Investigation of two-frequency Paul traps for antihydrogen production
Nathan Leefer, Kai Krimmel, William Bertsche, Dmitry Budker, Joel Fajans, Ron Folman, Hartmut Haeffner, Ferdinand Schmidt-Kaler
IInvestigation of two-frequency Paul traps for antihydrogen production
Nathan Leefer,
1, 2, ∗ Kai Krimmel,
1, 3
William Bertsche,
4, 5
Dmitry Budker,
1, 3, 2, 6
Joel Fajans, Ron Folman, Hartmut H¨affner, and Ferdinand Schmidt-Kaler
1, 3 Helmholtz-Institut Mainz, Mainz 55128, Germany Department of Physics, University of California at Berkeley, Berkeley, CA 94720 QUANTUM, Institut f¨ur Physik, Johannes Gutenberg-Universit¨at Mainz, Mainz 55128, Germany University of Manchester, Manchester M13 9PL, UK The Cockcroft Institute, Daresbury Laboratory, Warrington WA4 4AD, UK Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel (Dated: October 14, 2016)Radio-frequency (rf) Paul traps operated with multifrequency rf trapping potentials provide theability to independently confine charged particle species with widely different charge-to-mass ra-tios. In particular, these traps may find use in the field of antihydrogen recombination, allowingantiproton and positron clouds to be trapped and confined in the same volume without the use oflarge superconducting magnets. We explore the stability regions of two-frequency Paul traps andperform numerical simulations of small samples of multispecies charged-particle mixtures of up totwelve particles that indicate the promise of these traps for antihydrogen recombination.
INTRODUCTION
The measurable properties of hydrogen ( H ) and an-tihydrogen ( ¯ H ) atoms are expected to be identical aspostulated by the combined charge (C), parity (P), andtime (T) reversal symmetry [1]. One of the most promis-ing tests of this symmetry is the precise comparison of theoptical and microwave spectra of hydrogen and antihy-drogen. The spectrum of hydrogen has been extensivelystudied [2, 3], but precise measurements for antihydrogenare complicated by the small quantities of ¯ H availableand the technical complexity of the experimental appa-ratus [4]. The efficient production and trapping of coldand neutral antimatter systems is therefore a topic ofgreat interest.Production of antihydrogen requires the ability to trapantiprotons and antielectrons (positrons) in the same vol-ume. The state of the art is dominated by Penning traps,where a constant homogeneous magnetic field and inho-mogeneous static electric field allow for confining par-ticles of mass m and charge Q . The ALPHA experi-ment [5, 6] and the ATRAP experiment [7, 8] rely ona variation of a Penning trap for initial particle con-finement. Penning traps have the advantage of robusttrapping for a wide range of charge-to-mass ratios, whilealso facilitating a high charge density of positrons for effi-cient three-body recombination. A large trap volume andsuperconducting magnet creates a high magnetic trapdepth ( 1 K) for the resulting neutral antiatoms. A lim-itation, however, is the inability to trap the oppositelycharged particles in equilibrium in the same volume dueto the use of a DC potential for confinement along theaxial trap direction. Recombination is achieved by inject-ing antiprotons into the positron cloud [9]. The resultingantiatoms are typically created with energy above themagnetic trap depth, and most antiatoms are lost during recombination. Typical yields in the ALPHA appara-tus are several trapped antiatoms per attempt every ≈ Q m := Q/m in a three-dimensional trap. Inorder to achieve such three-dimensional confinement,the stability parameters of electrons — or in our casepositrons — would need to be worked out. For antihy-drogen formation, we approach the problem of simulta-neous three-dimensional particle confinement of antipro-tons and positrons with the idea of a two-frequency Paultrap. This trap design is aimed to combine the stabil-ity parameters of both particles and would also allow forcharge overlap inside the trap.A Paul trap provides a dynamical trapping potential inall three space directions, and works for positive and neg-ative charges equally well. The problem arises from thevastly different charge-to-mass ratio of antiprotons (¯ p )and positrons (¯ e ). The stability of a Paul trap is charac-terized by dimensionless stability parameters a and q [17],which are related to the static and dynamic amplitudes, a r X i v : . [ phy s i c s . a t o m - ph ] O c t FIG. 1. Survey of various trap geometries that can realizethe potential indicated in Eq. (1). Particularly interesting areplanar all-rf Paul traps indicated by the geometry in lowerright. Such a geometry is suitable for miniaturization withmodern atom chip technology [21, 22]. Atom chip technologymay then also support deep traps for the produced neutralantihydrogen. respectively, of the confining potential. Both parametersscale linearly with the charge-to-mass ratio, Q m . A isstable for 0 < q < . a ≈
0, with optimaltrapping achieved around q = 0 .
5. A trap optimized fortrapping antiprotons will have an effective q ≈
900 forpositrons and is fully unstable.A Paul trap optimized for positrons with large Q m istheoretically stable for antiprotons, but suffers from poorequilibrium charge overlap. Particle confinement is char-acterized by the pseudopotential U ∝ m ( a + q )Ω r ,where m is the particle mass, Ω is the frequency of thetrap potential, and r is the distance from the trap cen-ter [18]. If antiprotons and positrons confined in the sameregion thermalize due to the Coulomb interaction and a ≈
0, the characteristic cloud radius of antiprotons willbe a factor of (cid:112) m ¯ p /m ¯ e ≈
45 larger due to the depen-dence of q on Q m . The larger cloud radius of antiprotonswill also make them more susceptible to anharmonicitiesof the trapping potential. We note that the ASACUSAcollaboration reported work for several years on a large-volume, superconducting resonant-cavity Paul trap forantihydrogen production [19]. More recent reports in-dicate the intention to use this trap for spectroscopy ofantiprotonic helium, ¯ p He + [20].In this paper we discuss features of a two-frequencyPaul trap with an infinite, perfect quadrupole poten-tialthat allows simultaneous confinement of antiprotonsand positrons and allows the antiproton and positroncloud sizes to be matched. Trap frequencies are chosensuch that positrons are confined by the high-frequencycomponent of the trap potential and protons are primar-ily confined by the low-frequency component. This allowsthe pseudopotentials for antiprotons and positrons to beadjusted independently. Our work was partially inspiredby the preliminary discussion of two-frequency Paul trapsin Ref. [23]. TWO-FREQUENCY PAUL TRAP
The quadrupole potential of a two-frequency Paul traptakes the form V ( t, r ) = ( V + V cos Ω t + V cos Ω t ) ( x + y − z )2 r , (1)where r is a geometric scale for the trap. We will choosefrequencies such that the fraction Ω / Ω is a number η ≥
1. The potential can be created by a system of hyperbolicelectrodes with cylindrical symmetry, or approximatedby more practical geometries as indicated in Fig. 1.In the initial discussion only motion along the x-direction is considered. From the symmetry of the po-tential these results will also hold for the y-direction, andmay be extended to the z-direction by scaling stabilityparameters by a factor of −
2. The equation of motionfor a charged particle in the potential of Eq. (1) can bewritten¨ x ( τ ) + ( a − q cos 2 η − τ − q cos 2 τ ) x ( τ ) = 0 , (2)where τ = Ω t/ q , = − Q m V , Ω r , (3) a = 4 Q m V Ω r (4)are low-frequency ( q ), high-frequency ( q ), and DC ( a )Mathieu parameters. The time derivative indicated by¨ x ( τ ) is with respect to the time variable τ . Equation (2)is a specific example of a Hill differential equation: asecond order, linear differential equation with periodiccoefficients. Qualitative discussion
Setting q = 0 recovers the well known Mathieu equa-tion for a single-frequency trap. A trapped particle un-dergoes high-frequency motion at multiples of the trapdrive frequency, Ω , in addition to a slow macromo-tion at a secular frequency of ω = (1 / a + q / .In an optimal trap a ≈
0, and the secular frequencyis ω ≈ q Ω / (2 √
2) [17]. If we operate the trap at q ≈ , q ≈ . η , large enough so that the secular oscilla-tion frequency ω (cid:29) Ω . In this regime it is possible totreat a non-zero q as a slowly varying DC term in addi-tion to a .We now consider Eq. (2) from the perspective of twocharged particles, A and B , with opposite charges andmasses m A < m B . To facilitate the discussion we in-troduce the notation q A,B , and a A,B to distinguish trapparameters for the light particle, A , and the heavy q = q = q = ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● q s e c u l a r f r equen cy ( Ω ) - - ( Ω ) FIG. 2. Scaling of the secular frequency as a function of q for B in units of Ω , for η = 5. The values of q foreach curve are indicated on the plot. The value of a waszero for all calculations. The solid lines indicate the ex-pected secular frequency extracted from the pseudopotentialof Eq. (5). Filled circles are estimated from a fast Fouriertransform (FFT) of the numerical integration of Eq. (2). Theinset shows the calculated FFT used to extract secular fre-quencies for q = 0 . , q = 0 .
02. Vertical lines indicate thedriving frequencies. The lowest-frequency peak is the secularfrequency. We do not include Coulomb interaction here butdo so in Eq. 18. particle, B . An important observation is that q B , =( m A /m B ) q A , . The same relationship holds for a , al-though this DC parameter will be set to zero for mostof the manuscript.If the high-frequency confinement is optimized for thelighter particle A and for η < m B /m A , the secular oscil-lation frequency of the heavy particle B due to q B willbe slower than Ω . Therefore we must also consider thedynamic pseudo potential of q B for B . We may write theeffective potentials experienced by each particle as [17] U A ( x ) = 18 m A (cid:18) a A + ( q A ) (cid:19) Ω x , (5) U B ( x ) = 18 m B (cid:18) a B + ( q B ) η + ( q B ) (cid:19) Ω x , (6)where we typically choose η to be large enough that q A is close enough to zero that its effect on particle A can be ignored. The validity of the pseudopotential ap-proximation is discussed in detail in Refs. [24, 25]. Thepseudopotential for particle B in Eq. (5) can be arrivedat by alternately considering the limiting cases where q B , go to zero. When q B = 0, Eq. (2) can be rewritten withanother time transformation τ → τ (cid:48) = Ω t/ η factor multiplying q B . We confirmed the scal-ing of these pseudopotentials by numerical determinationof the secular frequency, shown in Fig. 2. Equation (5)also illustrates the ability of a two-frequency trap to cre-ate overlapping yet independent potential wells for two U A ( x ) U B ( x )- -
10 0 10 20x ( arb. units ) po t en t i a l ( a r b . un i t s ) FIG. 3. Sketch of pseudopotentials for particles A and B assuming a frequency ratio of η = 200 and mass ratio, m B /m A ≈ q A = 0 . q A = 0 (solid lines) and q A = 0 . q A = 0 .
02 (dashed lines). The small perturbationof U A ( x ) by q A oscillates with frequency Ω as indicated bythe shaded band and will average to zero. charged particles with an appropriate choice of trap pa-rameters and frequency ratio, visually demonstrated inFig. 3. This opens the possibility for combining and sep-arating different species of charged particles with highefficiency.In the remainder of the manuscript we discard the su-perscript notation. Where relevant, it is assumed that q , and a refer to trap parameters for the lighter chargedparticle. Floquet Theory
To determine the stability of a two-frequency trap wedefine the vector u ( τ ) = [ x ( τ ) , ˙ x ( τ )]. Equation (2) maythen be written in matrix form˙ u ( τ ) = P ( τ ) · u ( τ ) , (7)where P ( τ ) = (cid:18) a − q cos 2 η − τ − q cos 2 τ ) 0 (cid:19) . (8)If η is a rational number it can be represented as an ir-reducible fraction m/n , where m and n are both integersand m ≥ n . In this case the matrix P ( τ ) has periodicity T = mπ such that P ( τ + T ) = P ( τ ).General closed-form solutions of Eq. (7) do not exist,however it is possible to use Floquet theory to make state-ments about the existence of bound solutions for par-ticular values of equation parameters. The existence ofbound solutions implies a stable trap. q q η = q q η = q q η = q q η = q q η =
13 0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.100.120.14 q q η = / FIG. 4. Stability diagram of a two-frequency Paul trap for q and q . Lighter shading indicates stable operating parameterregions. Frequency ratios, η , are indicated in the upper-left corner of each plot. A discussion of Floquet theory may be found in mostdifferential-equations texts, such as Ref. [26], and appli-cation of Floquet theory to Paul traps in Refs.[24, 25, 27].Here we simply state that the boundary of stability re-gions may be found by identifying parameters for whichthe solution x ( τ ) has periodicity T or 2 T . A general so-lution with period 2 T contains all solutions with period T , so we choose x ( τ ) = (cid:88) k c k e i km τ , (9)where the sum over k extends from −∞ to + ∞ . Equa-tion (9) and Eq. (8) lead to the identity (cid:88) k (cid:34) (cid:18) a − k m (cid:19) c k − q ( c k − n + c k +2 n ) − q ( c k − m + c k +2 m ) (cid:35) e i km τ = 0 . (10)The only way for Eq. (10) to hold for all τ is if eachelement of the sum satisfies this relation independently.Equation (10) may therefore be summarized as a matrixequation D · ...c k − c k c k +1 ... = 0 , (11) where the elements of the infinite matrix D are given by D ij = (cid:18) η a − k m (cid:19) δ i,j − q ( δ j,j − n + δ j,j +2 n ) − q ( δ j,j − m + δ j,j +2 m ) . (12) Stability diagrams
Equation (11) is equivalent to the statementdet( D ) = 0 . (13)Stable trap operating parameters can be identified byfinding parameters that satisfy Eq. (13). Although D isan infinite matrix, a matrix of size (10 m + 1) × (10 m + 1)centered around D was found to be a sufficient approx-imation for evaluating stability. We find that parame-ters where det ( D ) > D ) < D ) needs to be evaluated,and Eq. (13) does not need to be solved exactly. Us-ing larger matrices changes the stable area by less than0.1%. The matrix evaluated can still be large, but mostelements are zero and programs such as Mathematica or MATLAB have efficient tools for computations withsparse matrices.The stability diagrams in q , q space are shown inFig. 4 for integer and rational frequency ratios. Thesediagrams show a structure of unstable resonances thatincrease in density with the frequency ratio. Near the q axis these unstable features correspond to a parametricresonance condition between the secular oscillation fre-quency ω of the particle due to q and the frequency q q1 q q1 FIG. 5. A direct comparison of the a) matrix determinantand b) numerically calculated stability diagrams for η = 5.The unstable resonances are finely resolved in the left diagramand extend all the way to the q -axis, while in the diagramto the right the stability arms merge before reaching the q -axis. For further comparison between the two methods, theanalytic calculation took 15 s and the numerical calculationtook 9700 s, a difference of nearly three orders of magnitude.Regarding the scaling of the computation time, both methodstake significantly longer for larger η . Ω . These resonances become infinitely thin, but ex-tend all the way to the q axis. For large frequency ra-tios this structure indicates that a damping mechanismwill be necessary for long-term stable operation of a two-frequency trap. The stability region for rational numbersshares general features with the closest integer ratio dia-grams, with a significantly denser resonance structure ascan be seen in Fig. 4.It is important to note that stability for the light par-ticle does not guarantee stability for the heavy parti-cle. Stability calculations can easily be evaluated forboth particles, however simultaneous stability can be ob-tained with a general guideline. Revisiting particles A and B : if m B /m A (cid:29) η , stable values of q and q forthe light particle are also stable for the heavy particle if q η m A /m B < .
9. This can be seen by setting q = 0 inEq. (2) for the heavy particle, and recovering the regularMathieu equation with the transformation τ = τ (cid:48) η . NUMERICAL SIMULATIONS
In the previous section we require the determinant ofthe matrix D in Eq. (11) to be zero. With this methodwe are forced to approximate an infinite matrix with alarge but finite representation.Another issue of the matrix determinant solution isthat we are bound to rational frequency ratios η , whichmeans that in practice an irrational frequency ratio canonly be approximated by rounding to a nearby rationalnumber. While the matrix method works for any rationalnumber in principle, even short decimal numbers requirethe evaluation of impractically large matrices. For exam-ple, finding stable regions for η = 3 . q q a ) q q b ) q q c ) FIG. 6. Stability diagrams for η = e , an irrational num-ber, without damping or coupling. In a), the entire simu-lated ( q , q ) range is shown. Red boxes in a) and b) di-agrams indicate regions plotted in the b) and c) diagrams,respectively. The diagram in b) magnifies the region of q ∈ [ . , . , q ∈ [ . , .
15] and the diagram in c) magnifiesthe region of q ∈ [ . , . , q ∈ [ . , . e and thus may continueinfinitely. the evaluation of a 314160 × Mathematica . The numericalmethod then not only makes solutions with an irrational η possible, but also allows for modifications of the equa-tion of motion to incorporate effects such as damping,multiple interacting particles, real trap geometries, orelectrical noise that heats the particles.We evaluate the equation of motion for a time inter-val [ τ , τ ], during which the amplitudes of a particle’soscillation A at τ and A at τ are either of the sameorder of magnitude or escalate to a difference of manyorders of magnitude for, respectively, stable or unstable( q , q ) combinations. This time interval’s length has tobe chosen in respect to the available computation power.Simultaneously the precision with which we can deter-mine stability increases with the length of this time in-terval due to the fact that a solution close to the borderof stability takes a long time to diverge. We compromisedbetween a short solution time and precision by choosing∆ τ = τ − τ < / Ω ). The computerhad 64 GB RAM and 10-core processor, running simula-tions for a time interval of [0 , q q a ) q q b ) q q c ) q q d ) q q e ) q q f ) FIG. 7. (top row) Stability diagrams with damping b for η = 45, evaluated with numeric integration. They correspond to b = 0 (a)), b = 0 . b = 0 . q region and the maximum of stable q .(bottom row) Stability diagrams of a charged particle in a two-frequency Paul trap with η = 45 and an additional magneticfield B along the z-axis (see text), calculated with the matrix determinant method. Values of the magnetic field parametercorrespond to p = 0 . p = 0 . p = 0 . p so that at q = 0 the maximum value of q increases, while the maximum value of stability for q decreases. We define the parameter s ( τ , τ ) to determine whethera particular combination of ( q , q ) is stable. This stabil-ity parameter is evaluated by integrating the square ofthe solution over [ τ , τ ] and dividing it by the intervalslength ∆ τ : s ( τ , τ ) = 1∆ τ (cid:90) τ τ x ( τ ) dτ. (14)We choose s ( τ , τ ) < q , q ) as stable. This condition catches oscillationsthat are close to the parametric resonances and have largeamplitudes. For s ( τ , τ ) > τ , but ultimately the value of this condition is arbi-trary and chosen based on available computing resources.This stability parameter s ( τ , τ ) is evaluated by numeri-cally integrating the coupled equation s (cid:48) ( τ ) = 2 x ( τ ) / ∆ τ simultaneously with the equation of motion. This sig-nificantly reduces computational overhead. Using theserules to decide if a particular combination of ( q , q ) isstable or not, we can modify the original equation of mo-tion and consider extensions to the stability question. Numeric integration vs. matrix determinant
Without damping and coupling the numerical and ma-trix determinant methods agree, but the ability to treatother cases — in general modifications to Eq. (2) — andthe fact that we can use irrational frequency ratios η iswhat sets the numerical method apart from the matrixdeterminant method. A downside of these numerical cal-culations for differential equations is the larger amountof time it takes to do a calculation, and the lower preci-sion around sharp features. While the analytic solutionstake around five minutes to yield results, the numericalsolutions take anywhere from three to over ten hours.The stability diagrams from numerical integration andmatrix determinant calculations are compared in Fig. 5.The thin resonances extending to the q -axis are less visi-ble in the numerically calculated diagrams due to the lim-ited ( q , q ) resolution and finite integration time. Theseresonances are infinitely thin near the q -axis — whichmeans hardly resolvable — and evolve over large timeperiods that would require excessive computation powerto evaluate.Stability diagrams for irrational values of η share gen-eral features with diagrams for nearby integer values of η ,but contain complex structures of instability that appearto have a fractal nature of scale-invariant complexity asshown in Fig. 6. Equation of damped motion
Adding a damping term to Eq. (2) yields a new differ-ential equation that can now model effects such as lasercooling or coupling of the particles’ mechanical motionsto a cold resonant circuit [28, 29],¨ x ( τ ) + 2 b ˙ x ( τ ) + ( a − q cos 2 η − τ − q cos 2 τ ) x ( τ ) = 0 , (15)where b = β/m Ω and β is the damping parameter.Equation (15) is numerically integrated and the stabil-ity assessed by the same threshold as in Eq. (14). Theeffect of damping terms is pictured in Fig. 7. Equations of motion with magnetic field
For antihydrogen production the ion trap must be op-erated in the presence of a magnetic field to trap theresulting neutral particles. A magnetic trap uses an in-homogeneous magnetic field to create a potential well. Asa first approximation of the effect of the magnetic fieldon the charged particles, we consider the effect of a uni-form magnetic field B along the z-axis of the trap. Dueto the Lorentz force the x- and y-motion are no longerindependent and we get a pair of coupled equations,¨ x ( τ ) − p ˙ y ( τ ) + ( a − q cos 2 η − τ − q cos 2 τ ) x ( τ ) = 0 , (16)¨ y ( τ ) + p ˙ x ( τ ) + ( a − q cos 2 η − τ − q cos 2 τ ) y ( τ ) = 0 , (17)where p = (2 B Ze ) / (Ω m ) is a dimensionless magneticfield parameter related to the cyclotron frequency. Sta-bility can be evaluated using the matrix determinantmethod after making a coordinate transformation to aframe rotating with frequency p/ p are shown in the sta-bility diagrams of Fig. 7. In general the magnetic fieldincreases stability due to the radially directed Lorentzforce. This effect has been observed in single-frequencyPaul traps combined with Penning traps [15]. ANTIHYDROGEN PRODUCTION IN ATWO-FREQUENCY PAUL TRAP
As discussed previously, a benefit of a two-frequencytrap for charged particles with vastly different charge-to-mass ratios is the independent control over the trap-ping potential for each species. In a single-frequency trapthe heavy particle experiences a much weaker trappingpotential, resulting in a larger volume of confinement and poor overlap with the light particle cloud. Addinga low-frequency field allows heavy charged particles tobe compressed without affecting the light charged par-ticles. This also opens the possibility of independenttransport of charged species within the same volume. Us-ing a nested electrode structure the two charged speciesmay be initially trapped in different volumes and thenmerged. This is advantageous for antihydrogen produc-tion, where the current procedure is to trap antiprotonsand positrons in independent potential wells and theninject the antiprotons into the positron cloud.We ran numerical calculations of trapping that accountfor the inter-particle Coulomb interactions. For a collec-tion of N positrons and N (cid:48) antiprotons we introduce thevariables r i = ( x i , y i , z i ) and R k = ( X k , Y k , Z k ) to in-dicate the position vector for the i -th positron or k -thantiproton, respectively. The equations of motions thatwe solve are:¨ r i + ( a − q cos 2 η − τ (cid:48) − q cos 2 τ (cid:48) ) x i y i − z i =(18)Γ N (cid:88) j (cid:54) = i r i − r j | r i − r j | − Γ N (cid:48) (cid:88) j (cid:48) r i − R j (cid:48) | r i − R j (cid:48) | , ¨ R k − ρ ( a − q cos 2 η − τ (cid:48) − q cos 2 τ (cid:48) ) X k Y k − Z k =(19)Γ ρ N (cid:48) (cid:88) l (cid:54) = k R k − R l | R k − R l | − Γ ρ N (cid:88) l (cid:48) R k − r l (cid:48) | R k − r l (cid:48) | , where the ρ = m p /m e is the ratio of masses, and Γ is adimensionless Coulomb constant given byΓ = 4 e m e Ω l . (20)This constant is defined in cgs units ( e = 4 . × − statC), and the characteristic length scale l ischosen to be 10 − cm. This formulation of the equa-tions of motion is inspired by the work in Ref. [30], whereantihydrogen production in a single-frequency trap wasconsidered. In that work production of transient, clas-sically bound antihydrogen states was observed in nu-merical simulations for a single-frequency Paul trap op-timized for positron confinement. Reference [30] alsoclaims a trapping mechanism related to the attractionbetween trapped positrons and antiprotons. While theCoulomb interaction certainly provides a significant at-tractive force between antiprotons and positrons in closeproximity, we believe the effect to be overstated in [30]and they primarily observe the weak but still-significant ■ ■ ■ ■ ■ ● ● ● ● ● ■ ■ ■ ■ ■ ● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) η = q = q = q = ● positrons ■ antiprotons ■ ■ ■ ■ ■ ● ● ● ● ● ■ ■ ■ ■ ■ ● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) η = q = q = q = ● positrons ■ antiprotons ■ ■ ■ ■ ■ ● ● ● ● ● ■ ■ ■ ■ ■ ● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) η = q = q = q = ● positrons ■ antiprotons ● ● ● ● ●● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) ● ● ● ● ● ■ ● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) ● ● ● ● ● ■ ■ ■ ■ ■ ● ● ● ● ● r m s o r b i t r ad i u s ( μ m ) FIG. 8. Numerically calculated rms radii for various numbers of positrons (circle) and antiprotons (squares). The calculationis carried out for the same number of positrons and antiprotons inside the trap, up to five particles of each kind and withCoulomb interaction.Ions were assumed to have an average kinetic energy corresponding to a temperature of 4 K. Adding alow-frequency potential can significantly affect the cloud size of antiprotons, while leaving the positron cloud largely unaffected.Black symbols show rms radii for positrons and antiprotons without the low-frequency potential. Calculations were performedfor η = 7 ,
45 and 170. The bottom row shows the same data on an enlarged vertical scale. These plots show that for a frequencyratio η = 170 it is possible to almost match positron and antiproton cloud sizes (see Fig. 9). confining force of an infinite-range dynamic potential onantiprotons. This is demonstrated in Fig. 8, where asingle-frequency Paul trap optimized for positrons pro-vides weak confinement for antiprotons without positronsin the trap.In the first simulation, we consider a mixture composedequally of positrons and antiprotons. The equations arenumerically integrated for 2 to 10 particles in total.Theroot-mean-square (rms) radius, (cid:112) (cid:104)| r i | (cid:105) , of each parti-cle’s orbit in a simulation is evaluated and the average ofthese values over all particles is calculated. The simula-tion for each particle number is repeated 15 times witha randomly chosen set of initial velocities correspond-ing to a 4 K temperature. However, the temperatureof particles inside the trap during an experiment is notnecessarily 4 K, as heating mechanisms such as due tothe RF drive in the presence of nonlinearities and theCoulomb interaction in comparison to the resistive cool-ing time constant may be significant.The results of thesesimulations for positrons and antiprotons are shown inFig. 8. The values of q = 0 .
37 and q = 0 .
024 for thepositrons were chosen as stable operating regions for allthree frequency ratios and along all three axes. The trapdrive frequencies are chosen such that Ω / π = 600 MHzand η = 170. Results were calculated with and withoutthe q term, and clearly show that for large frequencyratios the antiproton cloud can be compressed consider-ably without having any significant effect on the positroncloud. In the second simulation we consider a mixture of 10positrons and two antiprotons using the same parame-ters as before. The simulation shows that recombina-tion is likely to occur at an enhanced rate with a two-frequency trap, due primarily to the increased overlapof the antiproton and positron clouds. Figure 9 showspositron and antiproton orbits with and without the low-frequency potential. The low-frequency potential doesseem to moderately affect the positron orbits, possiblydue to the increased rate of energy changing collisionsbetween positrons and antiprotons. The average rmsdistance between antiprotons and positrons is plotted inFig. 10.To quantify relative recombination rates, we calcu-late the energy of each reduced-mass positron-antiprotonpair, E ( t ) = 12 m e m p m e + m p ( v rel . v rel ) − Γ | r rel | , (21)where v rel is the relative velocity of a given positron-antiproton pair and r rel is the distance between them. Anegative energy corresponds to a classically bound state.A list containing the energy of every positron-antiprotonpair E ij ( t ) is calculated as a function of time, and thenumber of negative energies is recorded at every pointin the simulation. Figure 10 shows the cumulative num-ber of bound pairs produced during the simulation, withand without the low-frequency confining potential. Anon-zero q results in bound antiproton-positron pairs FIG. 9. Orbits for 10 positrons (light gray) and two antipro-tons (red) in a two-frequency trap with positron trap parame-ters q = 0 . η = 170, corresponding to Ω = 2 π × MHz and Ω ≈ π × . MHz , q = 0 (left) and q = 0 . q = 0 the antiproton orbitextends outside the plot boundaries. appearing 5 × more often than for q = 0. Charge density
The Coulomb interaction between positrons and an-tiprotons is conservative, and to create a bound statefrom initially unbound particles a third party must re-move energy from the system. The spontaneous emis-sion of photons is one possible mechanism, but is a slowprocess compared to the characteristic close-interactiontime for charged particles. The primary mechanism forantihydrogen production in the ALPHA experiment isa three-body scattering process that relies on a high-density positron plasma [31]. More than 10 positronsare trapped with a density of > ¯ e /cm [10].We estimate the achievable positron densities in a Paultrap by assuming a force balance between the Coulomband pseudopotential forces for a positron at the edge ofthe positron cloud, N e r = 12 mω r, where the equation is written in cgs units. The an-tiproton density is assumed negligibly low. This leadsto an expected charge density ρ ¯ e = (3 / π ) mω / ( e ).If we assume a trap drive frequency of Ω / π = 6 × Hz, and q = 0 .
37, the positron secular frequencyis ω/ π ≈
80 MHz and the maximum positron densityis ρ ¯ e ≈ × ¯ e /cm , significantly higher than in theALPHA experiment. This suggests that three-body re-combination is also a viable option for an all rf trap.Extrapolating the conclusions of our simulations fromseveral charged particles to several million charged par-ticles is not straightforward, for instance instability mayarise due to excitation of collective particle oscillations by r m s d i s t an c e ( μ m ) ( μ s ) c u m u l a t i v eboundpa i r s FIG. 10. (top) Plot of average rms distance betweenpositrons and antiprotons for positron trap parameters q = 0[black] and q = 0 .
024 [red]. With a second AC potential ap-plied to the trap, the maximum rms distance of an antiproton-positron pair decreases by an order of magnitude and the timeperiod of the two particles orbiting each other is reduced byroughly a factor of 10. The frequency ratio is η = 170 and q = 0 .
37. (bottom) Cumulative number of bound antiproton-positron pairs for the same trap parameters, (black) q = 0and (red) q = 0 . q . Note that the aver-age rms amplitude does not increase significantly, suggestingthat heating on the recombination timescale can be neglected. the dynamic potential. These problems may be avoided,however, by using more compact trap configurations, forinstance planar traps fabricated with atom-chip technol-ogy [21, 22]. These can localize charged particles moreprecisely and permit optical access without restrictionsenforced by large magnets. Increased overlap and local-ization would facilitate studies of other recombinationmechanisms, such as resonantly enhanced photoinducedrecombination [32–34]. Recombination in a smaller vol-ume may also simplify direct laser cooling of ground-stateantihydrogen that may be produced [35, 36]. Togetherthese methods may reduce the number of positrons andantiprotons necessary, and increase the precision and rateof experiments with trapped ¯ H . Heating, cooling and non-linear resonances in realPaul traps
Real Paul traps have effects that perturb thequadrupole potential we considered. Perfect hyperbolicelectrodes are hardly realizable and often approximatedby spherical surfaces. Those real trap geometries makefor a potential different from the perfect quadrupoleand add structures of instabilities to the stability dia-grams [37]. Mathematically, the additional instabilities0can be calculated as resonances of the secular frequencywith terms of the series expansion of the potential. Spacecharge effects of the particles being confined in a smallvolume of the trap also contribute to deviations fromthe assumed harmonic potential. The effects of non-linear resonances caused by deviations from the perfectquadrupole in Paul traps are well known for single fre-quencies, but have to be worked out for a two-frequencytrap.In our simulations using only up to ten particles wedo not see any signatures of these instabilities, but forlarger particle numbers inside the trap, the probabilityof collisions between particles is increased and can placethe momentum of the particles out of phase with thepotential. The particles then gain energy from the fieldinstead of being forced on a stable orbit. This energygain translates to a higher equilibrium temperature ofthe particle cloud and can have considerable influence onthe cloud size. This rf heating effect is dependent on theparticle number, T ∝ N / [38].How big the influence of rf heating, trap imperfectionsand (sympathetic) cooling in our trapping constellationis, is a topic of future investigations. SUMMARY
We have discussed the potential of two-frequency Paultraps for the simultaneous trapping of positrons and an-tiprotons for recombination to antihydrogen. Stable re-gions in the trap parameter space have been identifiedand confirmed using independent methods based on Flo-quet theory and direct numerical integration of the equa-tions of motion. Floquet theory provides stability mapsfor any rational frequency ratio, while numerical integra-tion provides stability maps of reduced precision for anypossible frequency ratio. Additional effects such as thoseof damping and magnetic fields were also investigated.We have further confirmed that two-frequency potentialsenable charged particles with very different charge-massratios to be trapped simultaneously in volumes of simi-lar size, a significant improvement over single-frequencyPaul traps. The influence of this control on the rate ofantihydrogen production is a topic of continued investi-gation.The feasibility of two-frequency Paul traps for antihy-drogen recombination is a topic that merits further study;a number of important questions need to be answered.What effect does micromotion in an rf trap have on theenergy spectrum of the produced antihydrogen? Howwill the trap electric fields contribute to ionization lossof Rydberg states? What trap depths can be achievedwith real electrode geometries? Can atomic ion speciesbe used for sympathetic cooling?Investigation of recombination dynamics in two-frequency traps can be pursued initially with ions such as Be + or Ca + and electrons. 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