Precise Measurements of the Masses of Cs, Rb and Na A New Route to the Fine Structure Constant

Transcription

1 Hyperfine Interactions 132: , Kluwer Academic Publishers. Printed in the Netherlands. 177 Precise Measurements of the Masses of Cs, Rb and Na A New Route to the Fine Structure Constant SIMON RAINVILLE, MICHAEL P. BRADLEY, JAMES V. PORTO, JAMES K. THOMPSON and DAVID E. PRITCHARD Research Laboratory of Electronics, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA; simonr@mit.edu Abstract. We report new values for the atomic masses of the alkali 133 Cs, 87 Rb, 85 Rb, and 23 Na with uncertainties 0.2 ppb. These results, obtained using Penning trap single ion mass spectrometry, are typically two orders of magnitude more accurate than previously measured values. Combined with values of h/m atom from atom interferometry measurements and accurate wavelength measurements for different atoms, these values will lead to new ppb-level determinations of the molar Planck constant N A h and the fine structure constant α. This route to α is based on simple physics. It can potentially achieve the several ppb level of accuracy needed to test the QED determination of α extracted from measurements of the electron g factor. We also demonstrate an electronic cooling technique that cools our detector and ion below the 4 K ambient temperature. This technique improves by about a factor of three our ability to measure the ion s axial motion. Key words: single ion mass spectrometry, ppb, cesium, rubidium, sodium, electronic cooling, feedback. 1. The fine structure constant If QED correctly predicts the g-2 anomaly of the electron, the fine structure constant, α, is known to 3.8 ppb [1, 2]. In order to test QED at this accuracy, and to remove theoretical uncertainties highlighted by the recent readjustment of the theoretical value of g-2 [3], other measurements of α at this level of accuracy are needed. Unfortunately, the next most precise measurements are far from this accuracy: α has been measured to 56 ppb using the AC Josephson effect [4, 5], to 37 ppb using a beam of neutrons [6], and to 24 ppb from the Quantum Hall effect [7] (see Figure 1). There is obviously a great need for a QED-independent determination of α at the ppb level: not only could it probe QED an order of magnitude more accurately, but it could possibly suggest the source of the discrepancies in these other measurements. Corresponding author. Currently at NIST, Gaithersburg, MD , USA. Groups at the Universities of Washington and Harvard plan to reduce this error to about 1 ppb.

2 178 S. RAINVILLE ET AL. Figure 1. Current measurements of the fine structure constant α with uncertainty below 100 ppb are plotted with respect to the 1998 CODATA recommended value [12]. The measurement of the g-2 factor of the electron combined with QED calculations yields a value of α with an uncertainty of (3.8 ppb) [1, 2]. The relative uncertainties of the other measurements are: 24 ppb from the Quantum Hall effect [7], 37 ppb from neutron beam interferometry [6], and 56 ppb from the AC Josephson effect and the magnetic moment of the proton [4, 5]. The method discussed here (this work) currently yields a preliminary value at 29 ppb. A route to α that appears likely to yield a value at the ppb level is opened by the relationship (in SI units) α 2 = 2R c 10 3 M p m p m e (N A h), (1) where m x and M x represent the mass of particle x in SI and atomic units respectively, and N A h is known as the molar Planck constant. The Rydberg constant, R has been measured to an accuracy of about than ppb [8], the ratio of the mass of the proton to the electron m p /m e is known to 2 ppb [9], and we have determined the mass of the proton M p (in atomic units) to 0.5 ppb [10] (Van Dyck et al. have recently reported a value of M p accurate to 0.14 ppb which agrees with ours [11]). The speed of light c is a defined constant. Thus a measurement of N A h, the molar Planck constant, at the ppb level can determine α to about 1 ppb. The molar Planck constant can be precisely determined from the quantum relationship between de Broglie wavelength and momentum: λ = h λv = h mv m = 103 N A h M. (2) Note that to determine N A h we need to measure the particle s atomic mass M as well as the wavelength of the light λ and the particle s velocity v. This simple

3 PRECISE MEASUREMENTS OF CS, RB AND NA 179 physics underlies the neutron-based measurement of α [13], where our value of the neutron s atomic mass [10] leads to the value for α mentioned in Figure 1. Chu s group at Stanford University has measured the recoil velocity v for Cs atoms absorbing photons of laser light at the D 1 line to 10 6 using an atom interferometer [14]. They now project an accuracy of 10 ppb using a vertically configured apparatus like the one used to reach ppb measurements of the local gravitational acceleration constant g. Hänsch s group at the Max-Planck-Institut in Garching has measured the wavelength of the cesium D 1 line (λ) with precision of [15]. Combining these results with our sub-ppb measurement of the atomic mass of Cs should give N A h to about 10 ppb. From Equation (1), this will lead to a new determination of α to about 5 ppb. The simplicity of the physics involved in this route to α recommends it as the preferred check of QED and the other α measurements. The most complicated physics is in the Rydberg constant, where the theory is more than adequate at the ppb level. By comparison, the quantized Hall effect may involve unknown solid state or sample geometry corrections, and ongoing programs to determine α from the fine structure of an atom involve new atomic energy level calculations with required ppt certainty, a much more uncertain proposition. The possibility of redundancy in the experimental determination of N A h would greatly enhance the confidence in determinations of α from Equation (1). The mass ratio m p /m e would be the only quantity without more than a single direct measurement at the ppb level (a recent value of m e /m12 C extracted from theory and boundstate electron g factor measurements in hydrogenic 12 C has confirmed the value to about 2 ppb [16]). This is not a trivial point since it would take a considerable weight of evidence to believe that disagreement between the QED and N A h determinations of α signifies some error in QED. 2. Experimental setup Our experimental apparatus and procedure for measuring ion mass ratios have been described earlier in the literature [17]. We will briefly outline here the general features of this experimental setup. We trap a single ion in an orthogonally compensated Penning trap (of characteristic size d = cm) [18, 19] placed in a highly uniform magnetic field of 8.5 T. The harmonic axial oscillation of the ion in the trap is detected with a superconducting resonant circuit (niobium coil) having a stable Q of about The heart of our recently improved detector is a DC SQUID which has a technical noise floor 10 times smaller than the RF SQUID it replaced. The trapping voltage is adjusted to match the ion s axial frequency ω z, with the resonance frequency of the detection circuit ( 160 khz). Typical frequencies for the other two normal modes of motion of an Ar + ion in our trap are 3 MHz for the trapped cyclotron frequency ω c, and 4 khz for the magnetron motion ω m. The cyclotron and magnetron modes are observed and cooled indirectly by coupling them to the axial mode with a tilted

4 180 S. RAINVILLE ET AL. quadrupole RF field applied on the split guard rings of our trap. With the proper amplitude time product, such a coupling pulse ( π-pulse ) can phase-coherently transfer all the energy of the cyclotron motion into the axial mode, in a manner qualitatively and formally equivalent to the Rabi two-state problem [20]. The absence of direct damping of the radial modes avoids the possibility of frequency shifts due to such coupling and means that the cyclotron resonance has practically zero linewidth. In addition, this procedure allows us to work with nearly ideal fields (pure quadrupole electric field and uniform magnetic field) which helps to keep the calculated systematic errors below a few parts in The trapped cyclotron frequency ω c is obtained by directly exciting the ion s cyclotron mode, allowing it to evolve in the dark for a delay time T, and applying a π-pulse. The axial ring down is then observed to extract the accumulated phase. Since phases are only defined between 0 and 2π, a series of measurements is made with different delay times T to determine the proper phase unwrapping. With a phase error of about ±12 degrees, a measurement of one minute gives a precision of From this, we can extract the free space cyclotron frequency ω c = qb/m (where q is the ion charge, B is the magnetic field strength, and m is the mass) using the invariance relationship (valid even if the trap is tilted with respect to the B field or not perfectly cylindrical): ω c = qb/mc = (ω c )2 + (ω z ) 2 + (ω m ) 2. (3) We don t directly measure the magnetron frequency ω m when taking data, but we use the relation ( ω m ω2 z ) 2ω c 4 sin(θ m), (4) where θ m (typically, < rad) is the measured angle between the magnetic field and the trap axis obtained by actually measuring the magnetron frequency once. The precision of a single ion cyclotron frequency measurement is limited by the short term fluctuations of the magnetic field. Finally, a mass ratio is determined by alternately measuring the cyclotron frequencies of two ions during one night (when the magnetic field fluctuations are reduced). A typical set of data is shown in Figure 2. The final precision on the mass ratio (typically, ) is currently limited mainly by the drift (few ppb per hour) of our magnetic field during the time it takes to make and isolate a new single ion. In this setup, the ions are produced in the trap by ionizing neutral gas with an electron beam. Obviously, this method is not selective and unwanted ions are also produced and trapped in the process. A crucial point for us is to be able to isolate the ion of interest as quickly as possible. The case of Cs 3+ was particularly difficult since the making of a single Cs 3+ would produce about 100 and 10 of the singly and doubly charged states respectively. Furthermore, the hydrocarbons C 5 H + 6 and C 3 H + x used as comparison species break apart into many unwanted fragments under the electron-beam bombardment. The overall effect of these factors was to increase

5 PRECISE MEASUREMENTS OF CS, RB AND NA 181 Figure 2. Typical night of data. The solid line is a second order polynomial fit to the field drift. The 360 bar shows the magnitude in Hz of a 360 error in phase unwrapping. Also shown is a bar representing the magnitude of a ppb change in frequency. the time to make and isolate a single ion to minutes from the 5 10 minutes required for previous measurements. 3. Results of the alkali measurements To determine the masses of the alkali atoms 133 Cs, 87,85 Rb, and 23 Na, we measured the free-space cyclotron frequency ratios r ω c2 /ω c1 listed in Table I. The reference ions were selected because of the similar mass to charge ratios (aiding in the reduction of systematic errors) and because we have previously measured the atomic masses of each of the consituent atoms. A cyclotron frequency ratio r of two different ions was determined by a run measuring a cluster of ω c values for an ion of type A, then for type B, etc. In a typical 4-hour run period (1:30 5:30 am when the nearby electrically-powered subway was not running), we recorded about 5 alternations of ion type (Figure 2). The measured free-space cyclotron frequencies exhibited a common slow drift. We fit a common polynomial (t) plus a frequency difference to the data. From this we obtained the frequency ratio r n and the uncertainty σ n for a single night. The average order of (t) was 3 and was chosen using the F-test criterion [21] as a guide. The distribution of residuals from the polynomial fits had a Gaussian center with a standard deviation σ resid = 0.28 ppb and a background ( 2% of the points) of non-gaussian outliers, as in our earlier measurements [10]. As before we chose to handle the non-gaussian outliers using a robust statistical method to smoothly deweight them [23].

6 182 S. RAINVILLE ET AL. Table I. Measured ion cyclotron frequency ratios, corrected for systematics A/B ω c /(2π) (MHz) Nights ω c [A]/ω c [B] 133 Cs +++ /CO (135) 133 Cs ++ /C 5 H (427) 87 Rb ++ /C 3 H (266) 87 Rb ++ /C 3 H (174) 85 Rb ++ /C 3 H (164) 85 Rb ++ /C 3 H (285) 23 Na + /C (076) 23 Na ++ /C (098) Figure 3. Example of Variation of Mass Ratio from Night to Night. A measurement of the neutral mass of 133 Cs is extracted from each night s run of cyclotron frequency ratio measurements and plotted in ppb relative to our final published value of the neutral mass of 133 Cs. The open and closed circles are from frequency ratio measurements of Cs +++ /CO + 2 and Cs ++ /C 5 H + 6, respectively. The error bars on each night s measurement are extracted from the low order polynomial fit to both ion s cyclotron frequencies and reflects the distribution of the cyclotron frequency measurements during that night. The shaded region represents the one sigma confidence interval arrived at in the final analysis. As shown in Table I and Figure 3, we measured each frequency ratio on more than a single night. For ratios involving Cs and Rb the measured ion mass ratios were distributed from night to night with a scatter larger than the uncertainty predicted from the statistical scatter within a single night (χν 2 5). By contrast χν for ratios involving Na. None of the earlier data taken using this apparatus [10] exhibited these excess night-to-night variations. A search for the source of these fluctations is discussed elsewhere [24] and was unsuccesful. To account for

7 PRECISE MEASUREMENTS OF CS, RB AND NA 183 Table II. Measured neutral alkali masses Species MIT mass (u) ppb 1995 mass (u) [22] ppb delta σ Cs (27) (3200) Rb (15) (2700) Rb (14) (2500) Na (28) (2300) this excess scatter, the uncertainties in the weighted average of the ion mass ratios involving Cs ++ /C 5 H + 6 and Cs+++ /CO + 2 were increased by factors of 2.6 and 2.2, respectively, so that χν 2 = 1. Since the Rb measurements all had similar m/q, we assumed that the night-to-night fluctuations involving the Rb ratios were drawn from a common statistical distribution. Therefore, we increased the uncertainties for the Rb ion ratios by a factor of 2.2 so that the overall Rb χν 2 was reduced to 1. For Na, χν so the uncertainties were not adjusted. After correcting for molecular binding and electron ionization energies [25], we obtained a set of neutral mass difference equations. We added to this the set of mass difference equations used to determine the atomic masses in [10]. A global least square fit to this overdetermined set of linear equations gave the neutral masses of the alkali metals (see Table II) with uncertainties σ od as well as the previously published neutral masses with χν 2 = The previously published masses were essentially unchanged [10]. Uncertainties in M[ 16 O] and M[H] (the only atoms other than 12 C in the ratios of Table I) contributed less than 0.1 ppb uncertainty to the alkali masses. The use of two distinct reference ions gave a check on systematics by providing two independent values for each neutral mass. For Rb and Cs χν 2 is less than 1. However, because of the larger uncertainty on M[Cs] from Cs ++ /C 5 H + 6 we quote a final uncertainty of 0.20 ppb (cf. σ od (Cs) = 0.16 ppb). For 87,85 Rb we quote σ od ( 87,85 Rb) as the final uncertainties. For the neutral masses from Na ++ /C + and Na + /C + 2, the statistical uncertainties are 0.09 and 0.07 ppb, respectively. The 0.2 ppb disagreement of the two values may be evidence for a systematic at the 0.1 ppb level. To reflect this we assigned M[ 23 Na] a 0.12 ppb uncertainty (cf. σ od (Na) = 0.06 ppb) which spans both independent measurements. Table II quotes the final values for M[ 133 Cs], M[ 87 Rb], M[ 85 Rb] and M[ 23 Na] obtained from the global least square fit to the overdetermined set of mass difference equations with uncertainties from the above discussion. Also included in Table II are the alkali masses from the 1995 mass evaluation [22]. Our values differ from the 1995 values by typically 1.5σ 1995, which suggests that the uncertainties on the masses from the 1995 evaluation were slightly underestimated. Our value for M[ 133 Cs] lies within the uncertainty of the recent measurement

8 184 S. RAINVILLE ET AL. of M[ 133 Cs] reported by the SMILETRAP collaboration [26] to 0.6 ppb. Using R [8], m p /m e [9], the preliminary value of the photon recoil shift [27], f D1 for the photon recoil transition [28], and our values for M[ 133 Cs] and M p we obtain α 1 = (40). This preliminary value is shown in Figure 1 with an uncertainty of 29 ppb. The method discussed here holds promise for many independent values of α at a precision of few ppb. 4. Electronic cooling We have recently developed techniques to cool the detector and ion below the 4 K ambient temperature of the coupling coil and trap environment. This is done with electronic cooling [29]. Not only does this technique greatly improve our signalto-noise, but it also reduces the amplitude of the thermal motion of the ion, an important source of error in our measurements. These significant improvements will be crucial in our efforts to improve our accuracy beyond In particular, we plan to make simultaneous measurements of the cyclotron frequencies of two different ions to alleviate the scatter due to magnetic field fluctuations between the measurements which are now the limit on our precision. Our first approach to do this will be to put two ions in the same trap at the same time. The essence of electronic cooling is to measure the thermal noise, phase shift the signal and then apply it to the coupling coil in such a way that it cancels the thermal excitation in the coil. The key is that our DC SQUID has technical noise much lower than 4 K and can measure the current in the coil well in a time shorter than the thermalization time of the coil (Q 0 /ω 30 ms). This feedback also decreases the apparent quality factor Q of the coil. Figure 4 shows the thermal noise of the coil at different gain settings. Analyzing these data, we find that the thermal energy in the coil, corresponding to the area under the peak, is reduced below 4 K by the factor Q/Q 0, as expected from the detailed solution of the circuit (assuming a parallel LRC coupling coil where the resistor R = Q 0 ω 0 L has the usual Johnson noise current). Another effect of this feedback is to effectively reduce the impedance presented to the ion by the detector, thereby increasing the damping time of the ion. This reduces the bandwidth of our signal, increasing our signal-to-noise ratio (the Johnson noise is a constant current/ Hz). This translates directly into a better ability to estimate the parameters of the axial oscillation of the ion. With this technique, we can now measure the phase of the cyclotron motion with an uncertainty as low as 5 degrees, which is more than a factor of 2 improvement. Our ability to determine the amplitude of the ion signal has also improved, again by more than a factor of 2, and we can measure the frequency of the axial motion with 4 times better precision. The better phase noise allows us to obtain the same precision on a cyclotron measurement in a shorter time. This will be very important in the future since we would have to acquire data for 10 minutes to reach a precision of with the previous phase noise ( 12 degrees), or 100 minutes for 10 12! We can

9 PRECISE MEASUREMENTS OF CS, RB AND NA 185 Figure 4. Thermal profile of the detector coil as a function of the quality factor Q adjusted with the gain of the feedback. The thermal energy in the coil (area under the peak) is proportional to Q/Q 0,whereQ 0 is the Q of the detector coil without feedback. This shows that the feedback does indeed reduce the thermal fluctuations in the coil. also use the improved signal-to-noise to reduce the cyclotron amplitude we use, which in turn reduces the frequency shifts due to relativity and field imperfections. Finally, this technique gives us the ability to arbitrarily select the damping time of the ion by changing the gain of the feedback. The improved detector opens some exciting possibilities for our experiment. For example, we can now probe and characterize the electrostatic anharmonicities of our trapping potential much more effectively. It also opens the door for us to very high precision at small mass-to-charge ratio, (e.g., 6,7 Li, 3 He, 3 H) where we used to suffer from excessively short ion damping times. Before every measurement, we cool the ion s motion by coupling it to our detection circuit until it comes into equilibrium with the detector. (Only the axial motion is coupled to the detector, but we cool the two radial modes using the mode coupling field mentionned in Section 2.) This remaining 4K motion of the ion adds vectorially to the displacement from our cyclotron drive pulses and hence prevents us from establishing an exactly reproducible amplitude and phase of motion with each excitation pulse. This effectively adds random noise to the phase we measure. Since it is the same Johnson noise that drives the ion s thermal motion and is added to the ion image current to form our detected signal, these two sources of noise both contribute to our measurement error (phase noise). Moreover, the thermal cyclotron amplitude fluctuations cause relativistic mass variations and also combine with field imperfections to introduce random fluctuations of the cyclotron frequency which can be of the order of few parts in After magnetic field fluctuations, this is the dominant source of noise in our measurements, and will be the main obstacle to higher precision in our simultaneous measurements of two

10 186 S. RAINVILLE ET AL. different ions. With the electronic cooling technique described above, the ion s motion should now come into equilibrium with the colder detector thereby greatly reducing these problems, and allowing a precision better that in a reasonable amount of time. The first practical result of this improved detector was a new calibration of our cyclotron amplitude. Knowing the absolute ion cyclotron radius is important to be able to calculate and correct (if necessary) the cyclotron frequency we measure for frequency shifts due to relativity and trap imperfections. Also knowing the ionion separation will be crucial when we make measurements on two ions in the same trap. Before this work, there was a factor of two uncertainty in our amplitude calibration, despite our having worked hard on three different methods to determine it. During the month of July 2000, we accurately measured the relativistic cyclotron frequency shift (a natural law) versus cyclotron radius for Ne ++ and Ne +++ ions using the feedback technique described above to maintain adequate S/N over a broad range of cyclotron radii. The measurements on each ion separately led to two independent amplitude calibrations with an uncertainty of 1.6% and 1.8%, respectively. The agreement between the two calibrations was about 3%. These two measurements were done at very different cyclotron frequencies so that any systematic error would affect them differently. Thus we feel confident that we now know the absolute amplitude of motion of our ions to about 3%, a tremendous improvement. Acknowledgements This work is supported by the National Science Foundation and a NIST Precision Measurements Grant. References 1. Kinoshita, T., IEEE Trans. Instrum. Meas. 46 (1997), Van Dyck Jr., R. S., Schwinberg, P. B. and Dehmelt, H. G., Phys.Rev.Lett.59 (1987), Kinoshita, T., Phys. Rev. Lett. 75 (1995), Williams, E. R. et al., IEEE Trans. Instrum. Meas. 38 (1989), Cohen, E. R. and Taylor, B. N., Rev. Mod. Phys. 59 (1987), Kruger, E., Nistler, W. and Weirauch, W., Metrologia 35 (1998), Jeffery, A. M. et al., IEEE Trans. Instrum. Meas. 46 (1997), Udem, T. et al., Phys.Rev.Lett.79 (1997), Farnham, D. L., Vandyck, R. S. and Schwinberg, P. B., Phys. Rev. Lett. 75 (1995), Difilippo, F., Natarajan, V., Boyce, K. R. and Pritchard, D. E., Phys. Rev. Lett. 73 (1994), Van Dyck Jr., R., Farnham, D., Zafonte, S. and Schwinberg, P., In: Trapped Charged Particles and Fundamental Physics, AIP 457, Asilomar, CA, 1998, pp Mohr, P. and Taylor, B., Rev. Mod. Phys. 72 (2000), Kruger, E., Nistler, W. and Weirauch, W., Metrologia 32 (1995), Weiss, D. S., Young, B. C. and Chu, S., Phys. Rev. Lett. 70 (1993), Udem, T., Reichert, J., Holzwarth, R. and Hansch, T. W., Phys.Rev.Lett.82 (1999), 3568.

11 PRECISE MEASUREMENTS OF CS, RB AND NA Haffner, H. et al., In: F. FusoandF. Cervelli(eds), Abstracts of 17th International Conference on Atomic Physics, AIP, Firenze, Italy, 2000, pp Weisskoff, R. M. et al., J. Appl. Phys. 63 (1988), Van Dyck Jr., R., Wineland, D., Ekstrom, P. and Dehmelt, H. G., Appl. Phys. Lett. 28 (1976), Gabrielse, G., Phys. Rev. A 27 (1983), Cornell, E. A., Weisskoff, R. M., Boyce, K. R. and Pritchard, D. E., Phys. Rev. A 41 (1990), Bevington, P. and Robinson, D., Data Reduction and Error Analysis for the Physical Sciences, 2nd edn, McGraw-Hill, Boston, Audi, G. and Wapstra, A. H., Nucl. Phys. A 595 (1995), Huber, P., Robust Statistics, Wiley, New York, Bradley, M. P. et al., Phys.Rev.Lett.83 (1999), Mallard,W.G.andLinstrom,P.J.(eds),NIST Chemistry WebBook, NIST Standard Reference Database No. 69, NIST, Gaithersburg, MD, 1999 ( 26. Carlberg, C., Fritioff, T. and Bergstrom, I., Phys.Rev.Lett.83 (1999), Young, B., Ph.D. thesis, Stanford University, Udem, T., private communication, Forward, R., J. Appl. Phys. 50 (1979), 1.