Theory of the coplanar-waveguide Penning trap

A novel planar Penning trap is presented, which results from the projection of the well-known three-dimensional cylindrical trap onto the surface of a chip. The introduced trap is also a coplanar-waveguide cavity, similar to those used in circuit quantum electrodynamics experiments with superconducting two-level systems. It opens up the possibility of integrating a single trapped electron, or geonium atom, into quantum circuits. The trap is an elliptical Penning trap, with the magnetic field parallel to the chip's surface. A design procedure is described, which permits the compensation of electric anharmonicities up to sixth order. This should render possible the observation of a single trapped electron and the accurate measurement of its eigenfrequencies, a sine qua non requirement for a useful planar geonium technology.


Introduction
A single electron in a Penning trap is known as a geonium atom [1], as coined by its inventor, the 1989 Nobel Prize winner H Dehmelt. It is an outstanding system for measuring fundamental constants and testing the laws of physics [2]. Examples include measurement of the electron's g-factor [3] and mass [4]. Electrons in cryogenic Penning traps have been proposed for implementing a quantum processor [5][6][7]. Initially motivated by the scalability requirements for quantum computation [8], a planar Penning trap has been conceived at the University of Mainz [9]. Trapping of electrons has been demonstrated with it, both at room temperature and with a cryogenic setup [10,11]. Recently, several optimized versions of the Mainz trap have been proposed at Harvard [12]. These should pave the way for the observation of a single electron, a primary goal not yet achieved with planar Penning traps [11].
In this paper, the coplanar-waveguide (CPW) Penning trap has been introduced. Its design is inspired by modern planar microwave technology. Superconducting MW-cavities have been built on a chip with very low losses, achieving quality factors in the range of Q ∼ 10 5 -10 6 [13]. The basis of those cavities is the CPW-transmission line, consisting of a central conducting strip and two outer ground-planes [14]. The mentioned high-Q resonators have triggered the field of circuit-quantum electrodynamics (circuit-QED) [15,16]. As explained in the following sections, the CPW-Penning trap originates from the projection of the well-known cylindrical trap [17] onto the surface of a chip. That projection naturally results in a finite section of the CPW-transmission line. This can function as a cavity or, alternatively, can be coupled to the mentioned circuit-QED resonators. Hence, the CPW-trap opens up the possibility of a geonium atom becoming a building block of future 'quantum circuits'. Electrons have already demonstrated practical applications in circuit-QED. An example is the recently achieved highcooperativity coupling of electron-spin ensembles-housed in ruby and diamond coatings in the chip-to superconducting CPW-cavities [18]. The aim is to provide those circuits with long-coherence quantum memories. Other systems have also been considered for the storage of quantum information and other circuit-QED applications, such as Bose-Einstein condensates [19] and polar molecules [20]. A detailed discussion of the possibilities of a geonium atom in superconducting quantum circuits goes beyond the scope of this paper; that will be treated in future publications.

Coplanar-waveguide (CPW) Penning trap
The genesis of the CPW-Penning trap can be described in the following way. Starting from a hypothetical cylindrical trap [17], an imaginary cut is performed along its surface, as shown in figure 1(a). The imaginary slit permits one to virtually 'stress' and 'flatten' the originally cylindrical electrodes. These are transformed into flat rectangular segments and then projected onto a plane. At both sides of the projected segments, two outer ground-planes are added, for shielding and equipotential reference. The magnetic field of the original three-dimentional (3D) trap, B = B ·û z , is left unchanged and is parallel to the surface of the chip. The obtained set of electrodes together with the ground planes, form a section of a CPW-transmission line, such as those used in circuit-QED [15,16]. The denomination CPW-Penning trap is chosen for this reason.
As sketched in figure 1(b), the basic CPW-Penning trap consists of five planar electrodes: the 'ring', two compensation electrodes and two end-caps.

Electrostatic trapping potential
For simplicity, we assume that the conducting chip housing the trap extends to infinity. With it, the calculation of the trapping potential is straightforward; details are explained in appendix A. The result is The symbols V r , V c and V e represent the applied dc-voltages to the ring, compensation electrodes and end-caps, respectively. This is sketched in figure 2. The functions f r , f c and f e depend only on the dimensions of the trap (see equation (A.2)). On the contrary, f gaps , which represents the contribution to the potential of all insulating gaps, depends on both the geometry and all the applied voltages (see equation (A.2)).

Example of a CPW-Penning trap.
The basic functioning of a CPW-trap is best illustrated by computing an example with equation (1). For this purpose, we choose l r = 0.9, l c = 2.0, l e = 5.0, η = 0.1 and S 0 = 7.0 (all in mm). The choice of this geometry is in principle arbitrary; however, this concrete example will serve, throughout this paper, to elucidate the main properties of the CPW-trap. The discussion will also be focused on a single electron at cryogenic temperatures.
For the example of figure 3, the voltages are (all in Volt) V r = −1, V c = −1.15 and V e = −4. These allow capture of electrons or any negatively charged particles around the position (x = 0, y = y 0 , z = 0). In contrast to the cylindrical trap, the end-caps cannot be grounded; actually, |V e | > |V c | |V r | must hold. This is required for the existence of an equilibrium position at some y 0 > 0 above the chip's surface. The continuous smooth curve in figure 3(a) shows the actual shape of the electric potential at y 0 ( 1.37 mm, for the example). The variation of φ along the vertical axes y is shown in figure 3(b).

2.1.2.
Equilibrium position of the trap. y 0 is determined by the equality ∂φ (0,y,0) ∂ y | y=y 0 = 0. If the insulating gaps are vanishingly small, η → 0, then f gaps → 0 (see equation (A.2)). With this approximation, the equation for calculating y 0 is We have introduced the tuning ratio T c = V c V r and the end-cap to ring ratio T e = V e V r . Equation (2) shows that the trapping height depends only on voltage ratios T c , T e → y 0 = y 0 (T c , T e ). This formal dependence also holds for the less restrictive case that η is small 'enough', η l r , l c , l e , S 0 (see the discussion in appendix A). Equation (2) cannot be solved analytically for y 0 ; only numerical values can be obtained.
The expansion coefficients are defined by . The symmetry of φ(x, y, z) alongû x andû z implies that all C i jk with odd i and/or odd k vanish. The C i jkcoefficients define to a great extent the performance of the trap. They (or equivalent ones) have been studied in detail for cylindrical [17], hyperbolic [21] and toroidal Penning traps [22]. Moreover, as for equation (2), if the slits between electrodes are small 'enough', then C i jk scale linearly with the ring voltage ⇒ C i jk = V r · c i jk , where c i jk = c i jk (T c , T e ) depend only on T c and T e .

Constraints on the C i jk -coefficients.
Plugging the series expansion (3) into the Laplace equation, ∇ 2 φ(x, y, z) = 0, the following equalities can be obtained: In the case of a 3D hyperbolic or cylindrical trap, the coordinates x and y are undistinguishable ⇒ C 200 = C 020 . Thus, equation (4) (left) reduces to C 200 = −2 C 020 . From it, the potential of an ideal classical Penning trap arises ⇒ φ = C 002 (z 2 − (x 2 + y 2 )/2). In the case of the CPW-trap, though, x and y are distinguishable and the curvatures C 200 and C 020 are not identical: C 200 = C 020 . Hence, the general form of the pure quadrupole potential, i.e. including terms only up to second order, is The ellipticity parameter is given by . In general, = 0, and the CPW-trap is therefore an elliptical Penning trap [23,24].

Ideal elliptical CPW-Penning trap
The motion of a particle in the ideal elliptical trap of equation (6) has been calculated by Kretzschmar [24]. The reduced cyclotron (ω p = 2πν p ), magnetron (ω m = 2πν m ) and axial (ω z = 2πν z ) frequencies of the trapped particle-with charge q and mass m-are When = 0, the usual expressions [25] for the frequencies of a 'circular' Penning trap are recovered. For the example of figure 3, the ellipticity is 0.41. According to equation (7), the frequencies of a trapped electron are ω p = 2π · 14 GHz, ω z = 2π · 28 MHz and ω m = 2π · 26 kHz. A magnetic field of B = 0.5 T is assumed, motivated by the suitability of the corresponding cyclotron frequency for circuit-QED applications [15].
The radial motion in an ideal elliptical trap is (x(t), y(t) − y 0 ) = (A p ξ p cos(ω p t) + A m ξ m cos(ω m t), A p η p sin(ω p t) + A m η m sin(ω m t)), where the amplitudes are given by and [24] The symbols E p and E m represent the cyclotron and magnetron energies, respectively. As shown by Kretzschmar, the orbit of the reduced cyclotron motion is only slightly affected by the ellipticity: ξ p η p 1. It very nearly follows the circular shape of conventional Penning traps. In contrast, the magnetron motion becomes an ellipse, where the orientation of the major and minor axes (along x or y) depends on the sign of . Moreover, that motion is stable for −1 < < 1 and, at the limit | | → 1, it becomes very slow, ω m → 0. In that case, the magnetron ellipse tends towards a line, with an increasingly wide major axes and a vanishing minor one (if → +1 ⇒ ξ m → ∞, η m → 0 and vice versa). For values | | 1, the magnetron becomes an unbounded hyperbolic motion and trapping is not possible [24].

Electrical anharmonicities
The ideal trap described in section 2.3 is only valid for vanishing amplitudes of the particle's motion. In real experiments the electric anharmonicities must be taken into account. These generate energy-dependent fluctuations/deviations of the particle's frequencies.

Frequency-shifts matrix
All anharmonicities, even and odd, up to the fourth order in the expansion of φ, 3 i + j + k 4 (see equation (3)), produce frequency shifts that scale linearly with the energies of the particle. Hence, they can be expressed in matrix form: Each perturbation to φ quad appearing in equation (3) delivers such a frequency-shifts matrix. In total, the CPW-cavity trap requires nine M i jk matrices, corresponding to each C i jk perturbative Hamiltonian. The expressions for all M i jk are given in appendix B. The overall frequency-shifts matrix is the sum of all of them: For a single electron captured in the example trap of section 2.1.1, with the voltages of figure 3 and B = 0.5 T, the overall frequency-shifts matrix is 3.2. Compensation of ν z E z : optimal tuning ratio The accurate measurement of the axial frequency is essential; in most cases the determination of the other electron's eigenfrequencies or spin state depends upon it [2,26]. Thus, M 2,2 = ν z E z is the most relevant and dangerous of all frequency shifts in M. In the example, it amounts to 203 Hz K −1 . Such a dependence of ν z on the axial energy-which is not constant but fluctuating [27]-would render the detection of the electron almost impossible, even at cryogenic temperatures.
According to (B.1) and (B.7), M 2,2 is given by the sum of M 004 2,2 and M 012 2,2 . Taking into account that ν m ν z ν p , we have M 012 2,2 is always positive, since it is proportional to the square of C 012 , while M 004 2,2 can be positive or negative, depending on the sign of C 004 . Hence, if an appropriate optimal tuning ratio can be found, such that the latter matrix element cancels the former, T  theoretically varies with the magnetic field; however, that dependence is negligible (∼ −2 × 10 −6 T −1 for the example).

Physical origin of M 012
2,2 M 012 2,2 is due to the slight dependence of the trapping height on the axial energy, y 0 = y 0 (E z ). Indeed, for vanishing energy (E z = 0) y 0 is the solution to the implicit equation C 010 (y 0 ) = 0. For E z > 0, that equation must be modified into C 010 (y 0 ) + z 2 · C 012 (y 0 ) = 0. Here, z 2 represents the time average of A 2 z cos 2 (ω z t). Thus, the real height, y 0 = y 0 + y, depends on the axial amplitude, and hence on E z . y can be estimated as follows (we assume the approximation C 012 (y 0 ) C 012 (y 0 )): At y 0 + y, the axial potential is modified with respect to y 0 . In particular, the E z = 0 axial curvature, C 002 (y 0 ), changes to C 002 (y 0 ). This subsequently forces the variation of ω z as a function of E z : The model described can be tested by computing numerically the radial motion of an electron in a real CPW-Penning trap, using φ(x, y, z) of equation (1) without approximations. The numerical calculation shows a vertical shift of the radial ellipse relative to the ideal one. An example, based upon the trap of section 2.1.1, is plotted in figure 4(b). This has been computed assuming E z = 4.2 K, E m = −E z (ν m /ν z ) and E p ∼ 0. The shift predicted by equation (14) amounts to y = −0.355 µm and is in good agreement with the numerical result of y = −0.325 µm. For low ellipticity, → 0 ⇒ η 2 m → 1 and C 002 2 C 020 → −1 (see equation (4)). In this case, equations (15) and (13) (right) become identical. The former is actually accurate only for moderate values of , below ∼ 0.5. Otherwise, it delivers shifts increasingly bigger than those predicted by the rigorous equation (13). In the limit → +1 ⇒ η m → 0 [24] and, as predicted by that equation, the effect of C 012 upon ν z is theoretically cancelled.

Compensation for different trapping heights
An optimal tuning ratio can be found within a continuous interval of trapping heights; however, it varies smoothly as a function of y 0 . This is shown in figure 5(a), where the plot of T opt c versus y 0 is presented. A 'useful' interval exists (0.8 y 0 2 mm for the example), where M 2,2 can be eliminated. Beyond the upper or lower bounds of that interval, the optimal tuning ratio does not exist.
As shown in figure 5(b), T c and T e can be tuned independently and multiple combinations can be found to obtain one particular trapping position. However, the compensated interval is determined by a univocal relationship, y 0 ⇔ (T e , T opt c ), as featured in that graph. It must also be noted that T e is the main parameter for changing y 0 , while 1.2 T c 1, and the latter is basically used for compensation.

Optimal trapping position y 012
0 . Figure 5(c) shows variation of C 012 with the trapping position. At y 0 0.95 mm, C 012 vanishes. In this case, the compensation of ν z E z also implies that C 004 will disappear. Hence, within the compensated interval, an optimal trapping position can be found, y 012 0 , for which C 012 = C 004 = 0.

Effect of ellipticity upon compensation
For a circular trap (such as the cylindrical or the Mainz/Harvard planar traps [9,12]), the symmetry of x and y implies that C 202 = C 022 . Thus, if C 004 = 0, the constraints of equation (5) force all other octupole terms, C 202 , C 022 , C 400 , C 040 and C 220 , to vanish simultaneously. For an elliptical trap, though, x and y are not indistinguishable: if C 004 = 0, that only implies that C 202 = −C 022 , but in general they will not disappear. The same argument applies to the case C 012 = 0 and the coefficients C 210 and C 030 . For these reasons, even when C 004 = C 012 = 0, the matrix M will always contain some elements different from 0. An ideal elliptical trap cannot be exactly approximated with a CPW-Penning trap. The non-vanishing elements are basically the shifts produced by E p (first column of M) and those due to E m (third column).
3.5.1. Shifts due to E p . It can be shown that for all y 0 within the compensated interval, the frequency shifts produced by the cyclotron and magnetron energies are of the same order of magnitude as in the example of equation (12). Thus, if E p is in the cryogenic domain (E p ∼ 4.2 K), the corresponding deviations of ν p and ν m are in the range of a few µHz, and of a few mHz for ν z . The resulting relative uncertainties in any of the frequencies are negligible. Increasing the magnetic field would make those shifts even smaller (see appendix B). Therefore, the first column in M can always be ignored.

Shifts due to E m .
The numbers in the third column of equation (12) are apparently more significant. However, applying sideband cooling [2,28,29], the magnetron energy takes the value E m = −E z (ν m /ν z ), which is in the range of a few mK (if E z ∼ 4.2 K). Since the magnetron energy does not fluctuate in time [2], the resulting frequency shifts represent mere offsets-which can be corrected for-without affecting the 'visibility' of the trapped particle (see section 3.8). In the example, the biggest relative offset would be ∼ 10 −7 for ν z , propagating to ∼ 10 −12 in ν c . Increasing the magnetic field, or the applied ring voltage, reduces further the third column of M (see appendix B).

First-order nonlinear anharmonicities
After eliminating the linear dependence of the axial frequency upon E z , nonlinear shifts might still be important, especially when y 0 is small. The next most significant even anharmonicities, whose effect can be calculated by first-order perturbation theory, are C 006 and C 008 . These produce the following quadratic and cubic shifts, respectively: For the example trap, these nonlinear shifts are shown in figure 6.

Second-order nonlinear anharmonicities
The next most significant odd anharmonicities, after those included in equation (3), are C 014 , C 212 , C 032 , C 410 , C 230 and C 050 ( ↔i + j + k = 5). The calculation of the corresponding frequency shifts, with rigorous second-order perturbation theory, would be extremely cumbersome. Instead, we employ the model presented in section 3.3. Following the derivation of equations (14)-(15), we obtain The shift predicted by equation (17) has a similar magnitude to that produced by C 006 (see figure 6 (a)). It must be taken into account when designing a CPW-trap. Note that equations (17) (and 18) vanish at y 012 0 . Equation (18) predicts a frequency shift scaling with the products E z · E p and E z · E m . In the former case, the shift is proportional to 1/ν 2 p ; hence, it is normally negligible. In the latter case, as already discussed, E m is very small and the corresponding value of ν z is also inappreciable. Thus, C 212 can be ignored. The same arguments apply to C 032 , which produces a shift very similar to equation (18) (ξ p,m must be simply substituted by η p,m ). The remaining fifth-order coefficients, C 410 , C 230 and C 050 , generate only deviations of ν p and ν m with products of E p and E m . Therefore, as explained in section 3.5.1 and 3.5.2, they can be ignored too. Finally, similar arguments apply to all sixth-order coefficients that have not been considered in section 3.6, namely C 222 , C 204 , C 024 , C 420 , C 402 , C 042 , C 240 , C 600 and C 060 ; they are all irrelevant.

Observation of a single trapped electron
With the dependence ν z = ν z (E z , E 2 z , E 3 z ) given in equations (13), (16) and (17), we are now in a position to perform a realistic simulation of the axial signal of a trapped electron. We assume the detection scheme employed in [30,31], where the signal appears as a shortcut (= the axial dip) of the resonance resistance of a parallel LC-circuit. The goal is to compare the actual signal in the real trap to the ideal one, with the purpose of estimating the 'relative visibility' of the former. The technical details are therefore unimportant; however, they can be found in [32,33]. The simulations are shown in figure 8. The curves are obtained by averaging the axial dip (with The detection LC-circuit is assumed to be perfectly resonant with the electron's axial frequency and with a resonance resistance of Z LC = 100 k . It is also assumed that the signal is picked up from one of the compensation electrodes. The difference in the widths of the ideal dips in panels (a) and (b) is caused by the variation of the induced signal with y 0 . In panel (b) the ideal dip and the real one at 4.2 K are indistinguishable.
) over a Boltzmann distribution of the axial energy [27]. Three different values of the axial temperature T z have been analysed. Figure 8 shows the reduction of the 'visibility' of the dips with increasing axial temperature. A random position, y 0 = 1.6 mm (but with optimized tuning ratio), has been chosen for plot (a). In this case, C 006 and C 014 produce the increasing deterioration of the dip, while C 008 is negligible (see figure 6). In plot (b), y 0 = y 012 0 = y 006 0 = 0.82 mm (see figure 7). Now, C 004 = C 012 = C 006 = 0; however, C 008 still diminishes the quality of the signal with T z . It can be concluded that the detection of a single electron at 4.2 K (or lower) should always be possible within the compensated interval. However, for increasing temperatures the nonlinear anharmonicities make its observation significantly more difficult, even for relatively modest values of T z . Besides fabrication and materials issues [12], those anharmonicities represent one of the main limits to the scalability of the CPW-Penning trap, since they inevitably increase for smaller y 0 .

Summary and conclusions
For the sake of clarity, the discussion has been illustrated with the help of a concrete example. According to it, the main properties of the CPW-Penning trap can be summarized as follows.
• The existence of a trapping interval where the linear dependence of the axial frequency on the axial energy can be eliminated by appropriately tuning the voltage of the compensation electrodes.
• The existence of a particular position within the compensated interval, y 012 0 , where the main anharmonicities, C 012 and C 004 , simultaneously vanish.
• The optimization of the compensation electrode, permitting the further elimination of C 006 at the optimal trapping position.
Although these properties do not universally apply to all conceivable CPW-traps (in particular, the existence of y 012 0 and/or y 006 0 ), they are indeed quite common: many examples can be presented-covering a very wide range of possible trap dimensions and aspect ratios-which do exhibit the characteristics listed. Thus, these might generally serve as a practical guidance in the process of designing a CPW-Penning trap for a particular application.
The potential integration of a geonium atom into future circuit-QED applications has been one of the main motivations for the development of the CPW-Penning trap. However, it offers many other interesting possibilities, such as the achievement of an extremely elliptical ( → +1) 'quasi-2D' regime for electrons in free space, similar to that of high electron mobility transistors, graphene and other systems. These and other options will be discussed in detail in future papers.
For simplicity, we assume that the slit between the ground planes and the central strip is infinitely small. Thus, the boundary conditions are independent of x → V (x , 0, z ) = V (z ). The integral along the electrodes delivers the following functions: In equation (A.2), the upper limit of the compensation electrode is z c up = l c + η + l r /2 and the lower one is z c low = η + l r /2. For the end-cap, the upper limit is z e up = l e + l c + 2η + l r /2 and the lower one is z e low = l c + 2η + l r /2. The gaps between electrodes modify the pure Dirichlet's problem into a much more complex one, involving also Neumann boundary conditions. A rigorous analytic solution (if possible) would demand the use of highly sophisticated mathematical techniques [34]. In order to avoid this mathematical complexity, we approximate the boundary conditions across any of the insulating gaps (which are a priori unknown) by a linear interpolation between the two dc-voltages applied to the corresponding electrodes: V (z ) = m z + n. Here m, n depend on the particular gap. The linear interpolation approximation is acceptable as long as the width of the gaps, η, is small compared to the dimensions of the electrodes (see [22] and references therein).
The result is The auxiliary function f 1 ( r , z , m, n) has the following form: