Continuous Stern-Gerlach effect and the Magnetic Moment of the Antiproton

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1 Continuous Stern-Gerlach effect and the Magnetic Moment of the Antiproton W. Quint a, J. Alonso b, S. Djekić b, H.-J. Kluge a, S. Stahl b, T. Valenzuela b, J. Verdú b, M. Vogel b, and G. Werth b a Gesellschaft für Schwerionenforschung, D Darmstadt, Germany b Institut für Physik, Johannes-Gutenberg-Universität Mainz, D Mainz Abstract The measurement of the magnetic moment (or g-factor) of the antiproton and of the proton is a sensitive test of CPT invariance. We discuss the possibility of applying the continuous Stern-Gerlach effect to detect quantum jumps between the two spin states (spin up and spin down) of the antiproton. The measurement will be performed on a single antiproton stored in a Penning trap. The g-factor of the antiproton is determined by measuring its cyclotron frequency and its spin precession frequency in the magnetic field of the trap. With the double Penning trap method the g-factor of the antiproton can be determined with an accuracy of 1 ppb. Key words: magnetic moment, antiproton, CPT PACS: 1 Introduction The measurement of the magnetic moment of the antiproton and of the proton is a sensitive test of CPT invariance [1]. Only a few ideas for a measurement of the antiproton magnetic moment exist. Heinzen and Wineland presented a scheme in which an antiproton in a Penning trap is cooled and investigated through image-charge coupling to a laser-cooled ion [2]. A procedure of exciting the cyclotron motion of an antiproton in a Penning trap by driving the spinflip transition as well as the anomaly transition was described in [3]. Another approach is the hyperfine measurement on antihydrogen atoms which has recently been proposed [4]. In this contribution we discuss the possibility of applying the continuous Stern-Gerlach effect to measure the magnetic moment of the antiproton with an accuracy of 1 ppb. Such a measurement would represent an improvement in accuracy by more than six orders of magnitude [5]. Preprint submitted to Elsevier Science 7 May 2003

2 2 Single Antiproton in a Penning Trap We envisage the following experimental scenario. A single antiproton [6] (or proton) is confined in a cylindrical Penning trap [7] at a magnetic field strength of 6 T. An antiproton confined in the trapping potential oscillates harmonically in the axial direction at a frequency ω z depending on the trap voltage. In the radial plane the particle motion is a superposition of the modified cyclotron motion at frequency ω + (2π 90 MHz) and a slow drift around the trap center (magnetron motion, ω ). If the trap and the vacuum enclosure of the apparatus are kept at liquid-helium temperature, the background pressure is below mbar and the storage time before antiproton annihilation is longer than several months [8]. The antiproton is resistively cooled close to the ambient temperature of 4 K by keeping its oscillation frequencies ω + and ω z at the resonance frequencies of high-q resonance circuits attached to the trap electrodes. The trapped antiproton is monitored via the currents which are induced in the trap electrodes by its oscillations. The magnetic field at the antiproton s position is determined through the cyclotron frequency ω c = (q/m)b, which can be calculated from the three measured oscillation frequencies using the relation [9] ω 2 c = ω ω 2 z + ω 2. (1) 3 Continuous Stern-Gerlach effect The determination of the g-factor of the antiproton results from a measurement of the spin precession frequency ω L (Larmor frequency) which is related to the g-factor by ω L = gµ N B, where µ N is the nuclear magneton. With the relation ω c = (q/m)b, the g-factor can be calculated from the two experimentally determined frequencies g = 2 ω L ω c. (2) For the measurement of the Larmor precession frequency ω L of the antiproton, we propose to use the continuous Stern-Gerlach effect which was applied for the first time by Dehmelt in the g-factor measurement of the free electron in a Penning trap [10]. Later on, this effect has also been utilized for electronic g-factor measurements on hydrogen-like ions [11 13]. The principle of the continuous Stern-Gerlach effect is based on a coupling of the magnetic moment µ of a particle to its axial oscillation frequency ω z in a Penning trap. This coupling is achieved by a quadratic magnetic field component ( magnetic bot- 2

3 tle ) superimposed on the homogeneous magnetic field B 0 of the Penning trap B(z) = B 0 + β 2 z 2. Due to the interaction of the z-component µ z of the magnetic moment with the magnetic bottle term the trapped particle possesses a position-dependent potential energy V m = µ z (B 0 + β 2 z 2 ), which adds to the potential energy V el of the particle in the electrostatic well. Therefore, the effective trapping force is modified by the magnetic interaction, and the axial frequency of the trapped particle is shifted upwards or downwards, depending on the sign of the z-component µ z of the magnetic moment. This axial frequency shift is given by δω z = β 2 µ z /mω z, (3) where ω z is the unshifted axial frequency. Spinflip transitions are induced by applying a radiofrequency drive at the Larmor frequency ω L (2π 250 MHz) to a trap electrode. The axial frequency jump for a transition between the two spin directions is ω z ( ) ω z ( )= 2δω z. The inhomogeneous magnetic field component is produced by a ferromagnetic ring electrode of the Penning trap. It is important to note that the cyclotron quantum number of the antiproton must not change during the detection of the spin direction. Otherwise, the changing orbital magnetic moment of the cyclotron motion would lead to an additional axial frequency shift and would obscure a spin transition. Therefore, at the antiproton s cyclotron frequency the trap electrodes must be electronically decoupled from the environment. In particular, no electronic resonance circuit must be connected to the trap in which the spin direction is determined ( analysis trap ). Alternatively, the antiproton can be cooled to mk temperatures by image-charge coupling to a laser-cooled ion in an adjacent trap minimum [2]. The experimental detection of a spinflip of an antiproton via the continuous Stern-Gerlach effect is more difficult compared to spinflip detection of an electron, because the magnetic moment of the antiproton is smaller by a factor of 658 than the electron magnetic moment. However, the axial frequency shift of an antiproton upon a change in the spin direction can be made visible by a proper choice of the strength of the magnetic bottle term β 2 and of the axial frequency ω z of the stored antiproton (see Equ. 3). Table 1 shows some experimental parameters of the measurements of the magnetic moment of the free electron [10] (first column), of the magnetic moment of the bound electron in hydrogen-like carbon 12 C 5+ [12] (second column), and of the magnetic moment of the bound electron in oxygen 16 O 7+ [13] (third column). The fourth column in table 1 lists a set of parameters which are suited for a measurement of the g-factor of the antiproton. 3

4 4 Double Penning Trap Method The strong magnetic inhomogeneity ( magnetic bottle ), which is required for spinflip detection of an antiproton via the continuous Stern-Gerlach effect, would severely limit the measurement accuracy to a level of about However, the application of the double Penning trap method will make it possible to reach a much higher accuracy in the determination of the g-factor of the antiproton despite the presence of the magnetic inhomogeneity. We have successfully applied this method for the measurement of the magnetic moment of the bound electron in hydrogen-like carbon 12 C 5+ [12] and oxygen 16 O 7+ [13] with an accuracy of about 1 ppb. The double Penning trap method is based on the following measurement procedure. The determination of the spin direction is performed in a potential minimum of the arrangement of cylindrical electrodes in which the magnetic field is made inhomogeneous as mentioned above ( analysis trap ). Spin transitions are induced at a different position a few cm apart in which the magnetic field is spatially homogeneous ( precision trap ). The antiproton is transported between the two potential minima by variation of the voltages applied to the different electrodes. While the antiproton is in the precision trap, its cyclotron frequency is non-destructively measured by the image-current technique in order to determine the magnetic field strength at its position. The excitation of spin-flip transitions and the measurement of the magnetic field in the precision trap take place simultaneously. Therefore, fluctuations in the magnetic field strength are cancelled to a high degree in the ratio of the two frequencies ω L /ω c. Finally, the antiproton is moved back to the analysis trap for detection of the spin direction. We expect that an accuracy of 1 ppb (or better) in the determination of the g-factor of the antiproton can be reached with the double Penning trap method. The corresponding measurement on the proton in the same apparatus will yield an important test of CPT invariance. 5 Conclusions and Outlook The application of the continuous Stern-Gerlach effect together with the double Penning trap method offers exciting possibilities for a high-precision measurement of the magnetic moment of the antiproton on the ppb-level or better. Such a measurement can be prepared off-line with a single trapped proton, which in itself is a measurement of fundamental interest. The measurement on the antiproton can performed at the present Antiproton Decelerator (AD) ring at CERN or at the low-energy antiproton area at the future GSI-accelerator complex which is presently being discussed and planned. 4

5 6 Acknowledgements We thank H. Häffner and N. Hermanspahn for their cooperation during many years. W.Q. thanks G. Gabrielse for stimulating discussions in the summer References [1] R. Bluhm, V. A. Kostelecky and N. Russell, Phys. Rev. D 57 (1998) [2] D. J. Heinzen and D. J. Wineland, Phys. Rev. A 42 (1990) [3] W. Quint, G. Gabrielse, Hyp. Int. 76 (1993) 379. [4] E. Widmann et al., Measurement of the antihydrogen hyperfine structure, Letter of Intent for AD, CERN/SPSC [5] K. Hagiwara et al., Phys. Rev. D 66 (2002) [6] G. Gabrielse, D. Phillips, W. Quint, H. Kalinowsky, G. Rouleau, W. Jhe, Phys. Rev. Lett. 74 (1995) [7] G. Gabrielse, L. Haarsma and S.L. Rolston, Intl. J. of Mass Spec. and Ion Proc. 88, 319 (1989); ibid. 93, 121 (1989). [8] G. Gabrielse et al., Phys. Rev. Lett. 65 (1990) [9] L. S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986). [10] R. S. Van Dyck, Jr., P. B. Schwinberg, and H. G. Dehmelt, Phys. Rev. Lett. 59, 26 (1987). [11] N. Hermanspahn, H. Häffner, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, G. Werth, Phys. Rev. Lett. 84 (2000) 427. [12] H. Häffner, N. Hermanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verdú, G. Werth, Phys. Rev. Lett. 85 (2000) [13] J. Verdú, T. Beier, S. Djekic, H. Häffner, H.-J. Kluge, W. Quint, T. Valenzuela, M. Vogel and G. Werth, J. Phys. B 36, 655 (2003). 5