Precision measurement aspects of ion traps

Transcription

1 Meas. Sci. Technol. 1 (1990) Printed in the UK REVIEW ARTICLE Precision measurement aspects of ion traps R C Thompson Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 282, UK Received 22 May 1989 Abstract. The use of ion traps in precision measurements of various kinds has been growing rapidly in recent years. This review attempts to survey work in this area. It starts with a brief description of the two main types of ion traps and how they work, and discusses the different methods available for detection of ions in a trap and for reducing their kinetic energy. The main part of the review deals with measurements of the magnetic moments (g-factors) of electrons, positrons and ions in traps; precision mass determinations, especially for rare isotopes produced in small quantities; and measurements of microwave and optical transition frequencies in ions, especially with applications to frequency standards in mind. The review concludes with a very brief sketch of some of the other main uses of ion traps to date, touching on the study of quantum jumps and ion crystals, and the measurement of the lifetimes of excited electronic states of ions 1, Introduction It is now more than 20 years since the first reviews of ion traps appeared (Dehmelt 1967, 1969). Since that time, the ion trapping technique has been applied to many different kinds of problems in physics, ranging from precise RF spectroscopy to studies of the anisotropy of space, and from the crystallisation of an ion cloud plasma at mk temperatures to the quantum mechanics of a single ion interacting with the radiation field. Clearly it is not possible in a short article to cover all these aspects of the technique, so this review concentrates on how various precision measurements may be effected using ion traps. An ion trap is an electrode structure which, by the application of AC and/or DC potentials, possibly with the addition of a magnetic field, is able to confine the motions of charged particles to a small region of space. These charged particles are generally electrons or singly charged ions; however, some of the earliest experiments were performed with charged aluminium particles (Wuerker et a1 1959) and currently there are proposals to make measurements on fully stripped and hydrogen-like uranium (Moore et a1 1988). In the rest of this review the word ion will be used to refer to any charged particle in a trap. What then are the advantages for precision measurements offered by ion traps? These are summarised briefly here but are dealt with more fully where they arise later in the review. (i) Ion traps provide a stable, well controlled environment which is free from many perturbations such as pressure effects or wall effects. There are in general perturbations caused by the trapping fields but these can be controlled and in some cases eliminated. (ii) In many cases measurements can be made on very small clouds or even a single particle. Also the observation time can be very long -in one case a single electron was trapped for ten months (Gabrielse et al 1985) -thus making the most efficient use possible of rare or exotic species and avoiding any broadening of spectral lines due to limited observation times (transit-time effects). (iii) Cooling techniques can be used to extract kinetic energy from the motion of confined particles. This eliminates the Doppler effect and localises the particles at the centre of the trap where the fields are smallest and best approach the ideal of a three-dimensional simple harmonic oscillator potential. (iv) The above considerations treat the ion trap as a convenient box in which the ions are held. However, the intrinsic properties of the trap itself are often used, especially the oscillation frequencies of ions in the trap potential. These are extremely well defined at the trap centre and depend in a simple way on the charge to mass ratio of the ions. They can also be employed for sensitive detection of the ions. The remainder of the review is structured as follows. Section 2 deals with the principles of operation of traps, as well as detection and cooling techniques. Sections 3,4 and 5 treat precision measurements of g-factors, inertial mass and transition frequencies respectively. Finally, other uses of traps are sketched in section 6 and future prospects for the technique are assessed in section /90/ IOP Publishing Ltd 93

2 R C Thompson 2. Principles of operation 2.1. Types of ion trap The operation of a trap which aims to confine ions by electromagnetic forces is limited by Earnshaw s theorem, which states that there cannot be a minimum of the electrostatic potential in free space. The best that one can do is to create a saddle-point in the potential, which is a minimum in one direction and a maximum in another. The lowest order potential of this form with axial symmetry is the quadrupole potential given by V = A(2z2 - r2) (1) where r2 = x2 + y2 and A is a constant. This potential can be generated in a vacuum by electrodes having the shape of the equipotential surfaces of equation (l), which are hyperboloids of revolution about the z axis. They consist of two end-caps (similar to hemispheres) and a ring (like a doughnut). (a) Penning trap. The potential given by equation (1) with A positive will trap a positive ion in the z direction but as there is no restoring force in the r plane this is not sufficient for a stable three-dimensional trap. However, if a magnetic field B is applied parallel to the z axis then the trajectory of a particle attracted towards the ring will become a closed orbit consisting of a fast cyclotron motion superimposed on a slow magnetron drift around the centre of the trap, arising from the electric field. This arrangement is called a Penning trap (Penning 1936) and results in the three oscillation frequencies (e.g. Brown and Gabrielse 1986): Axial w, = (4eVo/mR2)1 2 Modified cyclotron w, = wc/2 + (w,2/4-032) ~ Magnetron W, = 0,/2 - (0,214-0,212) l where the unperturbed cyclotron frequency is given by w, = eb/m. Here R2 = rg + 22; where 2r0 and 22, are the diameter of the ring and the separation of the end-caps respectively. Vo and B are the applied potential and magnetic field and e and m are the charge and mass of the ion. It should be noted that the magnetron motion is unstable, i.e. energy has to be supplied to it in order to reduce the size of the orbit. Space charge effects limit the number density of a spherical ion cloud to a maximum of n = 2.7 x 1015 (B/tesla)2/(M/amu) m-3. More detail can be found in Brown and Gabrielse (1986), Toschek (1984) and Wineland et a1 (1984). (b) Radiofrequency (RF) or Paul trap. The other main type of trap was developed from earlier linear quadrupole radio-frequency mass filters (Fischer 1959). It does not use a magnetic field but instead an oscillating potential is applied to the ion trap electrodes, generally in addition to a DC one. The effect is to cause ions to oscillate at the driving frequency R (the micromotion). However, as the field is inhomogeneous, the net force time-averaged over one cycle is non-zero so the ions act as if they were also moving in an effective potential set up by the RF field (the macromotion). Unlike a true potential, this pseudopotential can have a global minimum in space and thus give three-dimensional trapping. The pseudopotential only provides an approximate description of the exact Mathieu type equations of motion; detailed theory can be found in reviews elsewhere (Dehmelt 1967, 1969, Wineland et a1 1984, Toschek 1984). However, it is useful as a guide and it gives rise to oscillation frequencies: wp = 2e2Vi/m2Q2R4 + 2eUo/mR2 wt = 8e2Vg/m2R2R4-4eUo/mR2. The stability parameters a and q are defined by a = - 16eUo/mR2R2 and q = 8eVo/mR2R2. Stable trapping is only found for certain limited ranges of a and q which can be found in the reviews listed above. The standard operating conditions have a and q in the region of 0.0 and 0.4 respectively. Again, space charge effects limit the maximum density which can be achieved, as the ions fill up the pseudopotential well created by the trapping fields. This maximum density is typically 1013 m-3, roughly comparable to the number density of a gas at Pa. Both types of trap need to be placed in an ultra-high vacuum (UHV) system in order to avoid collisions with residual gas atoms which can eject ions from the trap. Generally a base pressure at least as low as lo-? Pa (lo- mbar) is required, although a light buffer gas may be introduced in an RF trap. Typical operating parameters for some traps are listed in tables 1 and Detection of ions The type of detection employed clearly depends on what particles are in the trap and the sort of experiment being performed. In some cases it is sufficient merely to know how many ions there are while in other cases one wishes to know something about the state they are in (e.g. the polarisation state, electronic state or motional state). (a) Electronic detection. If ions are moving between electrodes then image currents induced in an external circuit may be picked up and used for detection. This is employed either directly (the so-called bolometric technique, e.g. Wineland et al 1978) or with an external drive close to resonance which can be used in conjunction with phase sensitive detection to amplify the signal, especially for a single ion (Dehmelt and Walls 1968). This also makes it possible to determine the spin and motional state of a trapped electron via small shifts of the axial frequency (Brown and Gabrielse 1986). An alternative detection technique (e.g. Loch et a1 1987) makes use of the damping of an external oscillating circuit of high quality factor (Q) when it is connected across the endcaps. At resonance the axial oscillation of the ions is excited and consequently the amplitude of the oscillation in the circuit decreases as energy is extracted. The amount of damping can be related to ion number and the frequency at which it occurs determines the charge to mass ratio. 94

3 Precision measurement aspects of ion traps Table 1. Parameters of typical Penning traps. A selection of trap operating parameters is given for four different precision experiments which are referred to later in the text. These give some idea of the range of uses of ion traps. The symbols used are self-explanatory and are discussed in the text. A small number of entries have had to be estimated due to the absence of specific data in the publications. The references are: 1, Brown and Gabrielse (1986); 2, Bollinger eta/(1985); 3, Kluge (1 988). Species e- P+ 9~e+ 122CS+ vdc o 8.0 B (TI fo (") ; GHz 76 MHz 1.4 MHz 740 khz w,127l 64 MHz 10 MHz 190 khz 70 khz w, khz 660 khz 13 khz 3.3 khz n (m-3) x Total N Temperature 4K 4K <0.1 K ev p (Pa) < 10-8 < 10-8 <lo-? 10-7 elm (C kg-') 1.8 x x x x 105 Use g-factor mass 303 MHz mass Accuracy 4 x x x x 10-7 Detector External External cw laser Time of circuit circuit fluorescence flight Reference Section Table 2. Parameters of typical RF traps. As for table 1 except that the classic 1959 experiment with charged dust particles has been included for comparison. The references are: 1, Wuerker eta/ (1959); 2, Cutler eta/(1986); 3, Bergquist eta/(1987); 4, Blatt eta/(1983). + + Species AI particles lg9hg 198Hg Q12.n 2ro (") vdc a VAC 4 %I271 (41271 n (KI-~) Total N Temperature p (Pa) elm (C kg-') Use Accuracy Detector Reference Section 210 Hz Hz 13 Hz -4 x Eye khz khz 47 khz 3 x K (He) 4.8 x io GHz Lamp fluorescence MHz MHZ 1.4 MHz mk <lo-? 4.8 x io5 282 nm 3 x cw laser fluorescence ?1ybi 500 khz khz 58 khz 10" ev (He) 5.6 x GHz Pulsed laser fluorescence (b) Optical detection. For ions whose resonance wavelength is attainable using lasers this is probably the most sensitive form of detection. It involves simply irradiating the ions on resonance and looking for fluorescent emission either at the same wavelength or at a different one also emitted in decay from the excited state. This has been shown to be sensitive enough to give a good signal to noise ratio even for a single ion of barium (Neuhauser et al 1980), magnesium (Wineland and Itano 1981, Diedrich and Walther 1987) or mercury (Bergquist et a1 1987). It has been used in many experiments in conjunction with optical pumping (e.g. Blatt et a1 1983) or laser cooling (e.g. Bergquist et a1 1987) so that in these cases detection is closely coupled with other parts of the experiment. (c) Direct detection of ions. The first two techniques discussed are non-destructive: the ions remain in the trap after detection. Destructive detection is also possible: in this case the trap must be reloaded after each detection event. Typically in such measurements the ions are released from the trap and collected on a channel plate detector (Schnatz et a1 1986, Kern et a1 1988). This is clearly quite a sensitive method but requires a continuous supply of ions, which may not always be possible. If the time taken by the ions to reach the detector is measured, then it is possible to determine the motional 95

4 R C Thompson state of the trapped ions; thus the method can be used to detect resonance of a driving field with an oscillation of the ions in the trap (Schnatz et al 1986) Cooling techniques The depths of the potential wells in ion traps are usually a few ev or higher and trapped atomic particles tend to have fairly high energies (compared to kt at room temperature). There are a number of reasons why it is often necessary to remove energy from the ions and hence cool them. First, it reduces losses of ions due to their boiling out of the trap. Second, it localises them near the centre of the trap and increases the cloud density. Third, it may be used to remove energy coupled into the ionic motion by one of the detection methods discussed above or by the trapping fields in an RF trap, which cause RF heating. Fourth, for precision measurements the Doppler broadening and shift of spectral lines may need to be reduced. Finally, in some experiments ions enter the trap with very high energy (e.g. positrons from a j source; see Schwinberg et a1 1981a) and need to be strongly cooled in order to proceed with the measurements. (a) Buffer gas cooling. Perhaps the simplest way to cool ions is to introduce a light buffer gas into the UHV system (e.g. Blatt et a1 1983). At sufficiently low pressures (typically of order Pa for hydrogen and helium) the main effect is to damp the ionic motion and hence reduce the temperature. This only works in an RF trap - in a Penning trap any collisions will tend to increase the size of the magnetron orbit. Obviously the lowest temperature which can be achieved in this way is that of the buffer gas so this technique cannot generally be used to cool below room temperature. (b) Resistive cooling. In section 2.2 it was mentioned that the motion of ions in a trap generates image currents which can be used for detection. If a resistor R is connected across the end-caps then the image currents dissipate heat in the resistor and in this way the axial motion can be damped at a rate proportional to e2r/m (Dehmelt and Walls 1968, Holzscheiter 1988). The lowest temperature attained is limited by Johnson noise in the resistor; however, if the tuned circuit is immersed in liquid helium this cooling process can be made very effective, especially for electrons (e.g. Van Dyck et a1 1986). (e) Laser cooling. For atomic ions the most effective method of cooling is to make use of radiation pressure from a laser tuned just below resonance with a transition in the ion. Ions can then only absorb light when they move towards the laser, so they are slowed down a small amount by the momentum of each photon absorbed (in contrast, the re-emission process is isotropic). The cooling limit is generally given by ktmin = hy/2, where y is the radiative width of the excited state, as shown in numerous reviews (Stenholm 1986, 1988, Itano and Wineland 1982). For magnesium ions one photon (at 280nm) changes the ionic velocity by 60 mm s-l and it takes roughly 6000 absorptions to cool from room temper- ature to Tmin, which is about 1 mk. This cooling can take place in much less than one second and only requires a laser power of perhaps 10 pw due to the small size of a typical ion orbit. In a Penning trap the laser illumination has to be inhomogeneous in order to cool both magnetron and cyclotron motions simultaneously and this leads to a slightly more complicated situation (Itano and Wineland 1982, Thompson 1988). Laser cooling is a powerful technique and has been used extensively to produce and study ultra-cold ions in a room temperature apparatus. However, it is only suitable for a small number of ionic species, so sympathetic cooling has also been developed, where one ionic species is cooled by collisions with a second, laser-cooled species, also stored in the same trap (Larson et a1 1986). This should enable the technique to be more widely used in future, especially as sympathetic cooling perturbs the ion less than direct laser cooling. 3. Measurement of gfactors 3.1, The geonium system For an electron, the g-factor is defined as the ratio of its spin magnetic moment to the Bohr magneton, pb= ehf2m. This has been measured to a fractional precision approaching using a single electron in a Penning trap. This system is often called geonium (see Dehmelt 1988) as it consists of an electron bound to the earth via the trap potential. The geonium pseudoatom has four independent contributions to its energy: the cyclotron, axial and magnetron motions discussed in section 2.1 and the spin energy which depends on the spin orientation with respect to the magnetic field. As with more conventional atoms, an energy level diagram can be constructed to show the possible states (Brown and Gabrielse 1986) Measurement technique The principle of the technique is to measure both the cyclotron frequency of the electron (0, = eb/m) and its spin precession frequency (0, = gp,b/h = igw,) in the same magnetic field B. The ratio yields the g-factor. Two features make the experiment potentially very accurate. First, U, can be determined from the three oscillation frequencies in the trap to high accuracy from the relation w, = w;2 + w; + w,. This relation is exact even if the magnetic field is misaligned with respect to the trap axis and if the trap potential is not axially symmetrical (Brown and Gabrielse 1982, 1986). Higher order trap imperfections are tackled by working at trap centre (using cooling techniques) where they have less effect, and by adding extra compensation electrodes to fine-tune the potential to the correct form (these are shown in figure 1). Second, w, is not measured directly but rather the anomaly frequency CO: = w, - wb. As g - 2 is only of 96

5 Precision measurement aspects of ion traps IcmiO I I I I I ' l I I [ I 1 1 I I ' I I I Field emission point 2 I 1 2 Metalked strip Oiinl Tungsten Motor Gloss f t 1 i- is removed as described above, while the increase in magnetron energy compresses the orbit as required, due to the unstable nature of the magnetron motion. At exact resonance the axial motion decouples, and this feature can be used to give a precise determination of w,. The final aspect of this experiment is the use of a 'magnetic bottle', that is, an extra (small) magnetic field varying quadratically with rand z, to measure wb and mi. The field arises from a ring of ferromagnetic material incorporated into the ring electrode, and one effect is to make w, depend on the degree of excitation in each of the three motions and the spin state. Thus, if 0, is monitored while an RF drive is applied at a frequency close to ob, a shift of typically a few Hz is seen at resonance. The line shape is asymmetrical due to the one-sided distribution of energy in the axial motion (see figure 2), and can be fitted to determine wb, which is close to the sharp lowfrequency edge. Similarly, the anomaly frequency is found by monitoring 0, while irradiating close to wk. If a spin-flip transition occurs, then 0, jumps by a certain amount. The rate at which these jumps occur shows a resonance at wk. Figure 1. Scale drawing of an experimental Penning trap used for the study of electrons (Brown and Gabrielse 1986). The magnetic field is vertical and the two compensation electrodes can be seen as well as the three normal electrodes (two endcaps and one ring). order the anomaly frequency is small and need only be measured to an accuracy of say lo-* for a lo-" precision in g. We now consider the cooling mechanisms and the related question of detection of resonances. For the cyclotron motion the cooling is automatic and follows from the usual Larmor formula for an accelerating charge, yielding a decay constant for the energy: yc = 4e2w:/12m,mc3 which for wj2z = 160 GHz (B = 5.9 T) gives lkic = 0.08 s, corresponding to strong damping. However, for the other two motions the radiative damping is very slow, so another mechanism is required. The axial motion is cooled by coupling it to an external resistance (kept at about 10 K to reduce noise) as described in section 2.2. Using R of order lo5 R, lly, is typically 0.02 s. Although the damping is strong, the axial motion still has a quality factor Q of 10' or so, allowing precise measurement of its frequency. This is achieved by monitoring the voltage across R as an axial drive is applied and is swept across the resonance. The drive is in fact applied between one end-cap and the ring; the detection uses the signal between the second end-cap and the ring. We can now deal with the cooling of the magnetron motion. This is accomplished using 'motional sideband cooling' (Brown and Gabrielse 1986), where a field of angular frequency w, + w, is generated inside the trap. Absorption of a photon at this frequency puts energy into both the axial and magnetron motions. The axial energy ~~~ khz Figure 2. Experimental cyclotron resonance (Brown and Gabrielse 1986). The shift in axial frequency is shown as a function of applied RF drive in the region of wb (see text). 3.3 Results These experiments have been performed over a long period (Schwinberg et a1 1981a, b, Van Dyck et a1 1984, 1987) on both electrons and positrons. The main difference for positrons is that they are loaded off-axis from a "Na positron source into an auxiliary trap and cooled to low energies before transfer to the principal trap for the measurements. The following are the most recent results (Van Dyck et a1 1987): ge- = (86) g: = (86) These represent the most accurately known fundamental constants to date (see Cohen and Taylor 1987). The anticipated future accuracy is (Van Dyck et al 1988). The equality of the two g-factors is a I 97

6 R C Thompson test of CPT invariance (Particle Data Group 1984) and the absolute value can, for example, be combined with QED theory to deduce a new value for the fine-structure constant using the relation (Mohr 1988) a, = (g - 2)/2 = ~ (u/z) Similar experiments are possible for protons and antiprotons, though work with a single particle is more difficult due to the larger mass, which leads to weaker coupling to the external circuit. Resonances in a small cloud of protons have been measured (Van Dyck et al 1985) and a beam of 21 MeV antiprotons from the LEAR storage ring at CERN has been reduced in energy and captured in a trap (Gabrielse et a1 1986). However, precise measurements of the proton/antiproton magnetic moments have not yet been possible The g-factor of the hydrogen molecular ion The g-factor of a bound electron differs from that of a free electron by typically a few PPM due to relativistic and QED corrections. In the simplest possible molecule, H:, the g- factor depends on the rotational and vibrational state and has been calculated by Hegstrom (1979). A group at the University of Mainz is measuring the g-factor of Hi in a Penning trap, making use of the spin dependence of the collisional cross section of a polarised beam of sodium atoms with the ions inside the trap (Loch et al 1987). When spin flips occur, induced by an RF field, then the loss rate of the ions due to the collisions changes. Thus the spin-flip frequency can be found and compared with the cyclotron frequency in the trap. The latest result is (Loch et al 1988) g = (18) compared to a theoretical prediction of g = (8). Further improvements should allow the vibrational structure to be resolved, thus giving a more critical test of the theory, which shows that the value of g depends on the vibrational state of the molecule. 4. Measurement of mass 4.1. Principle of the technique The frequencies of the three motions in a Penning trap (see section 2.1) are directly related to the charge to mass ratio of the ions. Thus, if all the other quantities are known, the mass of the ions may be determined from the oscillation frequencies. In practice, they are combined in such a way as to eliminate the effect of the electric field, and give the unmodified cyclotron frequency. An accurate measurement of the magnetic field or a calibration using an isotope of known mass then yield the mass of the unknown isotope to high accuracy. Depending on the accuracy required, the cyclotron frequency is determined using the relation given in section 3.2 (independent of trap imperfections) or simply by adding wb and w, (exact only to first order in any imperfections). These measurements clearly have much in common with those discussed in section 3, although this section deals mostly with atomic ions rather than with electrons. Mass measurements are particularly important for unstable nuclei, where they enable binding energies to be determined more directly than is otherwise possible. It is here that many of the advantages of ion traps, such as the small sample size required and the long interaction times obtained, come into play. Lower resolution instruments such as the pulsed ion cyclotron resonance (ICR) mass spectrometer (e.g. McIver 1978) are based on similar principles but are of simpler design than precision Penning ion traps, using six plane plates arranged in a box structure as electrodes. The requirement here is for a resolution better than 1 amu to identify molecular ions, but much higher resolution is often not necessary. Such instruments can be used to follow the course of chemical reactions with a low pressure background gas with millisecond time resolution (McIver 1978). Higher resolution instruments have also been constructed and have been used for mass comparisons at the lo- level (Lippmaa et al 1985). A detailed discussion of these instruments is outside the scope of this review and the reader is referred to Hartmann and Wanczek (1982). See also March and Hughes (1989) for a discussion of the expanding area of ion trap mass spectrometry Mass measurement of rare isotopes A group at Mainz aims to measure the masses of strings of isotopes produced by the ISOLDE mass separator at CERN (Kluge 1988, Kern et a1 1988). The experiment has to be performed on-line as many of the half-lives are short (of the order of minutes or less). However, a trap designed to collect ions directly or indirectly from a highenergy beam is incompatible with a high-precision measurement which needs UHV pressures, homogeneous fields and near-perfect trap geometry. This problem is overcome with two intermediate steps before the ions reach the measurement trap. The first is a rhenium foil into which the ions from ISOLDE are implanted. The second is a so-called bunching Penning trap which is loaded by heating the rhenium foil to re-emit the implanted atoms which are ionised as they leave the surface. Finally, the ions are released from the bunching trap and travel down a transfer tube to the measurement trap which is located in a superconducting magnet at low pressure (< lo- Pa). This final transfer is very efficient, with up to 70% of the ejected ions being successfully captured in the second trap (Schnatz et al 1986). The mass measurement proceeds by applying an RF field to the ions at a frequency corresponding to w, (equal to wb + U,). This drives a two-quantum transition in the ions, with energy going into both the magnetron and the cyclotron motions. The resonance is detected by releasing the ions out of the trap into a drift tube. Those ions which have absorbed energy from the RF field have a larger cyclotron orbit and hence a larger orbital magnetic moment. When the ions reach the inhomogeneous part of 98

7 Precision measurement aspects of ion traps (i) Tritium mass. If the mass of the tritium nucleus 32 (3H) can be determined at this level of precision, then t ~,,,!,,,, 1,,,,!,,,,,,,,,!,,,,,,, j Frequency IHz) Figure 3. Mean time of flight of 12,Cs ions as a function of applied RF frequency (Kluge 1988). The resonance at lower frequency (higher mass) is attributed to the T,,, = 21 s isomer (see text). the field, the cyclotron energy is converted into longitudinal kinetic energy and those ions reach the channel plate detector earlier. Thus, the time-of-flight spectrum shows a minimum when the RF is resonant with the twoquantum transition. Typical linewidths are a few Hz, giving a resolving power of 3 x lo5 and a fractional accuracy of 2 x lo-'. The magnetic field has to be calibrated by measurements on a stable isotope of known mass. Figure 3 demonstrates the power of the technique. It shows a time of flight spectrum for 122Cs, which is known to have two half-lives (4.2 min and 21 s). There are two resonances differing in frequency by about 3 Hz (corresponding to 0.5 MeV). If the measurement is delayed, the left-hand peak disappears, so this must correspond to the shorter half-life. Thus, the masses of the ground and isomeric nuclear states have been resolved here in a mass measurement for the first time (Kluge 1988, Kern et a1 1988). This method needs typically 50 ions for each cycle but, as the detection process is destructive, between lo4 and lo5 detected ions are needed for a resonance curve (taking up to one hour to record). However, the trapping efficiency into the first trap is only of order so the total requirement is for about 10" ions (of the order of 1 pg). This could be improved considerably if, for example, the ISOLDE beam could be bunched in some way (see, for example, Moore and Gulick 1988) Other high-precision mass measurements It is now possible to make mass measurements of specific particles or isotopes to a precision of io-' or lo-'' (e.g. Van Dyck et a1 1988). These measurements allow the determination of fundamental constants such as the proton-electron mass ratio. This is achieved by storing electrons and protons alternately in the same trap and measuring the ratio of the cyclotron frequencies (Van Dyck et a1 1985). Other possibilities also arise of which we mention just two here. important extra information will be available in experiments which measure the 3H beta-decay spectrum with a view towards the determination of the antineutrino rest mass (Van Dyck et al 1988, Lippmaa et a1 1985). It is still not known whether neutrinos have a mass or not, and the answer to this question has important astrophysical consequences as well as being of fundamental interest. The ion trap mass measurement has to have an accuracy of < lo-' in order to determine the antineutrino mass, which may be only a few ev, with reasonable precision. At least three research groups are already attempting to make this measurement. (ii) Uranium Lamb shijt. In hydrogen-like uranium the ground state Lamb shift changes the mass by two parts in lo9 so if a mass comparison can be made for 238U92+ and 238U91+ to an accuracy of about lo-'', the Lamb shift could be 'weighed' to better than 10% (Moore et a1 1988). This would be very interesting as it would check the scaling of the Lamb shift with 2, whose dominant term is predicted to follow the relation E = (cr/n)[(z~)~/n~]f(za)mc~ where F(Za) is a slowly varying function of Za (Mohr 1988). It would be a very difficult experiment as the ionisation of the uranium would require high electron energies (over 100 kev to remove the last electron) and high currents (due to the small ionisation cross sections). Furthermore, extremely high vacuum (< Pa) is required to prevent recapture of electrons via collisions. 5. Measurement of transition frequencies 5.1. Introduction Atomic transition frequencies can in principle be measured to very high precision, but often there is a practical limit to the precision arising from perturbations on the atom, which either broaden or shift the resonances. Examples include the Doppler effect due to thermal motion, collisions with other particles, stray electric and magnetic fields and the effects of a finite interaction time. In ion traps many of these perturbations can be eliminated or at least reduced, so that the inherent high stability of suitable transitions in ions can be realised in experiments to measure the frequencies to high precision. In this way several ionic hyperfine splittings have been measured to accuracies approaching or even exceeding lo-'' (Werth 1982). It is seldom necessary to know splittings so accurately - even studies of the hyperfine anomaly in strings of isotopes of the same element only requires lo-' or so (Werth 1988) -but there are at least two important uses of such an experiment. These are either to measure tiny changes in such a transition frequency or to use the transition as a frequency standard in terms of which other, less precise, measurements can be made. An example of the first of these uses is a search for anisotropy of space by studying the variations in a transition frequency in trapped 'Be+ 99

8 R C Thompson ions as the orientation of the ions changes (Prestage et a1 1985). The same system (described in section 5.2 below) has also been used as an atomic frequency standard, and its performance was comparable to that of caesium beam standards (Bollinger et a1 1985). Most (but not all) of these precise atomic frequency measurements use laser cooling of the ions in order to eliminate the first-order Doppler effect. This is not necessary if the transition wavelength is greater than the amplitude of the ionic motion (Dicke 1953), which is the case for microwave transitions. However, the secondorder effect is still present at the lo-'' level for ion energies of the order of 1 ev and this can only be eliminated by cooling. Here there is a significant difference in the application of Penning and RF traps. In an RF trap the RF heating means that only a few ions at most can be laser cooled to very low temperatures (e.g. Bliimel et al. 1988). For larger clouds the RF fields and collisions heat the ions faster than the energy can be removed by the laser cooling (Blumel et al. 1989). Thus, precision experiments with laser cooling in RF traps are often limited to single ions. On the other hand, large clouds of up to a thousand or more ions may be easily cooled in a Penning trap (Bollinger et all985, Thompson et a1 1988, Plumelle et a1 1986) although the second-order effect increases with the size of the cloud (Wineland 1984). The other point to note about the Penning trap is, of course, the presence of the magnetic field B. This generally means that in order to measure a transition frequency v with high precision, a value for B has to be found such that v is independent of B to first order, or else field instabilities will cause severe problems. This in turn often imposes restrictions on the hyperfine structure of an ion if it is to be successfully employed in such a measurement. Perhaps the greatest quantity of work has been performed on a transition in 9Be+ which is certainly not an ideal ion in some ways but is convenient for laser cooling work in a Penning trap. The resonance transition in Be' is at 313 nm. This is conveniently generated by frequency doubling in an RDP crystal the light from a cw ring dye laser operating at 626 nm (Bollinger et a1 1983). The light is tuned to the low-frequency side of the 2s 2S1,2 (MI=-3,M ) to 2p 2P3,2 (-3,-3) transition, and the ions (roughly 1000) are optically pumped into the lower level of this transition (state A, say). This gives a high level of fluorescence as the ions are continuously re-excited by the laser. The field-independent clock transition is between the levels (- 3, i) (state B) and (- i,t) (state C) in the ground state at a field of 0.82 T (see figure 4(a)). It has a frequency of 303 MHz, and has been measured to a precision of 2 x The measurement proceeds as follows (Bollinger et al 1985). First the ions are prepared in state A using the laser cooling light as described. This also ensures that the ions are cold before each cycle. Then, with the laser off, microwaves are applied at the electron spin-flip frequency of - 24 GHz, transferring half of the population to state B. This is followed by the interrogation at the clock frequency which transfers some of the population to state C (if it is resonant). Finally the spin-flip frequency is used again, re-equalising the populations of states A and B, which results in more transfer out of state A, if the clock frequency was resonant. This is detected by measuring the fluorescence level when the cooling laser is 5.2. Microwave transitions Several measurements have been made on microwave transitions between hyperfine levels of ions confined in both RF and Penning traps. Ideally one looks for a heavy ion (so that the Doppler effect is small) with a large hyperfine splitting (to give high fractional precision Av/v for a given AV). Mercury satisfies these requirements well, but has only recently been laser cooled due to problems with the laser wavelength required (see section 5.3). However, optical pumping experiments on mercury with a conventional resonance lamp are possible due to a chance coincidence of isotope shifts in "'Hg' and "'Hg+ with the "'Hg' hyperfine splitting (Jardino et a1 1981, Major and Werth 1973). Here a cloud of "'Hg' ions in a trap is optically pumped, using light from a "'Hg resonance lamp, into one of the hyperfine levels. However, microwaves applied at the appropriate frequency equalise the populations of the ly9hg+ hyperfine levels, leading to a change in the level of observed fluorescence. This has recently been used to make a commercial frequency standard at 40.5 GHz stable to or so over a period of a year (Cutler et a1 1986). microwaves (312, -112) Frequency ihz) Figure 4. (a) Hyperfine structure of the 'Be' 2s 2S1,2 ground state as a function of applied magnetic field B. The clock transition (vl) is independent of 6 to first order at 6= T. (b) Signal obtained on the v1 field-independent transition for a 550 s interrogation time. The dots are experimental and the curve is a least-squares fit (Bollinger et a/ 1989a). 100

9 ions Precision measurement aspects of ion traps first switched on again, before it has pumped all the population back into the initial state A. The interrogation at the clock frequency uses the Ramsey method of separated oscillatory fields (Ramsey 1956). This means that two short (0.5 s) coherent pulses of microwaves are separated by a period of -20 s with no irradiation, and it results in an oscillatory lineshape, symmetrical about line centre with a 25 mhz width. The centre frequency has been determined to less than 0.1 mhz, allowing for systematic effects such as the second-order Doppler effect, magnetic field fluctuations and the rotation of the earth (Bollinger et a1 1985). More recently improvements have been made including the use of more efficient 7-c pulses for the population transfers and sympathetic cooling with laser cooled + Mg to keep the Be' ions cold during the whole cycle (Bollinger et a1 1989a). This enables the interval between irradiation to be raised to several hundred seconds, giving a linewidth of < 1 mhz. Preliminary estimates of the systematic fractional uncertainty in the measured frequency are < 10-l4. A recent curve is shown in figure 4(b). This experiment has been run as a frequency standard at 303 MHz (Bollinger et al 1985). A recent suggestion by Weinberg (1989) shows that the results of the experiment may be interpreted as evidence for lack of non-linearity of quantum mechanics at the lo-'' level. This test was made a factor of lo6 more sensitive by lengthening the interrogation time on the clock transition (Bollinger et a1 1989b). The same set-up has also been used to study spatial anisotropy (Prestage et a1 1985). One other application is to the study of a condensed state of ions in a trap, that is, a crystal-like structure formed if the ions are cooled sufficiently strongly (Gilbert et a1 1988, see section 6 below). It is not essential for microwave experiments in traps to use laser cooling. In Mainz a series of experiments have measured ground-state hyperfine splittings to high precision (see Werth 1982, 1985, 1988) using large clouds in RF traps at an ion energy of the order of 1 ev. These experiments use a pulsed laser for optical pumping and for detection via fluorescence. The second-order Doppler effect has to be allowed for and the results also extrapolated to zero magnetic field, where the M,=O to 0 transition is field independent to first order. One of these experiments has also been used as a frequency standard (Knab et a1 1985) Optical transitions In recent years it has become possible to narrow the linewidths of stabilised lasers to the khz region (e.g. Hough et a1 1984) and this has opened up the serious possibility of using such narrow bandwidth lasers in the study of optical transitions to metastable states in trapped ions. A frequency standard in the optical region has the clear advantage that a given fractional precision Av/v can be achieved in a shorter time than is the case for microwave standards, as v is a factor of lo4 or so higher. However, it is recognised that transferring this stability via a frequency chain of some sort into the microwave region may well be a problem. We concentrate here on an experiment on lg8hg' ions confined in a miniature RF trap. Such miniature traps were first used by Toschek and co-workers for the study of small clouds of Ba' ions (Neuhauser et a1 1978) and also for the first work on single ions (Neuhauser et a1 1980). However, although Ba' has some long-lived levels its energy level structure is not ideal for use as a frequency standard (see below). The Hg' experiments were performed initially on a small cloud of collisionally cooled ions (Bergquist et a1 1985). Up to 5 pw of laser radiation at the resonance line (194 nm) was produced by frequency doubling and mixing techniques (Hemmati et a1 1983) and this was used to detect the presence of the ions in their ground state. A second laser source at 563 nm also irradiated the ions and excited them by Doppler-free two-photon spectroscopy (see, for example, Thompson 1985) into a longlived 'D,,, state, thus reducing the level of 194 nm fluorescence seen. These initial experiments showed a linewidth in the spectra of 420kHz and their most interesting feature was the presence of sidebands on the two-photon resonance due to the secular motion of the ions in the RF field (at approximately 1 MHz), revealing this effect for the first time in optical spectra (Bergquist et a1 1985). More recently these experiments have been improved by using a single laser-cooled mercury ion in a similar trap (Bergquist et al 1987). Here the single-photon transition at 282nm to the same metastable level was excited using frequency-doubled radiation. Since after a period of exposure to this radiation with the 194nm beam off, the ion is left either in the ground state or in the metastable state, it suffices to turn the 194 nm radiation back on and to determine if there is or is not any immediate fluorescence, so the result of each 'cycle' is either a zero or a one. At the start of each cycle it is ensured that the ion is back in the ground state by measuring the level of fluorescence. In this way the twophoton excitation probability can be determined as a function of laser frequency. The results of this experiment are shown in figures 5(a) and (b). The first scan (a) shows the central carrier with two motional sidebands on each side while the high-resolution scan (b) shows just the central carrier and the first upper sideband at 1.45 MHz. The linewidth here is about 30 khz, which arises mainly from residual laser frequency fluctuations (the natural linewidth of the transition is just 2 Hz). It is possible to determine the temperature of the ion from the intensity of the first sideband, and this gives T = 1.7 mk, which corresponds to the theoretical Doppler cooling limit for Hg'. At this temperature the average vibrational quantum number (a) in the trap pseudopotential well is about 11. Even lower temperatures were obtained by subsequently switching the cooling from the 194nm resonance transition (with large natural linewidth) to the 282 nm transition, with small natural linewidth and hence a much lower cooling limit. In fact the transition rate on this transition is too low for the method to work so it has to be artificially enhanced by introducing a third 101

10 R C Thompson Frequency detuning [in MHz) belonging to a different electron core configuration, which appears to be extremely long-lived. This would lead to the effective loss of ions from the experiment if it could not be tackled in some way, although it has been suggested that this level itself could be used in a standard of some sort (Blatt et al 1989, Lehmitz et a1 1989). Similarly, barium has often been suggested as a possibility (see for example Toschek 1984) but although the laser cooling wavelengths are suitable and the metastable states have lifetimes of many seconds, the candidate transitions lie in the infrared region and there are also some complicated Raman-type resonances which are only now beginning to be fully understood (Sauter et a1 1988). Dehmelt (1982) has suggested a series of possible standards in elements such as Tl' and In+ where the 'So to 3P, transition might make an optical standard with a possible reproducibility of lo-'' (see also Nagourney 1988). However, such transitions are extremely difficult to probe with current laser technology and it may be some time before serious progress can be made in their study experimentally. 6. Other uses of ion traps Ib) 0.80, A 1 I " Frequency detuning (inkhz1 Figure 5. (a) Quantised signal showing the E2 allowed ZS1,2-2Ds,z transition in a single, laser-cooled l9'hgf ion, as a function of laser detuning at 282 nm. The recoilless absorption resonance is in the centre and two Doppler sidebands can be seen on each side. (b) High resolution scan showing carrier and first sideband. The linewidth is 30 khz at 282 nm and the relative intensities indicate T= 1.6 f 0.5 mk (Bergquist eta/ 1987). radiation field to empty the Dsj2 state. This shortens the lifetime and makes the temperature limit correspondingly higher, but still lower than the previous one. In this way (n) = 0.05 is achieved, corresponding to T = 47 pk. This is the lowest temperature yet achieved in an ion trap (Diedrich et a1 1989). The precision of measurements made on this system depend to a large extent on the laser linewidth achieved. The current linewidth of roughly 1 khz (Bergquist et a1 1989) represents a value of Av/v of lo-'' but of course the centre of the transition can be determined to a higher precision than this if the systematic effects are understood. If a linewidth of 2Hz could be achieved then a measurement accuracy of or so could be envisaged (Bergquist et a1 1989, Wineland et a1 1987b). Ytterbium is another possible candidate for an optical frequency standard and at least three groups around the world are working on this ion including the National Physical Laboratory (Barwood et a1 1989). The laser wavelengths required are easier to produce than those for Hg+ but the level structure is not ideal. In particular, it appears that one candidate metastable level decays not only to the ground state as expected but also to a level The environment inside ion traps is unique and therefore it is not surprising that unusual experiments have been performed in them, apart from precision measurements of the type discussed so far. For instance, it is possible to isolate a single atomic particle and observe it for a long period (Neuhauser et a1 1980, Wineland and Itano 1981, Nagourney et a1 1983, Diedrich and Walther 1987). If an ion is in its ground state it can be continuously excited into a resonance level and its fluorescent decay back to the ground state monitored with a photomultiplier, with a signal of up to lo5 counts per second. If however, the ion is excited into a metastable state somehow then the fluorescence must stop until it decays once more to the ground state. Thus the state of the ion can be monitored by the presence or absence of fluorescence and this enables the quantum jumps to and from the metastable state to be studied (Cook and Kimble 1985). Three experiments were performed within a short time of each other to verify this simple analysis (Sauter et a1 1986b, Bergquist et a1 1986, Nagourney et a1 1986). The results were as expected except that in one experiment anomalous results were obtained when two or three ions were trapped simultaneously (Sauter et a1 1986a, b). There appear to be many more double and triple jumps - where the fluorescence level changes by an amount corresponding to two or three ions jumping simultaneously - than would be expected from the known time resolution of the detection system. This effect appears to be absent in the other experiments (see for example Itano et a1 1988). For a review of this area see Blatt and Zoller (1988). The quantum jump experiments are not only of academic interest. If ion traps are to be used either for spectroscopy of ultra-narrow resonances or for frequency standards based on narrow transitions, then the tran- 102

11 Precision measurement aspects of ion traps sition rate is bound to be very low as the states involved In the Penning trap much larger clouds have been must be long-lived. Also, it is likely that very few ions are studied and in this case they always rotate due to the used in an experiment, in order to reduce perturbations magnetron drift, so the crystal structure cannot be seen to an acceptable level (see Wineland 1984). Therefore it is so clearly. Theoretical predictions using molecular dyof paramount importance that the maximum amount of namics simulations (Rahman and Schiffer 1986, Dubin information is extracted from any transitions which do and O'Neil 1988) show that there should be a definite occur. Studying the fluorescence on a second transition shell structure with little migration between shells but a in the manner described above can supply this inform- larger diffusion on each shell. This is indeed what is seen, ation with essentially 100% efficiency as the ion acts as although a structure of open concentric cylinders was an amplifier with very high gain (this method was used in also sometimes seen, which is not predicted theoretically. the work described in section 5.3; see also Dehmelt 1988). Clearly there is much of interest in these crystal Thus the complete study and understanding of quantum structures. It is interesting to note that very similar effects jumps is of importance to the future development of are predicted to occur in storage rings except that there frequency standards based on ion traps. the basic symmetry is cylindrical rather than spherical One other recent development has been the successful (Rahman and Schiffer 1988). These structures have not cooling of clouds of ions to the point where they yet been observed as it is more difficult to obtain low crystallise into a lattice with a spacing determined by a temperatures in storage rings. balance between the Coulomb repulsion of the ions and The point at which condensation occurs in traps is the trap potential well (see figure 6). This makes the generally expressed in terms of the coupling strength r, crystal lattice spacing typically 20 pm which is of course which is essentially the ratio of the coulomb repulsion enormous compared with conventional crystals. These between neighbouring particles to the kinetic energy of experiments have been performed in RF traps (Blumel et each particle. Solid-like behaviour is expected when r al 1988, Wineland et al 1987a) as well as in Penning traps exceeds 170 from molecular dynamics simulations of an (Gilbert et a1 1988). In an RF trap the crystal may take up infinite plasma. The values of r achieved in the experia fixed orientation in space, due to the slight imperfec- ments range up to several times this, so the results permit tions in the trap potential. Clouds of up to 100 ions have a comprehensive test of the theory and therefore a fuller been studied by Blumel et al (1988) and smaller clouds understanding of phase transitions in general. have been imaged (Diedrich et a1 1987, Wineland et a1 Finally it should be mentioned that ion traps have 1987a). The transition between the ordered and dis- been used for various lifetime measurements of metaordered states shows a number of interesting features stable states of ions (up to tens of seconds in Li' and such as hysteresis and bistability; simulations of the ions Ba'). These states cannot be studied in any other can reproduce most of these features (Bliimel et a1 1988). environment because the observation time has to be so This work also sheds light on the mechanism of RF long, but the situation is ideal for traps. The measureheating in traps and shows its relation to chaos (Blumel ments are reviewed by Knight in Van Dyck et a1 (1988). et a1 1989, Hoffnagle et al 1988). 7. Conclusion The use of ion traps for studies of the properties of ions is an area of rapid growth in physics. They have shown their value especially in precision measurements where the advantages of a stable, well controlled and perturbation-free environment are obvious. In the future it is expected that they will be used in the next generation of frequency standards, with projected reproducibilities of at least lo-'', and in future measurements of fundamental constants. However, it is not only standards laboratories and the like which make use of ion traps. There is much interest in the application of traps to the study of fundamental physical processes such as quantum jumps, laser-atom interactions, and phase transitions as well as to routine measurements of atomic masses, hyperfine structure and lifetimes. This is clearly a technique which will continue to be applied in many new areas in the future. Figure 6. Crystalline structure of seven 24Mg+ ions in an RF trap (Diedrich et a/ 1987). The mean distance between the ions is 23 pm. Acknowledgments The author has benefited from many conversations with colleagues at the National Physical Laboratory and 103

12 ~ 1969 ~ 1982 R C Thompson Imperial College, especially Professor P L Knight. This work was supported in part by the UK Science and Engineering Research Council. References Barwood G P, Bell A S, Klein H and Gill P 1989 Trapped Yb' as a potential optical frequency standard Proc. 4th Int. Symp. on Frequency Standards, ed. A de Marchi (Berlin: Springer) pp Bergquist J C, Diedrich F, Itano W M and Wineland D J 1989 Hg' single ion spectroscopy Proc. 4th Int. Symp. on Frequency Standards, ed. A de Marchi (Berlin: Springer) pp Bergquist J C, Hulet R G, Itano W M and Wineland D J 1986 Observation of quantum jumps in a single atom Phys. Rev. Lett Bergquist J C, Itano W M and Wineland D J 1987 Recoilless optical absorption and Doppler sidebands of a single trapped ion Phys. Rev. A Bergquist J C, Wineland D J, Itano W M, Hemmati H, Daniel H-U and Leuchs G 1985 Energy and radiative lifetime of the 5d96s22D,,2 state in Hg I1 by Doppler-free two-photon laser spectroscopy Phys. Rev. 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Commun Sauter Th, Gilhaus H, Neuhauser W, Blatt R and Toschek P E 1988 Kinetics of a single trapped ion: multistability and stimulated two-photon light force Europhys. Lett Sauter Th, Neuhauser W, Blatt R and Toschek P E 1986b Observation of quantum jumps Phys. Rev. Lett Schnatz H, Bollen G, Dabkiewicz P, Egelhof P, Kern F, Kalinowsky H, Schweikhard L and Stolzenberg H 1986 In-flight capture of ions into a Penning trap Nucl. Instrum. Methods A Schwinberg P B, Van Dyck R S Jr and Dehmelt H G 1981a Trapping and thermalisation of positrons for geonium spectroscopy Phys. Lett. 81A b New comparison of the positron and electron g factors Phys. Rev. Lett Rlchard Thompson is a Lecturer in Physics at Imperial College, London. From January 1983 to September 1986 he worked at the National Physical Laboratory in Teddington. Previously he was at the Clarendon Laboratory in Oxford, apart from a year and a half spent at the Kernforschungszentrum in Karlsruhe. West Germany. He obtained his first degree at Oxford in 1976 and his doctorate in atomic spectroscopy in