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5.1 Penning traps

A Penning trap is a combination of static magnetic and electric fields that can keep a charged particle localized within the trap for extremely long periods of time (for a review of Penning traps see [68]). A trapped particle moves in a number of different ways. The two motions relevant for Lorentz violation tests are the cyclotron motion in the magnetic field and Larmor precession due to the spin. The ratio of the precession frequency ws to the cyclotron frequency wc is given by
ws/wc = g/2, (49)
where g is the g-factor of the charged particle. The energy levels for a spin 1/2 particle are given by Es = nw + sw n c s where n is an integer and s = ± 1/2. For electrons and positrons, where g ~~ 2, the state n,s = - 1/2 is almost degenerate with the state n - 1, s = +1/2. The degeneracy breaking is solely due to the anomalous magnetic moment of the electron and is usually denoted by wa = ws- wc. By introducing a small oscillating magnetic field into the trap one can induce transitions between these almost degenerate energy states and very sensitively determine the value of wa.

The primary use of measurements of wa is that they directly give a very accurate value of g - 2. However, due to their precision, these measurements also provide good tests of CPT and Lorentz invariance. In the mSME, the only framework that has been applied to Penning trap experiments, the g factor for electrons and positrons receive no corrections at lowest order. However, the frequencies wa and wc both receive corrections [60]. At lowest order in the Lorentz violating coefficients these corrections are (with the trap’s magnetic field in the z-direction)

wec-= (1- ce00 - ceXX - ceYY)we,c0, ± (50) wea = wea,0± 2beZ + 2deZ0me + 2HeXY ,
expressed in a non-rotating frame. The unmodified frequencies are denoted by e,0 wc,a and the Lorentz violating parameters are various components of the general set given in Equations (32View Equation) and (33View Equation).

The functional form of Equation (50View Equation) immediately makes clear that there are two ways to test for Lorentz violation. The first is to look for instantaneous CPT violation between electrons and positrons which occurs if the bZ parameter is non-zero. The observational bound on the difference between wa for electrons and positrons is |w+a - w-a |< 2.4 × 10- 21me [97]. This leads to a bound on bZ of order bZ < 10 -21me. The second approach is to track wa,c over time, looking for sidereal variations as the orientation of the experimental apparatus changes with respect to the background Lorentz violating tensors. This approach has been used in [217] to place a bound on the diurnal variation of the anomaly frequency of e- -21 Dw a < 1.6× 10 me, which limits a particular combination of components of bm, cmn, and dmn Hmn at this level. Finally, we note that similar techniques have been used to measure CPT violations for proton/anti-proton and hydrogen ion systems [118]. By measuring the cyclotron frequency over time, bounds on the cyclotron frequency variation (50View Equation) for the anti-proton have established a limit at the level of -26 10 on components of cp- mn.

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