Paul–Straubel–Kingdon trap for true zero-point confinement of an individual ion and reservoir

Abstract

A modification of the Paul–Straubel trap previously described by us may profitably be operated in a Paul–Straubel–Kingdon (PSK) mode during the initial loading of an individual ion into the trap. Thereby the coating of the trap ring electrode by the atomic beam directed upon it in earlier experiments is eliminated, as is the ionization of an already trapped ion. Coating created serious problems as it spot-wise changed the work function of the ring electrode, which caused large, uncontrolled dc fields in the trap center that prevented zero-point confinement. Operating the Paul–Straubel trap with a small negative bias on the ring electrode wire is all that is required to realize the PSK mode. In this mode the tiny ring trap in the center of the long, straight wire section is surrounded by a second trapping well shaped like a long, thin-walled cylindrical shell and extending to the end-caps. There, ions may be conveniently created in this well without danger of coating the ring with barium. In addition, the long second well is useful as a multi-ion reservoir.

The Paul–Straubel trap previously described by us (1) contained a ring of radius R ≈ 3r₁ made from wire of radius r₁ supported by straight wires also of radius r₁. Especially using the modified cylindrical can with conical end pieces shown in Fig. 1, the trap may profitably be operated in a Paul–Straubel–Kingdon (PSK) mode during the initial loading of an ion into the trap to eliminate coating of the trap-ring electrode by the atomic beam directed upon it in earlier experiments. For small ring traps of radius R = 3r₁ this was a very serious problem, because spot-wise it changes the work function of the electrode, causing large, uncontrolled dc fields in the trap center in the case of a Ba beam. These fields pulled the ion out of the trap center into regions of finite rf fields and driven micromotion and thereby prevented true zero-point confinement in the lowest vibrational levels of the trap. Operating the Paul–Straubel trap with negative bias on the ring electrode wire that is small enough to reduce the depth of the radial pseudo potential (2) well inside the ring produced by the rf trapping field E_a by, say, no more than 50% is all that is required to realize the PSK mode in which the long, straight sections of wire form a Kingdon trap. Moreover, while in the pure Kingdon mode (4) an ion can still escape through the openings for the wire in the can, e.g., at D, ac field and, thus, pseudo potential Ψ∝E_a² are largest there in the PSK mode, effectively sealing the gap). Also, it is fairly obvious that for the modest negative bias to be applied to the wire, the trap center A must remain the point of lowest combined potential energy, because the repulsive pseudo potential vanishes there. Thus, the colliding beams from oven and electron guns can fill the whole trap with energetic ions that will remain trapped for a long time because the repulsive pseudo potential keeps them from colliding with the wire and ring (5). Oven and cathode are at ≈ −100 V, and electrons from the dark Ba-coated cathode are accelerated toward the shell at ground potential and on to the wire during the positive half-periods of the rf trapping voltage applied. As evaporation cooling lowers the average energy of ions and thermalizes them (2, 3), they may fill the ring trap with high probability provided that an operating voltage can be found high enough to create an effective trap at “A” in the face of the xy-defocusing negative bias but small enough not to form a high barrier, keeping any ions from even approaching the trap from outside. Approximate potential distributions along lines perpendicular to the plane of drawing intersecting the points A and B in Fig. 1 are shown in Figs. 2 and 3, with B located on the wire at x = x_B, y = 0, z = r₁.

Figure 1.

Paul–Straubel–Kingdon trap (schematic). Ions created near C by electrons from about the 100-V negative cathode bombarding beam atoms from the oven fill a thin cylindrical shell reservoir around the wire, stretching across the whole trap in the x direction. Under favorable conditions, some ions drop into the inner ring trap, whose center A at the origin marks the lowest value of the combined dc + pseudo potential. The drawing is in the xy plane; it is not to scale, and the x-axis coincides with the axis of the long support wires.

Figure 2.

Potentials along the z-axis perpendicular to the plane of the drawing of Fig. 1. Curves: (1), the combined dc and pseudo potential Φ(z) + Ψ(z) for the ring trap above the origin, point A in Fig. 1; (2), Φ(z) only, above A; (3), the combined potentials φ_L + ψ_L above B; and (4), only φ_L, above B.

Figure 3.

Same potentials along the z-axis as in Fig. 2 but for a rf trapping voltage that is higher by a factor of .

Clearly, gaining at least a semi-quantitative understanding of the potential distributions in the PSK trap would be very desirable. To this end we first study the dc potential in a similar but simpler, very long cylindrical trap of wire radius r₁ and inner shell electrode radius r₂. We attempt to do so by approximating the complex charge distribution over the equipotential surface of the wire electrode by three sections of discrete line charges plus two point charges for symmetrization. These sections are a line charge of constant density λ extending along the x-axis from −∞ to +∞, from which, with R₊ = R + r₁, the 2R₊ long center part has been cut out, two point charges of value λR₊ at the points x = 0, y = ±R₊, z = 0 returning the cut out charge, and a circular line charge of radius R and reduced constant density αλ with α to be determined by fitting the potential along lines through points A, B parallel to the z-axis to the boundary conditions. Obviously, the conductive ring electrode forces the potential on its surface to be axially symmetric about the z-axis, suggesting the reasonable assumption that this symmetry should also hold approximately in the center of the ring at the origin at point A as it does for the above charge distribution.

Choosing as a unit of length the wire radius r₁, we set the potential on the wire at B to ln z = ln 1 = 0 and on the shell electrode, approximated by a concentric cylinder of radius r₂ = 100 r₁, surrounding the wire electrode, we set the potential to ln z = ln 100 = 4.6, implying an applied potential difference U₀ = 4.6 units, from which potential distributions for other applied potential differences may simply be scaled. The fields associated with the sections are listed in Table 1. For the potentials, common constant factors have been omitted and constant additive terms adjusted to make the φ values vanish at the points specified.

Table 1.

Potentials of line charges and combinations used

Charge	Potentials	Zeros
Line	φL(z) = ln z	Point B
Gap line (including point charges)	φG(z) = ln[(z2 + R+2)1/2 + R+] − ln 2R+ − R+/2(z2 + R+2)1/2 + 1/2	Point A
Ring	φR(z) = −απR(z2 + R2)−1/2 + απ	Point A
ac line	ψL(z) = C[dφL(z)/dz]2	∞
ac gap line + ring	Ψ(z) = C[dφG(z)/dz + dφR(z)/dz]2	A, ∞

The true dc potential Φ(xyz) vanishes everywhere on the surface of the wire electrode and at B, whereas at the origin (A) it has a small value, Φ₀. Introducing Φ′(z) = φ_G (z) + φ_R(z), with Φ′(0) = 0, we initially neglect Φ₀ and require U₀ = φ_L(r₂) ≈ Φ′(r₂) to get first numerical values for α and Φ′(z) ≈ Φ(00z) = Φ(z) and the roughly right variation of Φ(xyz) for z near A. From this we reconstruct the variation of Φ(xyz) with x: Going from inside the ring at x = 2r₁ along the x-axis increases Φ(x00) from 0 to Φ₀ at x = 0 just as much as going along the z-axis from 0 to z = 2r₁/ would, assuming a pure, axially symmetric quadrupole potential ∝ (−x² − y² + 2z²). Thus, it then holds approximately Φ₀ ≈ Φ′(r₁) ≈ 2Φ′(r₁) and Φ(z) = Φ′(z) + 2Φ′(r₁) is an improved expression for Φ(z). To match the boundary condition again we now require

or

and must adjust α slightly to satisfy the last equation. This yields α = 0.4449. Potential values for our numerical example with U₀ = 4.6 are given in Table 2. All function arguments above are values of respective z-coordinates. In a standard Paul trap, the relation Φ₀ = U_0P/2 would hold for the dc radial hill. This yields for equal hill height U_0PSK/U_0P = 11, which is in good agreement with voltage-loss factors quoted in ref. 6 for Paul–Straubel traps.

Table 2.

Numerical example of dc potential values

Coordinate	Potential values
z	φL	φG	φR/α	2Φ′(r1)
0		0	0
1	0	0.0302	0.1612	0.2092
100	4.6052	3.0457	3.0474

The constant factor C in the rf trapping potential Ψ(z) has been adjusted to provide an axial trapping well with a force constant about 4 times larger than that of the dc well. Figs. 2 and 3 exhibit potential distributions that are computer-generated from the formulas in Table 1. Curve (3) in Fig. 2 shows the potential well above point B in Fig. 1, with the minimum at point M. In Fig. 1, in any plane containing the wire axis and, thus, also a plane containing the z-axis, the locus of the M points is a nearly straight line that is parallel to the wire and the x-axis. Closer study of this curve for the simple no-ring case lets us see an ion created on the M-locus at rest at point C in Fig. 1 as moving toward the trap center A and on to the other end of the trap, sliding like a sled periodically up and down along the well-shaped bottom line of a long, narrow potential valley of appreciable depth in the center. The coordinate z_M of the M points is determined by the balance of repulsive pseudo force ∝ dψ_L/dz ∝ z_M⁻³(ln r2)⁻² and attractive dc force ∝ dφ_L/dz ∝ 1/z_Mln r₂, yielding z_M² ∝ 1/ln r₂ or, calibrated with the data z_M = 4.6, r₂ = 100, yielding z_M² = 97/ln r₂. These formulas have been obtained for the potential in a long, cylindrical section of fixed radius r₂ for which φ_L = 4.6 ln z/ln r₂ holds for a constant applied voltage of 4.6 units. In a rough approximation they should also hold in the center and in conical end sections of the trap in Fig. 1. Thus, we have at the intersection of the minima locus with the conical shell section, z_M = r₂ = 7.1 and φ_L = 4.6. The pseudopotential contribution there may be determined as 1.2 from ψ_L ≈ L z_M⁻² (ln r₂)⁻² and its value 0.5 at x = 0, z_M = 4.6, yielding L = 224, for a total potential φ_L + ψ_L = 5.8 units at the intersection, whereas in the center its value is only 2, making use of data from Fig. 2 in both cases. Thus, the potential valley is 3.8 units deeper in the center. In the PSK case of primary interest according to curve (1) of Fig. 2, one flank of the valley opens up into a roughly round crater, centered on the z-axis, about 9 units wide and another 1.8 units deeper than the valley outside it. In Fig. 3, drawn for a rf voltage V₀ by a factor larger, this crater has developed a high barrier around it, preventing evaporation-cooled ions from dropping into it. Further increase of V₀ completely walls off the central Straubel trap region from the long valley above it, opening up the possibility of using the latter as a multiple-ion reservoir from which the single ion in the Straubel trap may be replaced once the need arises. These more quantitative results confirm the working of the scheme sketched in the first paragraph. The presence of a reservoir full of ions also holds the potential for picking out from it the rare isotopic ion of interest for injection into the ring trap. A rare isotopic ion created occasionally in the center of earlier ring traps by crossed atomic and electron beams was easily destroyed by them as well. On the other hand, a rare isotope in the Paul–Straubel ring that is selectively cooled there by a laser tuned to its isotope-shifted frequency will not easily be removed by only evaporation-cooled majority ions dropping into the ring trap from the reservoir.

The parameter α = 0.445 implies a negative excess charge q_e on the ring compared with 2R₊λ on the cutout line section, with q_e/2R₊λ= πα = 1.4, which, from distances ≫R, looks like a point charge, which helps us in visualizing a picture of the off-z-axis potential.

When rare isotopes are to be injected and only minimal oven charges are available, miniaturizing oven and cathode further and bringing them even closer to the very thin wire of about a 100-μm diameter is still possible.

Using baby powder charged merely by squirting it from the commercial plastic container in a mockup similar to Fig. 1 but in a cubic shell and modeled after a simple Paul trap in air (7), we had no difficulty in making this remote filling scheme work. Admittedly, however, strongly damping air replaced the modest laser damping and evaporation cooling.

Preliminary experiments by N.Y. with Ba ions have also demonstrated the scheme and, when finalized, will be published in a forthcoming paper.

ABBREVIATION

PSK

Paul–Straubel–Kingdon