An ultralow-noise superconducting radio-frequency ion trap for frequency metrology with highly charged ions
J. Stark, C. Warnecke, S. Bogen, S. Chen, E. A. Dijck, S. Kühn, M. K. Rosner, A. Graf, J. Nauta, J.-H. Oelmann, L. Schmöger, M. Schwarz, D. Liebert, L. J. Spie?, S. A. King, T. Leopold, P. Micke, P. O. Schmidt, T. Pfeifer, J. R. Crespo López-Urrutia
AAn ultralow-noise superconducting radio-frequency ion trap for frequencymetrology with highly charged ions
J. Stark,
1, 2, a) C. Warnecke,
1, 2
S. Bogen, S. Chen,
1, 3
E. A. Dijck, S. K¨uhn,
1, 2
M. K. Rosner,
1, 2
A. Graf, J. Nauta,
1, 2
J.-H. Oelmann,
1, 2
L. Schm¨oger,
1, 4
M. Schwarz,
1, 4
D. Liebert, L. J. Spieß,
1, 4
S. A. King, T.Leopold, P. Micke,
1, 4
P. O. Schmidt,
4, 5
T. Pfeifer, and J. R. Crespo L´opez-Urrutia Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Heidelberg Graduate School for Physics, Ruprecht-Karls-Universit¨at Heidelberg, Im Neuenheimer Feld 226,69120 Heidelberg, Germany State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Innovation Academy for PrecisionMeasurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071,China Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,Germany Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167 Hannover,Germany (Dated: February 5, 2021)
We present a novel ultrastable superconducting radio-frequency (RF) ion trap realized as a combinationof an RF cavity and a linear Paul trap. Its RF quadrupole mode at 34 .
52 MHz reaches a quality factor of Q ≈ . × at a temperature of 4 . Be + Coulomb crystals.
I. INTRODUCTION
Over the past decades, Paul traps have proven them-selves as indispensable instruments in physics and chem-istry, as well as wide-spread analytical applications.Their confinement of ions inside a zero-field environmentwith long storage times makes them especially suited forquantum computing and optical frequency metrology:
High trapping frequencies allow for recoil-free absorp-tion of photons, enabling quantum computation andquantum logic spectroscopy (QLS) by coupling elec-tronic and motional degrees of freedom of the ions. Cru-cially, this has paved the way for many fundamentalphysics studies with atomic systems (for a review see,e.g., Ref. 7), such as searches for a possible temporalvariation of fundamental constants or local Lorentzinvariance, that have been made possible by the ul-timate accuracy and low systematic uncertainties of Paultrap experiments. For such fundamental studies, highly charged ions(HCI) are very interesting candidates (see, e.g., Ref.15). Due to the steep scaling of their binding energieswith charge state, fine-structure and hyperfine-structuretransitions can be shifted to the optical range and be-come reachable for high-precision laser spectroscopy. a) Author to whom correspondence should be addressed:[email protected]
In addition, transitions between energetically close elec-tronic configurations at level crossings are found to be inthe optical range.
HCI have been proposed to testStandard Model extensions, as some HCI feature elec-tronic transitions with enhanced sensitivity to a possiblevariation of fundamental constants, or to probenew spin-independent long-range interactions using iso-tope shift measurements with the generalized King plotmethod.
Resulting from the compact size of theirelectronic orbitals, HCI feature reduced atomic polariz-abilities, small electric quadrupole moments, and sup-pressed field-induced shifts. This also renders thempromising candidates for next-generation fre-quency standards with suggested relative systematic un-certainties below 10 − . In addition, HCI offer forbiddentransitions in the ultraviolet (UV), vacuum ultraviolet(VUV) and soft x-ray regions, which allow the devel-opment of ion-based frequency standards with improvedstability. What used to be called ‘HCI precision experiments’were for several decades carried out with electronbeam ion traps (EBIT), ion-storage rings, and electron-cyclotron ion sources (e.g. Refs. 28–34). Due to the highmotional ion temperatures (
T > K) in those devices,the achieved relative spectral resolution merely reachedthe parts-per-million level. A few years ago, sympa-thetic cooling of HCI transferred from an EBIT into acryogenic Paul trap containing a laser-cooled Coulombcrystal of Be + ions brought down the accessible tem-peratures of trapped HCI from the megakelvin into the a r X i v : . [ phy s i c s . a t o m - ph ] F e b millikelvin range. Later, a pioneering experiment in aPenning trap performed spectroscopy on the forbiddenoptical fine-structure transition in Ar at 441 nm athigher ion temperatures ( T ≈ reach-ing a relative uncertainty of ∆ ν/ν (cid:39) . × − . Thepotential of HCI for optical frequency metrology was fi-nally unleashed with the recent ground-state cooling ofthe axial modes of motion ( T < µ K) of a two-ion crys-tal consisting of one HCI and one Be + ion and the sub-sequent application of QLS in a Paul trap to the afore-mentioned forbidden transition in Ar , reporting astatistical uncertainty of ∆ ν/ν (cid:39) − . One major effect that systematically limits the achiev-able accuracy in Paul trap experiments is the time-dilation shift caused by residual ion motion, which repre-sents a key problem for frequency standards based onsingly charged ions. To overcome this limitation re-quires a strong reduction of trap-induced heating rates,preserving small occupation numbers of the quantumharmonic oscillator throughout the interrogation time.In this paper, we present a new cryogenic Paul trap ex-periment providing ultrastable trapping conditions whichpromises an exceptionally high suppression of motionalheating rates. Its centerpiece is a novel radio-frequency(RF) ion trap realized by integrating a linear Paul trapinto a quasi-monolithic superconducting RF cavity. Theresonant quadrupole mode (QM) of its electric field gen-erates an ultrastable pseudopotential that radially con-fines the ions. Originally developed for experiments withHCI, which are expected to exhibit higher heating ratesthan singly charged ions, the technique can also beapplied to any other ion species.The cryogenic and vacuum setup builds upon the Cryo-genic Paul Trap Experiment (CryPTEx) at the Max-Planck-Institut f¨ur Kernphysik (MPIK) in Heidelbergand on the cryogenic design of Refs. 42 and 43 devel-oped at MPIK in collaboration with the Physikalisch-Technische Bundesanstalt (PTB) in Braunschweig forthe CryPTEx-PTB experiment, and is consequentlynamed CryPTEx-SC. The low-vibration cryogenicsupply provides mechanically ultrastable trapping con-ditions by decoupling external vibrations from the trapregion. An EBIT is added as a source for HCI, anda low-energy HCI transfer beamline connects EBITand Paul trap. Since we aim for direct frequency combspectroscopy of HCI in the extreme ultraviolet (XUV)range, a dedicated XUV frequency comb based on high-harmonic-generation inside an optical enhancement cav-ity has been set up and commissioned at MPIK. II. CONCEPT
The superconducting cavity (SCC) generating thepseudopotential for ion confinement integrates a linearPaul trap as shown schematically in Fig. 1. The reso-
Magnetic field Electric field(i)(vi) (ii)(v) (iii)(iv)
Figure 1. Principle of a cavity with an electric QM. A two-dimensional ideal LC circuit (i-iii) rotating around the spec-ified axis generates a three-dimensional dipole resonator (v)with coaxial cross section (iv). Four symmetrically arrangedresonator poles with alternating polarity (vi) then shape aquadrupole electrode structure inside a single resonant tank.Reprinted from Ref. 44. nance frequency ω of its electric QM is identical to thetrap drive frequency, Ω ≡ ω . The ions are confined ina superposition of the thereby generated 2D pseudopo-tential and an electrostatic potential along the third di-rection, which is configured by additional electrodes in-tegrated in the quadrupole electrodes (not shown in Fig.1). Each of the quadrupole electrodes consists of an outershell electrode with a cylindrical bore containing a coax-ial inner conductor separated by a narrow gap. These twoelements are opposite RF poles of the cavity, and theirsmall separation increases the lumped capacitance of thecavity and thus lowers its resonance frequency. Theirrelative RF phase is fully defined by geometry, which,given a manufacturing tolerance conservatively estimatedto ≤ µ m, suppresses phase differences between themto the level of ∆Φ ≤ . × − rad. In this way, a typicalsource of excess micromotion, which is otherwise difficultto compensate with wired trap configurations, is stronglyreduced. Quasi-monolithic resonators reach very high values ofthe quality factor Q , commonly defined as the ratio ofthe stored electromagnetic energy W to the dissipatedpower P d per RF cycle, Q = ω W/P d . In our case, theSCC strongly reduces resistive losses and increases Q .As this parameter also sets the time scale τ = 2 Q/ω for the decay of the stored electromagnetic energy, am-plitude and phase fluctuations are averaged over manycycles in our cavity, which generates very stable valuesfor the RF-voltage amplitude and the associated time-averaged pseudopotential.More importantly, Q determines the bandwidth of theRF cavity excitation spectrum, ∆ ω = ω /Q , and filtersout noise from the external RF drive for all frequenciesseparated by more than a few linewidths ∆ ω from itsresonance. This should result in spectrally narrow sec-ular frequencies and strongly reduced motional heatingrates of trapped ions. In particular, the motional modefrequencies ω i (cid:28) ω are well separated from the QM, | ω − ω i | (cid:29) ∆ ω , and residual RF noise at ω ± ω i or ω i , that would result in ion heating, is drastically sup-pressed.Other noise sources causing motional heating are alsostrongly cancelled. Johnson-Nyquist noise, the largestnon-anomalous heating source in room temperature se-tups, is greatly reduced in the SCC operating at 4 K.The electrostatic trap voltages are fed through low-passfilters that are held at a temperature of 4 K. Anomalousheating, depending on the distance r between ion andelectrode as typically r − to r − , has a microscopicreach and is strongly suppressed at the distance scale r = 1 .
75 mm of this trap.
III. DESIGN
Long ion storage times on the order of ten minutesare needed for QLS and frequency metrology experi-ments. Thus, experiments with HCI crucially dependon extremely high vacuum (XHV) conditions to suppresscharge-exchange reactions with residual gas. Here, thecryogenic trap environment reduces the pressure to levelsbelow 10 − mbar. The three main design require-ments for the ion-trap environment consisting of SCCand the surrounding cryogenic setup, shown in Fig. 2,are: (R1) multiple optical access ports to the trap centerfor lasers, external atom or ion sources, and detection offluorescence photons; (R2) efficient capturing and prepa-ration of HCI inside the trap to optimize the measure-ment cycle; and (R3), a high mechanical stability and lowdifferential contraction during cooldown to 4 K to avoidmisalignment.
A. Cryogenic setup
We use a pulse-tube cryocooler (Sumitomo Heavy In-dustries RP-082, specified with 40 W at 45 K and 1 Wat 4 K) connected to the cryogenic trap environmentby means of a low-vibration supply to refrigerate twonested thermal stages inside the vacuum chamber, wherethe outer stage shields the inner one from room tem-perature thermal radiation. Both are made of 99 . . the suppression is 30 to 40 dB between 60 Hz and 1 kHz,with a low-pass cut-off frequency around 0 . Twelve ports in the horizontal plane provide opticalaccess to the trap center (R1), for instance, for thelasers used for Be photoionization and Doppler cool-ing of Be + , as well as for the collimated Be atomicbeam produced by an oven connected to the trap cham-ber. Two ports along the trap axis serve for injection andre-trapping of HCI from the EBIT and are equipped withelectrostatic lenses inside the thermal stages and withmirror electrodes protruding into the monolithic tank(R2). Fluorescence from the trap center is collected witha cryogenic optics system consisting of seven lenses(UV fused silica and CaF ) relaying an image of theions through a 2 mm aperture at the outer temperaturestage. Here, an aspheric lens (UV fused silica) projectsit through a vacuum window onto the detection system,consisting of an electron-multiplying charge-coupled de-vice camera (Andor iXon Ultra 888) and a photomulti-plier tube. By adjusting the vertical position of this lens,the magnification can be set between 7 . . .
17% at the wavelength of the Be + Doppler-cooling transition at 313 nm, assuming sphericalemission of the ion.Narrow stainless-steel tubes mounted on the cryogenic-shield apertures restrict the solid angle of room-temperature radiation visible to the ion to 0 . π ,similar to our earlier cryogenic Paul traps. This alsolimits particle flux from room-temperature regions to thetrap, lowering there the residual gas density, suppressingHCI losses by collisions and charge-exchange reactions,and thus extending their storage time.
B. Superconducting cavity
A CAD model of the RF cavity is shown in Fig 3.The Paul trap quadrupole electrodes are an integral partof the resonator. Its electric QM radially confines theions as in a 2D-mass filter. Biased direct current (DC)electrodes trap the ions along the symmetry axis of thequadrupole. On its both ends, additional electrostaticmirror electrodes are used to capture injected HCI (R2).All conducting parts are made of high-purity, massiveniobium, a type-II superconductor with a critical tem-perature of T c = 9 .
25 K. For high mechanical stabil-ity (R3), the monolithic resonator tank supporting thequadrupole rods is machined from a single piece. Dur-ing cooldown, this suppresses differential contraction,which could lead to electrode misalignment. Sapphire a bcde f
100 mm
Figure 2. Section through the cryogenic trap setup: first (69 K, in orange) and second (4 K, in cyan) temperature stage; ( a )superconducting cavity, ( b ) imaging system, ( c ) detection system, ( d ) connection to cryogenic supply, ( e ) port to HCI transferbeamline, ( f ) electrostatic lenses. Detailed views of the cavity are shown in Fig. 3 and Fig. 4. Adapted from Ref. 44. is used as insulator material: Its small dielectric loss atcryogenic temperatures reduces RF power dissipa-tion inside the SCC, and its high thermal conductivity of230 Wm − K − at 4 K improves thermalization of elec-trodes and tank.
1. Superconducting cavity tank and optical access
The box-shaped cavity (220 × ×
114 mm ) consistsof the monolithic tank holding the coaxial quadrupole electrodes, the DC electrodes, and the electrostatic mir-rors, as well as the top and bottom lids sealing it. Theupper lid holds a chevron-shaped disk made of Nb thatgives optical access to the imaging system (see Fig. 4 andFig. 2) while suppressing RF emission, and thus cavitylosses. Its concentric rings and spokes transmit 83 .
2% ofthe light from the trap center that is emitted within asolid angle of Ω / π (cid:39) . .
99% lead wire with T c = 7 . Twelve narrow boresthrough the side walls of the tank give access to the
Inner electrodeInductive couplerQuadrupole electrodePb sealingMonolithictankLowerlid Optical access Mirrorelectrodeˆ x ˆ y ˆ z DC electrode
Figure 3. Cutaway drawing of the niobium RF cavity. Sap-phire insulators are shown in cyan. For details, see main text.Adapted from Ref. 44. trap center in the horizontal plane (R1). For suppressingRF leakage from the cavity, their diameter of 7 mm, or3 . λ ≈ .
2. Quadrupole electrodes
A critical cavity design parameter is the resonance fre-quency ω of its electric QM corresponding to the drivefrequency of the trap. In a linear Paul trap, the stabilityparameter for the radial motion of an ion with charge q and mass m is given by | q r | = 4 qV RF mr ω , (1)where V RF is the RF voltage amplitude. It defines theradial secular frequency ω r (cid:39) q r ω / √
8. For efficientground-state cooling of Be + ions, Lamb-Dicke param-eters well below 1 and thus high secular frequencies onthe order of ω r / π (cid:39) Since themaximum voltage V RF is technically limited, one obtainsan upper bound on the resonance frequency of about100 MHz.We introduce a coaxial geometry illustrated in Fig. 1,with each electrode having an inner and an outer conduc-tor as opposite poles of the cavity QM. Each of these con-ductors is electron-beam welded on one end to the tankwall (see Fig. 5), maintaining a superconducting connec-tion, while the opposite end is centered by sapphire in-sulators. Between them, a small gap of 300 µ m gener-ates a capacitance of ∼
230 pF in each electrode, result-ing in a total quadrupole capacitance of C QP ≈
928 pF,which is two orders of magnitude higher than with single-rod electrodes. This lowers the QM resonance frequency
QuadrupoleelectrodeCapacitivepickup Inner electrodeDC rod Lower lidMonolithictankUpperlidGridded lidˆ x ˆ y Figure 4. Radial section of the RF cavity through the trapcenter at z = 0. Adapted from Ref. 44. ω ∝ C − / into the desired range. We designed the elec-trodes and the cavity with finite-element method (FEM)simulations discussed in Sec. V A.To achieve a larger solid angle of the trap region to-wards the imaging system, the hyperbolic electrode ge-ometry of an ideal Paul trap was sacrificed and a blade-style geometry chosen instead (see Fig. 4). The taperedsection of the quadrupole electrodes has a tip radiusof r e = 0 . to thequadrupole potential below 8 × − for ion crystals witha radial diameter < µ m.
3. DC electrodes
For axial confinement of ions, eight DC electrodes areembedded inside the RF electrode structure, two of whichare symmetrically integrated in each quadrupole elec-trode around the trap center, as can be seen in Fig. 3
Smalltolerances Smalltolerances300 µmgapElectron beam weld
Figure 5. Cut through the coaxial quadrupole electrodes,with inner (yellow) and outer (blue) segments separated bya 300 µ m gap. The concentric elements are electron-beamwelded on one end to the tank and centered at the other onewith sapphire insulators (cyan). Reprinted from Ref. 44. DC supply rodSapphireDC electrode
Figure 6. Detailed viewof the DC electrodes(orange) embeddedwithin the quadrupoleelectrodes (only contoursindicated) and held bythree sapphire rods(cyan). Adapted fromRef. 44. with a detailed view in Fig. 6. The sliced electrodes aremounted under pre-tension and fixed in position usingthree sapphire rods of 1 . Electricalconnections are provided by long niobium rods of 1 . . z = 4 . | z | < µ m fromthe minimum, this yields a relative contribution of anhar-monic terms (up to the sixth order) to the axial potentialbelow 2 . × − .
4. Mirror electrodes
Retrapping of HCI inside the Paul trap will be imple-mented identically to the schemes described in Refs. 37and 63. The kinetic energy of the injected HCI is re-duced by sequential Coulomb collisions with laser-cooled Be + ions prepared beforehand in the Paul trap. De-pending on the initial kinetic energy and the dimensionsof the Be + crystal, thermalization and subsequent co-crystallization of the HCI requires about 10 − tran-sits through the trap center. This is realized by multiplereflections from the mirror electrodes mounted at bothends of the quadrupole structure (see Fig. 3). Their largeaxial separation of 155 mm allows the entire HCI bunch(length on the order of 10 mm) to enter the trap beforethe mirror electrode used for injection is switched to ahigher potential to close the trap. ab c x x x x ld s Figure 7. Horizontal section through the RF cavity at theheight of the upper coaxial electrodes. The detail views showthe RF couplers made of 2 mm thick niobium wire (green),which are soldered into SMA connectors (gold). The couplersare isolated from their mounts by PTFE sleeves (beige) andfixed to the cavity tank with aluminium holders (light grey).( a ) Microwave antenna: l = 57 mm, d = 5 . ∼ ◦ into the plane (not shown). ( b ) Inductive couplingto the magnetic field: x = 9 mm, x = 7 mm, x = 5 mm,and x = 14 . c ) Capacitive coupling to the electricfield: s = 4 .
5. RF coupling
Three types of RF couplers are installed at the cavity,as can be seen in Fig. 7. Coupling to its QM is realizedusing one capacitive pickup, which couples to the electricfield, and one inductive loop coupling to the magneticfield. The latter is used for the in-coupling of RF powerduring operation of the SCC. One end of the loop is RF-grounded to the cavity-tank wall. Reflected power fromthe loop is minimized by matching its impedance to theRF source. For this, we adjust its angle with respect tothe magnetic flux direction of the resonant mode.For monitoring the electromagnetic field inside the cav-ity we use a capacitive probe, which is weakly coupled tothe cavity. It can also be used to stabilize the RF-drivefrequency with respect to the quadrupole resonance.
In addition, a microwave λ/ S / ( F = 2) → S / ( F = 1) hyperfine transition in Be + at 1250 MHz is installed inside the cavity. It ismounted at an oblique angle to all trap axes as well asthe external quantization axis, thus coupling to all Zee-man components of the transition. For the commission-ing measurements presented below, the microwave an-tenna was replaced by a second capacitive coupler.
C. Electronics
The RF cavity rests on a copper platform, which iselectrically isolated from its 4 K stage, as shown in Fig.8, and can be biased for the deceleration of incoming HCI.It is connected for RF grounding to the 4 K stage by a
CopperPTFESapphireStainless steelNiobiumPlatform4 K stageRF ion trap 264 nF
Figure 8. Schematic drawing of the RF ion trap on the biasedplatform at 4 K. All parts of the cavity tank are individuallythermalized using solid copper links (blue). Reprinted fromRef. 44. capacitor set of 264 nF, corresponding to an impedanceof 18 mΩ at the quadrupole resonance. All electrical con-nections to the trap have a length of 2 m between their re-spective room-temperature vacuum feedthroughs and 4 Kconnectors. The wires are thermally anchored at bothcryogenic stages with 1 m length in between to reducethermal conduction. Electrical connections to the RFantennas use semi-rigid beryllium-copper coaxial cables(Coax Co., SC-219/50-SB-B) with PTFE dielectric. AllDC connections employ Kapton-insulated 200 µ m thickphosphor-bronze wires.We use two-way, single-stage low-pass filters (shown inFig. 9), identical to the ones described in Ref. 42, withcut-off frequencies of 30 Hz and 30 mHz, respectively, forsignals to and from the DC electrodes. They suppressnoise at the DC electrodes while protecting the DC powersupplies from RF pickup.Due to the small distance between the surroundingquadrupole electrode and each DC electrode includingits supply rod, a stray capacitance of about 15 pF (seeFig. 6) couples the quadrupole RF voltage to the DCelectrodes. In order to reduce the RF loss due to para-sitic capacitive coupling of the DC wires to RF ground,each rod is connected to its filtered DC power supply viaa 66 MΩ resistor. This lets the DC-biased electrodes alsooscillate at the full RF potential, which ensures strongradial confinement of transiting HCI and thus efficientretrapping (R2). To DC supplySinglefilter channel 4 . Figure 9. Schematic of the effective 4 K electronic filter circuitconnecting a single DC electrode to the external power sup-ply. Surface-mounted components are soldered onto a RogersRO4350B board grounded to the cavity tank. For details, seemain text. Reprinted from Ref. 44.
IV. CAVITY PRODUCTIONA. Material selection
We manufactured the cavity from a high-purity nio-bium ingot. The desired quality factor of Q = 10 − at a maximum electric-field gradient of 6 MVm − ismuch lower than the specified values for state-of-the-artsuperconducting cavities employed at accelerator facili-ties as, for example, the European X-Ray Free-ElectronLaser (EuXFEL). Thus, our acceptable impurity lev-els (Tab. I) are higher than there. In comparison withthe EuXFEL cavities, the concentration limits for N,C (both ≤
10 ppm) and Ta (500 ppm) are exceeded. Aresidual resistance ratio (RRR) was not specified by thesupplier. We obtain an upper limit for the RRR using thedependence of the Nb resistivity on the concentration ofsome impurities. The total effect (Tab. I) is a residualresistivity of 5 . × − Ω m at 4 . < >
300 specified for the Eu-XFEL.
B. Fabrication and surface preparation
Mechanical machining of metallic surfaces can contam-inate them, limiting the performance of Nb as an RFsuperconductor. The thickness of this so-called dam-aged or dirty layer depends on the manufacturing pro-cess, and varies between 50 and 200 µ m. Therefore, weinstead employed non-intrusive electrical discharge ma-chining (EDM) by wire at small material ablation ratesto suppress heating. This avoids gettering of hydrogen,oxygen, and other gases by Nb at temperatures above200 ◦ C. All parts were cut from the ingot using wireEDM in several steps with decreasing material ablationrate, yielding a well-defined contour with a smooth sur-face. Subsequently, all threads, holes, grooves and fitswere milled. After degreasing and cleaning, the cavityparts were sent to an external company for electropolish-ing. Due to our gentle manufacturing, only 50 µ m hadto be removed from all surfaces. Finally, all parts wentthrough several cycles of ultrasonic cleaning in isopropylalcohol, ethanol, and distilled as well as de-ionized water. Table I. Excerpt of the chemical analysis of the niobium ingotused in our cavity, as provided by the supplier. The materialwith a total purity > .
86% was purchased from CompanhiaBrasileira de Metalurgia e Minera¸c˜ao in Brazil.Impurity ∆ m/m (ppm) Impurity ∆ m/m (ppm)Nitrogen 18 Zirconium 1Oxygen 4 Tungsten 5Carbon 30 Tantalum 1233Titanium 7 Hydrogen 2Hafnium 2 Molybdenum 2Nickel 1 Iron 3
Monolithic tank Inductive couplerCapacitive coupler SapphireGroove for Pb sealingMirror electrode
Figure 10. Electropolished cavity with all lids removed. Forthe measurements presented here, the temporary RF couplersmade of copper that are shown in this image were later re-placed by niobium parts and the second inductive coupler wasremoved. Adapted from Ref. 44.
The cavity was assembled inside an ISO6 clean roomto avoid surface contamination. This was followed byelectron-beam welding of cavity tank and electrodes per-formed at a background pressure < . × − mbar.During this step, the cavity was kept closed except forthin tubes for evacuation of its interior. A total of 7 h ex-posure to clean-room air between cleaning and welding ofthe cavity parts was not exceeded. Finally, the DC sup-ply rods as well as the RF couplers were installed, and thetop and bottom lids were sealed using high-purity leadwire. A picture of the fully-assembled cavity without thelids can be seen in Fig. 10. V. FEM SIMULATIONSA. Simulation of the cavity resonant modes
We designed the cavity geometry for an electric QMwith a resonance frequency on the order of 10 MHz bymeans of FEM simulations of the electromagnetic eigen-modes using the commercial COMSOL MULTIPHYSICSsoftware and its RF module.At the start of the simulation, an automatic mesh isgenerated, resolving narrow regions, e.g. the gaps be-tween the coaxial electrodes. On this mesh, the resonancefrequencies and the eigenmodes are found by solving thewave equation on each mesh element with a plane-waveansatz. For nonlinear differential equations, a transfor-mation point is chosen to linearize the functions aroundthe given frequency.The simulations were performed for a cavity at T = 293 .
15 K in perfect vacuum and a perfectly con-ducting box as boundary condition. All cavity elements
Table II. Simulated eigenfrequencies of the four resonantmodes between 1 MHz and 100 MHz. The estimated uncer-tainties from mesh σ m and geometry σ g are added quadrati-cally to yield the total uncertainty σ tot . For details see text. ω / π (MHz) σ m (MHz) σ g (MHz) σ tot (MHz)34 .
851 0 .
009 0 .
002 0 . .
613 0 .
034 0 .
104 0 . .
651 0 .
036 0 .
102 0 . .
372 0 .
051 0 .
231 0 . were assigned identical material properties. Lackingknowledge of the dielectric characteristics of the nio-bium material utilized for cavity fabrication, the rel-ative permittivity and permeability were both set tounity. The electrical conductivity was maximized withinthe restricted computational resources and is given by σ = 5 . × Sm − . The simulated eigenfrequencies ofthe four resonant modes between 1 MHz and 100 MHz arelisted in Tab. II. They are obtained with five significantdigits and a relative simulation tolerance of 1 × − ,which is < .
851 MHz. Additional resonances at 58 .
613 MHz and58 .
651 MHz exhibit a dipole-like structure of the radialelectric field around the trap axis. At 70 .
372 MHz, allouter quadrupole electrodes have the same RF potential.Since the cavity confines the ions with the electric QM,excited with a narrowband RF signal, the other well-separated resonances are not further discussed.
1. Quadrupole resonant mode
The RF field amplitudes of the QM are shown in Fig.11. The electric field strength around the quadrupoleelectrodes has peak values in between the coaxial ele-ments and close to the trap axis, decaying towards thecavity walls (Fig. 11(a)). The outer coaxial electrodesshape the quadrupole electric field on the trap axis. Asexplained above, the simulation shows nearly identicalRF potentials for the DC electrodes and the quadrupolerods, with vanishing fields between them (Fig. 11(b)).Along the trap axis, the homogeneous distribution of theradial electric field causes a constant radial confinementstrength.The RF magnetic field inside the cavity (Fig. 11(c,d))is zero around the trap axis due to the geometry used,and its field lines, closed around the quadrupole struc-ture, lie in the ( xy )-plane. The peak values of theRF magnetic field are radially localized around thequadrupole electrodes close to the regions with maximumcurrent density on the cavity inner surfaces.The simulations (Fig. 11(a,c)) show only small leakageof electromagnetic energy through the optical ports, as a b0 . . . | ~ E | i n k V m − c d0 . . . . . | ~ B | i nµ T Figure 11. FEM simulations of the RF electromagnetic field peak amplitudes of the QM.
Top:
Electric field strength plottedin the radial plane at the axial trap center z = 0 ( a ) and in the horizontal ( yz )-plane through the trap center at x = 0 ( b ).Note that the scale is cut off below the maximum field strength of 10 kVm − to emphasize the changes around the trap axis. Bottom:
Magnetic field strength in the radial plane at z = 0 ( c ) and in the ( yz )-plane at x = 0 ( d ). Adapted from Ref. 44. RF fields do not penetrate deep into those openings dueto their comparatively large wavelengths. The numericalaccuracy of the simulations (five digits) does not allow de-termination of possible residual RF losses through thoseports.
2. Simulation uncertainties
Different systematic effects affect the simulation. Itsquality depends on the variable mesh element size ap-proximating the real geometry. The minimum size shouldbe smaller than the tiniest structures, while the maxi-mum should be smaller than the resonant mode wave-length. Thus, the mesh was refined down to a minimumsize of 51 µ m for convergence. For varying minimum ele-ment size in the region ≤ µ m, the simulation resultsshow fluctuations on the level of kHz for the QM and10 kHz for the other modes. The final result for eachmode listed in Tab. II is the average of these values, andthe largest deviation between any two frequencies is givenas systematic uncertainty σ m . Since the required computation time increases drasti-cally with simulation volume and mesh refinement, thecomplex cavity geometry had to be simplified. Elementssuch as insulators, RF couplers, threads, and lids at theouter surface of the cavity housing and in low-RF fieldregions only have a minor effect for the QM, and werethus removed. The influence of this simplification on theeigenfrequencies was estimated by comparing simulationswith a minimum mesh size of 0 . σ g in Tab. II. B. Simulation of the ion-trap potentials
To suppress higher-order contributions to the 3D har-monic trapping potential and optimize the geometriesof DC and quadrupole electrodes, we performed electro-static FEM simulations with the COMSOL AC/DC mod-ule in several steps. First, a mesh is generated with moredetail close to the trap center and a simplified geometry0stripped of insulators, DC supply rods, and inner coax-ial electrodes and the cavity tank replaced by a cuboidrepresenting its inner walls. All surfaces are modeled asperfect electric conductors, and Dirichlet boundary con-ditions are applied. The potential distribution within thesimulation volume is then obtained with a relative simu-lation tolerance of 1 × − by solving Gauss’ law.
1. Axial confinement
The simulated DC potential distribution in the hori-zontal plane around the trap center is shown in Fig. 12for a minimum mesh element size of 10 µ m. All elementsof the simulation geometry are grounded, except the DCelectrodes, which are biased to 1 . z ), which can be representedas Φ( z ) = C + C z + C z + C z + ... (2)around its minimum. Fitting this model on differentdata ranges ± ∆ z yields the expansion coefficients up to C listed in Tab. III. Higher-order terms cause a depen-dence of the axial eigenfrequency of a trapped ion on itsaxial position, and thus on its energy. The correspond-ing maximum frequency shift ∆ ω z due to the first twoanharmonicities, C and C , can be calculated followingRef. 72. Using the approximation ∆ ω z /ω z (cid:28)
1, it canbe expressed as∆ ω z ω z = 34 (cid:18) C C + 54 C C z (cid:19) z . (3)The shifts at maximum ion displacement z = ± ∆ z fromthe potential minimum at z = 0 are listed in Tab. III. − − − − − − aRadial position along ˆ x + ˆ y in mm A x i a l p o s i t i o n z i n mm .
00 0 .
18 0 .
36 0 .
54 0 . Potential in V . . . − − bPotential in VDataFitResiduals −
60 0 60 − . − . . . . cResiduals in nV Figure 12. Simulation of the DC electrode potential aroundthe trap center with 1 . a ) 2D potential in thehorizontal plane spanned by the trap axis z (dashed line) andthe direction ˆ x + ˆ y . ( b ) Line-out along the trap axis with asixth-order polynomial fit (see Eq. 2) in the range of ± µ mindicated by the gray region. ( c ) Fit residuals (note the dif-ferent scale). Reprinted from Ref. 44. Table III. Coefficients of the sixth-order polynomial fit (seeEq. 2) to the simulated trapping potential along the trap axis(see Fig. 12) for a DC electrode bias of U DC = 1 . ± ∆ z around the potential minimum. Theshift ∆ ω z /ω z (see Eq. 3) of the axial secular frequency fromlisted anharmonicities is given for the maximum displacement z within the fit range.∆ z = 0 .
25 mm 0 . . C (10 − V / mm ) 8 . . . C (10 − V / mm ) 1 . . . C (10 − V / mm ) − − . − . κ = C z /U DC . . . ω z /ω z (10 − ) 0 . . . For a single Doppler-cooled Be + ion ( T ≈ µ K) witha secular-motion amplitude of 120 nm at ω z / π = 1 MHz,the anharmonicities of the smallest fit range translate toa maximum frequency shift of ∆ ω z /ω z = 1 . × − .
2. Radial confinement
The simulated quadrupole-electrode potential in theradial plane around the trap center at z = 0 is shown inFig. 13(a) for a minimum mesh element size of 47 . µ m.Since quadrupole and DC electrodes are strongly RF cou-pled (see Sec. III C), all are set to common RF potentialsof ± . z ), the potential can be expanded in a mul-tipole series: Φ( x, y ) = A + (cid:88) n A n φ n ( x, y ) , (4) − . − . . . . − . − . . . . x in mm y i n mm −
50 0 50Potential in mV − . − . . . . x in mm −
50 0 50Residuals σ in µV − . − . . . . x in mm10 | φ | in nV Figure 13. Electrostatic simulation of the quadrupole-electrode potential at z = 0 for radial distances r ≤ . a ) Simulated potential distribution forquadrupole potentials of ± . b ) Residuals σ of thesixth-order polynomial fit (see Eq. 4). ( c ) Absolute value ofthe first anharmonic contribution φ of the sixth-order poly-nomial fit. Values ≤ . Table IV. Multipole-expansion coefficients of the quadrupole-electrode potential at z = 0 for quadrupole potentials of ± . n th order (see Eq. 4) to the simulation data fromFig. 13. The fit range is restricted to a radial distance of r ≤ . n = 2 6 10 A (10 − V / mm ) 3 . . . A (10 − V / mm ) 2 . . A (10 − V / mm ) − . . where the first three terms are given by φ = x − y ,φ = x − x y + 15 x y − y ,φ = x − x y + 210( x y − x y ) + 45 x y − y . This two-dimensional model is fitted to the data for es-timating deviations from the ideal quadrupole potential, φ . Results listed in Tab. IV show that only the next-to-leading order term φ contributes significantly. Ac-cordingly, the residuals of the sixth order polynomial fit(Fig. 13(b)) do not show contributions from higher mul-tipoles but rather reflect the mesh structure at this cutthrough the 3D simulation volume. The absolute value ofthe anharmonicity φ of the radial potential is plotted inFig. 13(c). Close to the trap axis ( r ≤ µ m for largerCoulomb crystals), its relative contribution to the radialpotential is below 8 × − . VI. RESONANT CAVITY QUALITY FACTOR
We characterize the SCC by determining its QM qual-ity factor at cryogenic temperatures. Two common meth-ods for this are cavity-ringdown and scattering-matrixmeasurements. In the former approach, the decay timeof stored energy following pulsed excitation is measured,yielding the quality factor of the corresponding reso-nance. Here, we instead employ the second technique,based on the bandpass behaviour of near-resonant trans-mission and reflection spectra. The power transmittedor reflected by the RF couplers in the cavity is describedusing the scattering-matrix ( ˆ S ) formalism, givingthe relation between the voltage amplitudes of incident( a i √ Z ) and reflected ( b i √ Z ) waves at different ports: (cid:126)b = ˆ S(cid:126)a with S ij = b i a j , (5)with Z being the transmission line impedance.Each cavity port i introduces losses parametrized by P d ,i = ω W Q − i at resonance. Including those, theloaded quality factor is given by Q − = Q − + (cid:80) n Q − n ,where the unloaded quality factor Q accounts only forcavity losses. For an isolated resonance at ω , the scat- − − − − − Frequency in MHz T r a n s m i ss i o n | S | DataSimulation34 . . . . . . . · − T r a n s m i ss i o n | S |
56 570 . . . . . · −
65 70 7501234 · − Figure 14. Transmission response function of the cavitydriven by the inductive coupler between 1 Hz and 100 MHzat room temperature as recorded with the capacitive pick-up(cyan), and simulated eigenfrequencies from Sec. V A (ma-genta). Reprinted from Ref. 44. tering between two ports i and j can be expressed as S ij ( ω ) = δ ij − (cid:112) Q ( Q i Q j ) − Q ( ω/ω −
1) + 1 . (6)We carried out the presented measurements with a vec-tor network analyzer (R&S ZVL3) driving the cavity withthe inductive coupler and probing it with the capacitivepickup. A broad transmission spectrum at room tem-perature is shown in Fig. 14. Three regions of increasedtransmission reveal the eigenfrequencies of the cavity. Bycomparing them with the simulations from Sec. V A, weclearly identify the isolated resonance around 34 .
383 MHzas the QM. The measured eigenfrequency deviates by ≈
468 kHz from the simulation result (see Tab. II). Thisdiscrepancy is much larger than the estimated uncertain-ties of experiment and simulation, which could be ex-plained by the effect of RF couplers and the externalelectronic circuit, both neglected in the simulation. Thedeviations of the other peaks from the simulation arelarger than for the QM as a result of the simplified ge-ometry. Narrow scans of the reflection and transmissionresponse function of the QM are displayed in Fig. 15.For determining the free, i.e. unperturbed parameters ofthe cavity, the input power was lowered to −
80 dBm and −
70 dBm for the transmission and reflection scans, re-spectively, and the data fitted with the model from Eq.6. In general, only the reflection measurement yields Q , while the transmission spectrum delivers Q . Fromthe reflection data, we obtain a resonance frequencyof 34 .
522 160 8(6) MHz and Q = 2 . × . Thecoupling constant of the inductive coupler, defined by k i = Q /Q i , is k = 2 . .
522 204(2) MHz and Q = 5 . × is measured.2 − − − − a R e fl ec t i o n | S | i nd B Data Fit Residuals − − − b T r a n s m i ss i o n | S | i nd B − . − . . . . − . . . Frequency − . in kHz − . − . . . . − Frequency − . in kHz Figure 15. Response function of the QM in reflection ( a ) andtransmission ( b ) at 4 . Compared with the result obtained in reflection, as thecapacitive probe is only weakly coupled to the cavity, k (cid:28)
1, losses by the inductive coupler dominate. Inthis approximation, the unloaded quality factor becomes Q (cid:39) Q (1+ k ) = 2 . × , in reasonable agreementwith the reflection analysis considering the simplified de-scription.The cavity can be impedance-matched to an external50 Ω signal generator, i.e. k = 1, to drive it efficiently.In this case, the reflected power vanishes at resonance,minimizing the input power needed for a desired intra-cavity power. Tuning the coupling strength is carriedout by adjusting the angle γ between the inductivecoupler and the magnetic field lines of the QM inside thecavity. Hereby, the transformed resistance of the LCR resonant circuit, R in = 12 ω N A B W Q cos γ, (7)depends on the number of windings N , the area A c ofthe coupler, the enclosed magnetic flux B , and the totalenergy stored in the cavity W . We tested this methodwith a normal-conducting prototype cavity and plan toapply it to the SCC. VII. OPERATION AS QUADRUPOLE-MASS FILTER
Since the SCC is designed to capture and store HCIfrom an external ion source, we first characterize the in-jection efficiency and HCI transmission by operating it asa quadrupole-mass filter radially confining the ion mo-tion as a function of the intra-cavity RF power. Weemploy an EBIT as an HCI source connected to the SCCthrough a transfer beamline (Fig. 16). A Heidelberg com-pact EBIT, operated at 1 .
145 keV electron-beam en-ergy produces argon ions in charge states up to q/ e = 16.After a charge-breeding time of about 100 ms, the HCIare ejected in bunches with a kinetic energy of about695 V × q . During transfer, the different charge-to-mass CDTColSLSLEBMCP S L S L P D T S L M C P E L E L M R RF iontrap M R E L E L M C P S K A P A P B O EBIT Electron beamHCIsBeryllium beamPhotoionization laserCooling and repumper laser
Figure 16. Scheme of EBIT and HCI transfer beamline con-nected to the SCC. C: Cathode, DT: Drift tubes, Col: Collec-tor, SL: Sikler lens, EB: Electrostatic bender, PDT: Pulseddrift tubes, EL: Electrostatic lens, MR: Mirror electrode, SK:Skimmer, AP: Aperture, BO: Beryllium oven. For details seetext. Adapted from Ref. 44. q/m species in the bunch separate according to their dif-ferent time-of-flight (TOF), and are detected after pass-ing the SCC with a microchannel plate (MCP) detector.The beamline is operated with static potentials optimiz-ing HCI transfer. Under the given conditions, the fastestions spend around 1 µ s inside the SCC, or approximately35 cycles of the RF field. Typical TOF spectra in Fig.17 show peaks from Ar to Ar ions. Their rela-tive amplitudes cannot be directly compared, since thesedepend on their specific EBIT yields, the beamline trans-mission for a given q/m , and the sensitivity of the MCPto different charge states and ion impact energies. The Ar Ar Ar Ar Ar Ar Ar pp M C P v o l t ag e i n m V
170 mV pp pp
12 12 . . . . . pp Figure 17. TOF spectra of Ar HCI measured with an MCPdetector behind the SCC as a function of stored RF powerquantified using the SCC pick-up voltage. Each spectrumrepresents the average of 192 extraction cycles. Reprintedfrom Ref. 44. a V o l t ag e s u m i n V Ar X + , X =161514131211100 80 160 2400 . . . . b Pickup voltage in mV pp N o r m a li ze d v o l t ag e s u m Figure 18. Transmission efficiency for different Ar HCI as afunction of the stored RF power quantified using the SCCpick-up voltage. Points represent the background-subtractedTOF peak voltage sum averaged over 192 extraction cycleswith their statistical uncertainties. Reprinted from Ref. 44. transmission efficiency for each individual charge statedepends strongly on the SCC RF power, measured withthe pick-up coupler. At high power, the strong radialconfinement of the ion motion improves the transmission,displayed in Fig. 18 as the integral of each q/m peak de-pending on the pickup voltage. The efficiency increaseswith RF power until it saturates for most charge statesabove 180 mV pp pickup voltage. As expected for sta-ble radial ion motion inside the SCC (see Eq. 1), highercharge states show better transmission already at smallRF power. This measurement proves the stable radialconfinement of HCI within the SCC. The next step willbe their deceleration and retrapping by Coulomb inter-action with trapped laser-cooled Be + ions. VIII. TRAPPING OF Be + IONS
Our first trapped-ion experiments with the setupsketched in Fig. 16 used Be + ions produced withinthe trap region by photoionization of Be atoms froman atomic beam. Because of their kinetic energies of ≈
140 meV, they are instantly captured by the SCC. Sub-sequent Doppler cooling brings their temperature downto the mK range. The effusive thermal Be beam em-anates from an oven heated to T = 1250 K that is lo-cated at 0 .
93 m from the SCC and separated from it bytwo differentially pumped vacuum stages. The Be beamis collimated to a diameter of 800 µ m at the trap cen-ter for avoiding surface contamination of the supercon-ducting electrodes. There, it crosses the photoionizationlaser at 90 ◦ , reducing the first-order Doppler shift tothe MHz range. Two-photon resonance-enhanced ioniza-tion proceeds through the 2s S → P transition at235 nm. With a 1 /e laser-beam diameter of 250 µ m atthe SCC center, we could load the trap at laser powers − . − . − . . . . × counts C a m e r a p o s i t i o n ˜ y i n mm − . − . . . . x in mm × c o un t s abc Figure 19. Images of Doppler-cooled Be + ion ensembles con-fined inside the SCC. ( a , b ) White bar: Scale of 200 µ m atthe focal plane of the camera. ( c ) Gaussian fits (magenta)to the projected intensity distribution of a single ion (orange)with standard deviations σ ˜ x = 26 . µ m and σ ˜ y = 40(2) µ mat the camera. Note the 2 × b ) and ( c ). above 80 µ W. For Doppler cooling of Be + we use thestrong S / ( F = 2) → P / transition at 313 nm. Therequired laser beam enters horizontally at an angle of30 ◦ to the trap axis, thus cooling both, axial and radialmodes. Perfect circular polarization would result in aclosed cooling cycle when using a co-linear bias magneticfield, but in the present experiments we used the Earthmagnetic field to define the quantization axis. Therefore,some population is optically pumped into the F = 1 hy-perfine sublevel of the ground state, requiring a separaterepumper beam detuned by 1 .
25 GHz from the coolingtransition, with the same polarization and propagationaxis as the cooling laser.Fig. 19 shows images of Doppler-cooled Be + Coulombcrystals, magnified M ≈
10 times as defined by thechosen position of the 40 K asphere. In these proof-of-principle experiments, the trap is typically operated withan RF input level of 21 dBm corresponding to RF ampli-tudes of V RF ≈ . Be + ions arestored at secular frequencies of ω z / π (cid:39)
209 kHz and ω r / π (cid:39)
409 kHz, calibrated by motional excitation ofa single Be + ion confined inside the SCC. Due to RFpower dissipation of some elements, the trap heats upby ∼ IX. CONCLUSION
We have introduced and commissioned a novel cryo-genic ion trap employing a superconducting cavity whichconfines ions within the RF field of its electric quadrupolemode. Its quality factor around 2 . × is two tothree orders of magnitude higher than values reportedfor normal and superconducting step-up resonatorsconnected to cryogenic Paul traps, and may be further in-creased in the near future, as losses due to trapped mag-netic flux inside the cavity walls or locally enhancedRF dissipation are eliminated. Proof-of-principle opera-tion showed a large acceptance for injected HCI and sta-ble confinement of laser-cooled Be + Coulomb crystals.The cavity bandpass strongly suppresses white noise fromthe RF power supply at the trap electrodes. Spectralcomponents at the secular frequencies ω i or their side-bands around the trap drive at Ω ± ω i , both of which cancause motional heating of the ion, are reduced by a factorof > . This should result in extremely small motionalheating rates below values reported for other cryogenicPaul trap experiments. Such small rates arounda few quanta per second are measured using sidebandthermometry, requiring preparation of the ions in theirmotional ground state. We will in the near future imple-ment the scheme described in Ref. 42 with a laser systemcurrently being developed.Using the recently established techniques fortrapping, cooling, and subsequent interrogationof HCI by QLS, this new apparatus, in combinationwith the XUV frequency comb, promises theimplementation of XUV frequency metrology based onHCI. DATA AVAILABILITY STATEMENT
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
ACKNOWLEDGMENTS
We acknowledge the MPIK engineering design of-fice led by Frank M¨uller, the MPIK mechanical work-shop led by Thorsten Spranz, and the MPIK me-chanical apprenticeship workshop led by Stefan Flickerand Florian S¨aubert for their expertise and the fab-rication of numerous parts as well as the develop-ment of sophisticated fabrication procedures of com-plex parts. For their technical support, we also thankThomas Busch, Lukas Dengel, Nils Falter, ChristianKaiser, Oliver Koschorreck, Steffen Vogel and PeterWerle. We acknowledge J. Iversen, D. Reschke, andL. Steder for support and discussions. This project re-ceives funding from the the Max-Planck Society, theMax-Planck–Riken–PTB–Center for Time, Constantsand Fundamental Symmetries, the European Metrol-ogy Programme for Innovation and Research (EMPIR),which is co-financed by the Participating States and fromthe European Union’s Horizon 2020 research and inno-vation programme (project number 17FUN07 CC4C),and the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) through the collaborativeresearch centre SFB 1225 ISOQUANT, through Ger-many’s Excellence Strategy - EXC-2123 QuantumFron-tiers - 390837967, and through SCHM2678/5-1.
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