A Bigger Ion Trap, for the Same RF Power

Migrating an ion from a surface trap with electrode-to-ion height $h_1=75$ µm to a geometrically scaled-up trap with $h_2=135$ µm (a factor of $1.8$) reduces the RF field gradient at the ion. The cost of this migration depends on which design quantity is held fixed:

• Holding the secular frequency ${\omega }_{\sec }$ fixed requires $V_{\mathrm{RF}}$ to scale as $(h_2/h_1{)}^2=3.24$ and $P_{\mathrm{RF}}$ as $(h_2/h_1{)}^4\approx 10.5$.
• Holding the Mathieu stability parameter $q$ fixed preserves $V_{\mathrm{RF}}$ and $P_{\mathrm{RF}}$, with ${\omega }_{\sec }$ reduced by the factor $h_1/h_2\approx 0.56$.

Both choices keep the trap inside the stable operating window ($q\lesssim 0.4$). The sections below derive the scaling laws from the Mathieu equation, apply them to the 75 µm → 135 µm migration with a scan over candidate drive frequencies, place the resulting operating point in the context of published Paul traps, and list operational caveats relevant to resonator retuning and anomalous heating.

1. Setup and notation

For a linear Paul trap, or a planar surface trap mapped onto its equivalent radial quadrupole, the notation is:

• $e$, $m$ — ion charge and mass (fixed throughout).
• $V_{\mathrm{RF}}$ — peak amplitude of the RF voltage delivered to the rails. The scope-quoted peak-to-peak is $V_{\mathrm{pp}}=2V_{\mathrm{RF}}$. Formulas below are written in terms of $V_{\mathrm{RF}}$; numerical results report $V_{\mathrm{pp}}$ where indicated.
• ${\Omega }_{\mathrm{RF}}$ — angular RF drive frequency. Numerical values quote the linear frequency $f={\Omega }_{\mathrm{RF}}/(2\pi )$.
• $r_0$ — characteristic distance entering the radial quadrupole expansion, ${\Phi }_{\mathrm{RF}}(x,y,t)=\frac{V_{\mathrm{RF}}\cos ({\Omega }_{\mathrm{RF}}t)}{2r_0^2}(x^2-y^2).$ For a 4-rod linear Paul trap, $r_0$ is the rod-to-center distance. For a 5-wire surface trap, $r_0^2=h^2/{\kappa }_{\mathrm{geom}}$, where $h$ is the ion-electrode height and ${\kappa }_{\mathrm{geom}}\in [0.05,0.3]$ is a dimensionless geometric efficiency set by the electrode layout. At fixed layout, $r_0^2\propto h^2$.
• $h$ — ion-electrode height for a surface trap. Two traps with identical layout but linearly scaled dimensions share the same ${\kappa }_{\mathrm{geom}}$.

Standing assumption. The scalings below treat ${\kappa }_{\mathrm{geom}}$ as constant when migrating from trap 1 to trap 2 — i.e., the new trap is a geometric scale-up of the old, not a different electrode topology. If the topology changes, ${\kappa }_{\mathrm{geom}}$ should be characterized in situ via an RF-tickle measurement of $q$ versus $V_{\mathrm{RF}}$ once an ion is loaded; the formulas then apply with the measured ${\kappa }_{\mathrm{geom}}$.

2. Derivation from the Mathieu equation

2.1 Equation of motion and the Mathieu form

The radial equation of motion along $\hat{x}$ is $m\ddot{x}=-e{\partial }_x{\Phi }_{\mathrm{RF}}=-\frac{e{V}_{\mathrm{RF}}\cos ({\Omega }_{\mathrm{RF}}t)}{r_0^2}x.$ Introducing the dimensionless time $\tau \equiv {\Omega }_{\mathrm{RF}}t/2$ casts this into canonical Mathieu form $\frac{d^{2}x}{d{\tau }^2}+[a_x-2q_x\cos (2\tau )]x=0,$ with $a_x=0$ when no DC quadrupole is applied. Matching the coefficient of $\cos (2\tau )$ identifies the stability parameter

$\boxed{q\equiv \frac{2e{V}_{\mathrm{RF}}}{m{\Omega }_{\mathrm{RF}}^2{r}_0^2}}$

The sign of $q$ alternates between $\hat{x}$ and $\hat{y}$; only $|q|$ governs stability, and bounded motion requires $|q|<q_{\mathrm{stab}}\simeq 0.908$. In practice one operates well below this bound, $|q|\lesssim 0.4$, so the adiabatic expansion below stays accurate.

2.2 Secular frequency

For $|q|\lesssim 0.4$ and $|a|\ll q^2$, the leading-order Mathieu characteristic exponent yields a slow secular oscillation at

$\boxed{{\omega }_{\sec }=\frac{{\Omega }_{\mathrm{RF}}}{2}\sqrt{a+\frac{q^2}{2}}\stackrel{a=0}{\to }\frac{q{\Omega }_{\mathrm{RF}}}{2\sqrt{2}}.}$

This fixes the ratio of secular to drive frequency, $\frac{{\omega }_{\sec }}{{\Omega }_{\mathrm{RF}}}=\frac{q}{2\sqrt{2}}\in [0,\frac{0.908}{2\sqrt{2}})=[0,0.32),$ so ${\omega }_{\sec }$ is always well below the drive. The fractional correction beyond this leading order is $\mathcal{O}(q^2/8)$; at $q=0.28$ it is $\sim 1\%$ and is neglected for design purposes.

2.3 Pseudopotential and trap depth

Split $x(t)=X(t)+\xi (t)$ into slow $X$ and fast $\xi$ at ${\Omega }_{\mathrm{RF}}$. The fast component obeys $\ddot{\xi }\approx -\frac{e{V}_{\mathrm{RF}}\cos ({\Omega }_{\mathrm{RF}}t)}{m{r}_0^2}X⟹\xi (t)=\frac{e{V}_{\mathrm{RF}}}{m{\Omega }_{\mathrm{RF}}^2{r}_0^2}X\cos ({\Omega }_{\mathrm{RF}}t)=\frac{q}{2}X\cos ({\Omega }_{\mathrm{RF}}t),$ so the micromotion-to-secular amplitude ratio at the trap rails is $q/2$. Averaging the full force over one RF period yields a slow equation $m\ddot{X}=-\partial \psi /\partial X$ with pseudopotential $\psi (r)=\frac{e^2{V}_{\mathrm{RF}}^2}{4m{\Omega }_{\mathrm{RF}}^2{r}_0^4}{r}^2=\frac{1}{2}m{\omega }_{\sec }^2{r}^2.$ Taking the depth at $r=r_0$ as the radial trap depth,

$\boxed{U_{\mathrm{dep}}=\psi (r_0)=\frac{e^2{V}_{\mathrm{RF}}^2}{4m{\Omega }_{\mathrm{RF}}^2{r}_0^2}=\frac{e{V}_{\mathrm{RF}}q}{8}=\frac{1}{2}m{\omega }_{\sec }^2{r}_0^2.}$

Conventions in the literature sometimes carry a factor of 2 from the $V_{\mathrm{RF}}$ vs $V_{\mathrm{pp}}$ choice or from different $r_0$ normalizations. With $V_{\mathrm{RF}}$ defined as the peak amplitude on a single rail and $r_0$ as set by the quadrupole expansion above, the three expressions are equivalent.

2.4 RF voltage and RF power

The trap is driven through a resonator (helical or quarter-wave) with effective shunt impedance $R_{\mathrm{sh}}=Q{Z}_0$, where $Q$ is the unloaded quality factor and $Z_0=\sqrt{L/C}$ is the characteristic impedance. For matched delivery of power $P_{\mathrm{RF}}$ to the resonator, $V_{\mathrm{RF}}^2\approx 2P_{\mathrm{RF}}{R}_{\mathrm{sh}}=2P_{\mathrm{RF}}Q{Z}_0,$ giving the working relation

$\boxed{V_{\mathrm{RF}}\propto \sqrt{Q{Z}_0{P}_{\mathrm{RF}}}⟺P_{\mathrm{RF}}\propto V_{\mathrm{RF}}^2}$

at fixed resonator geometry. Across different resonator builds, $Q$ and $Z_0$ depend mildly on ${\Omega }_{\mathrm{RF}}$; they are treated as constants for the leading-order scaling, and the correction is discussed in §7.

3. Closed-form scaling table

Holding ion species fixed (so $e$, $m$ fixed) and treating ${\kappa }_{\mathrm{geom}}$ as constant under scaling, the dependencies reduce to explicit power laws. Read each row as: if you change the row variable by factor $k$, the column quantity changes by $k^{\text{exponent}}$.

Vary ↓	$q$	${\omega }_{\sec }$	$U_{\mathrm{dep}}$	$P_{\mathrm{RF}}$
$V_{\mathrm{RF}}$ (others fixed)	$V^{+1}$	$V^{+1}$	$V^{+2}$	$V^{+2}$
${\Omega }_{\mathrm{RF}}$ (others fixed)	${\Omega }^{-2}$	${\Omega }^{-1}$	${\Omega }^{-2}$	—
$h$ (others fixed)	$h^{-2}$	$h^{-2}$	$h^{-2}$	—
$P_{\mathrm{RF}}$ (others fixed)	$P^{+1/2}$	$P^{+1/2}$	$P^{+1}$	—
At fixed $q$: ${\Omega }_{\mathrm{RF}}$	—	${\Omega }^{+1}$	${\Omega }^{+2}{h}^{+2}$	${\Omega }^{+4}{h}^{+4}$
At fixed $q$: $V_{\mathrm{RF}}$ required	—	—	—	$\propto (\Omega h{)}^2$

Two relations summarize the entire constrained-design problem at fixed $q$:

$\boxed{\frac{V_2}{V_1}={\left(\frac{{\Omega }_2}{{\Omega }_1}\right)}^2{\left(\frac{h_2}{h_1}\right)}^2,\frac{P_2}{P_1}={\left(\frac{V_2}{V_1}\right)}^2={\left(\frac{{\Omega }_2}{{\Omega }_1}\right)}^4{\left(\frac{h_2}{h_1}\right)}^4.}$

4. Application: 75 µm → 135 µm

4.1 Anchor the current operating point

With $f_1={\Omega }_{\mathrm{RF}}^{(1)}/(2\pi )=43$ MHz and $f_{\sec ,1}={\omega }_{\sec }^{(1)}/(2\pi )=4.2$ MHz, $q_1=2\sqrt{2}\frac{f_{\sec ,1}}{f_1}=2\sqrt{2}\cdot \frac{4.2}{43}\approx \mathrm{0.276.}$ This is well inside the safe operating window ($q\lesssim 0.4$). The migration analysis below holds $q_2=q_1$ throughout.

4.2 Voltage- and power-preserving operating point

With $h_2/h_1=135/75=1.8$, demanding $V_2=V_1$ in the constrained-design formula gives ${\left(\frac{{\Omega }_2}{{\Omega }_1}\right)}^2=\frac{1}{(h_2/h_1{)}^2}⟹{\Omega }_2={\Omega }_1\frac{h_1}{h_2}=\frac{43}{1.8}\text{MHz}\approx 23.9\text{MHz},$ and correspondingly, since ${\omega }_{\sec }\propto \Omega$ at fixed $q$, ${\omega }_{\sec }^{(2)}={\omega }_{\sec }^{(1)}\frac{{\Omega }_2}{{\Omega }_1}=\frac{4.2}{1.8}\text{MHz}\approx 2.33\text{MHz}.$ At this drive, $V_{\mathrm{pp}}=200$ V and $P_{\mathrm{RF}}=2$ W remain exactly the current values (to the accuracy with which $Q$ and $Z_0$ are independent of ${\Omega }_{\mathrm{RF}}$ across this band).

4.3 Scan of candidate drive frequencies

For any choice of ${\Omega }_2$, the required $V_{\mathrm{pp}}$ and $P_{\mathrm{RF}}$ at $q_2=q_1$ are $V_{\mathrm{pp}}^{(2)}=V_{\mathrm{pp}}^{(1)}{\left(\frac{{\Omega }_2}{43\text{MHz}}\right)}^2(1.8{)}^2,P_{\mathrm{RF}}^{(2)}=P_{\mathrm{RF}}^{(1)}{\left(\frac{{\Omega }_2}{43\text{MHz}}\right)}^4(1.8{)}^4,$ with $(1.8{)}^2=3.24$ and $(1.8{)}^4\approx 10.50$.

5. Literature comparison: where this design point sits

Table 5.1 reproduces a curated set of surface ion-trap entries from a recent review (electrode dimensions, RF specifications, $q$, radial secular frequency, ion-electrode height), with the two operating points of this work appended for direct comparison. Table 5.2 extends the scope across other Paul-trap geometries — linear blade, 4-rod, wheel, cryogenic surface, in-vacuum resonator — selected from published implementations.

5.1 Surface ion traps — review compilation

Observations from this table:

• The current operating point ($h=75$ µm, $\Omega /2\pi =43$ MHz, ${\omega }_{\sec }/2\pi =4.2$ MHz, $q=0.28$) coincides with row [33] to within 5% in secular frequency, at the same ion height and comparable $q$.
• The proposed operating point ($h=135$ µm, $\Omega /2\pi =23.9$ MHz, ${\omega }_{\sec }/2\pi =2.33$ MHz, $q=0.28$) lies between rows [21] and [34], both larger-$h$ designs with ${\Omega }_{\mathrm{RF}}$ in the 20–26 MHz band. Row [21] ($h=150$ µm, $\Omega /2\pi =25.8$ MHz) and row [34] ($h=230$ µm, $\Omega /2\pi =20.6$ MHz) place $\Omega /2\pi \sim 24$ MHz at $h\sim 135$ µm in the typical operating band.
• $q$ across the review spans 0.15–0.43. The value 0.28 is mid-range.
• $V_{\mathrm{RF}}$ across the review spans 50–223 V (peak). The value 100 V peak (= 200 V_pp) is also mid-range.

5.2 Other Paul-trap geometries — for breadth

6. Three candidate operating points

• (a) ${\Omega }_{\mathrm{RF}}/2\pi \approx 24$ MHz. $V_{\mathrm{pp}}=200$ V, $P_{\mathrm{RF}}=2$ W, $q\approx 0.28$, ${\omega }_{\sec }/2\pi \approx 2.33$ MHz. The unique choice that preserves both RF voltage and RF power. The resonator requires a new coil winding (more turns); see §7.
• (b) ${\Omega }_{\mathrm{RF}}/2\pi \approx 30$ MHz, with ${\omega }_{\sec }/2\pi \approx 3$ MHz. Requires $V_{\mathrm{pp}}\approx 315$ V and $P_{\mathrm{RF}}\approx 5$ W (≈ 2.5× the current power). Tractable with a standard helical resonator and within typical breakdown margins on surface-trap chips. Most of the anomalous-heating reduction from the larger $h$ is preserved.
• (c) ${\Omega }_{\mathrm{RF}}/2\pi =43$ MHz (preserving ${\omega }_{\sec }/2\pi =4.2$ MHz). Requires $V_{\mathrm{pp}}\approx 650$ V and $P_{\mathrm{RF}}\approx 21$ W. The voltage exceeds typical breakdown margins on bonded surface-trap chips and the power exceeds the thermal budget of most room-temperature helical resonators.

7. Operational notes and caveats

• Resonator retuning. The LC product scales by $(43/24{)}^2\approx 3.2$. Trap capacitance (~1–3 pF, dominated by the vacuum feedthrough and bonding pads) does not change with $h$. The fix is on the resonator coil: more turns or a longer winding to roughly triple $L$.
• $Q$ at lower frequency. Helical-resonator copper-skin surface resistance scales as $\sqrt{f}$, so $Q$ typically rises when moving from 43 MHz to 24 MHz. This contributes a modest increase in $V_{\mathrm{RF}}$ per $\sqrt{P_{\mathrm{RF}}}$ that is not included in the scaling table.
• Anomalous heating budget. Empirically the motional heating rate of surface traps scales as $\dot{\bar{n}}\propto h^{-4}{\omega }_{\sec }^{-1}$ (sometimes steeper than $h^{-4}$ for warm traps). $h:75\to 135$ µm gives a $1.8^4\approx 10\times$ suppression; dropping ${\omega }_{\sec }$ by $1.8\times$ gives back ~$1.8\times$. Net improvement is roughly $5$–$6\times$. This is usually the primary motivation for migrating to a larger trap.
• Geometric factor. The formulas assume the new trap is a linear scale-up of the old. If the electrode topology differs, ${\kappa }_{\mathrm{geom}}$ may differ and the $h^{-2}$ scaling acquires an additional dimensionless factor. After the first ion is loaded in the new trap, an RF-tickle measurement (scan $V_{\mathrm{RF}}$, measure ${\omega }_{\sec }$) yields the actual $q/V_{\mathrm{RF}}$ slope and refines the design point.
• Mathieu corrections. ${\omega }_{\sec }=q{\Omega }_{\mathrm{RF}}/(2\sqrt{2})$ is exact at $q\to 0$; the next correction is of relative size $q^2/8$. At $q=0.28$ this is ~1%, smaller than typical RF voltage stability. At $q\gtrsim 0.5$, the continued-fraction expression for the Mathieu characteristic exponent should be used in place of the leading-order formula.
• Voltage convention. $V_{\mathrm{RF}}$ in this document is the peak amplitude of the RF on a single rail. $V_{\mathrm{pp}}=2V_{\mathrm{RF}}$ is the scope-quoted peak-to-peak on that rail. If the trap is driven differentially ($\pm$ rails), the inter-rail $V_{\mathrm{pp}}$ doubles, but so does the field at the ion, and the $q$ formula's prefactor absorbs the difference. The convention should be stated explicitly when sharing values across laboratories.

8. Summary

For a geometric scale-up of a planar surface ion trap by a factor $h_2/h_1$, the cost in RF voltage and RF power depends entirely on the choice of conserved quantity:

$\frac{V_2}{V_1}={\left(\frac{{\Omega }_2}{{\Omega }_1}\right)}^2{\left(\frac{h_2}{h_1}\right)}^2,\frac{P_2}{P_1}={\left(\frac{{\Omega }_2}{{\Omega }_1}\right)}^4{\left(\frac{h_2}{h_1}\right)}^4\text{at fixed }q.$

Holding ${\omega }_{\sec }$ fixed forces ${\Omega }_2={\Omega }_1$, which yields $V_2/V_1=(h_2/h_1{)}^2$ and $P_2/P_1=(h_2/h_1{)}^4$. Holding $q$ fixed and allowing ${\Omega }_{\mathrm{RF}}$ to scale as ${\Omega }_2/{\Omega }_1=h_1/h_2$ yields $V_2=V_1$ and $P_2=P_1$, with ${\omega }_{\sec }$ reduced by $h_1/h_2$. For the 75 µm → 135 µm migration, the two extremes are a 10× increase in RF power versus a $1.8\times$ reduction in secular frequency, with a continuum of intermediate operating points enumerated in §4.3.