CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS

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1 ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 48 CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS GÜNTHER WERTH 1, HARTMUT HÄFFNER 1 and WOLFGANG QUINT 2 1 Johannes Gutenberg University, Department of Physics, Mainz, Germany; 2 Gesellschaft für Schwerionenforschung, Darmstadt, Germany I. Introduction II. A Single Ion in a Penning Trap III. Continuous Stern Gerlach Effect IV. Double-Trap Technique V. Corrections and Systematic Line Shifts VI. Conclusions VII. Outlook VIII. Acknowledgements IX. References I. Introduction In 1922 Stern and Gerlach succeeded in spatially separating a beam of silver atoms into two beams, utilizing the force exerted in an inhomogeneous magnetic field on the magnetic moment of the unpaired electron in silver. This so-called Stern Gerlach effect was the first observation of the directional quantization of the quantum-mechanical angular momentum, and represents a cornerstone of quantum physics (Stern and Gerlach, 1922). Apart from its historical role the effect has been used in numerous atomic beam experiments to determine the magnetic moments of electrons bound in atomic systems: An atomic beam enters a first inhomogeneous magnetic field where one spin direction is separated from the other. The polarized atoms then enter a region with a homogeneous magnetic field where they are subjected to a radio-frequency (rf) field which changes the spin direction. Finally, a second inhomogeneous magnetic field region analyzes the spin direction. Variation of the frequency w of the rf field and recording the spin-flip probability as a function of w yields a resonance curve. Together with a measurement of the field strength in the homogeneous magnetic field, the magnetic moment can be derived. Usually the size of the magnetic moment m is given in units of the Bohr magneton m B = eà/ (2m) and expressed by the g-factor: m = gsm B, (1) 191 Copyright 2002 Elsevier Science (USA) All rights reserved ISBN X/ISSN X/02 $35.00

2 192 G. Werth et al. [I where s is the spin quantum number. Accurate g-factor measurements represent a critical test of atomic physics calculations for complex systems (Veseth, 1980, 1983; Lindroth and Ynnerman, 1993). There has been an extensive discussion that the Stern Gerlach effect could be employed for neutral atoms only. For charged particles the magnetic force is overshadowed by the Lorentz force acting on a moving charged particle in a magnetic field. Proposals to use the acceleration of electrons moving along the field lines of an inhomogeneous B-field to separate the spin directions (Brillouin, 1928) have been discarded by Bohr (1928) and Pauli (1958) on the basis of Heisenberg s uncertainty principle. Nevertheless attempts have been made, however unsuccessful, to separate the two spin states in an electron beam by using the different signs of the force on the spin exerted by a longitudinal inhomogeneous magnetic field which results in a deceleleration or acceleration of the electrons (Bloch, 1953). Recently, new proposals have come up to perform Stern Gerlach experiments on electron beams which would in contrast to the analysis by Bohr and Pauli result in a high degree of spin separation under carefully chosen initial conditions (Batelaan et al., 1997; Garraway and Stenholm, 1999). Dehmelt has pointed out that confining a charged particle by electromagnetic fields provides a way to circumvent Bohr s and Pauli s reasoning (Dehmelt, 1988): the force of an inhomogeneous magnetic field on the spin of a particle which oscillates in a parabolic potential well leads to a small but measurable difference of its oscillation frequency for different orientations of the spin. Thus a precise measurement of the oscillation frequency gives information on the spin direction. Dehmelt et al. used this effect for monitoring induced changes of the spin direction of an electron by observing the corresponding changes in the electron s oscillation frequency. Since the sensitive electronics monitors the trapped particle s spin direction continuously, Dehmelt coined the term continuous Stern Gerlach effect for this technique. In a series of experiments he and his coworkers have applied this method to measure the magnetic moment of the electron and the positron, which culminated in the most precise values of a property of any elementary particle (Van Dyck et al., 1987). The experimental value g exp = (87) (2) agrees to 10 significant digits with the result of calculations based on the theory of quantum electrodynamics (QED) for free particles (Hughes and Kinoshita, 1999), g th = (687) (3) and provides one of the most stringent tests of QED. The theoretical result for the free-electron g-factor can be expressed in a perturbation series as ( ( a ) ( a ) 2 ( a g free =2 A 0 + A 1 + A 2 + A3 p p p ) 3 + A4 ( a p ) 4 + ), (4)

3 I] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 193 Fig. 1. Calculated expectation values for the electric field strength for hydrogen-like ions of different nuclear charge Z. (Courtesy Thomas Beier.) where a is the fine-structure constant. The coefficients A n in Eq. (4) have been calculated by evaluating the Feynman diagrams of different orders using a plane-wave basis set; they are of the order of unity. In contrast to a free electron, an electron bound to an atomic nucleus experiences an extremely strong electric field. The expectation value of the field strength ranges from 10 9 V/cm in the helium ion (Z =2)to10 15 V/cm in hydrogen-like uranium (Z = 92) (Fig. 1), and gives rise to a variety of new effects. The largest change of the bound electron s g-factor was analytically derived by Breit (1928) from the Dirac equation: g Breit = 2 3 (1+2 ) 1 (Za) 2 2 (1 1 3 (Za)2). (5) The conditions of extreme electric fields also necessitate changes to be made in the methods of calculations for the QED contributions to the electron s magnetic moment. In a perturbative treatment a series expansion in (Za) is made in addition to that in a. The expansion parameter Za, however, is at least for large Z no longer small compared to 1. In addition, the expansion coefficients can be large. For small values of the nuclear charge Z the perturbation expansion may give reliable results, and calculations were performed which include terms up to order (Za) 2 (Grotch, 1970a; Close and Osborn, 1971; Karshenboim et al., 2001). For more accurate theoretical predictions, non-perturbative methods have been developed where the solutions of the Dirac equation for the hydrogen-like ion rather than those for the free case are used as a basis set (Beier et al., 2000). The most recent summary of the status of QED for bound systems has

4 194 G. Werth et al. [I Fig. 2. Contributions to the g-factor of a bound electron in hydrogen-like ions for different nuclear charges Z. (Courtesy Thomas Beier.) Table 1 Theoretical contributions to the g-factor in 12 C 5+ Contribution Size Reference Dirac Theory Breit (1928) QED, free (all orders) Hughes and Kinoshita (1999) QED, bound, order (a/ p) (9) Beier et al. (2000) QED, bound, order (a/ p) 2,(Za) 2 term (4) Grotch (1970a) Recoil (in Za expansion) (1) Yelkovsky (2001) Finite size correction Beier et al. (2000) been published by Beier (2000). A graphical representation of the bound-state contributions to the electron g-factor is shown in Fig. 2. In this contribution we describe an experiment which for the first time applies the continuous Stern Gerlach effect to an atomic ion (Hermanspahn et al., 2000). We measured the magnetic moment of the electron bound to a nucleus with zero nuclear magnetic moment, hydrogen-like carbon ( 12 C 5+ ). For the bound-state contributions of order a(za) n the existing calculations (Grotch, 1970a) deviate already for Z = 6 by as much as 10% from the non-perturbative calculations to all orders in (Za). The numerical results of the calculations for C 5+ are summarized in Table 1. The leading term comes from the solution

5 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 195 of the Dirac equation and deviates from the value g = 2 for the free electron (Breit, 1928). The next-largest part is the well-known QED contribution for the free electron (Hughes and Kinoshita, 1999). The bound-state contributions of order a (calculated to all orders in Za) are given with error bars which represent the numerical uncertainty of the calculations. The quoted uncertainty of the a 2 (Za) 2 term is an estimate of the contribution from non-calculated higherorder terms. Finally, nuclear recoil contributions have been calculated to lowest order in Za by Grotch (1970b), Faustov (1970) and Close and Osborn (1971). Recently, Shabaev (2001) presented formulas for a non-perturbative calculation (in Za), and Yelkovsky (2001) presented further numerical results. The nuclearshape correction was considered numerically by Beier et al. (2000), and recently (for low Z) also analytically by Glazov and Shabaev (2001). The sum of the different contributions leads to a theoretical value for the g-factor in hydrogenlike carbon of g theor ( 12 C 5+ )= (10). (6) II. A Single Ion in a Penning Trap The experiment is carried out on a single C 5+ ion confined in two Penning traps. In a Penning trap a charged particle is stored in a combination of a homogeneous magnetic field B 0 and an electrostatic quadrupole potential. The magnetic field confines the particle in the plane perpendicular to the magnetic field lines, and the electrostatic potential confines it in the direction parallel to the magnetic field lines. In our experiment we use two nearly identical traps placed 2.7 cm apart in the magnetic field direction. They consist of a stack of 13 cylindrical electrodes of 2r 0 = 7 mm inner diameter. The difference between the traps is that in one trap the center electrode is made of ferromagnetic nickel while all others are machined from OFHC copper. Figure 3 shows a sketch of this setup. The nickel ring distorts the homogeneity of the superimposed magnetic field in the corresponding trap while the field remains homogeneous in the other trap (see Fig. 3). As will become evident below, the inhomogeneity of the field is the key element to analyze the direction of the electron spin through the continuous Stern Gerlach effect. Therefore we call the corresponding potential minimum analysis trap while we call the one in the homogeneous magnetic field precision trap. Each trap uses five of these electrodes to create a potential well, which serves for axial confinement. We apply a negative voltage U 0 to the center electrode while we hold the two endcap electrodes at a distance z 0 from the center at

6 196 G. Werth et al. [II Fig. 3. Sketch of the electrode structure and potential distribution of the double trap. ground potential. The potential F inside this configuration can be described in cylindrical coordinates r, z, ö by an expansion in Legendre polynomials P i : F(r, ö) = U 0 2 ( r ) i C i Pi (cos ö), (7) d i =0 where d 2 = ( z r2 0 / 2) / 2 is a characteristic dimension of the trap (Gabrielse et al., 1989). Two correction electrodes are placed between the center ring and the endcaps. The coefficient C 4 which is the dominant contribution to the trap anharmonicity can be made small by proper tuning of the voltages applied to the correction electrodes. Essentially then the potential depends on the square of the coordinates, and is a harmonic quadrupole potential F = U 0 2 z 2 r 2 / 2 d 2. (8) We optimize the trap by changing the voltages on the correction electrodes until the ion oscillation frequency is independent of the ion s oscillation amplitude as characteristic for a harmonic oscillator. With this method we can reduce the

7 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 197 Fig. 4. Ion oscillation in a Penning trap. dominant high-order term C 4, the octupole contribution, to less than The ion s frequency in the harmonic approximation is then given by qu0 w z = Md 2. (9) Radial confinement is achieved by the homogeneous magnetic field directed along the trap axis. This results in three independent oscillations (axial, cyclotron, and magnetron oscillation) for the ion motion, as depicted in Fig. 4. The fast radial oscillation frequency of the ion in the Penning trap is a perturbed cyclotron frequency w c. It differs from the cyclotron frequency w c = q M B (10) of a free particle with charge q and mass M because of the presence of the electric trapping field, and is given by w c = w c w c 4 w2 z 2. (11) It can be expressed also as w c = w c w m, (12) where w m is the magnetron frequency, a slow drift of the cyclotron orbit around the trap center, given by w m = w c 2 w 2 c 4 w2 z 2. (13)

8 198 G. Werth et al. [II For calibration of the magnetic field we use the cyclotron frequency of the trapped ion. It can be derived either from Eq. (12) or more reliably from the relation wc 2 = wc 2 + wz 2 + wm, 2 (14) since this equation is independent of trap misalignments to first order (Brown and Gabrielse, 1986). In this case the measurement of the magnetron frequency w m is required in addition to a measurement of w c and w z. The traps are enclosed in a vacuum chamber placed at the bottom of a helium cryostat at a temperature of 4 K and located at the center of a superconducting NMR solenoid. The helium cryostat provides efficient cryopumping. As an upper limit we estimate the vacuum in the container to be below mbar. The estimation was derived from the measurements on a cloud of highly charged ions, whose storage time would be limited by charge exchange in collisions with neutral background particles. We observed no ion loss in a cloud of 30 hydrogenlike carbon ions stored for 4 weeks. Together with the known cross section for charge exchange with helium as the most likely background gas at 4 K we obtain an upper limit of mbar for the background gas pressure. The magnetic field of the superconducting magnet is chosen to be 3.8 T. At this field strength the precession frequency of the electron spin is 104 GHz. Microwave sources of sufficient power and spectral purity are commercially available at this frequency. We load the trap by bombarding a carbon-covered surface with an electron beam. This process releases ions and neutrals of the element under investigation as well as of other elements present on the surface. Higher charge states are obtained by consecutive ionisation by the electron beam. We detect the ions by picking up the current induced by the ion motions in the trap electrodes. For this purpose superconducting resonant circuits and amplifiers are attached to the electrodes. Upon sweeping the voltage of the trap, and thus the ions axial frequencies, the ions get in resonance with the circuit and their signal is detected. Figure 5 shows such a spectrum, where we identified different elements and charge states. We eliminate unwanted ion species by exciting their axial oscillation amplitude with an rf field until the ions are driven out of the trap. Ions of the same species have different perturbed cyclotron frequencies in the slightly inhomogeneous magnetic field of the precision trap, because they have different orbits. Therefore, for small ion numbers, single ions can be distinguished by their different cyclotron frequencies. Figure 6 shows a Fourier transform of the induced current from 6 stored ions. We excite the ions cyclotron motion individually and thus eliminate them from the trap until a single ion is left. Typical cyclotron energies for signals as shown in Fig. 6 are of the order of several ev. In order to reduce the ion s kinetic energy we use the method of resistive cooling which was first applied by Dehmelt and collaborators (Wineland and Dehmelt, 1975; Dehmelt 1986). The ion s oscillation is brought into resonance

9 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 199 Fig. 5. Mass spectrum of trapped ions after electron bombardment of a carbon surface showing different charge states of carbon ions as well as impurity ions (a) before and (b) after removal of unwanted species. with the circuits attached to the electrodes. The induced current through the impedance of the circuit leads to heating of the resonance circuit, and the ion s kinetic energy is dissipated to the surrounding liquid helium bath (Fig. 7). This leads to an exponentially decreasing energy with a time constant t given by t 1 = R, (15) Md2 where R is the resonance impedance of the circuit. For the axial motion we use superconducting high-quality circuits (Q = 1000 at 1 MHz in the precision trap and Q = 2500 at 365 khz in the analysis trap). With the resonance impedances of R = 23 MW (analysis trap) and 10 MW (precision trap) the cooling time constants are 80 ms and 235 ms, respectively. For cooling the cyclotron motion we employ a normal-conducting circuit at 24 MHz (Q = 400) with a resonance impedance of 80 kw. Here we reach cooling time constants of a few minutes. Figure 8 shows the exponential decrease of the induced currents from the ion oscillations as the result of axial cooling. q2

10 200 G. Werth et al. [II Fig. 6. Fourier transform of the voltage induced in one of the trap electrodes from the cyclotron motion of 6 stored 12 C 5+ ions. The inhomogeneous magnetic field of the trap causes ions at different positions to have slightly different cyclotron frequencies. Fig. 7. Principle of resistive cooling. We do not cool the magnetron motion in a similar way because it is metastable: in the radial plane the ion experiences an electrostatic force towards the negatively biased center electrode. Ion loss is prevented by the presence of the magnetic field. Thus the potential energy is an inverted parabola. Therefore reduction of the ion s magnetron energy results in an increase in the magnetron radius. It is, however, essential to reduce the magnetron radius because of the magnetic field inhomogeneities. This is achieved in a well-defined way by coupling the magnetron motion to the axial motion by a radio-frequency field at the sum frequency of both oscillations (Brown and Gabrielse, 1986; Cornell et al., 1990). In the quantum-mechanical picture for the ion motion, the absorption of a photon from this field increases the quantum number of the axial oscillation by 1 while that of the magnetron oscillation is decreased by 1. An analysis of the absorption probabilities in the framework of a harmonic oscillator

11 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 201 Fig. 8. Exponential energy loss of the axial motion of trapped C 5+ ions by resistive cooling. yields that the quantum numbers tend to equalize. This leads to the expectation value E m for the magnetron energy, ( ( ) E m = àw m km + 2) 1 = àwm kz = w m ( ) àw z kz + 1 w m w 2 = E z. z w z (16) The axial oscillation is continuously kept in equilibrium with the cooling circuit and we thus reduce the magnetron orbit to about 10 mm. The mean kinetic energy of a single ion is often expressed in terms of temperature. This is justified by the statistical equilibrium of the ion and the resonant circuit. The statistical motion of the electrons in the resonance circuit causes Johnson noise in the trap-electrode voltages, which in turn leads to varying energies of the ion as a function of time (Fig. 9). Extracting a histogram of the cyclotron energies results in a Boltzmann distribution (Fig. 10) with a temperature of 4.9K close to the temperature of the environment. Calculating the temporal autocorrelation function of the energy gives, as expected, an exponential (Fig. 11) with a time constant well in agreement with the measured cooling time constant. In order to calibrate the magnetic field at the ion s position with high precision from Eq. (14) the three oscillation frequencies have to be measured. Because of their different orders of magnitude (w c/ 2p = 24 MHz, w z / 2p = 1 MHz, w m / 2p = 18 khz) the required precision is different. w c is determined from the Fourier transform of the current induced in a split electrode. Figure 12 shows that the relative linewidth of the resonance, well described by a Lorentzian, is of the order of 10 9 and the center frequency can be determined with an accuracy of In order to obtain sufficient signal strength the energy of the cyclotron oscillation has to be raised to about 1 ev. Due to the inhomogeneity of the

12 202 G. Werth et al. [II Fig. 9. Noise power of the induced voltage in a trap electrode from the cyclotron oscillation of a single trapped ion while its frequency is continuously kept in resonance with an attached tank circuit. Fig. 10. Histogram of the probabilities for cyclotron energies. The curve can be well fitted to a Boltzmann distribution, giving a temperature of 4.9(1) K.

13 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 203 Fig. 11. The time-correlation function of the noise in fig. 10 shows an exponential decrease. The time constant of 5.40(7) min corresponds to the time constant for resistive cooling of the cyclotron motion. Fig. 12. High-resolution Fourier transform of the induced noise at the perturbed cyclotron frequency. A Lorentzian fit gives a fractional width of magnetic field this changes the mean field strength along the cyclotron orbit. This has to be considered in the final evaluation of the measurements. The axial frequency w z is determined while the ion is in thermal equilibrium with the resonance circuit. At a given temperature the thermal noise voltage in the impedance Z(w) of the axial circuit, given by U noise = 4kT Re[Z(w)] dn, (17) excites the ion motion within the frequency range dn of the ion s axial resonance. This motion in turn induces a voltage in the endcap electrodes, however at a phase difference of 180º, as can been shown by modeling the system as a driven harmonic oscillator. Consequently the sum of the thermal noise voltage and

14 204 G. Werth et al. [II Fig. 13. Axial resonance of a single trapped C 5+ ion. The noise voltage across a tank circuit shows a minimum at the ion s oscillation frequency. Fig. 14. High-resolution Fourier transform of the axial noise near the center of the resonance frequency of the axial detection circuit. the induced voltage leads to a reduced total power around the axial frequency of the ion. This appears as a minimum in the Fourier transform of the axial noise as shown in Fig. 13. A spectrum with a resolution of 10 mhz (Fig. 14) shows that the center frequency can be determined to about 24 mhz. A different approach to explain the appearance of a minimum in the axial noise spectrum was taken by Wineland and Dehmelt (1975) considering the equivalent electric circuit of an oscillating ion in the trap. The magnetron frequency w m is measured by sideband coupling to the axial motion. If the ion is excited at the difference between the axial and magnetron frequencies, the ion s axial

15 II] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 205 energy increases, leading to an increased current at the ion s axial frequency. We detect this as a peak in the detection circuit signal. The uncertainty in this frequency determination is below 100 mhz. Imperfections in the trap geometry may change the motional frequencies. These changes have been calculated by different authors (Brown and Gabrielse, 1982; Kretzschmar, 1990; Gerz et al., 1990; Bollen et al., 1990). Considering only an octupole contribution to the trap potential, characterized by a coefficient C 4 in the potential expansion (5), these shifts amount to [ ( ) ] 2 3 w 2 E z 3 E c +6E m, (18) w c ( ) [ 2 1 wz = C 4 3E z + 3 ( ) ] 2 wz E c 6E m, (19) qu 0 w c 2 w c [ ( ) ] 2 wz 6E z 6 E c +6E m. (20) Dw z w z = C 4 1 qu 0 Dw c w c Dw m w m = 1 qu 0 w c E z, E c and E m are the energies in the axial, cyclotron and magnetron degrees of freedom. The tuning of the trap potential results in a coefficient C 4 as small as For the energies of the motions in thermal equilibrium the corresponding frequency shifts are below and need not be considered here. The coefficient C 6 of the dodekapole contribution to the trapping potential has been calculated to an accuracy of 10 3 for our trap geometry. The frequency shifts arising from this perturbation scale with (E/qU 0 ) 2 and are negligible here. The residual inhomogeneity of the magnetic field in the precision trap arising from the nickel ring electrode in the analysis trap 2.7 cm away changes the value of the oscillation frequencies of an ion with finite kinetic energy as compared to the ion at rest. A series expansion of the B-field in axial direction, gives frequency shifts Dw c w c = Dw z w z = Dw m w m = 1 mw 2 z 1 mw 2 z 1 mw 2 z B z = B 0 + B 1 z + B 2 z 2 + (21) [ B 2 1 B 0 àw c ( wz w c ) ] 2 E c + E z +2E m, (22) B 2 1 [E c E m ], B 0 àw c (23) B 2 1 [2E c E z 2E m ]. B 0 àw c (24) The size of the inhomogeneity term B 2 is measured by application of a bias voltage between the endcap electrodes. This shifts the ion s position in the axial

16 206 G. Werth et al. [III direction by a calculable amount, and the cyclotron frequency is measured at each position. We obtain B 2 =8.2(9) mt/mm 2. The shift in the perturbed cyclotron frequency which is of most interest here is dominated by the axial energy E z, and amounts to Dw c/ w c = for an axial temperature of 100 K. III. Continuous Stern Gerlach Effect The g-factor of the bound electron as defined by Eq. (1) can be determined by a measurement of the energy difference between the two spin directions in a magnetic field B: DE = hn L = gm B B, (25) where n L is the Larmor precession frequency. We induce spin flips by applying magnetic dipole radiation which is blown into the trap structure by a microwave horn. For the detection of an induced spin flip we follow a route developed in the determination of the g-factor of the free electron (Dehmelt, 1986; Van Dyck et al., 1986): the quadrupole potential of the Penning trap depends on the square of the coordinates (6) leading to a linear force acting upon the charge of the stored ion. Considering the force upon the magnetic moment of the bound electron by the inhomogeneous field in the analysis trap we get F = (m B). (26) The nickel ring in the analysis trap creates a bottle-like magnetic field distortion which can be described in first approximation by [ z ÀB = ÀB 2 r 2 ] 0 +2B 2 ê zàr. (27) 2 The odd terms vanish in the expansion because of mirror symmetry of the field. The corresponding force on the magnetic moment in axial direction is F z = 2m z B 2 z, (28) which is linear in the axial coordinate. It adds to the electric force from the quadrupole trapping field acting on the particle s charge. Since both forces are linear in the axial coordinate the ion motion is still described by a harmonic oscillator (Fig. 15). The axial frequency, however, depends on the direction of the magnetic moment m with respect to the magnetic field: w z = w z dw z = w z0 + m zb 2 Mw z0. (29) The value of B 2 in our set-up was calculated using the known geometry and magnetic susceptibility of the nickel ring electrode. We also determined it

17 III] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 207 Fig. 15. Axial parabola potential for an ion in a quadrupole trap including the magnetic potential for the spin-magnetic moment in a bottle-like magnetic field. The strength of the potential depends on the spin direction. Upper curve: spin down; lower curve: spin up. Fig. 16. Axial frequencies of a single C 5+ ion for different spin directions. The averaging time for each resonance line was 1 min. experimentally by applying a bias voltage between the endcap electrodes of the analysis trap and measuring the cyclotron frequency of the ion at different axial positions. The calculated and experimental values for B 2 in our experiment agree within their uncertainties of 10% and yield B 2 = 1 T/cm 2. For hydrogen-like carbon the frequency difference dw z / 2p between the two spin states amounts to 0.7 Hz at a total frequency of w z0 / 2p = 365 khz. As evident from Fig. 16, the axial frequency can be determined to better than 100 mhz. Fig. 17 demonstrates that after 1 min. averaging the expected frequency difference between the two spin states becomes obvious. Driving the spin-flip transition, we can distinguish the two possible axial frequencies, 0.7 Hz apart as calculated from the trap parameters. Varying the frequency of the microwave field and counting the number of induced spin flips per unit time yields a resonance curve as shown in Fig. 18. The shape of this resonance is asymmetric due to the inhomogeneity of the magnetic field. The general shape of the Larmor resonance in an inhomogeneous magnetic field has been derived

18 208 G. Werth et al. [III Fig. 17. Center of the axial frequency for a single C 5+ ion when irradiated continuously with microwaves at the Larmor precession frequency showing two distinct values which correspond to the two spin directions of the bound electron. Fig. 18. Number of observed spin flips per unit time vs. the frequency of the inducing field. The solid line is a fit according to Eq. (29). by Brown (1985) as a complex function of the trap parameters, the ion s energy and the field inhomogeneity. However, assuming that the ion s amplitude z(t) is

19 IV] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 209 constant during the time 1/ Dw L, the line profile is given by a d-function averaged over the Boltzmann distribution of the energy: c(w L )= 0 de d = Q(w L w L0 ) Dw L (w L w L0 ( 1+ ee z mw 2 z { exp w } L w L0. Dw L )) kte E/kT (30) Here Q(w L w L0 ) is the step function, which is 0 for w L < w L0 and 1 for w L > w L0, and e is a linewidth parameter so that the Larmor frequency depends as w L = w 0 (1 + ez 2 ) on the axial coordinate. A least-squares fit of this function to the data points of Fig. 18 yields the Larmor frequency with a relative uncertainty of 10 6 (Hermanspahn et al., 2000). This is sufficient to measure the binding correction to the g-factor in C 5+. The bound-state QED corrections for C 5+, however, are and were not observed in this measurement. IV. Double-Trap Technique The limitation in accuracy of the experiment described above stems from the inhomogeneous magnetic field as required for the analysis of the spin direction via the continuous Stern Gerlach effect. In fact the inhomogeneity of the field was chosen to be as small as possible, but still large enough to be able to distinguish the two spin directions. We obtained an improvement of three orders of magnitude in the accuracy of the measured magnetic moment by spatially separating the processes of inducing spin flips and analyzing the spin direction (Häffner et al., 2000). This is achieved by transferring the ion after a determination of the spin direction from the analysis trap to the precision trap. The voltages at the trap electrodes are changed in such a way that the potential minimum in which the ion is kept is moved towards the precision trap. The transport takes place in a time of the order of 1 s, which is slow compared to any oscillation period of the ion and is therefore adiabatic. Once in the precision trap, the ion s motional amplitudes are prepared by coupling the ion to the resonant circuits. We then apply the microwave field to induce spin flips. After the interaction time, typically 80 s, and an additional cooling time, the ion is moved back to the analysis trap. Here the spin direction is analyzed again. In principle one measurement of the axial frequency would be sufficient to determine whether it has changed by 0.7 Hz as compared to the value before transport into the precision trap. If, however, the ion is not brought back with the same radial motional amplitudes to the analysis trap, the axial frequency may have changed by as much as 1 Hz. This is because of the magnetic moment connected with the cyclotron and magnetron motion. To circumvent this problem

20 210 G. Werth et al. [IV Fig. 19. Determination of the spin direction in the analysis trap after transport from the precision trap. A change in axial frequency of about 0.7 Hz indicates that the spin was up (left) or down (right) when the ion left the precision trap. we induce an additional spin flip in the analysis trap to determine without doubt the spin direction after return to the analysis trap. Figure 19shows several cycles for a spin analysis. The total time for a complete cycle is about 30 min. While the ion is in the precision trap its cyclotron frequency w c =(q/m) B is measured simultaneously with the interaction with the microwaves. This ensures that the magnetic field is calibrated at the same time as the possible spin flip is induced. The field of a superconducting solenoid fluctuates at the levelof on the time scale of several minutes. Figure 20 shows a measurement of the cyclotron frequency of the ion in the precision trap over a time span of several hours. Every 2 min the center frequency of the cyclotron resonance was determined. The change in cyclotron frequency has approximately a Gaussian distribution with a full-width-at-half-maximum of This may impose a serious limit on the precision of measurements as in the case of high-precision mass spectrometry using Penning traps (Van Dyck et al., 1993; Natarajan et al., 1993). However, the simultaneous measurement of cyclotron and Larmor frequencies eliminates most of this broadening. Using Eqs. (10) and (25) we obtain the g-factor as the ratio of the two measured frequencies g =2 w L m w C M. (31) The mass ratio of the electron to the ion can be taken from the literature. In our case of 12 C 5+, Van Dyck and coworkers (Farnham et al., 1995) measured it with high accuracy using a Penning trap mass spectrometer. We measure the induced spin flip rate for a given frequency ratio of the microwave field and the simultaneously measured cyclotron frequency. When we

21 IV] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 211 Fig. 20. Distribution of magnetic field values measured by the cyclotron frequency of a trapped ion in a period of several hours. Data were taken every 2 min. The distribution is fitted by a Gaussian with a full width of Fig. 21. Measured spin-flip probability vs. ratio of Larmor and cyclotron frequencies. The data are least-squares fitted to a Gaussian. plot the spin flip probability, i.e. the number of successful attempts to change the spin direction divided by the total number of attempts, we obtain a resonance line as shown in Fig. 21. The maximum attainable probability is 50% when the amplitude of the microwave field is high enough. To avoid those saturation effects we take care to keep the amplitude of the microwave field at a level that the maximum probability for a spin flip at resonance frequency is below 30%. In addition we can take saturation into account using a simple rate-equation model.

22 212 G. Werth et al. [V In contrast to the single-trap experiment the lineshape is now much more symmetric. For a constant homogeneous magnetic field in the precision trap the lineshape would be a Lorentzian with a very narrow linewidth determined by the coupling constant g to the cooling circuit. However, the observed lineshape can be well described by a Gaussian. The fractional full width is This reflects the variation of the magnetic field during the time the ion spends in the precision trap which is of the same order of magnitude (see Fig. 20). The line center can be determined from a least squares fit to V. Corrections and Systematic Line Shifts The main systematic shifts of the Larmor and cyclotron resonances arise from the fact that the field in the precision trap is not perfectly homogeneous. As mentioned above, the ferromagnetic nickel ring placed 2.7 cm away in the analysis trap causes a residual inhomogeneity in the precision trap. The expansion coefficient from Eq. (21) gives B 2 = 8mT/ mm 2, three orders of magnitude smaller than in the analysis trap. Therefore we still have to consider an asymmetry in the line profile. Performing such an analysis gives a maximum deviation as compared to the symmetric Gaussian fit of In addition, the inhomogeneity of the magnetic field causes a shift of the line with the ion s energies. In order to obtain a sufficiently strong signal of the induced current from the cyclotron motion in the precision trap, the ion s energy has to be raised to about 1 ev. This finite cyclotron energy has a large magnetic moment and thus shifts the axial frequency as compared to vanishing cyclotron energy even in the precision trap by about 1 Hz. To account for this shift we grouped our data of the spin flip probabilities according to the different axial frequency shifts in the precision trap corresponding to different cyclotron energies, and extrapolated the ratios w L / w C to zero cyclotron energy (Fig. 22). We find a slope of D(w L / w C )/E C = 1.09(5) 10 9 ev 1. Other systematic shifts Fig. 22. Extrapolation of measured frequency ratios to vanishing cyclotron energy.

23 VI] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 213 Table 2 Systematic uncertainties (in relative units) in the g-factor determination of 12 C 5+ Contribution Asymmetry of resonance Electric field imperfections Ground loops in apparatus Interact. with image charges Calibration of cyclotron energy Sum Relative size are less important: From the residual imperfection of the electric trapping field (C 4 = 10 5 ) we calculate a shift of the cyclotron frequency of Of the same order of magnitude are frequency shifts caused by changes of the trapping potential due to ground loops when the computer controls are activated. The interaction of the ion with its image charges changes the frequencies by , but can be calculated with an accuracy of 10%. Relativistic shifts are of the order of at typical ion energies, but do not contribute to the uncertainty at the extrapolation to zero energy. A list of uncertainties of these corrections is given in Table 2. The quadrature sum of all systematic uncertainties amounts to The final experimental value for the frequency ratio w L / w C in 12 C 5+ is w L w C = (19)(13). (32) The first number in parentheses is the statistical uncertainty from the extrapolation to vanishing cyclotron energy, the second is the quadrature sum of the systematical uncertainties. Taking the value for the electron mass in atomic units (M( 12 C) = 12) from the most recent CODATA compilation (Mohr and Taylor, 1999)wearriveatag-factor for the bound electron in 12 C 5+ of g exp ( 12 C 5+ )= (10)(44). (33) Here the first number in parentheses is the total uncertainty of our experiment, and the second reflects the uncertainty in the electron mass. VI. Conclusions A comparison of the experimentally obtained result of Eq. (33) to the theoretical calculations presented in Table 1 shows that the bound-state QED effects of

24 214 G. Werth et al. [VII order a/ p in hydrogen-like carbon are verified at the level of Bound QED contributions of order (a/ p) 2 are too small to be observed. The nuclear recoil part has been verified to about 5%. It is believed that uncalculated terms of higher-order QED contributions do not change the theoretical value beyond the presently quoted uncertainties. Taking this for granted we can use experimental and theoretical numbers to determine a more accurate value for the atomic mass of the electron, since this represents by far the largest part in the total error budget (Beier et al., 2001). Using Eqs. (6) and (33) we obtain from Eq. (31) the electron s atomic mass as m = (4). (34) This is in agreement with the CODATA electron mass (Mohr and Taylor, 1999) based on a direct determination by the comparison of its cyclotron frequency to that of a carbon ion in a Penning trap (Farnham et al., 1995): m = (12). (35) VII. Outlook The continuous Stern Gerlach effect, using the frequency dependence of the axial oscillation on the spin direction of an ion confined in a Penning trap when an inhomogeneous field is superimposed, is a powerful tool to measure magnetic moments of charged particles with great precision. This accurate knowledge of magnetic moments is very important for tests of QED calculations. The g 2 experiment on free electrons by Dehmelt and coworkers (Van Dyck et al., 1987) was a first example, followed now by the first application to an atomic ion. The method described above is applicable to any ion having a magnetic moment on the order of a Bohr magneton, provided it can be loaded into the trap. For a given axial frequency and magnetic inhomogeneity B 2, the frequency splitting depends as 1/ qm on the mass M of the ion and its charge state q (Fig. 23). This will impose technical limitations when working with heavier hydrogen-like ions. Currently the stability of the electric trapping field limits the maximal resolution of the axial frequency measurements: a jitter of the trapping voltage by 1 mv, typical for state-of-the-art high-precision voltage sources, induces frequency changes of 100 mhz for trap parameters as in our case. However, materials with higher magnetic susceptibilities than nickel, such as Co Sm alloys, produce a larger magnetic inhomogeneity and therefore a larger frequency splitting, allowing to proceed to heavier ions. In addition, the induced magnetic inhomogeneity scales with the cube of the inverse radius of the ring electrode. Thus a reduction in size of the analysis trap increases the

25 VII] CONTINUOUS STERN GERLACH EFFECT ON ATOMIC IONS 215 Fig. 23. Difference in axial frequency for two spin directions in a bottle-like magnetic field for various hydrogen-like ions. The parameters B 0 = 3.8T, B 2 = 1 T/cm 2, and w z / 2p = 365 khz are those of our experiment. The frequency difference scales linear with the magnetic field inhomogeneity B 2. frequency difference for the two spin states significantly. This would also have the advantage that it reduces the amount of ferromagnetic material placed in the analysis trap, and so helps to improve the homogeneity in the precision trap. To further improve the homogeneity in the precision trap the distance between the two traps can be increased. Finally, shim coils may be used to make the field in the precision trap more homogeneous. We believe that we can maintain the presently achieved precision with other ions as well, and hope to even increase it when we apply some of the measures for improvement. This would result in a more significant test of higher-order bound-state QED contributions since they increase quadratically with the nuclear charge (see Fig. 2). The method can also be applied to more complicated systems: a measurement of the electronic g-factor in lithium-like ions would test not only the QED corrections in these systems but also correlation effects with the remaining electrons which change the g-factor significantly. When applied to hydrogen-like ions with non-zero nuclear spin the transition frequencies between spin states depend on the nuclear magnetic moment. Measuring the different transition frequencies yields the magnetic moment of the nucleus. This would be of special interest, because all nuclear magnetic moments so far have been determined using neutral atoms or singly ionized ions. The effective magnetic field seen by the nucleus in these systems differs from the applied magnetic field by shielding effects of the electron cloud. In a measurement on hydrogen-like ions this shielding is strongly

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