Atomic physics (2005)

16This phase, and the other similar phases that arise at each interaction, lead to an o set in the final output which is not important. But these phases must remain constant in time, otherwise the interference ‘washes out’.

17In a three-grating interferometer only a fraction of the amplitude goes in the required direction. Note, however, that the simple treatment of standing waves assumed the ‘thin’ grating approximation, but often the interaction between the matter waves and light takes place over a su ciently long distance that Bragg di raction occurs (as in crystals).

The phase factor depends on the relative phase of the two laser beams.16 These two states separate, as shown in Fig. 11.7, and this first region corresponds to a beam splitter for matter waves. After a free-flight through a distance L the atom enters the middle interaction region where it undergoes a π-pulse that acts on both arms of the interferometer to swap the states |1, p ↔ |2, p + 2 k . (In this apparatus the transit time for the atom to pass through the laser beams determines the duration of the Raman interaction.) The paths come back together after a further distance L and the final π/2-pulse acts as the beam splitter that mixes the wavepackets to give interference. The complete π/2–π–π/2 sequence gives a Mach–Zehnder interferometer. A comparison of Figs 11.3 and 11.7 shows that the Raman scheme resembles the Mach–Zehnder interferometer more closely than the three-grating device; the Raman scheme does not direct any amplitude in unwanted directions and the middle interaction region in the Raman interferometer acts just like a mirror to change the direction (transverse momentum) of both paths through a small angle.17

A Raman pulse and standing light wave give the same opening angle between the arms for a given wavelength of laser light and both

schemes use light whose frequency is detuned from the atomic transition to avoid spontaneous emission. A crucial di erence between these methods arises in the detection. The three-grating apparatus, with standing waves or nano-fabricated structures, distinguishes the two outputs by their di erent directions. Therefore the three-grating devices require a highly-collimated atomic beam at the input whose angular divergence is less than the angle between the two output directions θdi . The output channels of the Raman scheme are the two di erent states |1, p and |2, p + 2 k , as shown in Fig. 11.7; thus experiments only need to determine the final state of the atom, e.g. using a laser beam that excites a transition from |2 to another state that gives fluorescence for atoms in state |2 but not for those in |1 .18 This means that Raman interferometers use more of the atoms from a given source because they do not need to have tight collimation.19 Although the flux of atoms does not a ect the size of the phase shift given by eqn 11.10, the strength of the measured signal determines how precisely that phase shift can be measured, i.e. the interferometer measures a smaller fraction of a fringe if the signal-to-noise ratio is higher.20 Thus the type of Raman interferometer shown in Fig. 11.7 measures rotation more precisely than a three-grating device.

11.6Conclusions 257

18As in the atomic fountain described in Section 9.9.

19The creation of the two Raman beams with a well-defined frequency di erence, and other technical details, are described in Section 9.8.

20This argument assumes that it is purely statistical fluctuations (noise) that limit the precision, not systematic shifts.

11.6Conclusions

Matter-wave interferometers for atoms are a modern use of the old idea of wave–particle duality and in recent years these devices have achieved a precision comparable to the best optical instruments for measuring rotation and gravitational acceleration. We have seen examples of experiments that are direct analogues of those carried out with light, and also the Raman technique for manipulating the momentum of atoms through their interaction with laser light, as in laser cooling. Laser cooling of the atom’s longitudinal velocity, however, only gives an advantage in certain cases (see the section on further reading).21 Similarly, the high-coherence beams, or atom lasers, made from Bose condensates do not necessarily improve matter-wave devices—in contrast to the almost universal use of lasers in optical interferometers. Partly, this arises because of the interactions between the atoms themselves, as discussed in the derivation of the nonlinear Schr¨odinger equation in Chapter 10, which lead to phase shifts that depend on the atomic density. So far interferometry experiments that use BEC have been performed to find out more about the condensate itself, rather than as instruments for precision measurement of physical quantities. The interaction of atoms with the periodic potential produced by a standing wave gives a lot of interesting physics, in addition to the di raction described here, and we have only scratched the surface of atom optics.

21The Ramsey fringes produced by atomic fountain clocks arise from interference of the internal (hyperfine) states of atoms, but in this chapter the ‘atom interferometer’ has been reserved for cases where there is spatial separation between the two arms.

22This journal is a useful source of similar articles.

The review in Contemporary physics22 by Godun et al. (2001) surveys the field of atom interferometry at a level suitable for undergraduates, including important applications, such as the precision measurement of gravitational acceleration g, that have not been included here. The monograph Atom interferometry edited by Berman (1997) is a rich source of information on this subject.

This chapter describes the principal methods of building ion traps and a few of their many applications in physics. The examples illustrate the extremely high-resolution spectroscopy possible with microwave and laser radiation. The theme of precision measurement in an environment with very few perturbations continues in the next chapter on quantum computing—an application that has stimulated a new wave of research on trapped ions.

12.1The force on ions in an electric field

Charged particles in electromagnetic fields experience much larger forces than neutral atoms. An ion with a single charge e = 1.6 × 10−19 C in an electric field of 105 V m−1 experiences a force

This electric field corresponds to 500 V between electrodes which are 5 mm apart.1 In comparison, a neutral atom with a magnetic moment of one Bohr magneton in a magnetic field gradient2 of dB/dz = 10 T m−1 experiences a force of magnitude

We also see this large di erence in a comparison of trap depths. In a trap operating with a voltage of V0 = 500 V, singly-charged ions have a maximum ‘binding’ energy of order 500 eV. 3 This trap depth corresponds to the kinetic energy at a temperature of 6 ×106 K. This is more than enough to trap ions, even if the ions do receive a large recoil kick during the ionization process—to load an ion trap experimenters send a weak (neutral) atomic beam through the trapping region where an electron beam ionizes a few of the atoms by knocking an electron o . These ions created by electron bombardment have much greater kinetic energy than the thermal energy of atoms at room temperature (equivalent to only 1/40 eV). It would be unwise to try to be more precise about the typical energy of an ion since it depends on the voltage used. In contrast, a magnetic trap for neutrals has a maximum depth of only 0.07 K. This was estimated in Section 10.1 by taking the magnetic energy µBB for B = 0.1 T, e.g. the force in eqn 12.2 over a distance of

1This assumes electrodes in the form of a parallel-plate capacitor. Although ion traps have a di erent geometry, this still gives a reasonable estimate and shows that the electrostatic force gives strong trapping for a voltage readily available in the laboratory.

2This value is typical of magnetic traps with coils wound with copper wire. Superconducting magnets give higher gradients.

3We consider ions with a single positive charge +e such as Mg+, Ca+ and Hg+, since few experiments use species that acquire additional electrons to give negative ions. Section 12.8 deals with highly-charged ions.

4The theorem dates back to the nineteenth century and James Clerk Maxwell discussed it in his famous treatise on electromagnetism.

5The derivation of this equation from

the Maxwell equation div D = ρfree assumes ρfree = 0 and a linear isotropic

homogeneous medium in which D =r 0E with r constant. Ions are usually trapped in a vacuum, where r = 1.

6It takes just a few lines of algebra to prove this from Laplace’s equation.

12.2Earnshaw’s theorem

Earnshaw proved that: A charge acted on by electrostatic forces cannot rest in stable equilibrium in an electric field.4

Thus it is not possible to confine an ion using a purely electrostatic field. Physicists have invented ingenious ways around this theorem but, before describing the principles of these ion traps, we need to think about the underlying physics. The theorem follows from the fact that an electric field has no divergence in a region with no free charge density, div E = 0.5 Zero divergence means that all the field lines going into a volume element must come out—there are no sources or sinks of field within the volume. Equivalently, Gauss’ theorem tells us that the integral of the normal component of E over the bounding surface equals the volume integral of div E, which is zero:

Hence E · dS cannot have the same sign over all of the surface. Where E · dS < 0 the electric field points inwards and a positive ion feels a force that pushes it back into the volume; but E · dS > 0 somewhere else on the surface and the ion escapes along that direction. A specific example of this is shown in Fig. 12.1 for the field produced by two equal positive charges with a fixed separation along the z-axis. Midway between the charges, at the point labelled P, the electric fields from the two charges cancel and the ion experiences no force, but this does not give stable equilibrium. The argument above holds true, so the electric field lines around the point P cannot all be directed inwards. When slightly displaced from P, a positive ion accelerates perpendicular to the axis, whereas a negative ion would be attracted towards one of the fixed charges. This behaviour can also be explained by the fact that the point P is a saddle point of the electrostatic potential φ. The electrostatic potential energy eφ of the ion has the same form as the gravitational potential energy of a ball placed on the saddle-shaped surface shown in Fig. 12.2—clearly the ball tends to roll o down the sides. In this alternative way of looking at Earnshaw’s theorem in terms of electrostatic potential rather than the fields, stable trapping does not occur because the potential never has a minimum, or maximum, in free space.6

12.3The Paul trap

The analogy with a ball moving on the saddle-shaped surface shown in Fig. 12.2 provides a good way of understanding the method for confining ions invented by Wolfgang Paul. The gravitational potential energy of the ball on the surface has the same form as the potential energy of an ion close to a saddle point of the electrostatic potential. We assume here a symmetric saddle whose curvature has the same magnitude, but opposite sign, along the principal axes:

where x = r cos Ωt and y = r sin Ωt are coordinates in a frame rotating with respect to the laboratory frame of reference. The time average of this potential is zero. Rotation of the saddle shape around the vertical axis turns the unstable situation into stable mechanical equilibrium, and

Fig. 12.1 The electric field lines between two equal positive charges. Midway between the charges at the point P the electric fields from the two charges cancel. At this position the ion experiences no force but it is not in stable equilibrium. At all other positions the resultant electric field accelerates the ion.

Fig. 12.2 A ball on a saddle-shaped surface has a gravitational potential energy that resembles the electrostatic potential energy of an ion in a Paul trap. Rotation of the surface about a vertical axis, at a suitable speed, prevents the ball rolling o the sides of the saddle and gives stable confinement.

7Although we shall not analyse the mechanical system in detail, it is important to note that the wobbling motion is not entirely up and down but has radial and tangential components. Similarly, an object that floats on the surface of water waves does not just bob up and down but also oscillates back and forth along the direction of propagation of the waves, so its overall motion in space is elliptical. The discussion only applies for mechanical systems where friction has a negligible e ect so that the ball slides smoothly over the surface.

8The a.c. field does not change the ion’s average total energy because .the work done on the ion given by F · r cos(Ωt) sin(Ωt) averages to zero over one cycle—the force and velocity have a phase di erence of π/2.

12.3.1Equilibrium of a ball on a rotating saddle

In the mechanical analogue, the rotation of the saddle at a suitable speed causes the ball to undergo a wobbling motion; the ball rides up and down over the low-friction surface of the rotating saddle shape and the ball’s mean position only changes by a small amount during each rotation.7 The amplitude of this wobbling increases as the ball moves further from the centre of the saddle. For this oscillatory motion the time-averaged potential energy is not zero and the total energy (potential plus kinetic) increases as the object moves away from the centre. Therefore the mean position of the object (averaged over many cycles of the rotation) moves as if it is in an e ective potential that keeps the ball near the centre of the saddle. We will find that an ion jiggling about in an a.c. field has a similar behaviour: a fast oscillation at a frequency close to that of the applied field and a slower change of its mean position.

12.3.2The e ective potential in an a.c. field

To explain the operation of the Paul trap, we first look at how an ion behaves in a.c. electric field E = E0 cos(Ωt). An ion of charge e and mass M feels a force F = eE0 cos(Ωt), and so Newton’s second law gives

It has been assumed that the initial velocity is zero and r0 is a constant of integration. The field causes the ion to oscillate at angular frequency Ω with an amplitude proportional to the electric field. From this steadystate solution we see that the forced oscillation does not heat the ions.8 (These simple steps form the first part of the well-known derivation of the plasma frequency for a cloud of electrons in an a.c. field, given in most electromagnetism texts.) The following section describes an example of this behaviour in which the amplitude of the electric field changes with position E0(r).

12.3.3The linear Paul trap

one diagonally opposite and the a.c. voltage V = V0 cos (Ωt) is applied between the two pairs. Despite the fact that the voltages vary with time, we first find the potential by the usual method for electrostatic problems. The electrostatic potential φ satisfies Laplace’s equation 2φ = 0 (because div E = 0 and E = − φ). A suitable solution for the potential close to the z-axis, that matches the symmetry of the voltages on the electrodes, has the form of a quadrupole potential

The coe cients a0 and a2 are determined from the boundary conditions. There are no terms linear in x or y because of the symmetry under reflection in x = 0 and y = 0. The terms in x2 and y2 have opposite signs, and the variation with z is negligible for rods much longer than their separation 2r0. The potential must match the boundary conditions

To solve Laplace’s equation we have simply made a reasonable guess, taking into account the symmetry. This is perfectly justified since the uniqueness theorem says that a solution that fits the boundary conditions is the only valid solution (see electromagnetism texts). The usual

9This ignores the finite size of the electrodes and that the inner surfaces of the electrodes would need to be hyperbolic, e.g. a surface given by x2 −y2 = const., to match the equipotentials. However, by symmetry this potential has the correct form for r r0, no matter what happens near to the electrodes.

264 Ion traps

10This method would not be appropriate for shorter wavelengths, e.g. microwave radiation with frequencies of GHz.

11The two-dimensional quadrupole field between the four rods looks superficially like the quadrupole field with cylindrical symmetry in Fig. 12.1 and the analogy with a rotating saddle applies to both. A comparison of the potentials for the two cases in eqn 12.9 and eqn 12.23 shows that they are di erent. Note also that the electrostatic potential oscillates ‘up and down’ rather than rotating as in the mechanical analogy.

12The Mathieu equation arises in a variety of other physical problems, e.g. the inverted pendulum. A pendulum is normally considered as hanging down from its pivot point and undergoing simple harmonic motion with a small amplitude. In an inverted pendulum a rod, that is pivoted at one end, initially points vertically upwards; any slight displacement from this unstable position causes the rod to fall and swing about the stable equilibrium position (pointing straight down), but if the pivot point oscillates rapidly up and down then the rod remains upright whilst executing a complicated motion—the rod can make quite largeangle excursions from the vertical direction without falling over. The mathematical textbook by Acheson (1997) gives further details of the complexities of this fascinating system and numerical simulations can be seen on the web site associated with that book.

13This corresponds to the motion of an ion in a trap that has no d.c. voltage. In practice, ion traps may have some d.c. voltage because of stray electric fields but this can be cancelled by applying a suitable d.c. voltage to the electrodes (or additional electrodes near the four rods). The solution of the Mathieu equation with ax = 0 is discussed in the book on ion traps by Ghosh (1995).

14More detailed mathematical treatments of the Mathieu equation can be found in Morse and Feshbach (1953) and Mathews and Walker (1964).

method for solving an electrostatic problem applies even though the voltage on the electrodes changes because at the radio-frequencies (used in ion traps) the radiation has a wavelength much greater than the dimensions of the electrodes, e.g. a wavelength of 30 m for Ω = 2π ×10 MHz.10 The potential energy eφ of an ion has a saddle point in the middle of these electrodes that looks like the saddle shape shown in Fig. 12.2—a potential ‘hill’ in the x-direction and a ‘valley’ in the y-direction, or the other way around.11 From the gradient of potential we find the electric

field

E = E0 (r) cos (Ωt)

The equation of motion in the x-direction is

A change of variable to τ = Ωt/2 leads to

This is a simplified form of the Mathieu equation:12

with ax = 0.13 It is conventional to define the parameter in front of the oscillating term as 2qx (in anticipation of this e has been used for the ion’s charge), where

We look for a solution of the form

The arbitrary constant A gives the angular frequency of the overall motion and B is the amplitude of the fast oscillation at close to the driving frequency. The justification for choosing this form is that we expect an oscillating driving term to produce a periodic solution and substitution of a function containing cos Aτ into the equation leads to terms with cos Aτ cos 2τ .14 Substitution into the equation (with ax = 0) gives

We shall assume that A 1, so that the function cos Aτ corresponds to a much slower oscillation than cos 2τ , and also that the amplitude B 1 (both of these assumptions are discussed below). Thus the terms

proportional to cos Aτ cos 2τ dominate on each side, and equating their coe cients gives −4B = 2qx or

This amplitude of the fast oscillation is consistent with the result for a uniform electric field in eqn 12.6.15 This fast oscillation is called the micromotion. To determine the angular frequency A we consider how the mean displacement changes on a time-scale longer than the micromotion:

where qx is defined in eqn 12.14.17 We assumed that qx 1 but it turns out that this approximation works better than 1% for qx 0.4 (Wuerker et al. 1959). Since τ = Ωt/2 the mean displacement undergoes simple harmonic motion at an angular frequency given by

12.3The Paul trap 265

15Equation 12.10 shows that the com-

ponent of the electric field in this direction is E0 (r) · ˆex = −V0x/r02.

16The term 4AB sin Aτ cos 2τ timeaverages to zero.

17An arbitrary initial phase θ0 has been included to make the expression more general but this does not a ect the argument above.

or ωx 0.3 Ω. For a radio-frequency field oscillating at Ω = 2π ×10 MHz the ion must have a radial oscillation frequency ωx 2π × 3 MHz. If we choose ωx = 2π × 1 MHz (a convenient round number) then eqn 12.19 gives the numerical values V0 = 500 V and r0 = 1.9 mm for trapping Mg+ ions.18 To get this high trapping frequency the ion trap has electrodes closer together than we assumed in the introduction. By symmetry, the same considerations apply for motion in the y-direction, and so we define a radial frequency ωr ≡ ωx = ωy . The Paul trap has a sharp transition from stable trapping to no trapping in the radial direction when qr equals the maximum value of qx in eqn 12.20. Paul used this feature to determine the charge-to-mass ratio e/M of the ions and hence perform mass spectroscopy—generally the charge state is known (e.g. it is e or 2e, etc.) and hence M is determined.

So far we have only described confinement in the xy-plane. There are several ways to extend trapping to all three directions. For example, Fig. 12.3(b) shows a trap with two additional electrodes at z = ±z0 that repel the ions. For positive ions both of these end-cap electrodes have the same positive voltage to give a field similar to that shown in Fig. 12.1, with a minimum in the electrostatic potential along the z-axis—in the radial direction the static potential has a negligible e ect compared to the a.c. trapping. When the linear Paul trap has axial confinement weaker than that in the radial direction, i.e. ωz ωx = ωy , the ions tend to lie in a string along the z-axis with only a small micromotion

18In comparison, neutral atoms in magnetic traps oscillate at frequencies in the range 10–1000 Hz.