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The Isolated Electron

The Isolated Electron

The Isolated

From a single electron trapped in an artificial of macroscopic size a property of the electron called the g factor can be measured with unexcelled accuracy

by Philip Ekstrom and David Wineland

he electron is a remarkably sim- novel technique in which a single elec- With these properties it might seem ple of . It has tron is confined for weeks at a time in a that the electron could be understood Tand an electric ; it spins (or “trap” formed out of electric and mag- in terms of a simple mechanical mod- seems to ) with a bed amount of netic fields. In effect the electron and the el. The electron could be imagined as a angular , and it has a mag- confining apparatus make up an atom material particle with some definite size netic moment, so that an external mag- with macroscopic dimensions and an ex- and shape and with a negative electric netic exerts a twisting on traordinarily massive nucleus. Because charge distributed over its surface. The it. These four quantities constitute all the apparatus rests ultimately on the particle would be required to spin about the known properties of the electron; earth the nucleus might even be identi- an axis, much as the earth does, with a once their values have been established fied with the earth itself, and so the arti- specified and unvarying quantity of an- there is nothing more to be said about ficial atom has been given the name geo- gular momentum. the electron. nium, the earth atom. The of the In this model the of It can be argued that the electron is single electron in geonium is different the electron has a straightforward expla- even simpler than this tabulation of from that in an ordinary atom, and it nation. The rotation of a body with a properties might suggest. The reason is is more accessible to manipulation and charge on its surface necessarily entails that the four quantities are not all inde- measurement. For these reasons it was a circulation of , or in pendent; instead any one of them can be possible to measure the g factor with other words an . The cir- derived from the other three. For exam- an uncertainty of less than one part in culating current gives rise to a magnet- ple, the magnetic moment of the elec- 10 billion. ic field in the electron just as it does in tron is related to its mass, charge and the windings of an electromagnet. Both spin by a constant of proportionality hen the electron was discovered in the spin and the called the g factor. In the modern theory W 1897 by J. J. Thomson, it was im- magnetic moment can be represented by of the value of the g factor can mediately obvious that the particle has vectors, or arrows, lined up along the be calculated with great precision, and mass and electric charge; indeed, Thom- spin axis. By convention (and because it can also be measured experimentally. son made his discovery by measuring the charge of the electron is negative) Such measurements are therefore a sen- the ratio of mass to charge. It was also the vectors representing the spin and sitive test of the theory. Both the calcu- assumed, at least implicitly, that the the magnetic moment in opposite lation and the measurement are difficult, electron has the familiar mechanical directions. but they have been refined to such an properties characteristic of all bulk mat- Continuing to reason in terms of this extent that the g factor of the electron ter, such as a definite size and shape. A mechanical model, one would expect is now known to greater accuracy than magnetic moment and an intrinsic quan- the magnetic moment to be proportion- any other . The values tity of spin angular momentum were al to the electric current circulating at derived from theory and from experi- added to the list of properties some the perimeter of the spinning electron. ment are in accord to the last decimal years later during the development of The current should be directly propor- place known. the theory, following a pro- tional to the total electric charge and A series of experiments that has been posal made by Samuel A. Goudsmit to the rotational velocity. The rotation- under way for some years at the Univer- and George E. Uhlenbeck. The spin and al velocity in turn should be directly sity of Washington has recently culmi- magnetic moment were required in or- proportional to the spin angular mo- nated in a new measurement of the g der to explain certain features observed mentum and inversely proportional to factor, which has set a new record for in the spectrum of emitted or ab- the mass of the electron. The relation accuracy. The experiments employed a sorbed by . among these quantities, along with some numerical factors needed to keep the system of units consistent, can be ex- pressed succinctly in an : e p=-gqamc S. TRAP FOR AN ELECTRON captures the particle in a special configuration of electric and magnetic fields The electron occupies the central cavity formed by the two cap and the ring , which are machined to a mathematically determined shape: they are hyper- Here p (the Greek letter mu) is the mag- boloids Two additional electrodes, the guard rings, compensate for imperfections in the elec- tric field. The entire apparatus, which is about an inch and a half in diameter, is immersed in netic moment of the electron, s is thz liquid and inserted into the core of a superconducting magnet The electron is bound by spin angular momentum, e is the charge, the combination of static electric and magnetic fields in the trap, much as an electron in an m is the mass and cis the in atom is bound to the nucleus. Here the part of the nucleus is played by the apparatus, or even a . The constant of proportion- by the earth, on which the apparatus rests, and so the atom is called geonium, the earth atom. ality in the equation is the g factor, a 105 i h provide an explicit procedure for calcu- i VECTOR lating the numerical values of the vari- OF SPIN ous properties of the electron. The re- ANGULAR lation between the electron's magnetic MOMENTUM (s) moment and its spin, charge and mass takes the same form it had in the me- chanical model, and the constant of pro- portionality in the equation is the sameg factor. Moreover, a first approximation ! still yields a value of g = 2. That is

only an approximation, however, and I an infinite series of progressively small- , er corrections can be applied to it. Find- CHARGE (e) VECTOR I OF MAGNETIC ing the exact value of g by this method MOMENT (p) is impractical because the entire series of terms would have to be included in I 11 the calculation, but any finite precision can be attained by evaluating a finite PROPERTIES OF THE ELECTRON are electric cbarge (e), mass (mh spin angular momen- tum (s) and magnetic moment (p). A naive mecbnicd model conceives of the electron as a number of terms. spinning disk witb electric cbarge distributed on its perimeter. It is tbe current of circulating charge tbat gives rise to the magnetic moment, and the moment can therefore be derived from eginning in 1950, a series of experi- I tbe otber tbree properties. "be relation is defined by an equation in wMcb a constant of propor- B mental measurements of the g fac- tionality called tbe g factor appears, Tbe minus sign in tbe equation signifies tbpt tbe moment tor was undertaken by H.. Crane and a is opposite to tbe spin vector. The modern tbeory of electrons does not dow sucb a pictorial number of his colleagues and students at representation, but tbe relation of tbe magnetic moment to the spin still bolds, and g can be CPE the University of Michigan. What was with is witb equal culpted great precision. In tbe geonium experiments g measured precision. actually measured was not g but the I discrepancy between g and 2, and so I the project came to be known as the I dimensionless number. The g factor de- negative charge of the electron into a g-minus-2 experiment. Over the next termines how large a magnetic moment finite volume. The smaller the volume, two decades a succession of increasingly will be generated by a given amount of the larger the needed. Accord- precise values was reported by Crane, t spin, charge and mass. The minus sign ing to this scheme, the quite small mass Arthur Rich and their collaborators [see that precedes the g factor signifies that or energy of the electron implies that it "The g Factor of the Electron," by H. R. the magnetic moment is antiparallel to should have a rather large size. Experi- Crane; SCIENTIFICAMERICAN, January, I the spin angular momentum. ments in which electrons are scattered 19681. During the same period theorists In this mechanical model the value by other , however, effective- worked to evaluate progressively more assigned to g depends critically on the ly measure the sue of the electron, and complex terms in the infinite series. The j shape and size of the electron and on they indicate that the radius must be ex- ensuing duel between theory and exper- I how the electric charge is distributed ceedingly small. Indeed, all experimen- iment was punctuated by a few errors over its surface. The first experimental tal data gathered so far are consistent and subsequent corrections, but by 1972 i values of g came from the same spec- with the idea that the electron is a point the calculated and measured values of g troscopic observations that suggested particle, entirely without extension. The agreed to better than eight significant the existence of an intrinsic spin angu- arguments presented here then predict figures. An incidental benefit of this lar momentum. Those observations indi- that the electron mass is infinite, a mani- competition was the development of cated that g is approximately equal to 2. fest absurdity. new methods both for theoretical calcu- Some years later a more refined version Still another reason for doubting the lations and for experiments. It seemed of the quantum theory, formulated by accuracy of the mechanical model, and that each additional decimal place de- P. A. M. Dirac to be consistent with the that of the Dirac theory as well, derives manded some technical innovation. special theory of relativity, made pos- from refined measurements of the g fac- In the Michigan experiments a beam sible a precise theoretical calculation tor of the electron. Experimental evi- of electrons was passed through a mag- of 8; the value predicted was exactly 2, dence has shown that g is not exactly 2 netic field in an apparatus where the ori- in agreement with the observational evi- but rather is greater than 2 by about entation of the electrons' spin axis (or dence available at the time. .I percent; in other words, its value is magnetic moment) could be determined roughly 2.002. The Dirac theory could both before the passage and after. The iewing the electron as a rigid, rotat- not accommodate such an adjustment. applied field had two effects on the elec- Ving body is somewhat naive; after In the 1940's these problems were re- trons: it caused their spin axes to precess all, the motion of the particle must be solved by abandoning the mechanical and it caused the electrons themselves described by the laws of quantum me- model of the electron and devising a to describe a circular or helical orbit. chanics, where the notions of size and new and more abstract theory, quan- If g were exactly 2, the of the velocity cannot even be defined beyond tum electrodynamics. In quantum elec- spin precession and the orbit frequency a certain level of precision. Indeed, the trodynamics the electron is allowed to would be equal. The experiment mea- model has grave flaws, some of which be a dimensionless and its sured the difference between the fre- were recognized only days after it was mass is allowed to be infinite, at least quencies and hence determined g - 2. first proposed. For example, it turns out in principle. Surprisingly, there is a The accuracy of this method is limit- that the rotational velocity at the sur- mathemJtica1 procedure (called renor- ed mainly by the statistical of an face of the electron is greater than the malization) that cancels this infinity and experiment done with many interacting speed of light. Another source of diffi- recovers the observed, finite properties electrons and by deficiencies in the ex- .culty is the size attributed to the elec- of the electron, including the g factor. perimenter's knowledge of what hap- tron. The mass or energy of an electri- It is far from obvious how a particle pens to an electron during the mea- cally depends inversely with zero radius can have spin angular surement. Each electron moves through on its size. One reason this is so can momentum a magnetic moment. If 1 or the apparatus at high speed and spends be understood by noting that energy is offers no con- much less than a second in the magnetic required in order to pack the repulsive sistent mental picture, however, it does field. During that interval its trajectory - - can be ajtered by inhomogeneities in the CYCLOTRON MAGNETIC FORCE RESULTANT and by encounters with other elec- trons in the beam. The Michigan group continues to improve the experiment, seeking to control these uncertainties. An alternative strategy for improving the measurement of g is to confine a single, slow-moving electron for an ex- tended period in an environment that is simple enough to be analyzed precisely. This is the plan adopted in the geonium experiments done at the University of RON Washington. Th_e goal of these measure- UNIFORM MOMENTUM ments-a goal that can be approached CONFINEMENT OF AN ELECTRON in a uniform magnetic field results from the action but never attained-is to isolate an elec- of the field on the moving electric charge. The electron is subject to a force directed perpendic- tron from all extraneous influences and ular both to the field and to the particle’s direction of motion. The continuous application of make it hold still for examination. The the force diverts the electron into a circular path, which is called the cyclotron orbit The size long-term confinement of free electrons of the orbit increases in discrete steps as the energy and speed of the electron increase. By this mechanism the electron is constrained to revolve in a finite region of the field, but it is not yet was conceived more than 20 years ago fully confined; it can still drift parallel to the field lines, so that its trajectory becomes a helix. by Hans G. Dehmelt of Washington, and the work described here was carried out under his direction. The techniques ton is defined as ’s constant, h, the electron must always keep moving, for preparing and observing an isolated multiplied by e/4~mc.The other two however, a magnetic field can at least electron were developed mainly by us, quantities that determine the energy lev- confine its motion to a specified region. although our work was founded on ex- els are the strength of the external field The confinement results from the inter- periments with trapped clouds of elec- and the g factor. Any measurement of g action of the field with the charge of the trons done by our predecessors, Fred L. ultimately calls for a measurement of particle. If the electron is moving per- Walls and Talbert S. Stein. We have this energy difference in a context where pendicular to the field, a force arises that since left the project, but the single-elec- the strength of the applied field can also pushes the electron perpendicular both tron techniques have been refined fur- be determined. In practical experiments to its ofiginal direction of motion and ther by Robert S. Van Dyck, Jr., and even the need to know the field strength to the field. The result is to bend the Paul B. Schwinberg, who have mea- precisely is dispensed with. electron’s trajectory into a circle. sured the g factor with great accuracy. An electron at rest is a convenience, As the energy and speed of the elec- but it is a fictitious one; the uncertain- tron increase, so does the diameter of o understand the behavior of the ty principle of en- the circular orbit. Indeed, the speed and T electron in the geonium atom it is sures that no particle as light as the the orbital diameter change in concert helpful first to consider a simpler, ideal- electron can hold still for very long. If in such a way that the time required to ized system: an electron at rest in a uni- form magnetic field. Such a field can be represented by flux lines that are all par- allel and evenly spaced. The energy of this stationary electron depends on the orientation of its magnetic moment with respect to the external field. The energy is minimum when the moment and the field are parallel and maximum when they are antiparallel. It turns out that these two orientations of the electron, which are commonly called “spin down” and “spin up,” are the only ones possible. All intermediate orientations, where the magnetic mo- ment would be perpendicular to the field or would cross it obliquely, are forbid- den by the rules of quantum mechanics. (To be precise, the rules apply to the orientation of the spin axis and only in- directly to the magnetic moment. Since the two vectors always point in precise- ly opposite directions, however, the ori- entation of the electron can be defined with reference to either one of them.) Because an electron has only two pos- sible orientations a stationary electron in a uniform magnetic field has just two distinct energy levels. The difference in generated by the applied to the trap electrodes is superposed on energy between the levels is the product the uniform magnetic field. Because the electron is negatively charged the electric field exerts of three quantities. One of these quan- a force on it that is directed opposite to the field lines. The force has a vertical component, and tities is a combination of fundamen- so an electron that drifts toward either of the cap electrodes is pushed back toward the mid- plane. lo general the electron will overshoot the midplane and so will oscillate parallel to the tal constants called the , axis of the trap. The electric field also has a radial component, which the electron to- which defines the between an ward the ring electrode. Radial motion is opposed, however, by the action of the magnetic field, applied magnetic field and the magnetic which deflects the electron into a circular orbit called the magnetron orbit Hence in the com- moment of a particle. The Bohr magne- bined electric and magnetic fields the electron is fully captive, but its motion is complicated.

complete one circuit is almost constant. these combined states into two side-by- the constant that determines the spacing In other words, the period of the elec- side energy ladders. One ladder repre- between rungs on a single energy ladder. tron’s .motion is constant (or nearly so) sents all the states with the spin up and For typical values of the applied mag- regardless of its speed or energy. The the other ladder includes all those with netic field both of these are, constancy of the orbital period is spoiled the spin down. Within each ladder the in the region of the electro- only by an effect of : at rungs correspond to successively larger magnetic spectrum. The difference be- high speed the mass of the electron in- orbits, and the rungs are labeled with the tween them, which is called the anomaly creases and so the period of revolution principal , n. The spac- frequency, induces the double transition becomes slightly longer. For low-energy ing between any two rungs is the same where the spin and the orbital diameter electrons, however, the relativistic cor- on both ladders, but one entire ladder is change simultaneously. The anomaly rection is small. The of displaced with respect to the other by an frequency is therefore proportional to an electron in a magnetic field is called amount equal to the energy needed to g - 2. Since g exceeds 2 by only about the cyclotron motion because it is em- flip, or reverse, the spin. Since this ener- .1 percent, the anomaly frequency is ployed in a cyclotron to accelerate elec- gy is roughly equal to the rung spacing, smaller than the cyclotron-transition fre- trons or other charged particles. one ladder begins about a rung higher quency by a factor of about 1,000. At this point quantum mechanics in- than the other. tervenes again to modify the motion of Three kinds of transition between he procedure for measuring the g the electron. la classical, or pre-quan- states are possible in this system. Sup- Tfactor can now be given in terms tum, physics an electron circulating in pose the electron initially has its lowest- of frequencies. Two quantities a magnetic field can take on any arbi- possible energy: its spin is down, so that must be determined: the cyclotron-tran- trary energy and so can have an orbit its magnetic moment is parallel to the sition frequency and the anomaly fre- of any diameter. In the quantum theory field, and its cyclotron orbit is the small- quency. The actual values of both fre- only certain discrete and orbital est one allowed, with n equal to 0. The quencies depend on the strength of the sizes are allowed. There is a smallest- electron therefore occupies the bottom magnetic field and on the value adopted possible orbit, followed by a sequence rung of the lower energy ladder. A first for the Bohr magneton, and so they can- of larger orbits associated with increas- transition would leave the spin orienta- not be predicted beyond the accuracy ing energies; orbits of intermediate size tion fixed but would enlarge the orbit to with which these quantities are known. are forbidden. Each of the allowed or- the next allowed size; this transition cor- In the ratio of the two frequencies, how- bits is specified by a quantum number, responds to climbing one rung of the ever, both of these factors cancel and n, which can only assume integer values same energy ladder, to the n = 1 state. the only significant quantities remaining in the series 0, 1, 2, 3 and so on. The energy required to effect the change are the proportionality constants that Because the cyclotron orbits are IS equal to the spacing between rungs define the level spacings in the ladder quantized a diagram of the allowed en- and hence is given by 2 times the Bohr diagram. For the anomaly frequency ergy levels of the electron takes the form magneton times the field strength. that constant is g - 2 and for the cyclo- of a ladder. The bottom rung corre- This level having been reached, an- tron-transition frequency it is exactly 2. sponds to the smallest orbit of lowest other transition might flip the spin of the Hence the ratio of the two frequencies energy, and each successive rung repre- electron without altering the orbit, so equals (g - 2)/2. Multiplying by 2 and sents the next-largest orbit and the next- that the electron would move from the adding 2 gives the value of g itself. highest energy state. What is most im- n = 1 level of the spin-down ladder to This method of finding the g factor portant, the spacing between rungs is the corresponding n = 1 level of the may seem needlessly roundabout. Why almost constant (and it would be exact- spin-up ladder. The energy required in not just measure the spin-flip transi- ly constant if relativistic effects could be this case is g times the Bohr magneton tion frequency, which is directly pro- ignored). The energy difference between times the field strength. The third possi- portional to g, and have the desired val- rungs is equal to 2 times the Bohr mag- bility is for the orbit to change size by ue in a single step? One part of the an- neton times the strength of the applied one unit and for the spin to flip simul- swer lies in the substantial advantage to magnetic field. These are the same quan- taneously. For example, the electron be found in working with a dimension- tities that figure in the energy difference might step from the n = 1 rung of the less ratio. Suppose the spin-flip frequen- between the down and the up spin states, spin-down ladder directly to the n = 0 cy were measured to an accuracy of with one significant change: the g factor rung of the spin-up ladder. These levels eight decimal places. To find the g factor is replaced by the exact integer 2. The almost coincide; the energy required to that frequency must then be multiplied similarity of the two formulas suggests a flip the spin to the antiparallel orienta- by the Bohr magneton, by the strength procedure for measuring g, or at least tion is almost made up by the energy of the applied magnetic field and by for measuring the ratio of g to 2, without liberated when the orbit shrinks by one Planck’s constant. The eight decimal knowing to great precision the strength unit. Only the small difference in energy places would remain significant only if of the magnetic field or the value of the between these quantities need be sup- all these factors were known to at least Bohr magneton. All that is necessary is plied. It is equal to (g - 2) times the the same accuracy. Taking the ratio of to measure, under the same conditions, Bohr magneton times the field strength. two frequencies eliminates all the extra- the energy spacing between the spin The energy required for these changes neous quantities, so that the precision of states and the spacing between succes- of state is conveniently expressed in the frequency measurement alone deter- sive orbital states. Dividing the one en- terms of the frequency of electromag- mines the accuracy of the result. ergy by the other gives a ratio where the netic radiation that induces the tran- There is another argument in support Bohr magneton and the field strength sition. Any frequency can be convert- of the indirect method. Because g is appear in both the numerator and the ed into energy units by multiplying by known to be very close to 2, deriving the denominator, so that they cancel. Planck‘s constant, but for our purposes value of g from a measurement of g - 2 it is not even necessary to make the con- yields a large divident in accuracy. If n practice there is no way to “turn off’ version. The frequency itself can be al- (g - 2)/2 is known to eight significant I the spin while measuring the orbit lowed to stand for the energy. The fre- figures, then multiplying by 2 and add- size, nor can the electron be persuaded quency needed to flip the spin is propor- ing 2 yields a value of g accurate to to stay still while the energy of its spin tional to g, the constant that determines 11 significant figures. This seeming par- orientation is determined. Any real state the difference in energy between spin adox of accuracy obtained at no cost of the electron has an energy deter- states. In the same way the frequency of can be resolved in a number of ways. mined both by its orbit and by its spin radiation that changes the cyclotron or- One explanation is that the anomaly fre- direction. It is convenient to segregate bit size by one unit is proportional to 2, quency is 1,000 times smaller than the

109 , spin-flip frequency, so that a measure- electron is not trapped by the magnetic perposed on that orbital velocity there ment with a given fractional accuracy field alone. In a uniform magnetic field is now an axial oscillation parallel to yields a much smaller uncertainty in the electron cannot migrate across the the magnetic-field lines. In addition the terms of cycles per second. Another ap- field lines, but there is nothing to prevent interaction of the electric and the mag- proach is to point out that in any mea- it from drifting parallel to the field and netic fields gives rise to a third mode of surement of the anomaly frequency the thereby escaping. Something is needed motion. The electric field not only sup- cyclotron-orbit level spacing of exactly to seal the ends of the trap; what we plies a restoring force that confines the 2 is subtracted by the physical system chose at the University of Washington electron along the symmetry axis but itself and not by the experimenter; as a was an electric field superposed on the also has a radial component that tends result only the small discrepancy from 2 magnetic field, in a configuration called to push the electron away from the cen- need be measured. The situation of the a . ter of the trap and toward the ring elec- experimenter is somewhat like that of The electric field is formed by - trode. This radial force might allow the a surveyor asked to lay out a course ages applied to three electrodes, name- electron to escape if it were not for of 1,001 meters. The distance could be ly a cap at each end of the trap and a the magnetic field, which overcomes it measured meter by meter, but high ac- ring that girdles the midplane. Both caps through the same mechanism that gen- curacy could be achieved more easily are given a negative electric charge and erates the cyclotron rotation. The radial with a chain known to be exactly one the ring is given a positive charge. In force acting on the electron is converted kilometer long. That distance could then the resulting combination of electric and by the magnetic field into a circumferen- be subtracted from the total and only magnetic fields the electron is confined tial drift, so that the electron is deflected one meter would have to be measured in three dimensions. If it drifts horizon- into a circular path. with high precision. tally, the magnetic field diverts it into This last perturbation imposed on the a circular orbit. If it wanders vertical- electron’s trajectory is called the magne- he procedure adopted at the Uni- ly, parallel to the magnetic-field lines, tron motion, after the kind of micro- I Tversity of Washington for measuring it is reflected by the electric field on ap- -generating device in which it plays 1- the g factor can be summarized as fol- proaching the repulsive, negative charge an important part. The magnetron orbit lows. First an electron is captured in a of the cap electrode. is much larger than the cyclotron orbit, uniform magnetic field and then its mo- In general an electron reflected from but the electron moves around it much tion in the field is analyzed in order to one cap electrode will overshoot the slower. (“Large” and “slow” are appro- determine two quantities: the frequency midplane and then be repelled by the priate here only for purposes of com- of applied radiation that changes the or- field at the opposite end of the trap, only parison. The electron typically executes bit size and the frequency that induces to overshoot the midplane once again. some 3 5,000 magnetron revolutions per a simultaneous change in orbit size and In this way an oscillation along the sym- second, but in each of those revolutions spin orientation. From the ratio of these metry axis of the trap can arise. If the it completes 1.4 million cyclotron or- frequencies g can be found through triv- frequency of the oscillation is to be in- bits.) The magnetron motion slightly ial arithmetical operations. This synop- dependent of the amplitude, the elec- modifies all the energy states of the elec- \ sis does indeed set forth the principles trodes must have a certain shape. They tron, shifting the levels in the ladder dia- I of the experiment, but it is rather like must be hyperboloids with cylindrical gram. The shifts are small, however, and the famous recipe for elephant stew that symmetry around the axis of the trap. they can be calculated as well as mea- > begins, “Take one medium elephant.. ..** The addition of the electric field com- sured experimentally. Hence in a preci- ,. Most of the work has been left out, and plicates the motion of a trapped particle. sion measurement of the g factor it is much of the adventure. The electron continues to execute the possible to correct for the effects of the 1 One deficiency of the experiment as rapid circles around magnetic-field lines magnetron motion. it has been described so far is that the that constitute the cyclotron motion. Su- The trajectory of the electron in the

SIGNAL SIGNAL SIGNAL GENERATOR GENERATOR t I W

ELECTRODE ELECTRODE CAPACITANCE CAPACITANCE

VOLTAGE

EMPTY TRAP TRAP WITH ELECTRON

SIGNAL SIGNAL SIGNAL DETECTOR DETECTOR DETECTOR

EQUIVALENT CIRCUIT defines a configuration of elementary canceled in the external circuitry. An electron in the trap has the same devices that would reproduce the electrical properties of the trap. The effect as a small capacitance and a large inductance connecting the motion of the electron is observed by applying a -frequency sip- cap electrodes. Through these phantom circuit elements a small sig- nal at the top of the trap and detecting the transmitted part at the bot- nal reaches the detector. What actually happens is that the signal ap tom. When the trap is empty, the electrodes are coupled only by small plied to the top electrode drives the ele&oo’s axial oscillation, which capacitances. The signal tmnsmitted through these capacitances is induces a current of the same frequency in the bottom electrode,

113 properties that allow the electron signal to be isolated and monitored. Under typical operating conditions the inductance of the equivalent circuit representing the electron is equal to about 10,000 henries, a value that would be associated with a large coil of wire; indeed, such a large inductance can- not ordinarily be obtained in a high- frequency circuit. Of course, the coil of wire does not exist; the inductance merely reflects the small inertia of the trapped electron. What really happens inside the trap is that the electron re- sponds to the applied radio-frequency signal by oscillating in synchrony with it and thereby induces a replica of the sig- nal in the bottom electrodes. There is a strict limit on how much power can be transmitted in this way: if the oscilla- tions of the electron grow too large, the particle strikes the electrodes and is lost. One might say the resonant circuit burns out easily. A sensitive receiver is needed to detect even the largest signals the 5 10 15 20 25 trapped electron can handle. a TIME (MINUTES) We first observed clear evidence of single trapped electron by detecting the RECORD OF SEVEN ELECTRONS confined in the trap was made by detecting the current small induced currents. After pumping induced by the axial oscillation. The applied signal was adjusted to an amplitude near the maxi- to obtain a good vacuum in the trap a mum the electrons could tolerate. As the electrons escaped one at a time seven reductions in the small number of electrons from a hot induced current were observed. At a smaller amplitude an electron can be confined for weeks. filament were loaded into it. A radio- frequency was applied to the top of the trap and was adjusted to an Penning trap can thus be analyzed into currents in the trap electrodes. The cur- amplitude near the maximum the elec- three components. The electron gyrates rents can be detected in the circuitry to trons could tolerate. The signal reaching rapidly in small loops (the cyclotron which the electrodes are connected. the bottom of the trap was then moni- motion), and the center of these loops The electrical characteristics of the tored. After several minutes the output progresses slowly around a larger circle Penning trap can be analyzed in terms diminished a little, then it held steady (the magnetron motion). At the same of an equivalent circuit: a network of for several more minutes and dropped time the electron is vibrating back and elementary devices such as resistors, ca- again. Over the course of about half an forth along the axis of the trap, perpen- pacitors and inductors that could re- hour there were seven reductions in the dicular to the plane in which the cyclo- place the trap and mimic all its proper- magnitude of the received signal, all of tron and magnetron orbits lie. The total ties. In this kind of analysis the trap is about the same magnitude, and after the motion is analogous to that of a carnival imagined as a black box whose inner last reduction the signal had returned to ride called the Tilt-a-Whirl, where a car- workings the experimenter does not the background level characteristic of riage pivots in a small circle on a turn- know; he sees only the three terminals the empty trap. What we were observ- table that rotates in a larger circle while corresponding to the two cap electrodes ing, of course, was the loss of individual at the same time riding over a series of and the ring electrode. electrons from the population of seven undulations. that had evidently filled the trap initial- In what sense can this system be con- hen the trap is empty, the equiva- ly. In subsequent experiments we found sidered an atom, warranting the name W lent circuit consists of small ca- that if the driving voltage was reduced geonium? It is atomlike mainly in that pacitors connecting the three electrodes. when only one electron remained, that the electron is bound to a field of force If a high-frequency signal is applied to last electron would stay in the trap and like that of an . The mo- the top of the trap, across the upper cap could be observed for many weeks. tion of the electron within this field is electrode and the ring, some small rem- quantized, and transitions between vari- nant of the signal will be conveyed to the he permanent confinement of a sin- ous modes of motion correspond to defi- bottom of the trap by the capacitance T gle electron was an exciting result nite frequencies of radiation. In other between the cap electrodes. The trans- and great fun, but it was only the first words, the trapped electron has a spec- mitted signal can be detected by a re- step toward measuring the spectrum trum, which can be studied in the same ceiver connected between the lower cap of geonium and hence measuring g. A way as an atomic spectrum is studied. and the ring. number of improvements in the appara- It is all very well to create an artifi- The capture of a single electron dra- tus were needed before precision mea- cial atom, but if the atom is to be of matically alters the equivalent circuit. surements could be undertaken; Van any use for measurements, there must When the trap holds an electron, the Dyck and Schwinberg joined us in mak- be a means for observing its spectrum. trap acts as if a resonant circuit consist- ing some of these changes, and later they A single atom cannot emit or absorb ing of a capacitance and a large induc- took charge of the project. much radiation; indeed, the quantity is tance were connected in series between The geometry of the electric field so small that it can hardly be detected, the two cap electrodes. Although the turned out to be critical, and to obtain much less measured accurately. It turns contribution of these phantom circuit better control over it a new trap was out there is a more practical way to ob- elements to the transmission of the sig- constructed with five electrodes instead serve the electron’s motion; the key is nal is weaker than that of the trap ca- of three. The extra two elements, which the axial vibration. which indiices small nncitancp the new nath hac dictinrtive are rinm ctatinnpd in the oanc hptwppn I fie original cap and ring electrodes, al- employed to cool the trap to four de- in the spin orientation. The frequen- low small defects in the field to be com- grees . The cooling reduced ther- cies at which these changes take place pensated for electronically. The hot fil- mal fluctuations in the motion of the were what we had set out to determine. ament, which had tended to spoil the electron and helped in maintaining a Only the frequency of the axial oscilla- vacuum whenever a new batch of elec- high vacuum. These practical details, tion can readily be observed, and in a trons was loaded, was replaced by a which may seem mundane in the tell- uniform magnetic field that frequency field-emission point. The entire appa- ing, absorbed much of the several years is essentially independent of the cyclo- ratus was redesigned to fit in a tube of effort that went into the project. tron orbit and the spin direction. The about an inch and a half in diameter, so solution we eventually adopted was to that it could be inserted in the bore of a ne modification of a fundamental devise a means of coupling the axial fre- powerful superconducting magnet. The 0 nature was needed: in the original quency to the energy of the electron. magnet had to be cooled with liquid he- apparatus there was no way to detect The coupling was accomplished by in- lium, and a separate bath of helium was changes in the cyclotron orbit size or serting a ring of (a ferromagnet-

SPIN-FLIP TRANSITION n= 3

n= 3 n= 2

n= 0 n= 0 SPIN-DOWN SPIN-UP SPIN DOWN SPIN UP LADDER LADDER

CYCLOTRON-ORBIT TRANSITION n= 3 Ir i. n= 3 n= 2 Ii 11 I n= 2 n= 1

n= 1 n= 0 7 T 1ucu 28 n= 0 n= 1 SPIN-DOWN SPIN-UP SPIN DOWN SPIN DOWN LADDER LADDER

COMPOUND SPIN-FLIP AND CYCLOTRON-ORBIT TRANSITION n= 3

n= 3 n= 2

n= 2 n= 1 11E 1 L r n= 0 ‘7 n= 1 i: n= 0 SPIN-DOWN SPIN-UP I SPIN DOWN SPIN UP LADDER LADDER IC ENERGY STATES OF THE ELECTRON depend on the size magnetic field (B); the proportionality constant different for each of is ‘c the cyclotron orbit (given by the value of tbe quantum number n) kind of transition between states. If the electron flips its spin from and on the orientation of the particle’s spin vector (denoted “down” down to up, tbe energy change is proportional to g. If tbe size of the or ‘‘up“). For each spin orientation there is a series of almost equal- cyclotron orbit increases by one unit, the energy needed is propor- ;1 ly spaced energy levels corresponding to the various cyclotron orbits. tional to 2. A compound transition in which the spin Rips and the cy- : 11 The diagram of the two series resembles a pair of ladders, which are clotron orbit is enlarged simultaneously requires an amount of ener- displaced vertically by slightly more than one rung. The energy in- gy proportional to tbe difference between these factors, or g - 2. Be- I_ tervals between states are proportional to the strength of the applied cause the value of g is dose to 2 this is a small quantity of energy. 117 I ic metal) in the midplane of the trap, tude of the axial frequency shift changes ladder of cyclotron states. Because of encircling the original ring electrode. with each change in the cyclotron orbit these random thermal disturbances the The nickel ring had the effect of bow- size or in the spin state. The distortion of electron is constantly moving up and ing the magnetic field at the midplane, the magnetic field must be kept small or down whichever ladder it happens to so that it was somewhat more intense the energy levels under examination will be on. The resulting graph of energy near the ends of the trap; the resulting be too greatly modified, but only a small as a function of time resembles a crop configuration is called a magnetic bot- shift in the axial frequency is needed in of grass: the root level corresponds to tle. If there were no axial oscillation, order to make the changes in quantum the lowest rung on the ladder and the so that the electron remained in the state observable. Typically the axial os- blades of grass are energy excursions midplane, the magnetic bottle would cillation has a frequency of 60 mega- to higher rungs. Any measurement of have no effect. On each excursion away hertz and the frequency shift caused by the energy levels must be made against from the midplane, however, the slight the magnetic bottle is a few hertz. a background of this thermal . convergence of the magnetic-field lines In an ideal experiment the electron The measurement is made by careful- tends to reinforce the effect of the would maintain a steady motion in a ly searching for the frequencies that electric field in pushing the electron particular cyclotron orbit and a partic- work the largest change on the rapidly back toward the central plane. As a re- ular spin orientation until the experi- fluctuating pattern. First a signal with sult of this additional restoring force menter supplied the energy needed to a frequency near the cyclotron-transi- the frequency of the axial oscillation is change the state. A graph of the energy tion frequency is supplied and the aver- increased. of the electron as a function of time age height of the “grass” is monitored. The total strength of the interaction would therefore show a series of precise The applied signal is then tuned slow- between the electron and the distorted steps, all taken under the experimenter’s ly through a narrow range of frequen- magnetic field depends on the energy control, and would be flat between the cies. When the frequency corresponds of the cyclotron orbit and on the spin steps. A real experiment is quite differ- exactly to the interval between rungs orientation. The influence of the field is ent. Even at four degrees K. the thermal on the ladder, the electron absorbs ener- greater when the orbit is larger or when agitation of the electron has an average gy strongly and climbs at least several the spin is up. For this reason the magni- energy equivalent to a few rungs on the rungs. Thus the cyclotron-transition fre- quency is the frequency that gives the tallest grass. In the measurement of the anomaly frequency the background of thermal noise is less disruptive. The transition to be observed in this case is a compound one (both the spin and the orbit must change), and it is much less likely to hap- pen spontaneously. In general the spin does not flip unless the experimenter does something to cause it to flip, and so all the electron’s thermal transitions are normally confined to just one of the lad- ders in the energy diagram. To induce the lateral transition between ladders a strong signal must be supplied. How does the experimenter know the orientation of the spin at any given mo- ment, or in other words how does he know which ladder the electron is on? The needed information can be found in the energy diagram itself: the spin-down ladder, it will be remembered, extends about one full rung lower than the spin- up ladder. The root level of the grass therefore lies at a lower energy when the spin is down. The anomaly frequency is the frequency that changes the root level most often. When both the cyclotron-transition frequency and the anomaly frequency have been measured, it remains only to divide, multiply by 2 and add 2 in or- der to calculate g. The two frequencies can be determined precisely, even in the noisy background, because both the res- onances are very sharp, that is, a small change in the applied frequency causes a MAGNETIC-FIELD LINES large change in the rate at which transi- tions are observed. The instrument with MAGNETIC BOTTLE is created in the trap by a ring of nickel (a ferromagnetic metal), which which the frequencies are generated slightly distorts the uniform imposed magnetic field. Because of the ring the field lines are is ultimately based on a cesium atomic bowed outward at the midplane and effectively converge toward the ends of the trap. This ta- pering of the field contributes an extra restoring force to the axid oscillation of the electron. , but even the absolute calibration The magnitude of the force depends on the size of the cyclotron orbit and on the spin ori- of the instrument does not limit the entation, so that changes in those quantities give rise to slight shifts in the frequency of the accuracy of the measurement. Because axial oscillation. As a consequence of this coupling of the three modes of motion, the ener- both frequencies are generated by the gy state of the electron can be determined by monitoring the frequency of the axial oscillation. same instrument and only the ratio mat- ters, it is only the relative accuracy over CHANGES IN the range of frequencies employed that CYCLOTRON-ORBIT SIZE contributes to the uncertainty. AS it turned out, both of us had left be geonium project by the time the ap- paratus was ready for precision mea- surements. We learned of the first re- sults through a telephone call from Van t Dyck. Soon after he first detected transi- t i tions at the anomaly frequency he was m able to announce an improved value for > 0 the g factor, and he has since refined the z technique to obtain an accuracy of 10 3 (3 decimal places. The best value of g is W now 2.0023 193044, with a probable er- E -I ror of less thanane unit in the last deci- 9 mal place. Major improvements in ac- curacy are still to come, through further a refinements in the method. The theoreti- .. , TIME j cal value, calculated according to the Y prescription of quantum electrodynam- CHANGES IN SPIN ORIENTATION ics, is in exact agreement and has about the same uncertainty. RECORDING OF THE FREQUENCY SHIFT in the axial oscillation yields a graph that resembles a crop of In response to thermal fluctuations the size of the cyclotron orbit The value of g cited above is a sta- grass. changes continually, causing rapid and random excursions in the uipl frequency; these are the tistically weighted average of measure- blades of gper. Reversclls of spin orientation are much mer, and they are observed as a change ments made with several electrons, but h the mot level of the grass. ”be g factor is determined by irradiating the electron with elec- separate results for individual electrons tromagnetic energy and measuring two frequencies: the frequency that induces changes in the also agree to within about one unit in the orbit size and thereby yields the tallest grass, and the frequency that Bips the spin (with a si- last decimal place. The finding of agree- multaneous change in orbit size) and hence gives the highest rate of changes in the root level. ment from one electron to the next is hardly surprising, but it should not be dismissed as trivial. tering an ordinary electron, since the Dyck is now attempting an improved Fundamental to the idea of an ele- and the electron would then measurement of the ratio of the elec- mentary particle of matter is the as- annihilate each other. These problems tron mass to the mass. Deh- sumption that all particles with the same have been solved, however, by Schwin- melt has suggested that a single heavy name are perfectly identical. In this arti- berg. The g factor of the positron is ex- confined in a miniature trap might cle we have spoken repeatedly of “the pected to be exactly equal to that of the serve as an optical frequency standard electron,” meaning an archetype that electron. This expectation will soon be that would be free of certain defects is reproduced in innumerable identical given a stringent experimental test. common to present standards. Toward copies everywhere in the . On The techniques developed for the this end Dehmelt and W. Neuhauser, the assumption that all copies are iden- study of geonium can also be applied M.Hohenstatt and P. E. Toschek of the tical the experimenter makes measure- to the measurement of properties oth- University of Heidelberg have isolated ments with any electron he finds at hand er than the g factor, and with particles and even photographed a single in the laboratory, or with any collection other than the electron and the positron. ion in a trap similar to the Penning trap. of them, and reports his results as in- For example, the compensated Penning The common goal of all these undertak- formation about the properties of “the trap could be operated as a high-resolu- ings is to see the simplest elements of electron.” Some might call this policy tion mass , an instrument nature in their simplest setting, where presumptuous; in any event it can do that determines the ratio of a particle’s there is the greatest how of understand- no harm to test its validity. Although mass to its charge. By this mdans Van ing them. - there is much indirect evidence suggest- ing that all electrons are identical, geo- b nium offers one of only a few opportun- ities to carry out a direct test. The logical sequel to the measure- ment of the electron g factor is a mea- surement of the corresponding number for the positron, the positively charged of the electron. Earlier in- vestigations of the positron g factor, car- ried out by Rich and his colleagues at Michigan, have been hampered by the fact that are much less plen- (9-2)=1.1596522 x 10-3 [EIGHT SIGNIFICANT FIGURES] tiful than electrons. With the methods 2 devised at the University of Washing- ton the measurement could be made, at = 2 [@+I] + 2 = 2.oo23193o44 [iiSIGNIFICANTFWRES~ least in principle, with a single positron, which would be trapped to form a single atom of antigeonium. The experiment is CALCULATION OF THE g FACTOR begins with the measured values of the cyclotron certainly not without challenge: the pos- transition frequency (WJ and the anomaly frequency (wa), which is equal to the difference be- tween the spin-flip frequency (wJ and y,. Both frequencies depend on the Bohr magneton itrons are emitted with high energy in (p~)and on the strength of the imposed magnetic field (B); Planck’s constant (/I) also enters the of certain nuclei, the equation. In the ratio of Y, to Y, these quantities cancel exactly; what remains is the ratio and they must be slowed down and in- (g - 2112. Multiplying by 2 and adding 2 gives g itself; moreover, if the ratio is measured to an stalled in a trap without ever encoun- ~ccurrcyof eight significant figures, the value of g can be determined to 11 significant figures.

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