromotion are minimized: For an electric stray field of given strength, the amplitude and kinetic energy of the micromotion scale ∝ Ω−2. Finally, the heating due to fluctuating patch potentials scales ∝ ω−s1 ∝ Ω−1 [30]. On the other hand, high trap frequencies become increasingly technically challenging. The required drive power, which equals the power dissipated in the trap, scales ∝Ω4 and even under UHV conditions electric breakdown can occur.
From a technical point of view an important distinction between two different frequency domains needs to made: Tank circuits can be designed from conventional lumped components or as a special quarter wave resonator (“helical resonator”). The first regime, where standard coils and capacitors may be used, ranges up to about 25 MHz. This (soft) limit can easily be understood by estimating the minimum inductance and capacitance the circuit will exhibit. Straight wire of 1 mm diameter has an inductance of about 1.5µH per meter, so it will be hard to keep the total inductance below, say, 2µH. If this wire is one centimeter above a grounded surface (e.g. the vacuum vessel) it will further contribute with a stray capacitance of about 15 pF per meter. The capacitance of the trap will typically exceed 5 pF so the total capacitance easily adds up to more than 20 pF. The resonance frequency of this “minimal”, (20 pF, 2µH), LC circuit isfr= 1/2π
√
L C= 25 MHz. Both ion traps described are operated around Ω∼2π10 MHz, so only “lumped circuit” designs will be discussed.
Wiring
In order to minimize ohmic losses radio frequency litz5 should be used where ever possible, since the skin depth of copper at 10 MHz is only 21µm. However, this is only possible out- side the vacuum chamber, since litz is not suitable for UHV due to possible virtual leaks. In the tank circuits described below, the replacement of all solid wire outside the vacuum
5RF litz is different from ordinary litz: Here, every single strand is insulated to minimize losses from skin
2.2. TANK CIRCUITS 27
by litz resulted in a two times higher Q factor. As shown above, the capacity of the cir- cuit should be minimized, too. To this end, special attention should be paid to maximize the distance of the RF carrying wires to any grounded surface. UHV feedthroughs, espe- cially multipin feedthroughs can have annoyingly high capacitances (10 pF-100 pF). When connecting the trap, care should be taken that the wiring does not introduce unintentional phase-differences between opposing electrodes. This could be caused by unequal lengths or differing stray capacitances. Even small phase-differences cause sizeable micromotion that cannot be compensated. Fig. 2.9 shows the wiring of the endcap trap.
Inductors
From all components, the primary coil is the most critical. In most cases the quality factor of the coil will limit the total quality of the circuit, which is the reciprocal sum of theQfactors of the components 1/Q =P
i1/Qi. Even for “optimum” coils as described below it is hard to exceed a Q of, say, 500. In comparison, the Q value of a ceramic capacitor is typically around a few thousand. Designing a high Q coil requires paying attention to the following: No suitable ferrites exist in the range of frequencies above∼10 MHz, so the coil should be an air coil6. Eddy currents caused by the current itself (skin effect) and by the magnetic fields of a neighboring winding (proximity effect) increase the ohmic losses. At low frequencies the proximity effect can be neglected and close winding is favorable because this simply minimizes the length of the wire. At high frequencies an increasingly loose winding is preferable. The distinction between “low” and “high” is involved, but for the application described here the proximity effect dominates and is up to ten times larger than the skin effect. Further, the self capacitance of the coil should be taken into account. It increases linearly with diameter and aspect ratio (α =l/D, where l is length andD diameter) and causes the coil to have a self resonant frequency which marks the maximum frequency up to which it is useful. Broadly speaking this shows that the coil should be short, loosely wound and of large diameter, where the latter is limited by the self resonant frequency and/ or practicability. In [34] the optimum geometry has been determined, with the following result assuming constant inductance and volume: The optimum aspect ratio is α= 0.7, the optimum winding density, defined as the ratio of wire diameterdto wire spacing s(center to center)δ =d/s, is δ= 0.6. Empirically, for coils made of solid round copper wire, a quality factor of
Q∼41Dpf (2.11)
may be expected, where the diameter D is in units of centimeters and the frequency f is given in MHz. The use of RF litz will lead to a higherQ. The inductance of a coil with such a geometry can be estimated to within 10% by
L∼ πµ0Dn
2
4.6 (2.12)
wherenis the number of windings.
Capacitors
Capacitors should have low loss (sometimes called “tanδ”) and withstand high voltages. For radio frequency applications silvered mica and certain ceramics (e.g. “C0G”) are the most
6
suitable dielectrics. Vacuum capacitors have the lowest losses, but are bulky and expensive. In general theQfactors are so high (a few thousand), that the exact choice is uncritical.
Resistors
High voltage carbon film resistors turned out to be most suitable for this application. Other resistor types (e.g. metal film, wire wound) can have rather large stray capacitances and therefore short cut the RF. In addition, together with their stray inductance cumbersome resonances can occur.
Coupling
To achieve impedance matching the source needs to be coupled to the tank circuit appropri- ately. Two coupling methods have been used: First, the RF signal wire is soldered to the primary coil directly (“galvanic coupling”). By choosing the position (close to ground cor- responds to weak coupling and vice versa) the coupling strength can be tuned continuously. The advantage of this method is its mechanical stability. On the other hand it is not easy in practice to identify the optimum position. A second method is to place a current loop around the coil. Its magnetic field will have some overlap with the magnetic field of the coil and hence provide an inductive coupling. After choosing roughly the right diameter, the coupling can be fine-tuned by twisting the angle of the loop relative to the axis of the coil. This allows to achieve an extremely good coupling, limited only by the sensitivity of the measurement method.
Methods to optimize the coupling include maximizing the voltage on the trap electrode, or measuring the reflected power e.g. using a directional coupler. The first method is very easy to implement but is not very accurate and requires paying special attention to the perturbations introduced by the voltage probe. Signal generators and amplifiers are quite sensitive to reflected power, so measuring it using a directional coupler is by far the superior method.
Characterization
The tank circuit is then conveniently characterized as follows: First, the inductance of the coil is measured e.g. by attaching a well known capacitor to it and determining the resonance frequency. In a second step the trap is attached and the resonance frequency ω0 and -width ∆ω0 are measured, from which the quality factor Q = ω/∆ω0 is inferred. Thereby the coupling should be minimized. The quality factor in the impedance matched case (“critical coupling”) will then be exactly 1/2 of the measured valueQcrit =Q/2. Care should be taken that the measurement does not perturb the tank circuit, e.g. by using an ultra low capacitance probe or an antenna. From the measured values of L, ω0 andQthe remaining parameters are calculated as follows: C = 16Q 2 crit−1 16Lω2 0Q2crit (2.13) R = q Lω0 Q2 crit−161 (2.14)
2.2. TANK CIRCUITS 29
2.2.3. Results