Implementation of new techniques for high precision g factor measurements

Transcription

1 Implementation of new techniques for high precision g factor measurements Dissertation zur Erlangung des Grades "Doktor der Naturwissenschaften" am Fachbereich Physik der Johannes Gutenberg Universität Mainz Slobodan Djekic geb. in Vukovar Mainz, im Juni 2004

2 Berichterstatter der Arbeit: Prof. Dr. G. Werth Prof. Dr. G. Huber Datum der mündlichen Prüfung:

3 Abstract In this thesis experimental work is described which aimes at future improvements of current experiments at the University of Mainz to determine the g factor of the bound electron in hydrogenlike ions. These experiments use a single ion confined in a Penning trap. The main results are: measurement of the temperature of a single ion in different degrees of freedom, implementing of the novel detection technique ("peak technique") for the axial ion oscillation, and new phase sensitive spin detection technique. Measurement of the temperature of a single C 5+ ion was performed using two independent methods. Both of them, via well defined magnetic and electric field imperfections, were in a very good agreement and gave us an axial temperature of approximately 60 K. Also a direct measurement of the axial tank circuit resulted in a temperature of around 50 K, well above the ambient temperature of 4 K. Our experimental result leads to the conclusion that the problem is the input noise of our cryogenic preamplifiers. In the near future we are planning to modify the preamplifiers and reduce their noise temperature to the temperature of the surrounding liquid helium. Another important result is implementing of the novel technique for the ion axial motion detection in a Penning trap via observing an ion signal peak instead the ion signal dip. Implementation of this technique reduces the necessary time for the ion detection by a factor of 10 compared to the previously used bolometric technique introduced by Dehmelt and coworkers [Deh68, Deh73]. It makes a single ion preparation significantly faster and also Penning trap optimization much faster and easier. Furthermore, the peak technique gives us the possibility for different improvements of the g factor measurements like introducing a new phase sensitive technique. The main obtained result is a new phase sensitive spin detection technique which allows us to measure very small spin flip jumps, which was impossible with the previous frequency detection technique. Using this technique we were able to overcome the Fourier limit by a factor of 8, and observe axial frequency shifts of the order of 80 mhz, that is around factor of 3 better than before. This is a necessary requirement for planned measurement on ions of higher Z like Ca 19+, as well as for future experiments on protons and antiprotons. With the further improvement of the stability of the power supply for the trap electrode potential and cryogenic electronics for the ion decoupling we should determine a spin flip by a factor of faster than before. In such a short time all magnetic and electric field instabilities are much smaller, and consequently the accuracy of future measurement will improve.

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5 Contents Abstract iii 1 Introduction and Motivation 1 2 Theoretical Background 3 3 Experimental Setup Superconducting Magnet and Cryostat The Penning trap Penning trap with hyperbolical electrodes Compensated Penning trap Penning trap with cylindrical electrodes Double Penning trap Electronics Cryogenic electronics Room temperature electronics Interaction between the ions and electronics Experimental Procedure Loading particles Cleaning particles Single ion preparation Cooling particles Axial cooling Cyclotron cooling Magnetron cooling Penning trap optimization Tuning ratio optimization Orthogonality determination Spin detection Measurement and Results Temperature of a single ion in a Penning trap Temperature as a physical quantity Implications of the ion temperature on the accuracy of g factor measurements Cyclotron temperature measurement Axial temperature measurement via B 2 term Axial temperature measurement via C 4 term

6 vi CONTENTS Direct noise temperature measurement Discussion and results A peak technique for axial motion detection in a Penning trap Measurement principle Fast ion resonance detection Fast tuning ratio optimization via the peak detection technique Phase sensitive spin detection Fourier limit restriction Basic principles Measurements and results Conclusion and Outlook Conclusion Outlook A Magnetic field fluctuations in Penning traps 69 A.1 Results of the magnetic field stabilization A.1.1 Technical details on the pressure stabilization system A.1.2 Dependence of magnetic field on pressure A.1.3 Dependence of magnetic field on temperature A.2 Conclusion

7 Chapter 1 Introduction and Motivation In our group at the Johannes Gutenberg Universität Mainz, Penning traps have been used for determination of the bound electron s g factor in hydrogenlike ions to the order of 10 9 [Qui95, Häf00b, Ver04a]. This is presently considered as the best test of bound state quantum electrodynamics calculations. Simultaneously with our measurement significant progress has been made in Bound State Quantum Electrodynamics (BSQED) calculations and at present the quoted uncertainty of the theory is three times smaller than the experimental one [Yer02, Sha02]. It is therefore challenging to reduce the uncertainty of the ongoing experiments. At the same time attempts are being made to extend the measurement to different systems [Wie99]. On the one hand it is planned to go to higher-z ions [Gil01], since the QED-effects scale with Z 2 and thus higher-z will serve for better tests. On the other hand going to lower-z ions will result in an improved value of the electron mass [Bei02], which at present has been derived from a comparison of the experimental and theoretical value for the g factors of 16 O 7+ and 12 C 5+. At low-z ions such as 4 He + or 24 Mg + the theoretical uncertainties of the QED calculation are significantly smaller. Another challenging experiment, which is currently in preparation in our group, will extend g factor measurements to the bare proton. In that experiment the proton s magnetic moment is supposed to be determined with ultra high precision. All these projects require improvement of the present setup. The limitation in accuracy of the past experiment is imposed by several facts: 1. As will be outlined below, the detection of induced spin flip transitions of a single ion requires detection of small frequency differences of the order of a few 100 mhz of a total frequency of several 100 khz. This requires extremely stable operating conditions. 2. The axial temperature of the ion is presently of the order of 70 K, which is significantly higher than the ambient temperature of 4 K in our setup. This leads to line broadening and asymmetry. 3. The required measurement time for a spin flip detection is at present several minutes. During this time the magnetic field may change its value due to unavoidable fluctuations. Motivation for this doctoral thesis has been to work on these three limitations and to improve the present apparatus to the level suitable for measurements of the bare proton s g factor, and the g factors of Ca 19+ and U 91+. For this purpose we

8 2 Introduction and Motivation introduced a novel state-of-the-art technique for spin flip observation by means of a very sensitive motional phase shift detection. It has several advantages comparing to the previous one where we used a well established axial frequency detection scheme. The most important upgrades are: 1. Possibility to observe an axial frequency shift of only mhz. In the near future, by using GaAs switches in the cryogenic region, this shall be further improved to the order of 10 mhz. 2. The spin flip detection time is reduced by a factor of 20. This means that the influence of magnetic field fluctuations becomes smaller by roughly an order of magnitude. 3. The total time for performing an experiment for the g factor measurement of a new ion species will be 1-2 months presumably, instead of previously 8-10 months, saving therefore manpower and financial expenses by a factor 5-6. Furthemore we have to deal with a number of technical problems. Long and short term fluctuations of the magnetic field caused by ambient temperature and pressure variations may lead to uncertainties in our results and have to be reduced to the minimum amount. Stabilization of the magnetic field has been partially completed and it is shortly documented in the appendix of this thesis.

9 Chapter 2 Theoretical Background Several different types of experiments on highly charged ions are currently carried out in order to test the validity of quantum electrodynamics in strong nuclear fields. It is crucial to find systems where uncertainties in the nuclear description do not restrict the testing ground. This is a serious limitation in recent experiment on the Lamb shift on the hyperfine splitting in hydrogenlike ions [Stö00]. A good candidate, which fulfills the above requirement, is the bound-electron g factor in hydrogenlike ions. Measurements of the g factor on the bound electron in hydrogenlike carbon and oxygen ions, performed in a Penning trap in our laboratory, represent at present the best test of the bound state quantum electrodynamics theory in a strong electric field. The electric field created by the nucleus is of order V/cm and it modifies the vacuum field of the bound electron, changing in such a way properties of the electron [Wer94]. The first value for the g factor of the free electron has been given by Dirac g Dirac = 2, (2.1) but there is a slight difference from that value due to the presence of the selfinteractions with the radiation field. The different contributions to the g factor of the bound electron in hydrogenlike ions are the relativistic correction, radiative corrections, and nuclear corrections. Relativistic correction. term has been calculated by Breit in 1928 [Bre28], and nowdays is usually called Breit term [ ] 1 (Zα) g j = 2 2, (2.2) 3 α is the fine structure constante (1/α = 137, ), and Z is the atomic number. The Breit term is the largest bound state correction of the electron s g factor for hydrogenlike ions. For example the g factor of hydrogenlike uranium (U 91+ ) is reduced by the Breit term by 15%. Radiative corrections of the free electron. The first calculation of a radiative correction of the free electron s g factor was performed by Schwinger. His term α/2π describes the virtual emission and absorption of a photon by

10 4 Theoretical Background the electron, it is shown in Fig The higher-order QED terms account for the virtual emission and absorption of several photons and for the vacuum polarization. The accuracy of the QED calculations of the free electron s g factor is limited via accuracy of the fine structure constant α. The theoretical value of the free electron s g factor is experimentally confirmed on a level of [Deh90a]. Figure 2.1: Feynman diagram describing the QED contribution to the g factor of the free electron of order α/π. The straight line is the propagator of the free electron, the wavy line is the photon propagator, and the triangle denotes the magnetic field. Bound state radiative corrections. In addition to the radiative corrections of the free electron, the g factor of the bound electron in the 1s 1/2 state is modifed by bound state QED corrections. Six Feynman diagrams contribute to the g factor of the bound electron on the one-loop level. The three graphs a, c, and e in Fig. 2.2 describe the self-energy corrections, and the graphs b, d, and f represent the vacuum-polarization corrections to the bound state g factor. The Feynman graphs were evaluated [Yer02] for hydrogenic ions with different nuclear charge Z in non-perturbative calculations which include all orders in Zα. These new theoretical calculations are an exciting advancement in the field of bound-state quantum electrodynamics, results for the C 5+ and O 7+ are given in Table 2.1. The comparison of the new theoretical results with our high-accuracy measurements of the g factor of the bound electron in hydrogenic ions provide one of the most stringent tests of the theory of quantum electrodynamics in very strong electromagnetic fields. Nuclear corrections. There are several nuclear effects that influence the g factor of the bound electron in hydrogenlike ions and have to be considered in the theoretical calculations: a) nuclear recoil correction due to the finite nuclear mass. It has been calculated by V.M. Shabaev [Sha02] to all orders in αz, b) finite nuclear size effect c) nuclear polarization, i.e. the virtual excitation of nuclear energy levels. Compared to other bound-state QED tests, for example the Lamb shift and the hyperfine splitting in hydrogenlike ions, the g factor of the bound electron in hydrogenlike ions is less sensitive to the details of the nuclear structure. The nucleus influences the g factor of the bound electron only via the dependence of the electronic

11 5 Figure 2.2: Feynman diagrams describing the bound state QED contributions to the g factor of the electron of order α/π. Graphs a, c, and e represent self-energy corrections, and graphs b, d, show vacuum-polarization corrections. The double line represents the propagator of a bound electron in a hydrogenlike ion [Qui01]. wave function on the nuclear properties. However, as it is known [Bei00], investigations of the QED effects in high-z hydrogenlike systems are strongly restricted by an uncertainty due to the nuclear size effect. One can see clearly in Fig. 2.3 that for the U 91+ ion the finite size correction is of the same order as the α/π contributions. In particular, the investigations of the hyperfine splitting in heavy ions [Sha01] showed that the QED effects can be probed only at a specific difference of the hyperfine splitting values in hydrogenlike and lithiumlike ions, where the nuclear structure effects can be significantly reduced. In the case of the bound electron g factor the role of the nuclear structure effects is not so crucial as in the case of the hyperfine structure splitting. It turns out, however, that investigations of the g factor of lithiumlike ions seem also important since one may expect that the uncertainty due to the nuclear size effect can be significantly reduced in a combination of the g factors of hydrogenlike and lithiumlike ions.

12 6 Theoretical Background Description C 5+ O 7+ Dirac value (point) 1, , Fin. nucl. size 0, , Free QED α/π 0, , Bind. SE α/π 0, (1) 0, (1) Bind. VP α/π 0, , QED (α/π) 2 0, (3) 0, (6) Nuclear Recoil 0, , Total (theoretical) 2, (3) 2, (6) Experimental 2, (10)(44) 2, (15)(44) Table 2.1: Individual contributions to the 1s-electron g factor in hydrogenlike carbon and oxygen taken from [Yer02]. The labels SE and VP stand for self-energy and vacuum-polarization. Comparison of the total theoretical value and our experimental value shows agreement inside one standard deviation. The biggest error in experiment, 44 in second brackets, comes from the electron s mass uncertainties. contribution to the g-factor relativistic L-S-coupling BS-QED corrections 1 st order in estimation for -term of BS-QED corrections nuclear recoil volume of nucleus nuclear charge number Figure 2.3: Relativistic and QED contributions to the g factor for different nuclear charge numbers. With the vertical line hydrogenlike oxygen ion is denoted.

13 Chapter 3 Experimental Setup Our experimental setup consists of a superconducting magnet with a vertical bore, a cryostat connected to the Penning trap and the corresponding room temperature and cryogenic temperature electronics (Fig. 3.1). The total height of our setup is 210 mm. A combination of liquid nitrogen and liquid helium cryostats is placed in the magnet bore. On the bottom side of the helium cryostat, in a very good thermal contact, the cryogenic electronics have been placed. In the geometrical center of the main superconducting coils, where the magnetic field is most homogeneous, and in thermal contact with the cryostat is a vacuum chamber made of oxygen free high conductivity (OFHC) copper with the Penning trap inside. At the top of the setup is the so called "hat" which consists of an insulation vacuum chamber to which room temperature electronics boxes, an inlet port for microwaves, and a helium gas evaporation tube have been connected. 3.1 Superconducting Magnet and Cryostat Superconducting Magnet In our experiment we use a standard Nuclear Magnetic Resonance (NMR) superconducting magnet from Oxford Instruments with a maximum magnetic field of 6 T. The magnet has a vertical bore of 127 mm diameter. The cryogenic system of the magnet consists of two dewars, a liquid helium dewar which has a volume of 29 l, and a liquid nitrogen dewar of 84 l. Refilling time for helium is around 17 days, and for nitrogen 14 days. The magnetic field strength in our case is B 0 = T and it corresponds to the Larmor frequency of 105 GHz for hydrogenlike carbon, which has been chosen such that it is in the frequency range of an available microwave setup. A magnetic field is created by a cylindrical superconducting main coil, but there are also other 9 smaller, noncylindrical correction coils for the magnetic field optimization. Cryostat Our liquid helium cryostat has a cylindrical helium dewar with 5 l volume, in which one has to refill liquid helium every 5 days. The evaporation rate of helium when our cryogenic preamplifiers work is 22% per day, and in the case when the cryogenic electronics is turned off around 16% per day. The low consumption is achieved by

14 8 Experimental Setup Figure 3.1: Experimental setup main parts are: superconducting magnet, cryostat with the helium and nitrogen dewars, vacuum chamber with the Penning trap, cryogenic and room temperature electronics. very careful shielding of the helium dewar. The first layer is a radiation shield directly at the dewar surface. Then we have an additional aluminium tube of 2 m length which is cooled by evaporating helium to approximately 20 K. Therefore it is avoided that the cold part of the apparatus has a direct optical contact with the liquid nitrogen area, and the thermal radiation is reduced according to Stefan-Boltzmann s law by a factor of 230. Finally the outer part of the cryostat is a nitrogen reservoir with a 25 l volume, which represents a 77 K shield. The evaporating rate of the nitrogen is so high that we have to refill the reservoir every 36 h. The helium gas evaporation tube, as one can see in Fig. 3.1, is connected to the "hat". Due to the constant flow

15 3.2 The Penning trap 9 of helium gas through it, a precooling is achieved, reducing the heat load from the room temperature region. This stainless steel tube serves also as a holder for the whole apparatus including the helium dewar, cryogenic electronics and ultra high vacuum (UHV) chamber with the Penning trap. A constant temperature and length of the evaporation tube represent extremely important parts of the experiment since thermal insulation of the cryogenic parts from room temperature and a fixed position of the Penning trap play a crucial role in the stability of the magnetic field 1. For providing thermal insulation a turbomolecular pump stand for pumping the space between the magnet bore and the cryostat is installed. A pressure of 10 4 mbar by turbo pumping is achieved. After filling liquid nitrogen and helium into the cryostat, the insulation pressure reduces to 10 7 mbar via the cryopumping effect. On the other hand the working pressure inside the Penning trap vacuum chamber is estimated to be lower than mbar. Since there is no vacuum gauge that works in that range we used our ions inside a trap for a testing. After creation a cloud of 30 ions we measured a number of remaining ions during 4 weeks time. With the known cross section for charge exchange with the residual gas in the chamber and no ion losses in the measurement time, a simple calculation gave us an upper limit for the trap pressure. 3.2 The Penning trap The original idea of the Penning trap is almost 70 years old. It dates from 1936 when F.M.Penning published a paper on increasing the sensitivity of ionization vacuum gauges by using an axial magnetic fields [Pen36]. There he explained: With a magnetic field of sufficient strength, electrons, leaving the cathode, will miss the anode and return to the cathode - thereby reducing the anode current to zero, when the magnetic field is increased beyond a certain value....if there is a sufficient number of a gas molecules in the chamber, an electron can collide with these molecules. If it looses energy in these collisions the return to the cathode is impossible and the electron will describe a significantly longer path before eventually impinging onto the anode... After more than one decade, a proper theoretical description of confinement mechanism has been given by J.R.Pierce [Pie49]. He explained that it is possible to obtaine a pure sinusoidal motion of electrons trapped in a combination of quadrupol electric and strong magnetic field. Finally in the late sixties H.G. Dehmelt used the name Penning trap for such a device consisting of the hyperboloidal electrodes with a static electric field and a static magnetic field pointing along the symmetry axis. Single charged particles stored in Penning traps have been used for some of the most precise measurements in science [Dyc86, Dyc87]. The phenomenal accuracy of a Penning trap magnetic moment measurement comes from the ability to precisely measure the cyclotron frequency of single ions trapped in a highly uniform magnetic field, together with the Larmor precession frequency [Pau90, Gab95, Far95]. 1 This will be discussed in details in Appendix A.

16 10 Experimental Setup The ideal Pennig trap consists of a strong, uniform magnetic field and an electric field which can be produced by a hyperbolical or cylindrical [Gab89b] set of electrodes Penning trap with hyperbolical electrodes The first design of a Penning trap was made with hyperbolical electrodes, the shape of electrodes is chosen as an equipotential surface of a quadrupole field, Fig If the frequency of the oscillation is to be independent of the amplitude, the electrodes must be hyperboloids with cylindrical symmetry around the axis. The particle is bound axially with the electrostatic potential given by U(r, z) = U 0 2d 2 (z2 r2 2 ), (3.1) here U 0 is the ring voltage, and d characteristic trap dimension d 2 = 1 2 (z2 0 + ρ2 0 2 ). (3.2) The constants z 0 and ρ 0 are the minimum axial and radial distances from the center to the electrodes, see Fig One can achieve a completely harmonic electric field only for perfectly machined and infinitely large hyperbolical electrodes, what turns out to be a difficult task, especially for the ultra high accurate g factor or mass measurement. So it is necessary to compensate the Penning trap i.e. to improve the harmonicity of the electric field. This can be done by putting correction or compensation electrodes between the ring and endcaps electrodes. That leads to constructing of a new generation of so called compensated Penning traps. z Upper endcap x z 0 0 y Ring Lower endcap Figure 3.2: Hyperbolical Penning trap consists of one ring electrode and two endcaps all of them having hyperboloidal shape.

17 3.2 The Penning trap 11 Ion motions in a Penning trap In an ideal Penning trap one can assume the magnetic field as homogenous and parallel to the z-axes of the trap B = B 0 u z, (3.3) it serves for the radial confinement of a particle. This magnetic field is superimposed to the quadrupole potential given by Eq The motion of a single ion in this combination of electric and magnetic fields (see Fig. 3.3) decomposes into three normal modes: an axial mode (at ν z ) along the magnetic field axis, and two radial modes: a reduced cyclotron motion (at ν + ) and an ExB magnetron drift (at ν ) perpendicular to it. They are described by harmonic oscillations at the frequencies ν z = 1 qu0 2π md, (3.4) 2 ν + = ν c 2 + ν = ν c 2 (νc 2 (νc 2 ) 2 ν 2 z 2, (3.5) ) 2 ν 2 z 2. (3.6) Axial z Magnetron B Reduced Cyclotron Figure 3.3: The total motion of a single ion in a Penning trap consists of three independent motions. One is a harmonical oscillation along the magnetic field axes - axial motion. The other two are the circular motions in the radial plane - cyclotron motion around the magnetic field lines and slow magnetron motion around the trap center.

18 12 Experimental Setup Here q is the ion charge, m the ion s mass, and ν c = 1 2π q m B 0 is the free space cyclotron frequency which is not directly accessible but it can be deduced from three measurable eigenfrequencies by using the so called invariant theorem [Bro82] ν c = ν+ 2 + νz 2 + ν 2. (3.7) The invariant theorem is independent of trap misalignments to first order [Bro86]. Considering our measured frequencies for the single C 5+ ion: ν + = Hz; ν z = Hz; ν = Hz, immediately a frequency hierarchy becomes obvious ν + ν z ν. It leads to the following dependence for the uncertainty in the cyclotron frequency, which serves for the magnetic field calibration: ν c = ν + ν c ν + + ν z ν c ν z + ν ν c ν 0, 9993 ν + + 0, 0375 ν z + 0, 0007 ν. (3.8) From Eq.3.7 we calculate the necessary frequency measurement accuracies to reach the final goal of a relative g factor determination better than one part in billion (1 ppb) as following: a) 24 mhz for the reduced cyclotron frequency, b) 640 mhz for the axial frequency, and c) 34 Hz for the magnetron frequency. Since the magnetron frequency introduces a negligible error we check it only once a while, for example after creation of a new ion. At present we can easily achieve an absolute accuracy in its determination better than 3 Hz. The axial frequency is measurable with our present methods with an accuracy of 200 mhz, that is still factor of two better that we need for the spin flip determination in case of O 7+. The only critical measurement represents the reduced cyclotron frequency, which actually imposes the main limitation of the final result in our g factor measurement Compensated Penning trap The compensated Penning trap very closely approximates the ideal Penning trap and in the past decades such traps have demonstrated remarkable ability to produce highly precise measurement of certain fundamental quantitites such as the electron s g factor and the electron-positron g factor comparison, the proton-electron mass ratio, the proton s atomic mass, antiproton to proton mass ratio, masses of nuclei far from stability. Although there were some earlier attempts to improve the harmonicity of the trapping potential in a Penning trap, the first dramatic improvement in the harmonicity of a hyperbolical Penning trap was achieved in 1976 in Dehmelts group [Dyc76]. They introduced extra compensation electrodes into the trap and via adjusting their potential they were able to tune out anharmonicities. That opened a room for the precise electron anomaly measurement. For the direct detection of the axial as well as of the cyclotron motion a proposal has been made in 1973 [Win73] with necessary conditions of trap minituarization in order to reach a sensitivity that allows single electron detection. Eight years later a hyperbolic ring electrode split into four equal quadrants was a final success in the same group [Dyc81]. This quadring design allows excitation to occur on two opposite quadrants and detection on the other pair which in effect form the plates of a capacitor externally tuned to the ion cyclotron frequency. The first theoretical investigation of

19 3.2 The Penning trap 13 the possible trap optimization was performed by G. Gabrielse who made a very detailed relaxation calculation of the electrostatic properties of a compensated Penning traps with hyperbolic electrodes [Gab83]. An optimal electrode configuration, which makes the harmonic oscillation frequency of a trapped particle independent of changes in compensated potentials, has been proposed. This means that adjusting the compensation potential (to tune out harmonicities) will, in principle, not change the harmonic-oscillation frequency ω z of a trapped particle at all. Of course, due to the technical limitations in machining and assembling of the trap one can only approach that ideal case. Nevertheless an optimal design makes measurements much easier and allows to achieve higher resolutions in the axial frequency [Dyc89] Penning trap with cylindrical electrodes In the mid eighties, G. Gabrielse proposed instead of a hyperbolical to use a cylindrical Penning trap, Fig He and his coworkers published a very detailed paper describing the properties of cylindrical traps [Gab89b]. Cylindrical Penning traps have two important advantages: 1. Cylindrical electrodes can be machined to greater accuracy with much less efforts and faster than hyperbolic electrodes. 2. It is easier to study theoretically the anharmonicity compensation for a cylindrical trap because the potentials can be calculated analytically. Figure 3.4: Cylindrical Penning trap consisting of cylindrical electrodes. It is also possible with the right geometry to tune out the first-order and reduce the second-order anharmonicities simultaneously. Cylindrical Penning traps with long, open-ended endcaps electrodes [Gab89b] have another great practical advantage: the open access to the interior of the trap, so it s easy to load particles and

20 14 Experimental Setup also introduce microwaves or laser beams. In addition, it allows for the construction of nested Penning traps, to combine particle clouds with charge of opposite sign Double Penning trap One interesting idea that has been employed by Dehmelt s group was using a double hyperbolical trap configuration [Sch81] for the comparison of the positron and electron g factors. The method of continuously loading positrons into a Penning trap involves using the double Penning trap. It consists of the storage trap in which they load the positrons and an experiment trap without a positron source and with a field-emission point located on axis. The problem was in fact that the holes required for positron trapping and centering would made impossible the compensation of the storage trap to allow precision measurements. By use of a second trap it becomes necessary to transport the positron between the traps through a channel drilled in two adjacent end-cap electrodes of the double trap combination. TRAPPING POTENTIAL B z { Q~2000 Q~1000 ANALYSIS TRAP Superconducting LC-circuits PRECISION TRAP LC-circuit: Q=400 Figure 3.5: Double cylindrical Penning trap, consisting of two compensated Penning traps placed above each other on the same axis. It allows to determine a spin flip in the Analysis trap and to measure the ion frequencies in the Precision trap. In our experiment we use a double cylindrical five pole compensated Penning trap, Fig According to their purposes we call them Analysis trap and Precision trap. Both traps consist of two end-caps, two correction electrodes and one ring

21 3.3 Electronics 15 electrode, all of them with the same diameter of 7mm. Electrodes in both traps excluding a ring electrode in Analysis trap are made of gold plated OFHC copper. The Analysis trap has a ring electrode made of nickel to create a nonuniform magnetic field in the center of the trap. This is the well known magnetic bottle configuration, which is necessary to detect a spin flip via the continuous Stern-Gerlach effect, [Ger22, Deh86], providing a big enough axial frequency shift that can be detected with our present detection technique. After analysing the spin direction, the ion is transported to the Precision trap having a three orders of magnitude more homogeneous magnetic field. There we are able to measure the ion s reduced cyclotron frequency and Larmor precession frequency with a very high accuracy [Häf00b], better than one part in Electronics Cryogenic electronics As it is mentioned above, the cryogenic electronics are placed at the bottom of the liquid helium dewar in good mechanical and thermal contact such that it reaches the working temperature of 4.2 K. It has several functions in our experiment: electronic non-destructive detection of a single ion, resistive cooling of the ions motions close to the environmental temperature of 4.2 K, and stabilizing the trap voltages with filter bands down to a very low residual noise level. The cryogenic electronics [Sta98] include superconducting resonance circuits and low-temperature amplifiers and filters Room temperature electronics The room temperature electronics [Sta98] are placed in the boxes connected to the hat of our apparatus (Fig. 3.1), and several 19 racks in the laboratory. The main tasks are the following: providing precise and well stabilized DC voltages to the trap s electrodes (necessary to maintain a well defined axial oscillation frequency), creation of excitation signals for the ions (axial, magnetron, cyclotron and combination of them), processing and analysis of ion signals, and providing of the Larmor microwave signals. For the trapping potential creation we use a high precision power supply with 4 channels in a range of ±30 V and a resolution of 4 µv. Two channels are connected to the ring and correction electrodes in the Analysis trap and two others to the ring and correction electrode in the Precision trap. For observation of the axial motion it is critical to have extremely stable voltage sources. Fluctuations in the voltage applied to the electrodes increase the linewidth and reduce the signal amplitude. For the ion transport between the traps we use a power supply in the range of ±10 V with a resolution of 5 mv. Finally for the electron gun we need a high voltage power supply with 1 V resolution in the ±2 kv range. The ion motion has to be excited and also the magnetron motion has to be cooled via several signal generators. All alternating voltage signals were galvanically decoupled. Since we observed ground loop problems in our past experiment, we decided to design new electronics which will be loop-free. Also the high voltage power supply was only manually controlled so far, but in a future experiment on hydrogenlike calcium we will introduce remotely controlled electronics.

22 16 Experimental Setup Interaction between the ions and electronics The main principle of the interaction between the ions and electronics in Penning traps has been described 30 years ago by Wineland and Dehmelt [Win75]. The idea is to exploit induced image charges of the oscillating ions. The ion has the axial motion parallel to the magnetic field and the cyclotron motion in a plane orthogonal to the magnetic field. Therefore we need two independent resonant oscillatory circuits for detection. In the axial motion the ion will induce an oscillating current between the trap electrodes: I = dq/dt = (q/d)ż, (3.9) where q is the ion s charge and ż is its velocity, see Fig Figure 3.6: Bolometric detection principle of the single ion in a Penning trap. The ion comes into equilibrium with the tank circuits (represented by resistor R LC ) and a modulation of the thermal noise in R LC takes place. In our Analysis trap a big superconducting coil forming a resonant circuit (tuned to 370 khz) is connected between the upper and lower correction electrode, see Fig In the Precision trap we have a smaller superconducting coil tuned to an ion axial resonance frequency of 910 khz. For the Analysis trap we achieve a quality factor Q = 1200, corresponding to a resonance resistance of R = 20 MΩ. The parameters for the precision trap are Q = 1000 and R = 10 MΩ. Both superconducting circuits are placed closely to the liquid helium dewar to keep them in thermal equilibrium with the 4 K bath, and to have the electrical connections as short as possible. In the preamplifier stages we use field-effect GaAs transistors which do not suffer from carrier freeze-out at 4 K. If we tune the trap voltage to such a value that the ion s axial frequency coincides with the resonant frequency of the circuit, the induced current leads to a voltage drop across the circuit of several nv for a single trapped ion. The axial frequency is observed as a minimum or a dip in the Fourier transform of the Johnson noise of the resonant LC circuit. In both traps a "trick" is applied in order to increase the signal to noise ratio: A signal, which is created by an inverse Fourier-transform, having a flat spectral distribution is added to the thermal LC circuit noise. The thermal noise component stays constant while the total voltage increases. Therefore the depth of the dip is increased and the signal to noise ratio improved. However this method is limited since

23 3.3 Electronics 17 Figure 3.7: Detailed diagram of the Mainz double Penning trap together with cryogenic electronics and all electrical trap connections. the ions axial energy is increased with the square of the applied voltage. Nevertheless, this method is widely exploited for trap optimization purposes, see Sec. 4.4.

24 18 Experimental Setup As in the case of the axial motion, the ion oscillation in the radial plane at the reduced cyclotron frequency ω + induces image charges in the trap electrodes. Detection of the cyclotron motion is performed only in the Precision trap since there the magnetic field is more homogeneous and ω + can be measured with high accuracy. The lower correction electrode has been splitted [Win73] in two segments to allow a quadrupolar excitation and detection of the ion signal by the induced current signal. Here for detection we use a shielded copper coil. The stability of the field, however, does not allow an easy variation of the detection frequency. Therefore a lower Q = 400 for the circuit at 24 MHz was chosen which corresponds to the C 5+ cyclotron frequency for the magnetic field of 3.8 T. For additional fine tuning of the cyclotron motion detection circuit we have additional capacitors which can be added by means of cryogenic GaAs switches. The minimal ion energy for observation is 0.1 ev in the cyclotron motion and during measurements we worked with typically 0.7 ev. It still allows us to observe a full width at half maximum (FWHM) of the cyclotron resonance of the order of 10 9, around 25 mhz.

25 Chapter 4 Experimental Procedure First we are going to explain how to load particles into a Penning trap (Sec. 4.1). In Sec will be described the process of cleaning and single ion preparation. After that one has to cool particle to the mev range, that will be elaborated in Sec For the high precision measurements in the Penning trap it is necessary to optimize the trap and create an ideally harmonic potential (Sec. 4.4). Only in a case when all previous steps have been accomplished successfully, the necessary sensitivity for observing a spin flip of a single hydrogenlike carbon or oxygen ion has been achieved (Sec. 4.5). 4.1 Loading particles In our present setup, shown in Fig. 3.7, there is an electron gun at the bottom of the apparatus below the Precision trap, which serves for internal ion creation. The electron gun consists of a cathode with a field emission point (FEP) made of tungsten, an acceleration electrode, an anode with a graphite target with a hole drilled through the center, and a reflection electrode at the top of the Penning trap as the last electrode in the whole setup. By applying a certain negative potential 1 to the FEP we create an electron beam of 5 na which starts to move in the strong magnetic field along the z-axis towards the acceleration electrode and anode. After flying through the whole setup the electrons are reflected from the negatively charged reflection electrode. At the FEP they are reflected again. Eventually, after several cycles, because of space charged effects, the electron beam increases its radial dimension so that it doesn t fit anymore through the hole in the anode and hits the target. Via such an electron bombardment atoms are deliberated out from the target surface. Subsequently ionization takes place by successive collision with electrons. Via changing the electrode potential we can move and store the created ions in one of our traps, see Fig The minimum potential depth for ion trapping is 3.5 V in the Precision trap and 1.9 V in the Analysis trap. In spite of the fact that the target is made of graphite, it is possible to observe ion spectra full of other elements, like oxygen, silicon, magnesium, sulphur, since the target contains many impurities. That is a disadvantage from the point of view of the ion cleaning process, but on the other hand it allowed us to measure the g factor of O 7+ without changing the target, what means without the tedious process 1 In our experiment on C 5+ we applied tipically 1.7 kv for a creation time of 4 s.

26 20 Experimental Procedure of opening the Penning trap vacuum chamber 2. Figure 4.1: Mass spectrum shows different ions in the Precision trap. It is one of the first spectra in which the amount of unwanted O 6+ and C 4+ ions is similar to the amount of C 5+. The maximum voltage that we were able to apply was limited by the cryogenic high voltage cables to 2 kv. That is sufficient for hydrogenlike carbon and oxygen ion creation, since it is well above the ionization potential for these ions. For the optimization of the creation parameters one should take a close look to the charge state distribution (CSD) inside ion clouds immediately after they ve been created. We accomplished that task via producing ion mass spectra in our Precision trap. The latter is obtained via ramping the trapping (ring) voltage in the range from 10 V to 20 V. During a linear voltage increase we observed the ion s signal in the axial degree of motion. In such a way we optimized the set of parameters for the cathode and anode potentials, and the creation time for getting the best charge state distribution of an ion cloud. The main problem was the appearance of unwanted C 4+ and O 6+ ions, which have a lower ionization potential than the wanted ions. These ions appear almost always with a high abundance as shown in Fig Under such conditions we have only a tiny chance to find a cleaning procedure that will have an efficiency sufficiently high to isolate a single C 5+ ion. After some time, we understood that the problem was in applying a too low potential on the cathode electrode (it was 1, 1 kv), so that the electron current was also too low and the necessary creation time for reaching a high hydrogenlike carbon abundance too long. With such parameters, given in Table 4.1, we obtained a big ion cloud with plenty of oxygen ions with low charge states, created in successive ionization processes. After finding good creation parameters, given in Table 4.2, we were able in a short time, without taking any spectra, to clean the ion cloud in a reproducible way and to isolate a single hydrogenlike carbon ion in the Precision trap. 2 The vacuum chamber has not been opened from the beginning of the experiment in All our measurements on carbon and oxygen were performed with the same FEP and the target.

27 4.2 Cleaning particles 21 Cathode potential Anode potential Creation time Ring voltage Tuning ratio 1.1 kv 900 V 8 s V Table 4.1: Creation parameters with too low cathode potential, leads to the charge state distribution with a similar amount of C 4+, C 5+, and O 6+ ions, see Fig Cathode potential Anode potential Creation time Ring voltage Tuning ratio 1.7 kv 400 V 4 s V Table 4.2: Optimal creation parameters for hydrogenlike carbon ion creation in our Precision trap. 4.2 Cleaning particles Immediately after ion creation in the Precision trap with the optimal parameters given in Table 4.2 we start with the cleaning procedure. It is removing of unwanted ions from the trap by means of exciting their motional amplitudes with an electrical dipole excitation. The procedure involves applying a notch signal, that consists of a white noise excitation of the axial motion in a broad range of frequencies, omitting a narrow window only around the axial frequency of the ion we want to isolate (C 5+ ). After excitation the trapping voltage is reduced. In such a way all the ions except C 5+ escape from the trap. Figure 4.2: For successful cleaning of the ion cloud by motional excitation a synthetic signal is used, which is produced by stored Waveform Inverse Fourier Transform (SWIFT). It consists of a strong white noise in a certain frequency span, except a gap around the wanted ions. In Fig. 4.2 is our first notch signal from February 2003 which had a width of 30 khz around 890 khz while the ion axial frequency was 917 khz. Because the C 5+

28 22 Experimental Procedure ion was at the right edge of the notch signal it was accidentally often excited together with the other ions and the cleaning was not perfect. One of the first dips of a cloud of C 5+ ions after a successful cleaning procedure is shown in Fig The first dip of a single ion is shown in Fig Figure 4.3: A bolometric dip of a big ion cloud in the axial tank circuit. According to the width of the dip there are more than 25 ions in the cloud. The full width at half maximum (FWHM) is 30 Hz. The resonance shows a dispersion character due to a slight mismatch between the ions resonance frequency and the center frequency of the detection circuit. For futher improvement it turned out to be useful to reduce in a first step the trapping potential (ring voltage) to a very low value of 5 V. This process happens without any cooling, such that most of the ions (from all ion s species) from a cloud are lost. We continued with this procedure further reducing the trap voltage to 4.5 V and 4.2 V and applying several, between 3 and 5, white noise excitations with the notch filter. Usually at the level of 4.2 V we were able to observe a pure cloud of C 5+ ions, Fig Single ion preparation With further cleaning up to 3.7 V one can achieve a reduction of the ion number down to a single carbon ion. This fact can be evaluated from the width of the dip that should be only 0.8 Hz at the axial frequency of 912 khz. In total, with the known creation and cleaning parameters we need around min to isolate a single ion. It was not always easy to properly clean our trap, very often we have a lot of noise coming from outside, Fig Since we have no hints what was the source of the noise we were forced to live with that and tried other method for cleaning and checking a total number of ions in the trap. Instead of observing a dip using the axial detection scheme, we used our cyclotron tank circuit to observe a peak in the cyclotron spectrum, Fig Because the magnetic field is not ideally uniform carbon ions with a different energy will have a different reduced cyclotron frequency and different ions can be observed well

29 4.2 Cleaning particles 23 Figure 4.4: The first dip of a single C 5+ ion observed after a new start of the experiment, the full width at half maximum is only 0.8 Hz. Many efforts were necessary to successfully clean O 6+ which was always the last remaining contaminant ion species in the trap. Figure 4.5: Graph obtained during cleaning procedure in the Precision trap. Without any magnetron sideband cooling, simple resistive cooling turned out to be sufficient. The whole cleaning procedure usually takes 20 to 30 minutes. separated when their number is not too big. In Fig. 4.7 there are at the beginning three ions. With carefully decreasing the trapping voltage we removed the two ions with higher energy. Only one C 5+ remains. Ions of other species have completely

30 24 Experimental Procedure Figure 4.6: Axial tank circuit signal full of noise coming from external sources. different frequencies so that they are outside of our cyclotron LC circuits, and thus may not be observed. Figure 4.7: Reducing the number of C 5+ ions by carefully reducing the ring voltage. Ions with higher energy appear at lower cyclotron frequencies because the magnetic field decreases slightly from the trap center. After some changes in our apparatus in August 2003 we moved the ion axial frequency to 902 khz. Then we improved the notch signal by using an arbitrary wave generator and performed proper positioning of the signal around the center of tank circuit, see Fig Since then we have a straightforward and routine cleaning

31 4.3 Cooling particles 25 of the ion cloud and producing a single ion within 15 min. Figure 4.8: A new SWIFT signal with changed frequency. After modifying of the apparatus, the ion axial frequency shifted to the new value of 902 khz. The most sensitive check of existence of unwanted ions is performing a measurement of the cooling time constant. This is done by successive measurements of the reduced cyclotron frequency of the highly excited ion, as described in Sec Cooling particles For performing ultra-precise measurement in the Pening trap it is necessary to force the trapped particle to move in a very small orbit. The inhomogenities in the electrostatic potential and in the magnetic field broaden and shift the observed lines if the orbits are large. The axial and cyclotron motions are cooled by the cryogenic LC tank circuits ("resisitive cooling") connected to the trap electrodes. The magnetron motion orbit must be cooled separately by means of a sideband cooling method Axial cooling For the ions axial cooling we use a "Bolometric technique" proposed by Dehmelt and Walls, [Deh68]. They consider a harmonically bound electron with ion charge q and mass m oscillating with frequency ω z along the z-axis, which is perpendicular to the surfaces of an infinite parallel-plane capacitor in which it is confined. The capacitor s plates are separated by a distance D and are connected by a resistor R. The oscillating ion induces a current flow through the resistor given by Eq The Joule heating of the resistor exponentially damps the ion s motion with a time constant τ = md2 q 2 R. (4.1) The effect of replacing the infinite-plane capacitor by a quadrupole ion trap, leads to an increase of D, the so called effective electrode distance by approximately 30%. The first use was made in a liquid nitrogen environment [Win73].

32 26 Experimental Procedure In our setup exist two LC circuits for the ion axial cooling, one of them connected to the correction electrodes in the Analysis trap and second one to the correction electrode in the Precision trap, Fig If the ion axial motion is in resonance with the LC circuits then a thermal equilibrium between ion and circuit establishes. The kinetic energy of the ion will be dissipated through the resonant impedance and ion will be cooled to the noise environmental temperature of t he LC circuit. Energy decrease of a single ion can be described by an exponential decay E = E 0 exp( γt), where γ = q2 R m D = 1 2 τ. (4.2) γ also represents the minimal observable linewidth of the axial frequency. Deviation from the exponential cooling behaviour occurs, for example, when the cooling force is not any more proportional to the kinetic energy of the ions. The reason for that is the ion frequency dependence on the energy which introduce changing of the resistance of the LC circuit at the ion frequency during the cooling process. So it is of particular importance for the optimal ion axial motion cooling to build a Penning trap in which the axial motion will be harmonical, independently on the ion energy, and also to optimize trapping potentials before starting the measurements, see Sec Cyclotron cooling For the cyclotron cooling we use an LC circuit of Q = 400 which resonant frequency is around the C 5+ reduced cyclotron frequency, ν + = MHz. Cooling time optimization is performed by adding capacitors using GaAs switches in the cryogenic region. The shortest cooling time constant τ = 10 min is obtained for a case when both GaAs switches are ON, see Fig It means that the ion oscillates at the center frequency of the cyclotron tank circuit. Switching off one of the GaAs switches means that the ion is moved from the center of the tank circuit and the cooling is slower, τ = 16 min. Switching off the other GaAs switch made cooling very slow, τ = 63 min. When both switches are OFF, then we have practically no resistive cyclotron cooling. The tank circuit is almost completely detuned, Fig The cyclotron cooling time constant measurement is a very efficient method for determing if there are any impurities inside the Penning trap after cleaning procedure, at first point we have in mind ions of different species. Since they have a different reduced cyclotron frequency they do not appear in the cyclotron tank circuit spectrum and so they are not directly observable. In Fig is shown a cyclotron cooling time constant measurement from which is obvious, by looking to the measurement points, that there is more than one ion in the trap. The ion (or ions) of other species exchange energy by mode coupling with the single carbon ion. In such a way they disturb its cyclotron motion frequency. After proper cleaning we made a final check of the number of ions. In Fig the exponential line fits very well to data points, we could say that the single ion preparation is successfully finished.

33 4.3 Cooling particles 27 τ = 10.7 τ = 15.9 τ = 62.6 τ = 774 Figure 4.9: Cyclotron cooling of C 5+ for different detunings of the cooling tank circuit. The shortest cooling time constant is obtained in the case when both GaAs switches are ON. It means that ion oscillates at the center frequency of the cyclotron tank circuit. Figure 4.10: Direct looking to the cyclotron cooling curve helps us to decide whether there is only one single ion or not. Here is very obvious that there is more than one ion in the trap.

34 28 Experimental Procedure Figure 4.11: The cyclotron cooling time constant is around 9 min and the exponential cooling curve fits the data very well. This indicates that only one single ion is inside the trap Magnetron cooling The magnetron energy is mostly potential, and the magnetron radius tends to increase in time since lower energy orbits are larger. Any dissipative process that removes magnetron energy from the particle increases the magnetron radius until the particle strikes the electrode and is lost from the trap. The cooling time constant is extremely long so that the magnetron motion is virtually stable. Because of the same reason a particle injected in a large magnetron orbit will remain in it until one applies some cooling. The external mechanism for such cooling was discovered in 1975, it is motional sideband cooling [Win75]. One has to apply an oscillating electric quadrupolar field at a frequency ν m + ν z to the split correction electrode, that will couple the magnetron and axial motions. A photon (ω m + ω z ) gets absorbed by the particle. This increases both the axial and the magnetron energy. The axial energy then is damped since the axial motion is coupled to the LC circuit. The magnetron motion gains energy ω m that increases its potential energy but decreases the magnetron orbit. Such magnetron cooling reduces the magnetron orbit sufficiently to fall into the most uniform magnetic field region in the center of the trap, and prevents cyclotron frequency shifts. There is a limit of the attainable magnetron energy (E m ) by a sideband cooling and it is related to the axial energy (E z ), [Bro86]. E m = ν m ν z E z. (4.3) Only when the magnetron energy is greater than this limit, the cooling rate is larger than the heating rate and the magnetron orbit decreases. In an ideal trap in which the magnetic field is exactly parallel to the z-axis and the electrodes have an exact

35 4.4 Penning trap optimization 29 cylindrical symmetry the magnetron frequency is given by ν ideal m = ν2 z, (4.4) 2ν c but the real magnetron frequency will be [Bro86] ν m ν ideal m ( θ2 1 2 ɛ2 ), (4.5) where θ is a small angle between the magnetic field and the axis of symmetry of the electric field and ɛ denotes deviations of quadratic order of the cylindrical symmetry of the potential. 4.4 Penning trap optimization For performing high precision measurements in a Penning traps it is of crucial importance to reduce all kinds of spatial and temporal instabilities in the applied magnetic and electric fields. The magnetic field fluctuations will be explained in the Appendix A. Here we concentrate only on electric field influences. The most critical sources of error in the ion frequency measurements are the anharmonicity and nonorthogonality of the trap, [Hüb97]. The electric potential U in the center of a Penning trap can be expanded in Legendre polynomials [Bro86] U = 1 2 U 0 ( r k C k Pk (cosθ), (4.6) d) k=0 where U 0 is the trapping potential and d is the characteristic trap dimension, Eq The Legendre polynomials P k (cosθ) are used because azimuthal symmetry is maintained [Gab89b]. Assuming a reflection symmetry across the z = 0 plane only the even k coefficients C k will be non-zero. Since C 0 is a constant it can be ignored. Only the lowest order terms in Eq are important for a particle trapped in the center of the trap. For the ideal quadrupole potential, hyperbolic trap, all terms are zero exept C 2 = 1. The axial frequency is given by for a particle with mass m and charge q Tuning ratio optimization ω 2 z = qu 0 md 2 C 2 (4.7) A real trap, either hyperbolic either cylindrical, will have a non-zero C 4 term owing to machining imperfections, misalignments and other factors as a wrong electrodes sizes. When that happens, the axial oscillation is anharmonic, and the oscillation frequency depends on the amplitude. The shift in the axial frequency is [Gab84] ω z = 3 ω z 2 C 4 C 2 E z qu 0 C 2, (4.8)

36 30 Experimental Procedure here E z is the axial energy of the particle. To tune out the influence of a non-zero C 4 and reach a good enough accuracy for the measurement of very small shifts in the axial frequency ω z, correction electrodes are introduced into most Penning traps. In our case the endcaps are grounded, the ring electrode is at the potential U 0 and a potential U c is applied to the correction electrodes. The potential in the trap will be the superposition U = U 0 φ 0 + U c φ c. (4.9) Close to the trap center, the solutions to the Laplace s equation φ 0, and φ c can be expanded as [Gab89b] φ 0 = 1 2 k=0 C (0) k ( r d ) k Pk (cosθ) (4.10) and φ c = 1 2 k=0 D (0) k From equations 4.6., 4.10., and follows C k = C (0) k ( r d ) k Pk (cosθ). (4.11) + D k U c U 0. (4.12) It is possible to adjust the two terms on the right side to cancel and to make C 4 = 0. Here we will explain only the dip detection scheme of minimizing trap anharmonicities. We did it manually, what is faster but with lower accuracy, as well as automatically, which is a slower and more accurate method. Manual adjustment via dip detection method According to the relaxation calculation the optimum tuning ratio (ratio between the correction and the ring electrode potential) is around Experimentally we obtaine a little bigger one since the trap has some imperfections that are not included in the calculation. The first possibility to monitor the quality of the trapping potential is measuring the depth of the dip in the power spectrum of the axial tank circuit from a stored single ion, see Fig A second possibility is to excite the axial motion by applying a SWIFT signal in the frequency domain. For an anharmonic trapping potential, and a bad tuning ratio, the axial frequency is shifted either to lower or to higher values depending on the sign of the C 4 coefficient. Beside the shift of the frequency there is a broadening in the width of the dip, depending on the applied voltage, and of course on the value of the C 4 coefficient. If one applies a too high excitation it can happen that the ion energy is so much increased, and the axial motion so anharmonic, that the dip becomes invisible. For the tuning ratio optimization in Fig we started with a too high TR so the ion dip with excitation has a higher frequency and a bigger FWHM. After several tries that took a similar time as in the first case we obtained the same value for the tuning ratio T R = with the same accuracy of 20 µunits. The optimum tuning ratio shown in Fig. 4.14, provides us a frequency shift, with the

37 4.4 Penning trap optimization 31 Figure 4.12: Manual tuning ratio optimization by observing the depth of the dip for different tuning ratios close to the optimal one. Averaging time for a single ion dip was around 1 min, in total the measurement took several minutes and the achieved accuracy was aproximately 20 µunits (1 µunits= 10 6 ) in the tuning ratio. Figure 4.13: Frequency shift and asymmetric dip shape for a high excitation energy. For the tuning ratio detuned by 440 µunits there is an obvious broadening of the ion dip when one excites the axial motion with 0.1 V. This voltage is attenuated by around 4 orders of magnitude before reaching the trap electrodes. standard axial excitation of 0.1 V, of only 0.1 Hz which is sufficiently good to reach an overal accuracy in the g factor measurement of less than 1 ppb and to observe easily a spin flip of hydrogenlike carbon and oxygen ions (see Sec. 4.5).

38 32 Experimental Procedure ν ν Figure 4.14: Optimal tuning ratio. The frequency shift of the ion dip with the axial excitation of 0.1 V in comparing with the ion dip without excitation is only 0.1 Hz Automatic adjustment via dip detection method The tuning ratio can be also optimized via our computer control system in the same way as during manual optimization. The ion axial motion frequency is measured three times in a row (in 3 min time) without excitation and then three times with the SWIFT excitation. From the difference of the average value from the measurements with and without excitation the computer program calculates a value for the change of the tuning ratio. After setting a new tuning ratio potential to the trap electrodes a new measurement cycle starts. Figure 4.15: Automatical tuning ratio optimization. Final accuracy is better than 10 µunits, but measurement takes more then one hour, comparing with several minutes obtained in manual method.

39 4.4 Penning trap optimization 33 The program stops when the difference between the ion axial frequency with and without excitation is smaller than 25 mhz in three successive measurements. From our previous knowledge of the trap orthogonality (see 4.4.2) and tuning ratio dependence we can obtain a fast convergence of the measurement routine. Nevertheless under the best conditions (without external electrical noise, and low fluctuations in the power supply) we need 45 min to optimize the tuning ratio with an accuracy of about 10 µunits. In Fig one can see that the optimum TR is slightly different from the previous one. The reason for that is a new ion inside the trap. It is necessary to slightly reoptimize the tunig ratio after every ion creation Orthogonality determination From Eq one can see that it is possible to achieve that the C 4 term from the compensation electrodes cancel the C 4 contribution from the endcaps and the ring by adjusting the compensation potential to U c = C(0) 4. (4.13) U 0 D 4 But adjusting U c to tune out C 4 will change C 2 and hence will change the axial ion frequency ν z. This means that we have to search for the new axial frequency every time when we change U c. However, if the trap is configured so that D 2 = 0, then from Eq follows that adjusting U c will have no effect on C 2. Such a trap is called orthogonolized trap, the axial frequency is independent on changes in U c. Figure 4.16: Axial frequency dependence on tuning ratio for a trap voltage of V. One should check this value from time to time since it could be slightly changed, and obligatory after creation of a new ion. Since different trap imperfections like patch effects, misalignments, and machining errors will change all of the C (0) k and D k coefficients from their calculated values,

40 34 Experimental Procedure we have to determine their values experimentally. We performe it just by changing the tuning ratio and measuring the ion axial frequency. In Fig is shown a linear dependence of the axial frequency on the tuning ratio. It is slightly higher than we expected from the calculation [Ver03b]. During the measurement for the best signal to noise ratio and the shortest cooling time constant it s necessary to place the ion frequency in the center of the axial tank circuit, so we have to know the dependence of the axial frequency on the ring voltage. This is shown in Fig This linear dependence also helps to perform a faster tuning ratio optimization. Figure 4.17: Axial frequency dependence on ring voltage for a fixed tuning ratio of T R = Spin detection For spin flip detection we use the "continuous Stern-Gerlach effect" as introduced by Dehmelt [Deh86] in the measurement of the g factor of a free electron. We introduce a quadratic inhomogeneity in the magnetic field - a magnetic bottle. The ring electrode in our Analysis trap (Fig. 3.7) is made of nickel and it distorts the total magnetic field near the center of the trap. So we have a quadratic dependence of the magnetic energy on the z-coordinate B = B 0 + B 2 z , (4.14) the odd terms vanish in the expansion because of a mirror symmetry of the field with respect to the z = 0-plane. The net effect of this bottle is a coupling of the magnetic moment of the stored particle to the axial oscillation, which produces a frequency shift in the axial resonance. The corresponding force on the magnetic moment in axial direction is F z = z ( µ B) = 2µ z B 2 z, (4.15)

41 4.5 Spin detection 35 which is linear in the axial coordinate. It adds to the electric force from the quadrupole trapping field acting on the particle charge. Since both forces are linear in the axial coordinate the ion motion is still described by a harmonic oscillator. The axial frequency depends on the direction of the magnetic moment µ with respect to the magnetic field ν z = ν z δν z = ν z0 + µ zb 2 mν z0. (4.16) A transition of the spin direction of the bound electron can be induced by a microwave field. At our magnetic field of 3.8 T the Larmor precession frequency is about 104 GHz. Because of the different sign of the magnetic potential for the two possible spin directions (up and down) we would have the resulting difference of the axial frequency. Taking into account our measured value of the quadratic magnetic component in the Analysis trap B 2 = 10 mt/mm 2 the calculated spin flip jump for the 16 O 7+ ion at axial frequency ν z Hz is ν z ( ) ν z ( ) = B 2 gµ 4π 2 B ( 1 m ν z, ) = 475 mhz. (4.17) 2 In our measurement we observed a very good agreement with the calculated value, Fig Figure 4.18: Spin flip in hydrogenlike oxygen in our Analysis trap. Measurement of the ion axial frequency of one spin orientation takes around 2 min, using the bolometric detection scheme. After that we shine a microwave into the trap and measure the frequency again. Ion signals have a shape different then the usual dip because the ion axial frequency was not coincide with the resonant frequency of the LC tank circuit. We chose it for better fitting and easier spin state observing.

42 36 Experimental Procedure If we vary the microwave field frequency around the expected Larmor frequency in the Analysis trap and plot the number of induced spin flips per number of tries we obtain a resonance curve, which is used to determine the g factor. The latest results of the g factor of the hydrogenlike oxygen are given in [Ver03b, Ver04a].

43 Chapter 5 Measurement and Results In this chapter the main results obtained in our Penning trap in previous years are presented. Different aspects and methods of the temperature measurement on a single ion are discussed in Sec In section 5.2 a novel technique for the detection of a single ion is decscribed. It allows us to make a very fast optimization of the tuning ratio and serves as a powerful tool for introducing the novel technique for detecting small ion motional frequency shifts. This new technique, for detecting phase shifts in the axial motion of a single ion, will be used to detect spin flips as explained in Sec Temperature of a single ion in a Penning trap Temperature as a physical quantity The concept of temperature is usually applied only to ensembles of particles, and not to one single particle. However, in the case of thermal equilibrium with the environment, if one measures the motional energy E of the particle repeatedly, the probability of obtaining a certain value for the energy goes with the exponential exp( E ), where k k B T B is the Boltzmann constant. This means it will follow a Boltzmann distribution to which one can associate a temperature. This is a very obvious example of the ergodic principle [McQ76, Eks57], which states that the result of a measurement of an observable performed over an ensemble of particles, if averaged, yields the same result that given by several consecutive measurements of the same observable on a single particle, time-averaged. If the system is indeed ergodic and there is a thermal equilibrium between the environment and the particle, this yields the possibility of utilizing an ion s trapping motion as a probe for thermometry. Dehmelt [Deh68, Deh73] has developed a simple scheme in which the translational temperature of a stored ion-gas cloud is monitored. It relies on the interaction of the stored ions with an resistance, connecting trap electrodes, and the analysis of the noise spectrum of an external resistor at a given temperature Implications of the ion temperature on the accuracy of g factor measurements A single charged particle in a Penning trap [Pen36] interacts via electromagnetic forces with the electrodes of the trap which are coupled to attached electronic circuits

44 38 Measurement and Results used for detection of the ion s trapping motions. The ion s movement in the trap induces image currents in the trap electrodes and thus in the attached electronics. Since the nature of the coupling forces is bidirectional, thermal fluctuations in the electronics due to thermal charge carrier motion also affect the ion s movement in the trap [Tur00]. The strength of this coupling can be expressed in terms of a time constant τ with which the exponential energy loss of an excited ion would take place. That time constant is for the first time calculated in the famous paper written by Dehmelt and Wineland [Win75] and it is given by τ = m q 2 D 2 R (5.1) where m is the ion mass and q is its electric charge. D is the so-called effective distance between the signal pickup electrodes and R is the corresponding resistance. Equilibrium between the temperature of the electronics and the ion s motion is reached after a time t τ. The motional amplitude of a trapped ion is a direct measure for the corresponding energy. In an ideal Penning trap [Bro86], the amplitudes of the three trapping motions for a single stored particle are not coupled and therefore every degree of freedom of the ion can independently be attributed with a different temperature. Trap imperfections as e.g. geometrical deviations from the ideal case will lift this degeneracy to a certain degree, however not necessarily strong enough to compensate for different strong energy dissipations into the system e.g. due to different external noise levels in the respective resonance circuits. Therefore, even in an imperfect trap, different thermal equilibria can develop in different motional degrees of freedom of the ion. In the present case, the axial and cyclotron degrees of freedom have distinct motional frequencies in resonance with precisely tuned circuits of high quality factors. Note, that it is the charge carrier temperature of the electrodes and the electronics that is predominantly relevant for the present investigations. Accurate knowledge of a single particle s motional temperatures in the Penning trap is relevant for high-precision measurements of any quantity that depends on the particle s dynamics. As pointed out in [Bro85, Ver04b], knowledge of the single ion s temperature is of importance e.g. for the lineshape of the spin flip resonance used for the determination of the particle s g factor and influences the precision of such measurements directly. Because of the nonvanishing B 2 term of the magnetic field in the precision trap, the Larmor frequency and, consequently, spin flip probability are functions of the ion s kinetic energies [Bro86] ω L = ω L (E +, E z, E ). (5.2) The ion cyclotron motion undergoes resistive cooling (Sec ), and follows an exponential cooling curve. The axial motion stays in thermal equilibrium with the axial tank circuit, thus the energy E z fluctuates according to a Boltzmann distribution exp( E z /k B T z ). The magnetron energy is constant and equals the magnetron cooling limit E = (ω /ω z )k B T z. Since ω z is almost two orders of magnitude bigger than ω, the influence of the magnetron energy upon the spin-orientation depending axial frequency is very small. Therefore the true spin flip probability is obtained as the sum of the probabilities for the ion to experience a spin flip for any of the energy pairs (E +, E z ). This has been in detail discussed in our recent paper

45 5.1 Temperature of a single ion in a Penning trap 39 [Ver04b]. To find out the importance of the axial temperature the simulated spin flip probability resonance has been calculated for different ion axial temperatures, see Fig Figure 5.1: The line shape of the spin flip probability curve is strongly dependent on the ion axial energy. For the higher axial energy resonance becomes more asymmetric and shifts to the higher value, from [Ver03b] Cyclotron temperature measurement The cyclotron temperature can be measured by coupling of the axial and cyclotron motions in an inhomogeneous magnetic field. As stated above the nickel ring electrode of our analysis trap produces a bottle-like magnetic field which can be characterized by a series expansion B(z) = B 0 + B 1 z + B 2 z , (5.3) Odd terms vanish because of the mirror symmetry of the field. The coupling of axial and cyclotron motions leads to an axial frequency shift ω z for a given cyclotron energy E + [Bro86]: ω z = E + mω z B 2 B 0 (5.4) In our Analysis trap is B 0 = 3.8 T and B 2 = 8.2(9) mt/mm 2. The corresponding shift in ω z has been determined experimentally by systematic excitation of the cyclotron energy to roughly 5 Hz per mev [Wer02, Häf03]. Figure 5.2 shows the distribution of the thermally fluctuating cyclotron energy E + as obtained from a repeated measurement of the axial frequency ω z in the Analysis trap. A least squares fit to a Boltzmann distribution yields a cyclotron temperature T + = 4.9 ± 0.1 K. This temperature is slightly higher than the ambience temperature of 4.2 K from the liquid He bath due to additional noise originating from the electronics, being attached to the traps.

46 40 Measurement and Results Figure 5.2: Measured distribution of the cyclotron energy E + yielding a cyclotron temperature of 4.9 ± 0.1 K, from [Häf00a] Axial temperature measurement via B 2 term This type of axial temperature measurement is based on repeated measurement of the axial energy for a given ambience temperature and the calculation of the mean value of the resulting distribution. This is not as straight-forward as in the cyclotron case discussed before, since the axial frequency in a Penning trap in first order does not depend on the value of the axial energy. The axial temperature measurement via B 2 term is based on a coupling of the cyclotron motion to the axial motion by an rf field at the frequency ω + ω z applied in the radial plane of the trap between different segments of the split correction electrode. The effect of this coupling is to make the quantum numbers of both oscillations identical. Then we have the relation E + ω + = E z ω z (5.5) between the cyclotron and axial energies [Bro86]. This can be used to determine the axial energy from a measurement of the cyclotron energy. As the cooling time constant of the axial motion is much smaller than the cyclotron cooling time constant, the axial energy fixes the cyclotron energy. The axial motion first reaches equilibrium with the environment by resistive cooling. To ensure the equivalence of the quantum numbers the sideband coupling is performed during a time of around 10 s, which is about 100 times longer than the corresponding energy exchange time. The cyclotron energy is measured as described in section by monitoring the axial frequency in an inhomogeneous magnetic field for a sufficiently long time of e.g. several hours, performing measurements every few seconds. This measurement was performed in the Precision trap, where the magnetic field can be also expressed by equation 5.3. But here the B 2 term is smaller by three orders of magnitude. The associated constraint is that one can no longer count only on the direct relationship

47 5.1 Temperature of a single ion in a Penning trap 41 between the cyclotron energy and the axial frequency shift E + = mω z B 0 B 2 ω z (5.6) to determine the axial energy. This is because the axial frequency shift depends on the value of the B 2 term, which in the magnetic bottle is big enough to make thermal variations of the cyclotron energy visible as axial shifts, but not so in the by three orders of magnitude more homogeneous B field. Thus, by external excitation the axial temperature has to be increased by a well-known factor for the axial frequency shift to be big enough to be resolved. This is achieved as follows: First, the ion motion is resistively cooled until it reaches a thermal equilibrium with the surroundings. This is the temperature associated to the minimum attainable axial energy for the current setup, denoted by T 0, and also the temperature that is aimed to be determined. Then an axial frequency measurement is carried out (ω zcold ) and a measurement of the background Johnson noise level (U n0 ) in a frequency window around this axial frequency is performed. The axial motion is excited to increase the corresponding temperature to a higher value T 1. Once this rise has occurred, the axial and cyclotron motions are coupled and a net energy transfer from the axial to the cyclotron degree of freedom takes place. Then the background Johnson level is measured with the applied excitation, yielding a value for U n1 typically 5 times bigger than U n0. Finally, a second axial frequency measurement is performed, yielding ω zhot. If one now substitutes the known parameters in equation (5.6) by its numerical values (m = 12 u, ω z = 2π khz, B 0 = T and B 2 = 8.2µ T/mm 2 ), it results in: E ω z 2π. (5.7) A value for the cyclotron energy is obtained with the measured quantity for the frequency shift ω z = ω zhot ω zcold. (5.8) By repeating this procedure many times one obtains the Boltzmann distribution of the cyclotron energy. Using the fact that the quantum numbers of both the axial and cyclotron motion are made equal by sideband coupling, a direct relationship between the mean values of both energy distributions can be found: < E z > ω z = < E + > ω +. (5.9) Since the mean value of the Boltzmann distribution is given by k B T and since it can be directly derived from an exponential fit to the experimental results, equation (5.9) becomes T 1 = ω z < E + > (5.10) ω + k B from where one determines the axial temperature of the ion during the excitation. The final step is to obtain a proportionality factor between T 1 and T 0. For this purpose, the Johnson noise levels at both temperatures have been measured. The expression that relates the Johnson noise level (U n ) to the temperature (T ) of a system is [Joh28] Un 2 = 4k B T B f R (5.11)

48 42 Measurement and Results where B f is the frequency bandwidth under observation. Therefore, the pursued ratio to finally obtain the initial axial temperature is Un 2 1 /Un 2 0 : T 0 = ω z < E + > Un 2 0. (5.12) ω + k B Un 2 1 The resulting curve for the distribution of the cyclotron energy upon axial excitation and mode couplings plotted in figure 5.3, yielding a mean value of < E + >= 4.03 ± 0.31eV (5.13) where the uncertainty is purely statistically determined by the least square fit. Figure 5.3: Plot of the Boltzmann distribution of the cyclotron energy. The next number needed for the calculation of T 0, is U 2 n 1 /U 2 n 0. Statistical fluctuations of this repeatedly measured quantity shows a Gaussian distribution, as can be seen in figure 5.4. It introduces the largest uncertainty in the final number for the axial temperature measurement. It amounts to a value of < ( U n 1 U n0 ) 2 >= 23.2 ± 1.3. (5.14) With these numbers and the axial and cyclotron frequencies (ω z = 2π khz and ω + = 2π MHz, with associated uncertainties too small in comparison to < E + > and U 2 n 1 /U 2 n 0 to be considered), one gets a value from equation (5.12) for the axial temperature of the single hydrogenic carbon ion of T 0 = 77 ± 6 ± 8K, (5.15) where the first uncertainty comes from statistical considerations and the second one from the uncertainty in the value measured for B 2 [Häf00a]. One can see that in figure 5.3 there are some points hinting at negative energies. The reason is that

49 5.1 Temperature of a single ion in a Penning trap 43 Figure 5.4: Distribution of the proportionality constant which defines the ratio T 1 /T 0. It yields a mean value of U 2 n 1 /U 2 n 0 = 23.2 ± 1.3. there are negative axial frequency shifts which make the sign of E + negative (see equation 5.7). These negative shifts arise from the fact that for the determination of the cyclotron energy we are assuming that the substraction of two Boltzmann distributions (corresponding to ω zhot and ω zcold ) gives another Boltzmann distribution. This is, of course, not true. However, the ratio in the two mean values for the axial distributions is so high, that the difference actually resembles a characteristic exponential, save for the small tail appearing for negative energy values. This explains the non-zero probability for energy states below zero in figure Axial temperature measurement via C 4 term A second possibility for an axial temperature measurement is making use of the fact that the trapping potential can be intentionally chosen not to be completely harmonic. The axial frequency will in this case depend on the axial energy and the lineshape of the ion s resonance signal will be non-symmetric. Then the electric potential Φ along the symmetry axis of the trap (i.e. the z- direction) has a nonvanishing octupole term C 4 in the expansion of the potential Φ(z, 0) = j=0 C j z j ; C j = 1 j! j Φ z j (0, 0). (5.16) In the present setup, the harmonicity of the trapping potential can be influenced by choice of the voltages applied to correction electrodes of the Penning trap placed symmetrically between the ring and endcap electrodes. This is expressed in terms of the so-called "tuning ratio" T R, which is the ratio of the voltages applied to the

50 44 Measurement and Results correction electrodes and the ring electrode, Sec The term C 4 appears due to the deviation from an ideal, infinitely hyperbolical trap. It is related to the difference between the optimal tuning ratio (i.e. the one to achieve a harmonic trap potential) and the one actually applied. For the present trap setup, this relation is calculated to be [Ver04b] C 4 = 1 ( mπ ) 4 4! (Am + B m tr), (5.17) L m=1,3,5 where L is the length of the cylindrical trap and A m and B m are tuning ratio parameters which are calculated by use of the given trap geometry [Ver04b]. The dependence of the axial frequency on the axial energy can be written as ω z = ω z,0 + β E z (5.18) where ω z,0 is the axial frequency for vanishing (E z = 0) axial energy. β is given by β = C 4 (C 2 ) 3ω z 2 8π q U (5.19) and describes the dependence of the axial frequency ω z on the axial energy E z in terms of the determined C 4. By fixing a non-optimized tuning ratio and measuring the axial resonance signal of the ion, the motional temperature can be extracted from the lineshape of the resonance signal. To that end, the lineshape for a fixed axial energy E z of the ion is convoluted with a Boltzmann distribution of that energy to describe the smearing out of the resonance due to thermal energy fluctuations. Thus, one starts with a lineshape 1 as given by [Häf00a] (ω z ω) U(ω) 2 (5.20) γ 2 /4 + (ω z ω) 2 where U(ω) is the measured FFT signal voltage and ω z is the ion axial frequency, γ is the frequency-dependent damping factor due to the impedance Z(ω) of the system defined by 0 q2 γ = md Z(ω) = 1 (5.21) 2 τ in analogy and with the same nomenclature as in equation (5.1). The convolution of equation (5.20) with a Boltzmann distribution yields U(ω) exp( E z k B T ) (ω (ω z,0 + βe z )) 2 γ 2 /4 + (ω (ω z,0 + βe z )) de z. (5.22) 2 Thus, by numerical inversion of equation (5.22) a value for the axial energy (and therefore temperature) can be found for a given (measured) axial resonance curve at a fixed detuning of the trap. Figure 5.5 shows a measured resonance curve together with the fitted curve according to equation (5.22) yielding an axial temperature of 62 ± 10 K and two simulated resonances for axial temperatures of 50 K and 100 K. Similar measure- 1 It holds only for the ion axial oscillation frequency close to the center of the tank circuit.

51 5.1 Temperature of a single ion in a Penning trap 45 Figure 5.5: Comparison of the measured ion axial resonance curve with simulated resonances according to Eq. (5.22) for axial temperatures of 50 K, 62 K and 100 K. ments have been performed for a number of values for the detuning. The combined result for the axial temperature is 69 ± 8 K. This method does not require the transport of the ion, but has the disadvantage that the numbers in equations (5.17) and (5.18) have been calculated for an ideal trap geometry and may vary due to any geometrical imperfections or charge accumulation on the electrodes. However, an accurate measurement by other methods will yield an experimental value for C 4 through equations (5.17) and (5.18) Direct noise temperature measurement Assuming the equality of the noise temperature of the resonance circuit attached to the trap and the ion inside the trap, a value for the ion s temperature can be obtained by a measurement of the noise temperature of the electronics [Bor74, Whi96]. To that end, the Johnson noise level voltage U n of the electronics has been measured by use of a Yokogawa SA2400 spectrum analyzer upon frequency downconversion of the signal. An absolute value of the noise level has been calculated by use of the corresponding attenuation factor between the resonance circuit and the spectrum analyzer. This number (i.e. the transfer function) has been determined by a combination of measured attenuation factors and calculations based on the known electronic properties of the relevant components. Figure 5.6 shows a measured resonance spectrum of the axial ion signal. The sharp peak marks the frequency at which the noise level voltage is to be determined. The expression that relates this Johnson noise level (U n ) to the temperature (T ) of a system is given by equation U 2 n = 4k B T B f R (5.23)

52 46 Measurement and Results Figure 5.6: Resonance spectrum during a noise temperature determination. The sharp peak marks the frequency at which the noise level voltage is to be determined. and can be solved to give T = U 2 n 4k B B f R (5.24) with the same nomenclature as above. The observation bandwidth has been chosen to be B f = 7.8 Hz. The resistance R of the resonance circuit is determined from the measured quality factor Q by R = Q (5.25) ω r C where C is the capacitance of the resonance circuit. Corrections to this value have to be made due to the existence of attached electronics which have been taken into account by a corresponding simulation. It yields a resistance R of 11.0±1.4 MΩ at a capacitance of C=23±3 pf. With this, the resulting value for the noise temperature is T = 52 ± 5 ± 3 K, where the first uncertainty is the statistical measurement uncertainty and the second one is due to the uncertainties in R and C used in the simulation Discussion and results We have presented different approaches to measure the temperature of a single ion confined in a Penning trap and have applied these methods to the cyclotron and the axial motion of the ion. While the measured cyclotron temperature of T + = 4.9 ± 0.1 K is in agreement with the liquid helium surrounding, the resulting value for the axial temperature is not. Measurements performed on a single, hydrogenlike carbon ion 12 C 5+ have yielded results which agree within the given uncertainties and give a combined value of T z = 69 ± 6 K. Since the different approaches are based on different aspects of the trapping, a combination of the presented methods can

53 5.2 A peak technique for axial motion detection in a Penning trap 47 be used for an experimental determination of certain trapping parameters, as e.g. trapping potential anharmonicity terms and electronic noise levels. A direct measurement of the electronic noise temperature of the electronics attached to the trap electrodes has yielded a value of T z = 52 ± 5 ± 3 K, in fair agreement with the value determined from the ion s motion. Such a deviation of the axial ion temperature from the liquid helium ambience temperature of about 4 K has been commented on before [Win75]. It is most likely due to noise in the electronic components used for axial signal detection. These are different from the electronics used for cyclotron signal detection mainly with respect to the quality factor of the resonance circuit and the subsequent mixing and amplification of the signal. The main problem was clearly the so called fictious resistance in the input stage of our cryogenic preamplifier of axial tank circuit. Since we use exactly the same FET (type: SONY 3SK 166) in the cyclotron tank circuit, where we measured the ion temperature of 5 K, we conclude that the 1/f characteristic 2 of the used FET causes the problem. Reducing the axial ion temperature will significantly improve the accuracy of the measurement through improving the spin flip resonance line shape which will be more narrow and symmetric [Ver04a]. Apparently, the residual coupling of the axial motion due to e.g. trap imperfections to the other degrees of freedom is not strong enough to produce a thermal equilibrium amongst all degrees of freedom at the axial temperature. This is already clear from the fact that the cyclotron temperature T + has been measured to be 4.9 K in agreement with the ambience temperature while the axial temperature T z is more than one order of magnitude higher. From the fact that the measured energy distributions obey a Boltzmann distribution it may be concluded that the system under investigation (i.e. the single ion in interaction with the trap electrodes and attached resonance circuits) is an ergodic system in the sense defined before. The ergodicity is connected with the requirement that the particle is able to access the whole available phase space within the measurement time, which apparently is the case here. This seems to be reasonable also if one realizes that the typical motional frequencies are more than six orders of magnitude higher than the inverse measurement time and the ion s motion is in principle unrestricted within the trapping volume, especially since the ion s spatial trajectory is not closed. 5.2 A peak technique for axial motion detection in a Penning trap For the precise determination of the axial oscillation frequency we have been using the dip in the noise spectrum of the axial tank circuit as explained in Section 3.3.3, and shown in Fig. 3.6, (bolometric detection [Deh73]). To obtain a sufficiently good signal to noise ratio we have to average over several minutes. In many cases it is however desirable to shorten this time significantly. As described below, exciting the ion at its resonant frequency leads to a peak in the noise spectrum which can be observed in shorter time. Due to the existence of trap anharmonicities (C 4 and B 2-2 The resonance frequency in the cyclotron circuit is by factor of 25 higher than in axial tank circuit.

54 48 Measurement and Results terms) it is not well suited to obtain high accuracy in the frequency measurement. It has, however, advantages in some applications: 1. A very fast search for the ion resonance frequency. 2. Fast tuning ratio optimization and regular checking after every refilling of helium. 3. Powerful tool for implementation of the phase sensitive spin detection (see Section 5.3) Measurement principle The charged particle s axial motion induces an oscillation current in the traps electrodes [Win75]. The capacitance of the correction electrode together with the superconducting detection coil form a tuned LC circuit ("tank circuit"). On resonance this circuit s impedance is purely resistive R = Q/ω z C and, if resonant with ω z, it will damp the axial motion to the circuit s temperature. The induced ion current creates a voltage drop on the LC circuit which is amplified first by a cryogenic low noise amplifier, then with the room temperature electronics. Finally, after FFT (Fast Fourier Transform) in a spectrum analyser the axial motion is visualized. Instead to drive the ion only indirectly with the thermal noise of the detection LC circuit, like in the case of the dip method, one can apply also an external excitation. Our first setup scheme was made for checking the applicability of the peak detection method in general, in June Excitation line modulator r.f. generator switch box 1 to the trap Detection line spectrum analyser switch box 2 mixer 1 out from the trap Figure 5.7: A block scheme of electronic connections for the preliminary axial peak detection method. We created a carrier frequency at the ion s motion (ν z = Hz, 2.6 V pp ) by using a r.f. generator (Stanford Research System DS 345) and modulated it with a rectangular noise spectrum of 400 Hz width, see Fig Between the Penning trap and the r.f. generator we added the signal switch box 1, realised in a simple way by means of relays. In the detection line the ion signal coming from the axial tank circuit is amplified, first in the cryogenic and then in the room temperature axial amplifiers. A second switch box chops the signal after frequency down conversion with mixer 1. The function of the two switch boxes is to synchronize the excitation/detection

55 5.2 A peak technique for axial motion detection in a Penning trap 49 Figure 5.8: Timing scheme of the axial peak detection method. The whole measurement cycle consisits of excitation time t exc, waiting time t wait, and detection time t det. timing such, that the FFT spectrum analyzer (Yokogawa SA 2400) never "sees" the excitation because it might be then temporarily blinded. The hole scheme is performed in cycles, having a repetition rate of 0.9 Hz, see Fig We were able to observe the axial peak of a single carbon ion in only 5 s with a very good signal to noise ratio. An improved axial peak technique has been introduced in August The main improvement was a more precise and reliable timing by using a commercial delay generator (Berkeley Nucleonics Corporation) BNC 555, see Fig. 5.9, which has independent channels A and B for the trigger pulses for the excitation and detection line. It replaced the less reliable electro-mechanical switch boxes and has the possibility to optimize both, repetition rate and waiting time between the ion excitation and detection of the ion signal. Using the arbitrary wave form software we created a burst signal and load it to the function generator. The produced signal made a frequency sweep in the range from 900 khz to 905 khz, covering the new ion axial frequency at ν z = Hz, lasting roughly 10 ms. The best S/N ratio was obtained for an attenuation of 90 db between the function generator and the axial LC circuit input line through which the signal was introduced into the Penning trap Fast ion resonance detection The possibility of a fast search for the ion axial motion frequency is very important for a fast preliminary trap optimization. After warming up the apparatus, changing the detection electronics or simply creation of a new ion, due to patch effects and other imperfections in the trap, the ion frequencies can be significantly altered. It usually meant that one has to spend quite some time to adjust the trap parameters for an ultra precise measurement. With our new axial peak technique we reduced the single ion detection time by

56 50 Measurement and Results Excitation line A Delay generator B function generator Detection line spectrum analyser attenuator from the trap to the trap Figure 5.9: Improved axial peak detection scheme. A delay generator has been introduced triggering the excitation and detection line instead of two switch boxes. a factor of 20. We were able to get a very clear signal of the axial motion of the hydrogenlike carbon ion in 5 s, as illustrated in Fig Figure 5.10: The first axial peak detected in the spectrum analyser span of 800 Hz obtained in only 5 s with a full width at a half maximum (FWHM) of 3.8 Hz. Later, we found out the possibility to observe the axial peak of a single ion with a completely detuned trapping potential. Even for the anharmonicity of the trapping potential which is µunits (in terms of the TR), we can see the peak in a few seconds. In Fig three single ion peaks are shown, separately observed in 5 s each, plotted in a 800 Hz span. These peaks are obtained for the same negative and positive TR detuning (2 000 µunits) and therefore placed symmetrically around the peak with the optimal TR due to nonorthogonality (D 2 term) of the trap, Sec A much worse signal to noise ratio and broadening of the signal shape is induced by the anharmonicity, i.e. C 4 term. For such an anharmonic trapping potential the ion dip (obtained via bolometric method) was not visible. During an averaging time of around 100 s, which is necessarry to observe a dip, the ion energy and trapping potential fluctuate so much that the S/N ratio is too low to observe an ion dip.

57 5.2 A peak technique for axial motion detection in a Penning trap 51 µ µ Figure 5.11: Comparison between three ion peaks: TR with the optimal value of C , TR with the negative value of C , and TR with the positive value of C Another useful feature of the peak technique is the possibility to see either the ion peak or the ion dip. This is achieved by adjusting the waiting time between the ion excitation and detection, see Sec We have even found the intermediate state (for the certain waiting time Fig. 5.8) in which both (peak and dip) were visible. In spite of the fact that we increased the ion energy via external excitation, to have a better S/N ratio, energy fluctuations which obey a Boltzmann distribution did not broad the ion dip since we need much shorter averaging time, Fig Figure 5.12: Dip of the single ion obtained in 8 s in a frequency span of 1.56 khz has a FWHM of only 3 Hz, this is slightly bigger than the measuring bandwidth of 1.95 Hz, but similar to the FWHM obtained for the peak shown in Fig

58 52 Measurement and Results Fast tuning ratio optimization via the peak detection technique The peak detection scheme serves as a suprisingly good tool for the fast optimization of the trap parameters. It turns out, that for a rough estimation of the tuning ratio it is better to observe peaks, rather than the "classical" dips, since the S/N ratio is much better. Therefore one can start with a completely bad TR (of order up to µunits). Then it is easy in only few steps, just by observing the shape of the peak, to decide whether there is a positive C 4 term, (broadening of the peak to the right side), or a negative C 4 term (broadening to the left side). Single peak detection takes 5 s. In less than 1 min one can observe 7 peaks and find out a good TR with an accuracy better than 400 µunits, Fig Figure 5.13: The first estimation with completely unknown TR can be quickly done via the peak detection scheme in less than 1 min with an accuracy better than 400 µunits. This allows to start a further optimization with the more sensitive methods. After the rough determination of the tuning ratio via observing the peaks, the fine adjusting is started by dip observation with the new detection technique. Now we need a twice longer averaging time of about 10 s but we achieved one order of magnitude higher accuracy in the TR optimization. Again in less than a 1 min it is possible to observe 5 dip signals of a single ion in steps of 100 µunits. This time a broadening of the peak is almost invisible, since the Boltzmann energy distribution does not clearly show up for such small detunings. However the depth of the signal around the optimal value of the TR helps to resolve the optimum with an accuracy better than 50 µunits, Fig

59 5.3 Phase sensitive spin detection 53 Figure 5.14: In only 50 s we could observe 5 ion s dips, varying the TR in steps of 100 µunits. In such a way we can determine the TR with an accuracy of order better than 50 µunits which is sufficiently good for starting the g factor measurement. 5.3 Phase sensitive spin detection As pointed out in Sec. 5.2 another purpose of testing and implementing the peak technique for the axial motion detection was introducing the phase sensitive spin detection technique for future experiments. This is a useful improvement for the spin detection of light ions like carbon and oxygen, but it represents a necessary condition for the detection of the spin flip in the case of heavier ions. With our present Penning trap, where the magnetic field inhomogeneity of the magnetic bottle in the Analysis trap is B 2 = 8.2 mt/mm 2 [Wer02], we have a change in the axial frequency caused by a spin flip of 700 mhz for 12 C 5+ and 460 mhz for 16 O 7+. Compared to our maximal accuracy of the axial frequency measurement of 250 mhz it was still possible, with great care, to complete the measurement. Already for 40 Ca 19+ the spin flip would be only 180 mhz, and therefore comparable to fluctuations of ±100 mhz in the axial frequency induced by voltage fluctuations of around 4 µv, from the present voltage power supply. For the future projects in which it is planned to measure the g factor of hydrogenlike uranium and of the bare proton, the frequency shift induced by a spin flip is so small that even with a much better power supply (what is difficult to design) and much higher magnetic bottle (which will introduce line broadening and reduce the accuracy of the measurement) one would not reach the necessary sensitivity in the axial frequency determination. Hence, we were forced to implement a substantially different detection scheme.

60 54 Measurement and Results Fourier limit restriction The classical Fourier limit for the fourier transform of a time depending signal in the bandwidth B is given by the simple expression T = 1 B, (5.26) where T is the time necessary for a signal frequency measurement and it is inverse proportional to the bandwidth B. Therefore to increase the accuracy f of a frequency measurement (what means reducing of the bandwidth B = f) one has to increase the measurement time. Since we are not interested in the absolute value of the frequency but rather in the frequency difference between the two spin states we can overcome the Fourier limit by measurement of a phase difference instead of frequency. The validity of Eq can be seen by the following measurements of the ion frequency (via peak technique) having a resolution of the spectrum analyzer, which is similar to the linewidth of the excited ion. The ion s axial peak linewidth as reported in Sec. 5.2 was 3.8 Hz and according to that we chose a span of 3125 Hz for our spectrum analyzer (Yokogawa SA 2400). Since the analyzer has 801 channels, the corresponding bandwidth (B) for the Fast Fourier transform (FFT) is 3.90 Hz. Consequently, from Eq. 5.26, the minimum time for observing the transformed signal is 250 ms, this is equal to the detection time t det. We performed a measurement of the maximum signal amplitude of the tank circuit for the repetition rate of 1 s and different waiting times. The result is shown in Fig Figure 5.15: Tank circuit amplitude signal shows for the 3125 Hz span of the spectrum analyzer the Fourier limit interval of 214 ± 26 ms. The Johnson noise of the tank circuit is at a level of 76 dbv. But during an interval that equals the classical Fourier limit just before external excitation (by the

61 5.3 Phase sensitive spin detection 55 burst signal) the tank circuit shows an excitation noise which was even higher then the peak height. For the chosen span of the 3125 Hz the data acquisition time is given by the Fourier limit of 250 ms Basic principles The main idea is instead of measuring a frequency difference to measure the phase change accumulated during a certain time interval for two different spin states. In Fig it is shown how the phase of the ion will evolve in time and after a sufficient period of time one can observe the integrated phase difference ϕ between the spin up and the spin down state. According to the Fourier limit one would need for example for a 1 Hz frequency difference a measurement time of 1 s, but here for the phase observation a good observability is gained with a phase difference in the order of 90 0, that means only 1/4 s. This already gives a speeding up of the experiment by a factor of 4. Actually the Fourier limit is not really broken, but just avoided: the absolute frequency information is not preserved in such a phase measurement, only small differencies in frequency are observable. It can be shown, that the differential frequency resulution, i.e. the capability to resolve frequency differences, is given by σ( f) = 1 T σ( ϕ) (5.27) 2π where σ( ϕ) denotes the measurement accuracy of the change in the measured phase difference ϕ. In the limit of maximal phase uncertainty σ( ϕ) = 2π this relation represents again the "classical" Fourier limit: σ( f) = 1 T σ( ϕ) = 1 2π T 2π 2π = 1 T. (5.28) In a real experiment, Eq has the consequence, that the measurement of the phase ϕ with any phase uncertainty, which is better than 2π will improve the accuracy of a f-determination by the improvement factor F F = 2π σ( ϕ). (5.29) This factor will be limited by the achievable signal to noise ratio and has to be determined experimentally. One should point out clearly that an improvement factor F bigger than 1 does not mean a breaking of the Fourier limit, but rather avoiding it. The drawback is indeed that the absolute frequency information gets lost. The experimental setup in which this measurement scheme is implemented, is essentially identical to the peak detection method. For the detection, instead of observing the amplitude of the peak (ion signal) on the spectrum analyzer, we take a look to the phase of the signal. This possibility is provided by the spectrum analyzer we used (Yokogawa SA 2400). The timing scheme is shown in Fig First the ion is excited for several ms (t exc ) with a r.f. electrical dipole field, this is shown in the upper line in Fig Then the ion is decoupled from the tank circuit, to avoid its cooling and deterioration of the S/N ratio. This waiting time (t wait ) can be in the order of 1 s or even longer. Then the ion is coupled again to the tank circuit, to detect its phase during the detection time (t det ). This detection time can be arbitrarily small, since it does not depend on the Fourier limit. In the

62 } 56 Measurement and Results slower spin up B faster phase shift spin down measurement time required to measure 1Hz frequency difference: only ¼ sec. ( classical Fourier limit: 1sec.) Figure 5.16: A small frequency change leads to an accumulated phase change ϕ. real experiment t det can be chosen to equal the time constant of resistive cooling, since the ion signal is damped within a few hundred ms resistively and a longer detection time would not increase the S/N ratio. Finally we allow resistive cooling to take place (t cool ) of the order of 1 cooling time constant. The total length of one measurement cycle is of the order of 1.5 s, and it is defined by a repetition rate of the delay generator, t rep = t exc + t wait + t det + t cool. Figure 5.17: The timing scheme of a detection cycle consists of excitation time, waiting period, detection time and then resistive cooling. For a successful phase sensitive spin detection it is clearly important to have a sufficiently high S/N ratio of the phase signal. Therefore, before starting the measurements one has to optimize the excitation energy of the ion. The aim is to obtain

63 5.3 Phase sensitive spin detection 57 Figure 5.18: Illustration of the expected phase measurements in a complex plane. Each point in the phase diagram represents a single measurement point. For a good S/N the ions have a big amplitude and a small phase spread. For a poor S/N ratio both the real and the imaginary part of the complex amplitude go to zero. a good S/N ratio with a reasonably small excitation to avoid to drive the ion into an anharmonic region. A good S/N ratio then allows an easier and faster detection. Optimization should be done via observing the S/N ratio for successively measuring the ion s phases. For the optimal excitation energy we expect a narrow distribution in a phase diagram far from the origin and with a small phase spread, Fig In the experimental setup an excitation burst signal is implemented, similar as explained in Sec. 5.2, consisting of a waveform which represents a swept sine from 900 to 905 khz. The reason to use a fixed waveform instead of a noise signal, is to provide a well defined timing and phase information. The detection line is the same as described before. The main difference is adding of a cryogenic switch, realised by means of a FET transistor, for the trap decoupling, see Fig The switch will (when it is closed) shortcut the tank circuit changing its Q value by three orders of magnitude from Q = 1000 to the unity. This practically means a total decoupling of the ion from the tank circuit. This is necessary for avoiding resistive cooling of the ion during the waiting period shown in Fig It would be disadvantageous if the ion was coupled (and therefore cooled) by the tank circuit continuously since it would already decrease the amplitude within the waiting time. In this case, the signal during the detection period would be rather small. It can be expected that the waiting time, in which the phase accumulation takes place, will be in the order of 1 2 s and therefore much longer than the cooling time constant Measurements and results Once we understood our limitations we started with the optimization of the experimental setup to check how close to the limit we could get. We decided to start the measurements without the possibility of decoupling the ion from the cooling LC circuit, since it would have implicated warming up the apparatus and introducing FET switches into the cryogenic region, and this would have caused two months of

64 58 Measurement and Results Figure 5.19: Experimental setup consisting of the timing (delay) generator, controlling a burst creation generator for excitation of a trapped ion, a motional decoupling facility with an cryogenic switch and the spectrum analyzer for the phase detection. delay. The block scheme of the phase sensitive spin detection, Fig. 5.20, is very similar to the one we used for the peak detection (Fig. 5.9). This time we had to pay attention not only to the optimized S/N ratio but also to a very precise timing schedule for all devices and syncronization of excitation and detection lines. At the beginning we connected the burst function generator and frequency mixer to the internal 10 MHz time base of the delay generator, see Fig Still it was not sufficient since the observed ion phase drifted in time in a non-predictable way. The reason turned out to be a non-precise timing in our spectrum analyzer. Therefore we fed an external clock of 10 khz frequency, produced by a SRS DS 345 frequency generator as a clock source to the internal base of the spectrum analyzer. Finally after complete synchronization of all relevant devices the ion phase, which we observed, became stable. All devices including the phase sensitive spin detection electronics are at room temperature. The only missing part from the proposed scheme in Fig was a cryogenic switch necessary for the motional decoupling. First results and basic tests After preparation of a single carbon ion 12 C 5+ we optimized the trap parameters using the peak technique. Typical obtained parameters are: a ring voltage of U ring = V, and a tuning ratio of T R = For these parameters the ion axial frequency is equal to the center resonant frequency of the tank circuit in the Precision trap. In order to apply the phase sensitive technique we monitored the ions phase stability and signal amplitude for different excitation levels created by the burst signal generator. The optimum has been found for the voltage of 2.9 V at the output