Fundamental physics with trapped ions

Contemporary Physics

ISSN: 0010-7514 (Print) 1366-5812 (Online) Journal homepage: http://www.tandfonline.com/loi/tcph20

Fundamental physics with trapped ions G. Zs. K. Horvath , R. C. Thompson & P. L. Knight To cite this article: G. Zs. K. Horvath , R. C. Thompson & P. L. Knight (1997) Fundamental physics with trapped ions, Contemporary Physics, 38:1, 25-48, DOI: 10.1080/001075197182540 To link to this article: http://dx.doi.org/10.1080/001075197182540

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Contemporary Physics, 1997, volum e 38, number 1, pages 25±48

Fundamental physics with trapped ions

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G. Zs. K. H ORVATH *, R . C. T H OM PSON and P. L. K N IG H T Ion traps allow us to study single quantum systems, cooled to the lowest vibrational state of motion within con® ning electric and magnetic ® elds. In this article, we describe the working principles of ion traps and the methods used to cool the ions ( principally laser cooling) . W e then discuss how these cold ions, held almost at rest in the trap, can be used to illuminate fundamental issues of quantum mechanics: quantum jumps, the quantum Z eno eŒect and the quantum statistics of photons scattered by the ions. Finally, we describe how several ions loaded in a trap are cooled into ordered quasi crystals. 1.

Introduction

Ion traps have been in use for the con® nement of atomic ions and other charged particles for more than 30 years (for an historical review see, for example, Werth (1985)). In this time the ion-trapping technique has developed into a mature technology with well known advantages for particular types of experiment. Of particular interest is the possibility of applying laser cooling to trapped ions in order to cool the ions to kinetic temperatures in the millikelvin range (or lower), and the ease of working with single ions. Both of these features make many exciting and novel types of experiment possible. In this article, we survey brie¯ y some aspects of the ® eld of ion traps. We do not attempt to give a complete review but concentrate on the use of ion traps for the study of fundamental physical phenomena (see also Schenzle (1996)). More complete reviews can be found elsewhere (for example Thompson (1990, 1993), Blatt et al. (1992) and G hosh (1995)). We start (section 2) with a discussion of how diŒerent designs of ion trap work, and then in section 3 we discuss the cooling techniques used in traps. Section 4 surveys the use of traps to study quantum phenomena, sections 5, 6 and 7 look at the dynamics of ions in Paul, Penning and combined traps and section 8 presents a brief conclusion. 2.

Ion traps

In this section we give an outline of the working principles and properties of ion traps. Ideally a trap for ions would A uthors’ address: Optics Section, Blacket t Laboratory, Imperial College, Londo n SW7 2BZ, U K. *Current address: Institut d’ optiqu e theoriqu e et appliqu eÂ e, BP 147, 91403 Orsay Cedex, F rance. 0010-7514/97 $12.00

consist of a three-dimensional electromagnetic potential well. U nfortunately, this cannot be achieved, because according to Earnshaw’s theorem (for example Jackson (1975)) it is not possible to generate a minimum of the electrostatic potential in free space. H owever, this limitation can be overcome in several ways, all of them having particular advantages and disadvantages. We shall limit our description to schemes based on the use of a quadrupole electric potential with cylindrical symmetry of the form: w (r , z) =

U0 (2z2 - r 2 ), R 20

(1)

where r 2 = x 2 + y2 , R 20 = r 20 + 2z20 is a geometrical constant and U 0 is a potential. Such a potential can be easily realized by three electrodes of hyperbolic section (® gure 1). The trap consists of two end caps held at the same potential and a ring electrode at a diŒerent potential. At the trap centre, the potential forms a saddle and the charged particles will be con® ned either in the radial plane or in the axial direction but will escape in the other direction. In the Penning trap this is overcome by the addition of a magnetic ® eld, while in the Paul trap an rf potential is applied to the electrodes.

2.1.

T he Penning trap

Con® nement in the Penning trap (Penning 1936) is achieved by adding a static magnetic ® eld to the electrostatic ® eld. When a potential U0 is applied between the electrodes the electric potential inside the trap is of the form of equation (1). For trapping positively charged particles, U0 must be such that the ring electrode is biased negatively with respect to the end caps. With the sign convention used in equation (1) this means that U0 must be positive. This potential has a saddle point at the trap centre, with a minimum in the axial direction and a maximum in the radial plane. The ion will 1997 Taylor & Francis Ltd


26

G. Z s. K. Horvath et al.

Figure 1. Truncated quadrupole trap electrodes: in the Penning trap the two end caps carry a positive potential and the ring electrode carries a negative potential (for trapping positively charged particles). The separation of the end caps is 2z 0 and the radius of the ring is r 0 .

Figure 2. Radial orbit of a single particle in quadrupole traps. (a) Penning trap with a magnetron radius larger than the modi® ed cyclotron radius. (b) Same but with the magnetron radius smaller than the modi® ed cyclotron radius. (c) Radial orbit in the pseudopotential approximation (the micromotion is neglected) of the combined trap (Paul side); the amplitude of the modi® ed cyclotron motion (clockwise) is smaller than the amplitude of the magnetron motion (anticlockwise).

therefore be con® ned in the axial direction but attracted towards the ring electrode. A static homogeneous magnetic ® eld along the z axis provides radial con® nement by forcing the particle’s motion into epicyclic orbits in the radial plane (® gure 2). The equations of motion for a single trapped ion of charge q and mass m are easily solved to give the following solutions: (2a) x(t ) = r + cos(!+ t + } + ) + r - cos(!- t + } - )

= - r + sin (!+ t + } + ) z(t ) = r z cos(!z t + } z )

y( t )

r - sin (!- t + } -

),

(2b) (2c)

where

= qB (cyclotron frequency),

(3)

2

=

(4)

2

= !2c -

!c !z

!1 !+

=

m 4qU0 (axial frequency), mR 20 2!2z

1 (!c + 2

!1 )(modified

(5) cyclotron frequency),

(6)

Figure 3. Oscillation frequencies of a single particle in a Penning trap as a function of the trap voltage. The frequencies are plotted for a magnesium ion in a ® eld of 1 T. See the text for the de® nition of the various oscillation frequencies. !-

= 1 (!c 2

!1 )(magnetron

frequency),

(7)

and r + , r - and r z are the amplitudes of the various degrees of freedom, and } + , } - and } z are their initial phases. The axial motion is simple harmonic with frequency !z . In the radial plane the motion consists of a superposition of two circular motions which results in an epicyclic orbit. The modi® ed cyclotron frequency !+ , mainly due to the eŒect of the magnetic ® eld alone, is under normal trapping conditions close to the true cyclotron frequency !c ; in fact we can write !+ = !c - !- . The slower magnetron orbit at frequency !- is simply the result of the E ´ B drift. The condition !- < !+ is always true and using typical trapping parameters we have !- < !z < !+ . The oscillation frequencies as a function of the applied potential are plotted in ® gure 3. Typical trapping parameters are B = 1 T, R 20 = 5 ´ 10- 5 m 2 , U0 = 10 V and for magnesium ions the oscillation frequencies are then

= 2p ´ !- = 2p ´ !+ = 2p ´ !z

285 kH z, 72 kHz, 566 kH z,

The H amiltonian for a single trapped particle can be written as (Itano and Wineland 1982) H

= 1 m!1 !+ r 2+ 2

-

1 1 2 2 2 m!1 !- r - + m!z r z . 2 2

(8)

We see that the energy associated with the magnetron motion is negative. That is, the magnetron motion is unbound and therefore ions con® ned in a Penning trap are in unstable equilibrium. This complication is the price paid in eŒectively circumventing Earnshaw’s theorem. This means that, in order to cool the magnetron motion, that is to reduce its kinetic energy or equivalently to reduce the amplitude of its motion r - , energy must be added to this



degree of freedom! As a consequence, ions in a Penning trap cannot be cooled by a buŒer gas, as this cooling technique relies on the removal of energy from the ions by collisions with a neutral gas. Laser cooling in the Penning trap is also made more di cult. Finally the Penning trap must therefore be operated in an ultra-high-vacuum (UH V) environment in order to limit collisions with background gas, as they would increase the canonical angular momentum of the trapped particles, increasing the amplitude of the magnetron motion. The di culties in machining hyperbolic electrodes and providing some access to the trapped ions has led to some electrode designs whose shapes are far from ideal. Classical designs have truncated hyperbolic or spherical electrodes with access holes drilled in the electrodes. In some designs an electrode is made of a metallic mesh to allow for collection of the ¯ uorescence light from the ions. Cylindrical traps having hollow end caps are commonly used with superconducting magnets. Similarly, a planar trap allowing a large solid angle for collection of ¯ uorescence light for operation in a classical electromagnet has been operated at Imperial College. H owever, all these designs lead to anharmonicities in the trapping electric potential. These can be compensated for by the use of compensation electrodes and/or by an adequate design. Other imperfections arise from the departure from cylindrical symmetry of the electric ® eld and from any misalignment between the electric ® eld axis and the magnetic ® eld. The presence of other ions in the trap also modi® es the motion of the ions. Their presence leads to an outward force, in addition to the trapping ® elds. This causes a modi® cation of the oscillation frequencies (space-charge eŒect) and limits the maximum density achievable in a Penning trap. Until the end of the 1970s the Penning trap was not widely used, most experiments being performed in the Paul trap. This was probably due to the need for a relatively high magnetic ® eld and the di culty of cooling ions in the Penning trap (laser cooling was only proposed in 1975 (HaÈ nsch and Schawlow 1975, Wineland and Dehmelt 1975) and actually used in 1978 (Wineland et al. 1978)). Ions have been stored in a Penning trap for several days (Wineland et al. 1978). G abrielse (1992) has even transported a few electrons trapped in a Penning trap over thousands of kilometres. Laser cooling has been used to cool trapped ions to temperatures lower than 1 K, allowing very high resolution spectroscopic measurements. Since there is no `rf heating’ for large clouds as there is in the Paul trap (see the following section), extremely large clouds can be trapped and cooled. It also allows the study of cold òne-component plasmas’ (normal plasmas contain ions and electrons, whereas a onecomponent plasma is made up only of a single species, here the trapped ions). The presence of a strong magnetic ® eld can be required for some experiments, making the Penning

27

trap the ideal environment. The use of a strong, highly stable and homogeneous magnetic ® eld has allowed the use of the Penning trap for extremely high-accuracy mass spectroscopy measurements. Some of these applications will be discussed in more detail later in this article.

2.2.

T he Paul trap

The majority of experiments performed in ion traps have used the Paul trap. It is generally easier to run a Paul trap than a Penning trap, it is also easier to laser cool in the Paul trap, and in particular the Lamb±Dicke regime can be more easily reached in the Paul trap (this is a state in which the motion of the cold ions is restrained to dimensions much smaller than the transition wavelength, leading to elimination of the ® rst-order D oppler eŒect; see section 3.3). The electrode con® guration of the ideal Paul trap is identical to that of the Penning trap. In this trap, an oscillating electric potential is applied between the ring and the two endcap electrodes, usually in conjunction with a static electric potential. The trap potential then has the form w (r , z)

( t)

= U0 + 2V0 cos2 r 0 + 2z0

(r 2 - 2z2 ).

(9)

The trap is stable in the axial direction and unstable in the radial plane for half the cycle and vice versa for the next half of the cycle. However, owing to the ® eld inhomogeneity, the force averaged over a period of the oscillating ® eld does not average to zero but is directed towards regions of weak ® eld, that is towards the trap centre. At trap centre, there is no ® eld variation; therefore there is no motion due to the rf ® eld and ideally a particle set at trap centre would remain at rest. The equation of motion for a particle of mass m and charge q is d2 x i + (a i - 2qi cos s )xi ds 2

= 0, i = r , z,

(10)

where - az

qz

= 2ar = 16qU20 ,

(11)

8qV0 , m R 20

(12)

t.

(13)

=-

m R0

s

2qr

=

=1 2

Equation (10) is a Mathieu equation and its solutions may be stable or unstable depending on the values of the coe cients a and q (McLachlan 1947). By choosing appropriate values for U0 , V0 and we can select values for a i and qi such that the motion of the ion will be stable in both radial and axial directions. Figure 4 gives the stability diagram, showing the regions of stability in the a± q parameter space. There are several stability regions where trapping is theoretically


28


Figure 4. (a) Stability diagram for the radial (Ð ) and axial (. . . ) degrees of freedom of the Paul trap. Note the two largest intersecting regions where stable trapping is possible. (b) Detail of the ® rst stability region. (c) First stability region in the combined trap. The stability parameters a Z and q Z , de® ned in equations (11) and (12), are proportional to the applied dc and rf potentials respectively.

possible; however, up to now, trapping has been demonstrated only in the ® rst stability region. In general the solution to the equations of motion is complicated but, when a i = 0 and qi ! 1, an approximate solution can be found. In this case the motion reduces to 1 x i (t ) = Ai C i, 0 cos( b i t + } i ) 2 1 1 + qi cos (1 + b i ) t + } i 4 2 1 1 (14) + qi cos (1 b i) t + } i , 4 2 where 1 2 2 (15) qi , b i » 2 We see that, for a i = 0 and qi ! 1, the motion can be described by a slow secular harmonic oscillation at frequency 1=2b i with, superimposed, a micromotion consisting of two fast harmonic oscillations at frequencies (1 ± 1=2b i ) but of much smaller amplitude. The secular motion can be obtained simply by averaging the eŒect of the rf ® eld over one period. This approach leads to a description of the secular motion in terms of a pseudopotential.

{

f

f

g

g}

The potential depth of the trap can be de® ned as the potential energy diŒerence between trap centre and the electrodes. Working in the pseudopotential approximation for typical trapping parameters (R 20 = 5 ´ 10- 5 m 2 , = 2p ´ 2. 57 MH z) the U0 = - 10 V, V0 = 200 V and potential depth for magnesium ions is approximately 10 eV. As in the Penning trap, ideal electrodes cannot be made to generate the potential described by equation (1). The stability regions are not substantially altered except for the appearance of instability lines (for example Wang et al. (1993)). Often spherical electrodes are used as they are easier to machine and for many applications small deviations from the ideal potential are of little importance. Just as for the Penning trap, the presence of other trapped ions modi® es the potential well seen by the ions. This has the eŒect of reducing the depth of the pseudopotential well and imposing an upper limit on the density of an ion cloud in the trap. The Paul trap was the ® rst ìon’ trap to be successfully demonstrated; in this ® rst experiment a trap was used to con® ne charged aluminium dust particles (Wuerker et al. 1959) following a proposal by Paul et al. (1958) who developed a trap for atomic ions (Fischer 1959). Storage



29

as negative because of its associated total negative energy. The zero crossing of the magnetron frequency corresponds to a change in the potential associated with it from a potential hill (!- < 0) to a potential well (!- > 0); the sense of rotation is similarly inverted from clockwise (!- < 0) to anticlockwise (!- > 0) as shown in ® gure 2, while the sense of rotation of the modi® ed cyclotron motion remains clockwise. The side where !- < 0 is therefore called the Penning side of the combined trap, while the other is called the Paul side. One important fact is that, on the Paul side, the energy associated with the magnetron motion is positive and as a consequence all problems associated with the magnetron motion in the Penning trap disappear. The equations of motion in a combined trap are

Figure 5. The combined trap oscillation frequencies for a wide range of trap parameters: (a), experimental data; (Ð ), ® tted theoretical oscillation frequencies. See the text for the de® nition of the various oscillation frequencies. (From Dholakia et al. (1992).)

times of several days have since been reported (Plumelle et al. 1980). In the absence of cooling, the storage time is limited by `rf heating’. This is a mechanism by which kinetic energy is coupled from the rf ® eld into the secular motion during ion ±ion collisions (Brewer et al. 1990a). Rf heating seems to be responsible for limiting the maximum achievable density and minimum temperatures for an ion cloud. Except for when a light buŒer gas is used to cool the trapped ions, the rf trap is generally operated in a U HV environment at 10- 6 Pa or less to reduce ion ±neutral collisions. Many experiments have been performed with relatively small clouds, and often even with single ions. 2.3.

The combined trap

A third type of quadrupole trap is the combined trap. This is simply a con® guration where all three ® elds of the Paul and Penning traps are present. The equations of motion for this trap have been derived classically (Fischer 1959) and quantum-mechanically (Li 1988). The mass-selective nature of the trap has been demonstrated experimentally by Fischer (1959) and the ion oscillation frequencies have been measured for various trapping parameters (Dholakia et al. 1992). Figure 5 shows the measured oscillation frequencies in the combined trap for a wide range of trapping parameters. On the left half of the diagram, the rf amplitude is kept constant while the magnetic ® eld is varied. On the far left (B = 0), we have a pure Paul trap. We see that the addition of a magnetic ® eld removes the degeneracy of the radial oscillation frequencies. On the right half of the diagram it is the magnetic ® eld which is kept constant while the rf amplitude is varied. The pure Penning trap is reached when the amplitude of the rf ® eld is zero, on the far right. The magnetron frequency in the Penning trap is represented

d2x + (a r - 2qr cos s )x ds 2

= 2 !c

dy , ds

!c dx d2y , + (a r - 2qr cos s )y = - 2 ds ds 2

(16a) (16b)

The motion along the z axis is the same as in the Paul trap. These two equations can be decoupled by changing to a frame rotating at !c =2. Thus the motion in this rotating frame is completely equivalent to the motion in the Paul trap if we substitute a r + g2 for a r , with g = !c = . In the laboratory frame, the oscillation frequencies are then !n, r

= ìï

î

!n, z

n±

= ìï

î

1 b 2 n±

r

± 1 b 2

1 g2 z

þ

üï

üï þ

.

,

(17) (18)

The main advantage of the combined trap over the Paul trap is that the regions of stability are larger in the combined trap. This means that larger ion densities should be achievable. Furthermore, ions of opposite charges and diŒerent masses can be trapped simultaneously (see section 7). 2.4.

L inear and ring rf traps

In many experiments the desired signal-to-noise ratio can only be achieved if several ions are trapped. In a conventional rf trap, only the trap centre is free from micromotion; if two or more ions are present, the Coulomb repulsion will push the ions into regions where they will experience the rf ® eld. This will lead to an increase in their kinetic energies, dramatically limiting therefore the spectroscopic resolution of the experiment. An easy way to circumvent this problem is to design a trap where the rf ® eld vanishes along a line instead of just at a single point. This leads to linear traps and ring traps, with the trapping potentials applied between opposite pairs of rod electrodes. Con® nement in the plane perpendicular to the trap axis is achieved in the same way as in the conventional Paul trap, with rf ® elds. In the linear trap, con® nement along the trap axis is obtained by adding a weak static electric ® eld along

30


the trap axis using ènd-cap’ electrodes at each end of the linear trap. In the ring trap, no such device is necessary as the electrodes are bent round to form a continuous loop. These traps have been experimentally demonstrated (for example R aizen et al. 1992, Waki et al. 1992). In particular, it has been possible to prepare strings of very cold ions along the axis of the traps with negligible micromotion. Arrays of cold ions in linear traps have considerable potential as logic gate elements in quantum computers (Barenco 1996).


2.5.

M iniature traps

In recent years there has been a strong interest in traps capable of achieving very strong con® nement. Such kinds of trap have important practical applications; with a strong con® nement the Lamb±D icke regime can be reached relatively easily. It also makes it possible to achieve resolved side-band cooling using allowed electric dipole transitions. This permits the rapid cooling of trapped ions to their motional ground state. By trapping two ions in such a trap, the distance between them can be made very small. When their separation becomes smaller than the wavelength of the laser-cooling radiation, cooperative eŒects such as superradiance are expected (DeVoe and Brewer 1996). With no static potential applied to the electrodes, the maximum secular axial frequency is given by (JeŒerts et al. 1995): !z

=

2 qV0 2 . R 0 0 908m

1=2 .

(19)

Strong con® nement clearly requires a large trapping voltage and a small trap. Large values of V0 are limited by electric ® eld breakdown or arcing. Small traps with hyperbolic or even spherically shaped electrodes are increasingly harder to machine as they become smaller. This problem can be overcome by using electrodes whose shape means that they can easily be scaled to very small sizes. However, as the electrode shape diverges from the ideal hyperbolic shape, the eŒective depth of the pseudopotential decreases. In this way part of the gain obtained from the ease of making small electrodes is lost. The challenge consists in ® nding the optimum shape of electrodes. JeŒerts et al. (1995) have measured a secular axial frequency of about 60 MHz for beryllium ions which have a linewidth of 19 MH z (JeŒerts et al. 1995). Another ® eld of potential application of miniature rf traps is cavity quantum electrodynamics experiments (Rempe 1993), where some experiments will required the con® nement of a single ion to within a node of a standing wave (Cirac et al. 1993). 3.

Cooling techniques

The cooling of trapped ions is of great importance in most experiments. When they are loaded into the trap, ions usually have temperatures of thousands of kelvin (corresponding to energies of up to a few electronvolts, compared

with the trap depth of typically 10 eV) and their density is very low. There are four good general reasons to cool an ion cloud. (1) Hot ions will be more easily lost through collisions with background gases than will cooled ions. In this way, cooling can dramatically improve storage times. (2) Cold ion clouds, as their spatial extension is smaller, will experience lower anharmonicities and inhomogeneities of the trapping ® elds. This is particularly important for mass spectroscopy experiments where masses are determined through the precise measurement of the ion oscillation frequencies. (3) If the detection method relies on the detection of photons scattered by the ions from a laser beam tuned close to resonance, it is clear that a cold cloud will scatter more photons than a low-density hot cloud. This is especially true for a single ion as the signal-to-noise ratio can become very small if the ion does not spend a signi® cant amount of time scattering photons. (4) Finally, cooling can drastically reduce the Doppler broadening, allowing extremely accurate spectroscopic measurements. In the best cases, the LambDicke regime can be reached where the ® rst-order Doppler eŒect can be completely eliminated. Although the ® rst-order D oppler eŒect can be eliminated using other methods such as twophoton absorption, the second-order (relativistic) Doppler shifts can only be reduced through cooling. In addition, cold ions can be of particular interest on their own, as their dynamics can be signi® cantly diŒerent from the dynamics of hot ions. For example, strong laser cooling has allowed the observation of a phase transition in small clouds of trapped ions in a Paul trap (Diedrich and Walther 1987b) and the study of strongly coupled plasmas in the Penning trap (Brewer et al. 1988). It has also made possible the cooling of a single ion to the quantum-mechanical ground state of the trapping potential (Monroe et al. 1995). We describe now a few techniques for cooling trapped ions. Laser cooling will be discussed in more detail as it is the most frequently used cooling technique. Laser cooling of neutral atoms was described in this journal by Foot (1991), and applications of cold atoms by Adams (1994). 3.1.

R esistive cooling and related techniques

R esistive cooling has the advantage of being applicable to any trapped particle. In fact for this reason it is mainly used to cool subatomic particles or ions whose energy levels make laser cooling di cult to achieve. It is best explained by considering the axial motion of a trapped particle. As the trapped particle moves along the z axis, it induces image charges on the end-cap electrodes. If we now electrically



connect the two electrodes, we shall observe an electrical current, which images the ion’s displacement. If a resistor is inserted in the circuit, the motion will be damped. The minimum achievable temperature is limited by Johnson noise in the resistor, which is temperature dependent. So the best results are obtained by placing the whole circuit in a liquid-helium bath. This method can be applied to the radial degrees of freedom by splitting the ring electrode into two parts. H owever, the magnetron degree of freedom cannot be cooled in this way as a reduction in its energy leads to an increase in its amplitude, that is an increase in its kinetic energy. The magnetron motion can be cooled by side-band cooling where cooling is achieved by coupling it to the axial motion (Wineland and D ehmelt 1974). Other related cooling techniques, such as stochastic cooling (Lagomarsino et al. 1991), active-feedback cooling (Dehmelt et al. 1986) and adiabatic cooling (Li et al. 1991), permit the cooling of all degrees of freedom.

3.2.

Collisional cooling

Collisional cooling is the simplest cooling method. Trapped particles can be cooled by collisions with a gas of lower temperature. The most common of these methods is buŒer gas cooling. This method relies on energy exchanges via collisions between the trapped ions and a light neutral buŒer gas introduced in the vacuum system at typical pressures of 10- 4 Pa (10- 6 mbar). The buŒer gas will quickly transfer energy absorbed from the ions through collisions with the vacuum vessel. Clearly the minimum achievable temperature is given by the vessel temperature. This method has the advantage of simplicity. H owever, there are drawbacks. The collisions between the cooling gas and the ions may lead to the loss of trapped ions through charge exchange mechanisms and they may also cause perturbations to the energy levels of the trapped ions. Another type of collisional cooling is sympathetic cooling. In this case a species of ions is cooled via collisions with another species of ions which is cooled by another means. This will typically be done when one wants to cool an ion for which there is no laser wavelength available for laser cooling. In such a case, collisional cooling can be done by simultaneously trapping a laser-coolable species of ions (Larson et al. 1986). Sympathetic cooling can also be used when one wants the ions to be kept cold but not to be perturbed directly by the laser radiation (for example Tan et al. (1995a)). As stated before, buŒer gas cooling cannot be applied to the Penning trap as it would result in an increase in the amplitude of the magnetron motion of the ions and their eventual loss. N evertheless, a scheme has been devised by Savard et al. (1991) where buŒer gas cooling is achieved in the Penning trap. The increase in the amplitude of the

31

magnetron motion is counteracted by coupling the magnetron motion to the modi® ed cyclotron motion by an azimuthal quadrupole ® eld at the sum frequency of the two motions; in this way the heating of the magnetron motion is avoided. 3.3.

L aser cooling

Laser cooling is by far the most eŒective method of cooling. It was proposed simultaneously by Wineland and Dehmelt (1975) for the cooling of trapped ions, and by HaÈ nsch and Schawlow (1975) for the cooling of free atoms. However, the ideal of cooling using resonant radiation was originally proposed more than 40 years ago by Kastler (1950).² This cooling method makes use of the radiation pressure force to slow down the motion of the particles. In this section we shall discuss this method and see how it can be applied to the cooling of trapped ions. 3.3.1. Physical mechanism. We consider a two-level atom moving with velocity va in free space. The energy diŒerence between the ground and excited states is ò !a . The atom is illuminated by a monochromatic laser of angular frequency !l , propagating in the direction opposite to the atom’s motion. The atom, moving at velocity va towards the laser, will see the laser frequency Doppler shifted to higher frequencies by kl va where kl = !l =c. If the laser is tuned below resonance by an amount corresponding exactly to this Doppler shift, the atom will be in resonance and will absorb a photon. In absorbing this photon, the atom’s momentum will be reduced by an amount ò kl . The re-emission of a photon by spontaneous emission will again modify its momentum. H owever, spontaneous emission is symmetrical in free space and thus over a large number of emissions the average eŒect of the spontaneous decay on the atom’s momentum is zero. Therefore over many absorption ±emission events the net eŒect on the atom’s momentum is a reduction by N ò kl , where N is the number of scattered photons. The resulting force can be written as F max

=ò

kl

C

2

(20)

if we assume that the transition is strongly saturated. This force is usually called the radiation pressure force or scattering force. In the case of a trapped ion, if the ion oscillation frequency is much smaller than the resonance linewidth C (weak binding limit), the absorption±emission process is similar to that of a free atom. As the ion moves forwards and ² Although he believed tha t the achievemen t of very low temperatur es using his lumino-frigorique eŒect was possible, he did not believe in practica l applications : Èven if we achieve the experimenta l condition s for coolin g by radiation , this eŒect will remain a scienti® c curiosit y rather than a practica l means of obtainin g low temperatur es’ (K astler 1950).


32


backwards in the trap, if the laser detuning is larger than the atomic linewidth, photons will only be scattered when the ion is moving towards the laser, reducing the amplitude of its motion. If the laser is close to but still below resonance, overall the motion will still be cooled, as there will be more photons scattered when the ion is moving towards the laser than away from it (this is termed D oppler cooling). Fluctuations deriving from the randomness of the direction of the scattered photons and the random time distribution of the scattering events result in a heating of the trapped ion. Although the mean value of the eŒect of the scattered photons on the ion’s momentum is zero, the mean square momentum change is diŒerent from zero. The emission process is in fact a random walk process in momentum space, each emission representing a single step. We shall therefore have a diŒusion of k p2 l proportional to the number N of emitted photons: k p2 l = N ò 2 k2 . The increase in kinetic energy will therefore be N ò 2 k2 =2m. We see that the discreteness and randomness of the scattering process result in a heating of the ion’s motion (recoil heating). Close to equilibrium, the heating rate will be approximately constant, while the cooling rate will depend only on the amplitude of the motion, as the cooling rate is given by the diŒerence between the heating when the ion is moving away from the laser and the cooling when it is moving towards the laser. The minimum temperature is reached when the mean amplitude of the motion is such that the Doppler cooling compensates exactly the recoil heating. For a given minimum temperature, that is a given mean amplitude of the ion motion, the cooling rate (and therefore the minimum temperature) can be increased by increasing the variation in scattering rate as a function of velocity. In general this variation simply depends on the resonance linewidth. As we shall see in the next section, the minimum temperature is simply proportional to this linewidth. H owever, there is a fundamental limit to this minimum temperature. The smallest variation in the ion kinetic energy is the recoil energy imparted to the ion by one scattered photon, and clearly the ion’s kinetic energy, under constant photon scattering, cannot be less than this recoil energy² . In the following section we calculate the cooling limit of a trapped ion in a harmonic potential. In this we follow the same approach as Itano et al. (1982). 3.3.2. Doppler cooling limit. In what follows, we consider a two-level ion. The energy diŒerence between the ground and excited states is ò !0 and the excited state decays to the ground state by a one-photon electric dipole transition at a rate C . The trapping potential is supposed to be harmonic with angular frequency !z , which is assumed to be much smaller than the natural linewidth C (weak-binding limit), ² For magnesium , the D oppler limit is about 800 m K and the recoil limit is about 5 m K .

which in turn is assumed to be much less than !0 . These assumptions are easily veri® ed for many experiments; for example for 24 Mg+ the Zeeman splitting in a magnetic ® eld of about 1 T is much larger than the linewidth of the transition 43 MH z. The Zeeman sublevels that we can use for laser cooling are 3p 2 P 3=2 (M J = - 32) ® 3s 2 S1=2 (M J = - 12), and the system can, to a good approximation, be considered as a two-level system. The motion of the ion is assumed to be well described classically in the nonrelativistic limit. The ion interacts with a monochromatic polarized laser beam of frequency !l and wave-vector kl = !l =c. F or simplicity we approximate the laser beam with a plane wave. By doing this, we neglect the dipole force (for example Foot (1991)). Let us consider a scattering event where a free atom moving at velocity v absorbs a photon from the laser beam. Conservation of momentum and energy gives

= mv+

m v9

(21)

ò kl,

1 9 2 1 (22) mv + ò !0 = mv2 + ò !l, 2 2 where v9 is the velocity of the atom after absorption. Combining these two equations we have (23) ò !l = ò !0 + ò kl . v + R ,

where R = ò 2 k2l =2m is the recoil energy, that is the kinetic energy that an ion, initially at rest, would gain on absorption of a photon. The presence of the recoil term is equivalent to assuming that the actual Doppler shift is not given by the velocity before absorption but by the average velocity before and after absorption of the photon: 1 9 (24) ò !l = ò !0 + ò kl . (v + v ), 2 U sually this recoil induced shift in the D oppler eŒect is negligible (for magnesium it is equal to 0.1 MHz). Similar equations can be written for the emission process. After averaging over the distribution of scattered photons ^ (k e ), we obtain for the mean change in kinetic energy per scattering event:

k

EK l

ê k

= 2R +

(25)

ò kl . v.

The ® rst term on the right-hand side represents the recoil heating, the factor 2 being due to the recoil from the two interactions, absorption and emission. In order to achieve net cooling, the second term (Doppler cooling term) must be negative; this is the case when the ion is moving towards the laser beam. If we now write the scattering rate as a function of velocity and expand the expression for small velocities, we ® nd for the mean kinetic energy of the axial motion: EK ,z

= 1 mk v2z l 2

=ò

fz

(C =2)2 + 8

!

2

!

,

(26)


where fz is a numerical constant of the order of unity. The temperature is given by the ratio of the mean scattering rate to the variation of the scattering rate with D oppler shift. Therefore the minimum temperature is reached when this ratio is minimized, and this is the case when the laser is detuned by half the linewidth below resonance ( ! = C =2):


E K , z, min

= 1 fz ò 8

C.

(27)

For magnesium ions the axial temperature T z , de® ned as 1=2kB T z = k E K , z l , has a minimum value of the order of 1 mK. This result is valid for vR ! vrms , that is R ! ò C . When R > ò C , the lowest temperature is given by the recoil energy R. The recoil limit represents the ultimate limit of laser cooling of a two-level atom using scattered ¯ uorescence. For typical ions, the allowed electric dipole transition used for laser cooling obeys R ! ò C . If one wants to reach the recoil limit using Doppler cooling, much weaker transitions must be used, for example electric quadrupole transitions; however, because their linewidth is very small, the cooling rate will be similarly small. In practice this approach is not applicable and other schemes must be employed. There are now new sub-Doppler cooling mechanisms that have been applied to cooling neutral atoms in optical molasses and traps. These mechanisms rely on optical pumping, light shifts and laser polarization gradients (for example Dalibard and Cohen-Tannoudji (1989)). Application of the mechanisms to trapped ions is discussed by Wineland et al. (1992) and Yoo and Javanainen (1993). These mechanisms have recently been applied to the cooling of the weakly bound degree of freedom of a single magnesium ion trapped in an rf ring trap (Birkl et al. 1994), where sub-D oppler temperatures were observed. These subD oppler schemes are limited by the photon recoil. Subrecoil `cooling’ mechanisms for neutral atoms have been demonstrated in recent years (for example Aspect et al. (1988) and K asevich and Chu (1992)). The ® rst of these mechanisms (Aspect et al. 1988) is not in the true sense of the word a cooling mechanism, as it is based on the trapping of a sub-recoil-velocity atom in a particular atomic state in which it does not interact with the laser cooling light. It is not clear whether this essentially passive method can be applied to trapped ions, although experiments by Toschek’s group (for example Siemers et al. (1992)) have similarities with this mechanism. The other subrecoil cooling method, based on resolved side-band Raman transitions (Kasevich and Chu 1992), has already been applied to the cooling of a single beryllium ion to the zero-point energy (92% of the time in the vibrational ground state) of a miniature Paul trap using resolved-side-band R aman cooling (Monroe et al. 1995). At these low temperatures the motion in a trap becomes quantized and one is more interested in obtaining states of motion containing precise numbers of quanta, that

33

is Fock states. Cirac et al. (1993) have shown that under appropriate conditions it would be possible to prepare a trapped ion in a F ock state of its motion. This has recently been demonstrated in experiments by the group at the N ational Institute of Standards and Technology (NIST) (Meekhof et al. 1996). Clearly, the evaporative cooling methods recently used to achieve Bose±Einstein condensation of rubidium atoms cannot be applied to the cooling of single particles. 3.3.3. Lamb-Dicke regime and side-band cooling. The harmonic motion of a trapped ion will frequency modulate the laser ® eld as seen from the particle’s frame of reference, and it will also modulate the emitted photon spectrum as seen by an observer. If the motion is described by z = z0 sin (!z t ), the electric ® eld in the centre of mass frame of the particle is E (t )

= E0

exp [i kz-

!0 t )

= E0

exp {i[kz0

sin (!z t )- !0 t

(

] ]}

(28)

This expression can now be expanded in a series of Bessel functions (no second-order D oppler shift taken into account). Taking the Fourier transform and then the modulus squared gives the intensity absorption spectrum and ® nally, taking into account the natural linewidth C of the resonance, we obtain I (!) = I 0

S

+¥

n= -

¥

|J n (kz0)|2

(C =2)2 . (29) (C =2) + [! - (!0 - n!z )]2 2

This absorption±emission spectrum is made up of a `carrier’ or recoilless line at !0 with equally spaced Doppler-eŒect side-bands at frequencies !0 = n!z . Figures 6(a) and (b) show such spectra with kz0 = 50 and kz0 = 1 respectively. U sually, we have C > !z and the spectrum resembles the expected line shape from a particle moving in a harmonic well, in the limit !z ! C , !0 . If kz0 < 1 the amplitude of the `carrier’ becomes larger than the side bands. This is called the Lamb±Dicke regime. (In a sense the Lamb±D icke regime is similar to the long-wavelength approximation, but at the trap level.) This regime takes place when the ion is spatially con® ned to within a fraction of one wavelength of the laser light. As the con® nement increases, the amplitude of the carrier increases. In the microwave domain (k » 1 cm), the Lamb ±D icke regime is very easily attained even with large trapped ion clouds. The temperatures required to con® ne the ions to less than, say, a millimetre are readily achieved with large clouds using buŒer gas cooling. In the optical domain, however, the low temperatures required mean that the motion of the trapped ion must be considered quantum-mechanically. In the case of strong binding, that is when !z > C , the carrier frequency is D oppler free to ® rst order as the motional side bands are resolved (see ® gure 6). In fact,


34


Figure 6. Absorption spectrum of an atom undergoing a harmonic motion; the frequency is in units of the oscillation frequency: (Ð ), calculated from equation (29). (a) Linewidth of 0. 1 and 10 times the oscillation frequency. The amplitude of the motion corresponds to k z 0 = 50. (b) Linewidth of 0. 1 and 2 times the oscillation frequency. The amplitude of the motion corresponds to k z 0 = 1.

cooling is required in this case only to increase the strength of the carrier relative to the side bands or to reduce the second-order (relativistic) D oppler eŒect. Clearly, the D oppler cooling picture used in the previous section is then no longer applicable, as the motion must be considered quantum-mechanically and the absorption spectrum is not continuous. In order to cool in this limit, the laser must be tuned on a side band, below resonance. For this reason the cooling mechanism is usually called side band cooling. In the limit kz0 < 1, side-band cooling is easily explained. Let us assume that the laser is tuned to the ® rst side-band below resonance, !0 - !z . The ion will therefore absorb a photon of energy !0 - !z and, on re-emission, the probability distribution of emission is symmetrical and maximum at !e = !0 . Therefore on average the energy loss will be ò !z . The temperature limit will of course be higher than the D oppler limit, as !z > C ; a calculation gives (Wineland et al. 1987a) ò !z 1 (30) > ò C. KB T min = 2 1n (!z =C ) 2 The strong binding is not particularly easy to achieve if one wants a reasonably substantial cooling rate. The problem resides in di culty in obtaining trap oscillation frequencies larger than the electric dipole transition’s linewidth. Most of these linewidths are in the range of tens of megahertz, while typical oscillation frequencies are in the range of 1 MH z or below. There is currently a trend towards high-oscillationfrequency traps in order to achieve the Lamb±D icke regime with allowed electric dipole transitions. As we saw in section 2.5, this mainly requires miniature traps. 3.3.4. Laser cooling in the Paul trap. Laser cooling of ions stored in a Paul trap was ® rst demonstrated in 1978 by N euhauser et al. (1978). In the pseudopotential limit, the

motion is simple harmonic. Cooling of all degrees of freedom can be achieved with a single laser beam passing diagonally across the trap if there is enough asymmetry in the radial plane to couple the motion along and perpendicular to the laser beam and if the radial and axial oscillation frequencies are diŒerent. For clouds, rf heating will limit the lowest achievable temperature to well above the single-ion minimum temperature. The cooling of hot clouds can take several seconds as the density is small and the ions will not initially spend much time interacting with the laser.

3.3.5. Laser cooling in the Penning trap. Laser cooling of trapped ions was ® rst demonstrated in a Penning trap (Wineland et al. 1978). As seen in the previous section, cooling in the Penning trap is achieved by removing energy from the cyclotron and axial degrees of freedom and adding energy to the magnetron motion. Cooling of the axial motion is similar to the cooling in the Paul trap and is achieved by detuning the laser below resonance. At thermal equilibrium, the cyclotron amplitude will be signi® cantly smaller than the magnetron amplitude (as shown in ® gure 2(a)). In order to cool the cyclotron motion, we want the laser to interact with the ion mostly when the cyclotron motion is moving the ion towards the laser. This again is simply achieved by detuning the laser below resonance. For the magnetron motion, we want the laser to interact mostly when the ion is moving away from the laser. This is done by spatially oŒsetting the laser beam to the side where the ions are receding from the laser. In this way all degrees of freedom can be cooled with a single laser beam detuned below resonance, oŒset from trap centre and oriented diagonally (to cool the axial motion). Side band cooling could also be done in the Penning trap. In this case, three lasers are necessary for simultaneous



cooling of all degrees of freedom. The laser cooling beams for the axial and cyclotron motions must be tuned to the ® rst side band below resonance while the laser for the magnetron must be tuned to the ® rst side band above resonance. In principle a single diagonally oriented laser, tuned to the side band at !0 - !+ - !z + !- would result in the simultaneous cooling of all three degrees of freedom; however, the absorption intensity of this side band is extremely small and would make this scheme impractical. We see that, in the side-band cooling limit, the laser cooling beams can be spatially homogeneous. However, this scheme will only work if the side bands are well resolved, which is not easy to achieve, especially for the magnetron frequency. The absolute maximum value for !- is !c =2; so for Be+ a magnetic ® eld of more than 25 T would be required! 4.

35

Figure 7. A three-level system with one ground state and two excited states, as used for the observation of quantum jumps. The 1± 2 transition is allowed and is strongly driven, with a rapid spontaneous emission decay from 2 back to 1; state 3 is metastable, and the transition from 1 to 3 is weakly driven.

metastable state is the termination of the bright ¯ uorescence from the strongly allowed transition. One absorption to this metastable `shelf’ state leads to the visible absence of very many photons from the strong transition. At the time, such

Quantum eŒects in trapped ions

Ion traps are unique in that they allow the isolation of a single atom to small, well characterized volumes. Using laser cooling it has even been possible to con® ne a single ion to regions of dimension less than a wavelength of its cooling transition. This has made possible the testing of many aspects of quantum theory for single strongly localized particles as opposed to large numbers where only averaged eŒects can be measured. 4.1.

Quantum jumps

In 1913, Bohr proposed the model of ìnstantaneous transitions’ between the internal states of an atom on the absorption of a quantum of light: quantum jumps. The idea of a sudden transition between internal atomic states was not very well accepted; indeed even SchroÈ dinger (1952) compared the notion of quantum jumps with the theory of epicycles, in a famous sentence, `We never experiment with just one electron or atom or (small) molecule. In thought experiments, we sometimes assume that we do; this invariably entails ridiculous consequences. In the ® rst place it is fair to state that we are not ex perimenting with single particles any more than we can raise Ichthyosauria in the zoo.’ Dehmelt (1975) had proposed employing a three-level cold ion as an àtomic ampli® er’ to detect transitions between very stable states. This idea involved detecting the ¯ uorescence from a laser-driven transition from the ground state to a strongly ¯ uorescing excited state; at the same time a second laser was tuned to drive transitions from the ground state to a second metastable state of interest. N ormally transitions to metastable states are very hard to detect because of their low ¯ uorescence rate but, if they could be detected, they would have the very narrow linewidth needed in metrology. H owever, in Dehmelt’s atomic ampli® er, the signature of the transition to the

Figure 8. (a) Fluorescence from a single Ba+ ion as a function of time. The abrupt jumps in ¯ uorescence are due to quantum jumps as the ion is shelved in the metastable state unable to participate in the ¯ uorescence until it returns (at random times) to the ground + state. (b) Quantum jumps from three Ba ions. (Reprinted from Optics Communications, Vol. 60, Sauter, Th. Blatt, R. Neuhauser, W., and Toschek, P. E. ``Quantum jumps’’ observed in the ¯ uorescence of a single ion, pp. 287± 292, 1986, with kind permission of Elsevier Science ± NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)


36


shelving experiments were not feasible with single ions. H owever, in 1980 Neuhauser et al. (1980) isolated for the ® rst time a single barium ion in a Paul trap. Interest in the D ehmelt shelving proposal was revived by a paper by Cook and Kimble (1985) who showed that an atomic random telegraph would be generated in the ¯ uorescence from a single trapped ion as the ion jumps in and out of a metastable state, removing it temporarily from participating in transitions to the strongly ¯ uorescing level. In 1986 three groups (Bergquist et al. 1986, Nagourney et al. 1986, Sauter et al. 1986a) observed quantum jumps in Ba + and H g+ . Since then, quantum jumps have also been observed in magnesium, calcium and indium ions (Hulet and Wineland 1987, U rabe et al. 1993, Gisin et al. 1993, Peik et al. 1994) and even in large molecules (BascheÂet al. 1995). Before describing the actual experiments, we shall brie¯ y explain what a quantum jump is (for example K night and G arraway (1996) and references therein). Let us consider a three-level atom, with ground state |1l and excited states |2l and |3l (® gure 7). The |1l ® |2l transition is strongly allowed, while level |3l is metastable. If we illuminate our atom with a laser tuned to the |1l ® |2l transition, the system will cycle between these states, emitting one photon per cycle. D etection of these photons will therefore mean that the atom is cycling between the |2l and |1l states. If we now apply a second weaker laser (i.e. with a smaller Rabi frequency) tuned to the |1l ® |3l transition, we expect that the atom will sometimes make a transition to level |3l . Once the atom is in this metastable state, the ¯ uorescence from the |1l ® |2l transition will of course cease and the observer will observe an abrupt stop in the ¯ ow of emitted photons. The ¯ uorescence recommences when the atom decays back to the ground level by spontaneous or stimulated emission. The presence or absence of ¯ uorescence light provides the observer with information about the internal quantum state of the atomic particle. Figure 8(a) shows a sample of the ¯ uorescence emitted by a single Ba + ion undergoing quantum jumps and ® gure 8(b) the same signal for three such ions. Quantum jumps have several practical applications and form the `read-out’ measurement scheme in atomic clocks based on transitions in a single trapped ion; they could even be used to prepare Fock states of the harmonic motion of a trapped ion (Cirac et al. 1993). F inally the observation of quantum jumps provides an easy way to count small numbers of trapped ions (see ® gure 8). The èntanglement’ between the ion’s internal state and its vibrational degrees of freedom form the basis of proposals to use trapped ions to realize quantum gates in quantum computers (Barenco 1996). Observation of quantum jumps in 24 Mg+ requires a strong magnetic ® eld in order to create a multilevel system by Zeeman splitting. The energy levels of 24 Mg+ in such a case are depicted in ® gure 9. Quantum jumps in 24 Mg+ have the particular feature of not needing a second laser to drive

Figure 9. Energy level structure of the S 1=2 and P 3=2 states of a + 24 Mg ion in a magnetic ® eld. (From Hulet et al, (1988).)

the weak transition. In the experiment by H ulet et al. (1988) in a Penning trap, laser cooling was done on the strong 2 S1=2 (mj = - 12) ® 2P 3=2 (mj = - 32) transition at about 280 nm. The laser polarization was set to allow only r transitions ( m = ± 1). Therefore only one other transition was allowed. However, the large laser detuning from this transition meant that the transition rate was very slow. From this upper mj = + 12 state, the ion can spontaneously decay to the mj = + 12 ground level. This state corresponds to the metastable state |3l of our three-level model. This spontaneous R aman transition removes the ion from the laser cooling cycle and thus the emitted ¯ uorescence will abruptly cease. Once in the (mj = + 12) ground state, the laser can drive the ion back into the (mj = - 12) state. However, since this requires another oŒ-resonance Raman transition, the ion will remain in the metastable state for relatively long periods, of the order of 10 ms (Hulet and Wineland 1987, H ulet et al. 1988). 4.2.

T he quantum Z eno eŒect

Quantum theory predicts that, to a good approximation, an unstable state coupled to very many decay channels decays exponentially, with a rate C given by Fermi’s golden rule which is time independent; that is to say the possibility P ( t ) of surviving a short interval of time t is P 1 ( t ) = 1 - C ( t ).

(31)

H owever, if this time interval t is made su ciently short, the approximations made in deriving Fermi’s golden rule are violated, and the transition probability varies with time quadratically (this can be seen by integrating the timedependent SchroÈ dinger equation over a short interval t);


37

so the survival probability then becomes


P 1( t) = 1 - a ( t)

2

(32)

,

where a is some constant. `Short’ here in fact means times less than the reciprocal of the frequency range of the accessible decay channels. In practice, these short-time deviations from `normal’ decay behaviour are very di cult or impossible to see because this frequency range is so broad (for example in a decay or in radiative spontaneous emission). N evertheless, Misra and Sudarshan (1977) noted that this had important implications for decay processes which are monitored by frequent repeated measurements. Imagine a very idealized model of such measurements in which a succession of very brief measurements of survival in the initial state are made. Crudely, the probability of surviving in an excited state after N such measurements each separated by t is P N (N

t ) = P N (T )

= (1 -

C

= [1 -

C

t )N

+

( )] » T N

N

exp (- C T )

(33)

where the total elapsed time is T = N t. However, if the separation time t between measurements is very short, we must replace the linear time evolution with a quadratic, and P N (N

[

t ) = P N ( T ) = 1 - a ( t )2

]

N

®

1.

(34)

That is, frequent measurement inhibits the decay; Misra and Sudarshan termed this the quantum Zeno eŒect by analogy with the famous paradox of classical motion of antiquity due to Zeno. The decay inhibition is also referred to as the quantum `watched-pot’ eŒect: frequent observation preventing change. To observe the quantum Zeno eŒect in a true decay is very di cult and has not been attempted, but coherent transitions between stable states do indeed depend quadratically on time (for small times); so the inhibition of a driven transition (with a quadratic time evolution) by frequent measurements is indeed observable, and such an experiment was performed in 1990 by Itano et al. (1990) with trapped Be+ ions. Plenio et al. (1996) have shown how Zeno experiments can be realized which investigate this transition between quadratic and linear short-time evolutions. Let us summarize the experiment performed by the NIST group. The three relevant energy levels are shown in ® gure 10. Initially, the cloud of approximately 5000 Be+ ions was prepared in the ground state |1l . Then a long pulse of microwave radiation resonant with the |1l ® |2l transition was applied to transfer all the ions to the upper state |2l (a pulse with exactly the correct integrated intensity will do this coherently with 100% e ciency and is then termed a p pulse). However, during the application of this p pulse a series of short pulses resonant with the optical transition |1l ® |3l was applied. These short pulses played the role of the measurement pulses of the Zeno eŒect. It was found that, as a result of these measurement pulses, the population

Figure 10. Partial diagram of the energy levels of a 9 Be ion in a magnetic ® eld. (From Itano et al. (1990).)

transfer to the upper state |2l was reduced. They interpreted this result as due to the repeated collapse of the wavefunction caused by the (short) measurement pulses. That is, the measurement pulses destroyed the coherence between levels |1l and |2l created by the microwave radiation, so that every ion was projected into one of these states. The evolution under the in¯ uence of the p pulse restarts from a pure state after each measurement pulse. Predictions from this Zenolike interpretation agreed well with the experimental results. H owever, since the experiment was performed, alternative interpretations have been put forward (for example Frerichs and Schenzle 1991). Essentially, they all show that the same result can be obtained without requiring the collapse of the wavefunction. There is therefore some argument as to whether the NIST experiment really was an observation of the original quantum Zeno eŒect for decaying systems. This experiment and the following debate have addressed the fundamental question of what constitutes a `measurement’ in quantum physics and, more importantly, whether it is possible to distinguish experimentally between diŒerent interpretations. Current work is in progress at Imperial College to investigate Zeno eŒects with a single trapped ion rather than a cloud. For each single-ion experiment, the ensemble decoherence interpretation is replaced by the random `quantum jump’ interpretation relevant to single realizations (Beige and Hegerfeldt 1996, Power and Knight 1996). 4.3.

Photon statistics

The unique possibility of observing single atomic particles for long periods makes the study of the statistical properties of the light scattered by a single ion possible. In particular, quantum eŒects such as photon antibunching and subPoissonian statistics (Loudon 1983, R empe 1993) are


38


Figure 11. (a) Intensity correlation of one, two and three ions. (b) Intensity correlation for a single ion for diŒerent laser intensities (the top trace has the highest and the bottom trace the lowest intensity). The data have been corrected for the eŒect of the micromotion and the background due to stray light has been subtracted. (From Diedrich and Walther (1987a).)

expected to be observed. Photon antibunching in the ¯ uorescence from a single atom (Kimble et al. 1977) describes the fact that a single atom cannot radiate two photons simultaneously. This is simply because, after emission of a photon, the atom must be re-excited before it can emit another photon. This means that the train of photons emitted is slightly more regular than entirely random light, as on average there is a delay of half a Rabi period between two consecutively emitted photons. Besides being antibunched, the light emitted by a single atom will also exhibit subPoissonian statistics. This can be understood as being related to the antibunched nature of the emitted light; the fact that the atom has zero probability of emitting two photons simultaneously leads to a photon count distribution whose variance is less than the mean count rate, which is therefore sub-Poissonian. Although photon antibunching and subPoissonian statistics are usually related, the occurrence of one does not always imply that of the other. These two properties of antibunching and sub-Poissonian counting statistics were veri® ed experimentally by Diedrich and Walther (1987a) on a single magnesium ion held in a Paul trap. In order to observe evidence of photon antibunching, they measured the second-order correlation function g(2) (s ), which is proportional to the probability of detecting a photon at time t + s given that a photon had been detected at time t. In normalized form this is given by ( ) g 2 (s ) =

< : I (t + s ) I (t ) : > < I (t ) > 2 ,

(35)

where the colons denote `normal’ ordering of ® eld operators

(Loudon 1983) such that photon creation operators appear to the left of annihilation operators. For thermal light, this function has a maximum at s = 0 and decreases for larger delays. This means that thermal light is bunched; photons tend to be emitted in clusters. For laser radiation the function is independent of s . However, for the light emitted from a ( ) single atom, g 2 (s ) tends towards zero for small values of s . This correlation function was measured using a H anbury± Brown±Twiss set-up (Loudon 1983) with two photomultiplier tubes monitoring the light re¯ ected from and transmitted by a beam splitter in coincidence. The signal from one tube starts a timer, while the signal from the other tube stops it. The distribution of the time intervals gives, after normalization, the joint counting correlation function and ( ) ( ) g 2 (s ). Figure 11(a) shows g 2 (s ) for one, two and three ( ) ions; photon antibunching is clearly evident as g 2 (s ) has its (2) lowest value at s = 0. The value of g (s ) increases with increasing number of ions as there is no correlation between photons emitted from diŒerent ions. The strong modulation of g(2) (s ) at larger values of s is due to the micromotion of the ions which Doppler modulates the ¯ uorescence. Figure ( ) 11(b) shows g 2 (s ) for a single ion for various laser intensities after subtraction of stray light contributions and compensation for the micromotion modulation. The same experimental set-up was used to observe the subPoissonian nature of the ¯ uorescence from a single ion, but in this case they measured the number of events in which two photons are detected within a time interval of 4.607 ns. The corresponding probability obtained was lower by 28 standard deviations from the value expected from a perfect Poissonian



5.

Figure 12. Polarization-sensitive detection of the light scattered by the two ions: (a) p -polarized light, showing interference; (b) r polarized light, showing no interference. Plots are shown as a function of the scattering angle w . (From Eichmann et al. (1993).)

distribution. This demonstrates the non-classical nature of ¯ uorescent light; no classical ¯ uctuating intensity has a mean square which is less than the square of the mean intensity.

4.4.

V ariation of Y oung’s interference experiment

Young’s original double-slit interference experiment is often regarded as the paradigm of wave±particle duality. Recently Eichmann et al. (1993) reported an experiment where the two slits were replaced by two strongly localized ions. In other words, they observed the interference eŒects in the light scattered by these two ions. The strong localization of the mercury ions was made possible by the use of a linear Paul trap. The ions were laser cooled on the 5d 10 6s 2 S1=2 ® 5d10 6p 2 P 1=2 transition by a 194 nm laser beam. This beam also served as the light source for the interference experiment. The ions were con® ned to about 30 nm in the radial plane and 60 nm in the axial direction, corresponding to a temperature of only twice the Doppler cooling limit of 1.7 mK. The particular internal structure of 198 Hg+ was used to illustrate the impossibility of obtaining the `which path’ information. The ground state (6s 2 S1=2 ) and the excited state (6p 2 P1=2 ) of 198 Hg+ are twofold degenerate with respect to the magnetic quantum number mj ; this means that the light scattered from a linearly polarized laser beam will be either p or r polarized. If the light scattered by the two-ion system is p polarized ( m = 0), the ® nal state of the two-ion system is identical with its initial state, that is one cannot determine which ion, assuming that this has a meaning, has scattered the photon; therefore we expect to observe interferences in the scattered light. Now, if the scattered light is r polarized ( m = ± 1), the ® nal state of the system is diŒerent from its initial state; in fact, one of the ions is in a diŒerent state and therefore it is, at least in principle, possible to determine which ion has scattered that r -polarized photon. Therefore, in this case, we do not expect to observe any interference fringes. The r - and p -polarized scattered light from the two-ion system is shown in ® gure 12 (Eichmann et al. 1993).

39

Dynamics of ions stored in a Paul trap

In section 2, we brie¯ y described the motion of a single ion con® ned in a Paul trap. Although the motion is relatively complex, there is an analytical solution. H owever, when several ions are present simultaneously in the trap, the motion has no general analytical solution. Although one can use the pseudopotential approximation, this approach is not strictly valid for the trapping parameters used in most experiments. Because of its simplicity and the possibility of using buŒer gas cooling, the Paul trap is often used for spectroscopic measurements. The temperatures achieved using buŒer gas cooling are relatively high, leading to large Doppler broadening. H owever, the con® nement of the trapped ions means that for microwave spectroscopy the Lamb±D icke regime is readily reached at room temperatures. In that case, the second-order Doppler eŒect (which arises from the relativistic time dilation experienced by a moving particle) is in general the largest source of error. Accurate knowledge of the ion’s velocity distribution could allow for corrections and improve the resolution of spectroscopic measurements. The dynamics of ion clouds in the Paul trap have been investigated both experimentally and theoretically by several groups. A successful model developed by Blatt et al. (1986) and Siemers et al. (1988) and veri® ed experimentally gives the spatial and velocity distributions of ion clouds. They consider the ion cloud in the Paul trap as being in a quasistationary equilibrium, that is the time-averaged kinetic energy and spatial distribution over one period of the driving ® eld are assumed to be constant. A random ¯ uctuation force is added to simulate the presence of a buŒer gas and the eŒects of collisions with the other ions. The spatial and velocity distributions were found to be G aussian, but with time-dependent variances, that is their widths varied at the drive frequency. These results were compared with experiments on trapped Ba + ions. The results were in good qualitative agreement with the theoretical predictions.

5.1.

Ion crystals in the Paul trap

In the presence of strong cooling, trapped ions can undergo a `phase transition’ to a regular structure. Such crystallization was ® rst observed by Wuerker et al. (1959) for charged aluminium particles con® ned in a Paul trap and cooled using a buŒer gas (® gure 13). More recently, with the emergence of laser cooling, crystalline behaviour in small clouds of ions has been observed by several groups. The work at N IST was performed with H g+ ions in a miniature Paul trap (Wineland et al. 1987b). The ions were cooled on the 5d 10 6s 2 S1=2 ® 5d10 6p 2 P 1=2 transition with 1± 2 m W of 194 nm radiation. Ordered structures of up to ® ve ions were observed. For a two-ion crystal, the inter-ion


40


Figure 13. Crystal of 32 aluminium particles con® ned in a Paul trap. (From Wuerker et al. (1959).)

separation was found to be 7. 5 ± 1. 1 m m compared with a predicted value of 8. 5 m m. Although the trap was designed to have axial symmetry, the preferred spatial alignment of the crystals indicated the presence of asymmetries, probably due to contact potentials. The vibrational structure of the two-ion crystal was investigated by measuring the absorption spectrum of a weak electric quadrupole transition at 282 nm of one of the ions. As the width of this transition is much smaller than any of the oscillation frequencies of the trapped ions, the absorption spectrum consists of a D oppler-free carrier and D oppler-generated side bands at multiples of the ions’ oscillation frequencies. This allowed the observation of the stretch vibration mode of the two-ion crystal at 31=2 times the single-ion oscillation frequency (® gure 14). The amplitude of the side bands of the absorption spectrum gave an eŒective temperature of less than 8 mK . The experiments realized at G arching by Diedrich and Walther (1987b) were performed on small clouds (2±50 ions) of magnesium ions. They monitored the ¯ uorescence from a small cloud of trapped magnesium ions as the laser frequency was tuned up to the resonance. When the cloud is hot, D oppler shifts will strongly broaden the ¯ uorescence spectrum. This was found to be the case. H owever, for low rf amplitudes, the ¯ uorescence spectrum jumped discontinuously from a wide spectrum to a much narrower spectrum (® gure 15). This sudden variation in ¯ uorescence was interpreted as a phase transition from a cloud to an ordered structure. This was con® rmed by directly imaging the ¯ uorescence emitted by the ions; the ions formed a solid-

Figure 14. Absorption spectra of the weak electric quadrupole transition for individual ions. (a) Spectrum for a single ion. (b) Spectrum of one ion of a two-ion crystal; the new absorption lines are due to the stretch vibration modes of the cyrstal. Here the labels r and z refer to oscillations in the radial and axial directions respectively; (r-z), etc, refer to combination frequencies. Primes indicate two ions moving in anti phase. (From Wineland et al. (1987b).)

like structure with all ions strongly localized to much less than the inter-ion spacing of about 23 m m (see ® gure 15). 5.1.1. Order-to-chaos transition. In 1988, H oŒnagle and co-workers from IBM reported similar results on Ba + ions. They concentrated on the study of the melting of the twoion crystal as the q parameter (which is proportional to the rf amplitude) is increased towards the boundary of the stability region of the Paul trap. Essentially they claimed that for a value of q = qc lower than the maximum stable value of q, the crystal underwent an order-to-chaos transition, that is the crystal melted before the single-ion boundary was reached (HoŒnagle et al. 1988). Their results were con® rmed by numerical calculations. BluÈ mel et al. (1989), on the contrary, claimed that there is no order-to-chaos transition when the q parameter is increased, and that the crystal will only melt at the boundary of the single-ion stability region. They supported their claims with numerical and experimental evidence. They explained the melting of the crystal, when it occurred before the stability limit was reached, as being due to collisions with background gas. Since then several papers on this controversy have been published. Vogt (1994a,b) showed, mainly from numerical


41


calculations, that the melting of the crystal as q is increased could be explained by an enhanced sensitivity of the crystal to displacements from the ideal crystal con® guration. In this way, small perturbations such as spontaneous emission would su ce to melt the crystal. This extreme sensitivity of the crystal was interpreted as being due to the presence of low-order nonlinear resonances. On their side, the IBM group has published several papers to support their interpretation. Their 1990 paper reports experimental evidence supporting their interpretation (Brewer et al. 1990a). They claim that, below qc , the two-ion system can exhibit transient chaos while, for values above qc , the chaotic phase is stationary. However, in both cases, it seems that a collision with a background neutral is necessary to induce the chaotic regime. Recently, another paper by the IBM group presents an interpretation of this order-to-chaos transition in terms of a boundary crisis (HoŒnagle and Brewer 1994). Finally, the latest paper to date on this subject by BluÈ mel (1995) shows that strong damping can actually result in the melting of the crystal. In particular, he shows that the stability regions are reduced by the presence of damping. It has recently been discovered that, in some regions of the stability diagram, a two-ion crystal would be stable only at an angle to the trap axis (Moore and BluÈmel 1994, H oŒnagle and Brewer 1995). This is rather surprising as, from the trap symmetry, one would expect stable solutions only in the radial plane or in the axial direction. N o doubt, the study of dynamics and laser cooling of two-ion crystals in the Paul trap will continue until these fundamental questions are fully understood (see Walther 1993). 6.

Figure 15. (a) Excitation spectra of about ® ve simultaneously trapped ions for increasing rf amplitudes; the top trace has the lowest amplitude. The regions where a crystalline structure was observed are indicated by arrows. (b) Crystalline structure of seven magnesium ions observed in the Paul trap. The mean distance between ions is about 23 m m (From Diedrich and Walther (1987b).)

Dynamics of ions stored in a Penning trap

Contrary to the Paul trap, the Penning trap does not use an rf electric ® eld for the con® nement of the charged particles. This has the important consequence that there is no rf heating. H owever, there are disadvantages: the presence of a magnetic ® eld means that only particles along the symmetry axis of the trap can remain at rest; particles away from this axis will rotate around it. As a result, clouds of ions trapped in the Penning trap cannot be at rest; they will always be rotating around the trap axis. In a sense, this is similar to the Paul trap, where even `crystallized’ clouds are never absolutely steady but `breathe’ at the micromotion frequency (see ® gure 13). In the Penning trap, a crystallized cloud will rotate around the trap axis, just as a solid body would. The impossibility of obtaining clouds of stationary ions in either trap is due to the combination of the Coulomb interaction and the trapping methods; in order to circumvent Earnshaw’s theorem, dynamic trapping methods are used, a time-varying electric ® eld for the Paul trap and a velocity-dependent force (from the magnetic ® eld) in the Penning trap. We can therefore consider the impossibility of obtaining steady crystals as a

42


consequence of Earnshaw’s theorem. The absence of rf heating in the Penning trap means that large clouds can in principle be crystallized using laser cooling. Because of the presence of the magnetic ® eld, this would result in a rotating crystal, although all statistical and thermodynamic properties of the solid phase would be present.


6.1.

Plasma studies in the Penning trap

A cloud of ions con® ned in a Penning trap can be considered as a (non-neutral) plasma. N on-neutral plasmas have many similarities with neutral plasmas; for example plasma waves or shielding are present in both types of plasma. Contrary to neutral plasmas, non-neutral plasmas can relatively easily be con® ned for long durations, in particular in Penning traps. As a consequence they can evolve into a state of thermal equilibrium. In contrast with neutral plasmas, they can be cooled to low temperatures without loss (in a neutral plasma, cooling would result in a recombination of charges). An important quantity which characterizes a plasma is the D ebye length (for example Jackson (1975)): k

2 D

= e 0 kB2T n0 q

or k

D

»

([

21. 8

[]

T K n0 107 cm-

1=2

3

)

]

m

m

(36)

where kB is Boltzmann’s constant, T the temperature of the plasma, n0 [107 cm- 3 ] is the plasma density and q is the particle charge. The Debye length characterizes the spatial scale of collective eŒects in plasmas. In the limit where the D ebye length is small compared with the plasma dimensions, the density is constant in the plasma interior and drops to zero at the plasma boundary over a distance of a few D ebye lengths (Prasad and O’N eil 1979). The density of a zero-temperature plasma in a Penning trap is related to its rotation frequency !r about the trap axis by the relation n0

= 2e 0 m2 r (!c !

q

!r ),

(37)

where m is the mass of a particle and !c is the true cyclotron frequency. This expression is valid only in the limit where the plasma dimensions are small compared with the trap dimensions such that image charge eŒects on the trap electrodes can be neglected. The plasma has the shape of an ellipsoid of revolution, with an aspect ratio (axial to radial dimensions) dependent on the rotation frequency of the cloud. This rotation frequency depends on the cloud density and the trapping ® elds. For a plasma of very low density, the eŒect of the cloud on the motion of a single ion can be neglected to a ® rst approximation; the cloud will, on average, rotate at either the magnetron frequency or the modi® ed cyclotron frequency depending on which motion has the largest radius. If the size of the cloud is reduced, for example by laser cooling, its rotation frequency will increase or decrease from the magnetron or cyclotron frequency respectively. The

maximum density is reached when !r = !c =2. This condition is usually referred to as Brillouin ¯ ow. At this point, the plasma behaves essentially like an unmagnetized plasma. `Penning’ plasmas have been studied experimentally only very recently. Most work has been done on trapped ions (for example Brewer et al. (1988), Bollinger et al. (1993) and D holakia et al. (1995)), although results have also been reported on electron and positron plasmas (for example Prasad and O’Neil (1979) and Weimer et al. (1994). The advantage of working with ion plasmas is that they can be laser cooled to very low temperatures and their density and rotation frequency can be, in principle, easily controlled by the laser. In addition, the ¯ uorescence from the ions can provide important information on the plasma characteristics. On the contrary, electron plasmas cannot be easily cooled below 4 K and have the disadvantage of having very high cyclotron frequencies. The `static’ properties of a 9 Be+ ion plasma have been extensively studied at N IST (Brewer et al. 1988). They measured the shape, rotation frequency, density and temperature of 9 Be+ plasmas under various trap potentials and laser-cooling con® gurations. The experiments were realized using two lasers: one was essentially used to laser cool the cloud, whilst the second laser was used as a probe. The ions were cooled with about 50 m W of power of about 313 nm radiation. The light scattered from this cooling beam was imaged onto a photomultiplier tube. The probe laser, of much lower power (much less than 1 m W) was tuned to a diŒerent component of the transition, from whose upper state the ions have a two-thirds probability of decaying to a diŒerent Zeeman component of the ground state. In this way ions in this `shelved’ state are taken out of the cooling cycle and do not ¯ uoresce any longer. The resulting decrease in the total ¯ uorescence from the cloud gives therefore some useful information on the dynamics and number of the `tagged’ ions. More usefully, by scanning the probe laser across the shelving transition, the rotation frequency of the cloud, and its temperature and density at the position of the probe beam can be determined. With the probe beam oŒ, the shelved ions were returned to the cooling cycle in about 1 s. The size and aspect ratio of the clouds were measured by spatially translating the probe beam. The measurements con® rmed the theoretical values relating the aspect ratio to the rotation frequency of the cloud. Measurement of the rotation frequency of the cloud at various distances from the trap axis did not, within the experimental error, reveal the presence of shear. The dynamic properties of `Penning’ plasmas have also recently been studied. The dynamics of such a plasma can be appropriately described by its eigenmodes of oscillation. These electrostatic modes can be described (in spheroidal coordinates) by two integers l, m with l > 1 and l > m (Dubin 1991). Interestingly, all these modes have an analytical solution for the case of cold Penning plasmas. The (1, 0) and



43

Figure 16. (a) Plasma rotation frequency as a function of the torque laser detuning. (b) Rotation frequency of the cloud at which the heating resonance was observed as a function of the single-particle axial frequency; (Ð ), condition at which the (2, 1) mode is expected to be excited. (From Heinzen et al. (1991).)

the two (1, 1) modes correspond to the three centre-of-mass oscillation modes where the plasma keeps its shape but its centre of mass oscillates at the single-ion oscillation frequencies. The l = 2 modes correspond to quadrupole deformations. F or example the two (2, 0) modes correspond to a `breathing’ mode (the axial and radial extents of the cloud oscillate in phase) and a stretching mode (the axial and radial extents of the cloud oscillate out of phase). The three (2, 1) modes are azimuthally asymmetric modes. For example, in one of these modes, the plasma spheroid is tilted with respect to the z axis and precesses about the z axis. The low-order modes have been studied experimentally at N IST (Heinzen et al. 1991, Bollinger et al. 1993) on Be+ plasmas. Most work was done on clouds of 1000±5000 ions. Cooling was provided by approximately 100 m W of 313 nm laser radiation tuned 30±50 MH z below resonance. Typical temperatures were in the range 5±200 mK , resulting in a D ebye length of less than 10 m m; this was much smaller than the plasma dimensions (100±1000 m m), while the trap dimensions were another order of magnitude larger. Consequently, these clouds could be considered as cold spheroidal plasmas. The rotation frequency of the cloud was controlled using another laser beam of much lower power (about 2 m W) tuned above resonance and oŒset from the trap centre so as to impart a torque to the plasma. The rotation frequency was measured by driving the electron spin-¯ ip transition at about 22 G Hz. Absorption of this microwave radiation was observed by a decrease in the ions’ ¯ uorescence. Measurement of the motional side-band frequencies allowed a determination of the rotation frequency to an accuracy of about 5 kHz. The ¯ uorescence emitted by the ions was imaged onto a photon-counting imaging tube. Initially, the cloud would be rotating at a frequency close to the single-ion magnetron frequency and the torque beam detuning !T set close to resonance. By slowly increasing the torque laser detuning, the torque applied by the weak beam would increase the plasma rotation frequency !r until, because of the Doppler shift, the cloud would nearly stop

interacting with this beam. Figure 16(a) shows a plot of the plasma rotation frequency !r as a function of the torque beam detuning !T . Interestingly, this curve exhibits a strong hysteresis, which was shown to be due to a heating resonance. Indeed, images of the ¯ uorescence from the ion cloud (see ® gure 16(a)) showed that in the lower branch the plasma was much hotter than in the upper branch. The size of the hysteresis was found to depend sensitively on the angle between the trap electric ® eld axis of symmetry and the direction of the magnetic ® eld. Only for ® eld misalignments of less than 0.01Êwas it possible to observe no hysteresis. The heating resonance was identi® ed as an excitation of the (2, 1) plasma mode. This mode has the same asymmetry as a tilted quadrupole. F igure 16(b) shows the very good agreement between this assumption and the experimental measurements. 6.2.

Crystalline behaviour in the Penning trap

We saw that strongly cooled ions con® ned in a Paul trap could arrange themselves into ordered structures. The ions’ micromotion limited the eŒectiveness of the laser cooling of large clouds, thus limiting the maximum size of the crystals to fewer than one hundred ions (DeVoe et al. 1989). The absence of micromotion in the Penning trap means that, in principle, there is no limit on the size of a `Penning’ crystal. Before reviewing the work done on the crystallization of `Penning’ plasmas, let us de® ne two quantities to characterize the `state’ of a cloud. The most commonly used quantity to characterize the phase state of a one-component plasma is the Coulomb coupling parameter C , which is de® ned as (for example Ichimaru et al. (1987)) C

=

q2 4p e 0 a s kB T

.

(38)

It represents the ratio of the average Coulomb potential energy of an ion to the ion’s thermal energy; a s is de® ned by 4=3p n0 a 3s = 1 with n0 the average density of the ion cloud. A


44


Figure 17. (a) Cloud density as a function of spherical radius for 100 ions; the Coulomb coupling is 140. (b) Polar plot of ion positions for the outermost shell in a cloud of 100 ions; the Coulomb coupling is 140. (From Dubin and O’Neil (1988).)

Figure 18. Images of the shell structure of partially crystallized `Penning’ plasmas. (a) A single shell containing approximately 20 ions. (b) 16 shells in a cloud of about 15 000 ions; the cloud is illuminated only by the probe beam. (c) Same cloud at a diŒerent trap voltage and with all beams on. (From Gilbert et al. (1988).)

weakly coupled plasma has C ! 1; in this case the Coulomb interaction is very small compared with the kinetic energy of the particles and can be treated perturbatively. A plasma is considered as strongly coupled when C > 1; at such values the system cannot be treated perturbatively and the plasma begins to exhibit features qualitatively diŒerent from a weakly coupled plasma. In particular, it has been predicted that, for an in® nite plasma, a phase transition to a solid state with a bcc lattice structure should occur at C » 178 (for example Ichimaru et al. (1987)). For a ® nite plasma of a few hundreds or thousands of ions, boundary conditions are expected to lead to a diŒuse phase transition to a solid-like structure (for example D ubin and O’Neil (1988)). In particular, higher values of C might be required and imperfections in the lattice will be present. Calculations show that plasmas may have to be larger than 60 interparticle spacings along their smallest dimension in order to exhibit a proper bcc structure (Hasse and Avilov

1991, D ubin 1989). This corresponds to more than 105 ±106 particles; however, it is not clear whether such clouds would exhibit a clear phase transition. Numerical simulations of the dynamics of small plasmas (up to several hundred ions) in the Penning trap have been carried out by Dubin and O’Neil (1988). They found that, for moderate values of C (2±100), the cloud density showed an oscillatory variation as a function of the spherical radius. At higher values of C , the visibility of the modulation of the density increases until the minima reach zero (® gure 17(a)). At that point the ion plasma separates into concentric shells, with little diŒusion between shells but relatively high diŒusion within each shell. In other words, the plasma is solid like radially but liquid like azimuthally and polarly, a rather odd composite material. F or even larger values of C the diŒusion rate within the shells decreases and the formation of a two-dimensional, approximately hexagonal lattice is observed on the shell surfaces (see ® gure 17(b)). All these numerical calculations were done using isotropic damping forces. Experiments at N IST have subsequently veri® ed the presence of these shell structures (Brewer et al. 1986, G ilbert et al. 1988, 1989, Bollinger et al. 1990). They worked on 9 Be+ clouds con® ned in a cylindrical Penning trap. The setup was very similar to the set-up that they used for the plasma mode experiments. The cloud rotation frequency and temperature were measured, as in earlier experiments, using a probe beam. The ¯ uorescence in the axial direction was imaged onto a photon-counting imaging tube, with a resolution of about 5 m m. Clouds from 20 to 15 000 ions revealed a shell structure (® gure 18). The number of shells as a function of the number of ions corresponded well to the theoretical approximate prediction: (N/4)1=3 shells for an Nion cloud. Coulomb coupling constants in excess of 300 were measured. By tagging various parts of the partly crystallized plasma with the probe laser they were able to verify that diŒusion was smaller between the shells than within them. However, they were unable to observe complete crystallization of their plasmas. The lowest



temperatures were obtained for the smallest clouds. As boundary eŒects are larger in smaller clouds, higher values of C are required in order to observe a complete crystallization. The di culty lies in achieving low temperatures in large clouds. The most recent work at NIST on this topic was the observation of Bragg scattering patterns from large clouds of ions in a Penning trap undergoing strong laser cooling. It was possible to observe several rings of scattered light around a resonant laser beam which was passed through the ion cloud. The spacings of the rings were consistent with those expected for a bcc lattice and from these measurements the inter-ion spacing and the degree of crystallization of the cloud could be determined (Tan et al. 1995b). 7.

Dynamics of ions stored in a combined trap

In section 2 we brie¯ y described the motion of a single ion con® ned in a combined trap. The ion oscillation frequencies were measured in early work at Imperial College by applying a small additional oscillating voltage to the trap electrodes (Bate et al. 1992). The frequency of this signal is scanned and, when resonance is reached, the cloud absorbs energy and the ¯ uorescence signal from the laser-cooling beam changes. H owever, this method is invasive and measures only the centre-of-mass oscillation modes. Another innovative and non-perturbing method was then developed (Dholakia et al. 1992). This is based on a statistical analysis of the ¯ uorescence emitted by the ions. Essentially, it consists of measuring the ¯ uorescence photon ±photon correlation. This is of course similar to ( ) the measurement of the intensity correlation function g 2 (s ) discussed earlier (Diedrich and Walther 1987a), although the time scales involved for measuring the ion oscillation frequencies are much larger. The idea behind the photon±photon correlation technique is that any modulation present in the ¯ uorescence level will show up in the correlation data. This is true even when the ¯ uorescence level is so low that hardly any modulation can be detected in the ¯ uorescence signal itself. Experimentally these correlation data are obtained by building a distribution of time intervals between consecutively detected photons. This technique relies on the presence of modulation in the ions’ ¯ uorescence. U nder the right circumstances, the motion of the ions will modulate the ¯ uorescence level as their motion takes them in and out of the laser beam and their velocities shift them in and out of resonance. We can therefore expect the ¯ uorescence to be modulated at the ions’ oscillation frequencies. These frequencies should then show up in the correlation data. This technique has the advantage of measuring the true ions’ oscillation frequencies and not the cloud’s centre-of-mass frequencies. This has allowed for example the measurement of space-charge

45

shifted oscillation frequencies in the Penning trap (Dholakia et al. 1993). This technique is also expected to allow the observation of small crystals in the Penning trap. Figure 5 shows the experimentally measured oscillation frequencies in the combined trap for a wide range of parameters. The surprising feature is the presence of an avoided crossing between the axial and modi® ed cyclotron frequencies (see inset). Recent work with the combined trap has included the construction of a trap which is designed to hold simultaneously both electrons and protons (or equivalently positrons and antiprotons) by a group at Garching (Walz et al. 1995). This follows earlier suggestions by various workers (for example Li and Werth (1992)). It has been demonstrated that this trap can con® ne ions and electrons at the same time, and this is likely to be of use in the generation of antihydrogen, which requires collisions between positrons and antiprotons. Storing them in the same volume is more e cient than, for example, colliding two beams together as it gives more opportunities for suitable collisions to occur. 8.

Conclusions

We have demonstrated how laser-cooled trapped ions can be used to demonstrate a wide range of basic physics, from the observation of quantum jumps to the determination of the properties of strongly coupled microplasmas. Trapped ions have many applications, including their utilization in new atomic clocks where the avoidance of D oppler eŒects and the like have the potential for unrivalled time standards. They have already led to new insights into the way that quantum mechanics deals with the interactions of single particles, and there are very bright prospects for further exciting work in the next few years. Acknowledgements We are grateful to the U K Engineering and Physical Sciences Research Council and the European Union for the support of this work. One of us (G.Zs.H .K .) would like to thank the H eÂ leÁne et Victor Barbour Foundation in G eneva and the Swiss National Science Foundation for support and the CVCP for an ORS award. References Adams, C. S., 1994, Atom optics, Contemp. Phys., 25, 1±20. Aspect, A., Arimondo, E., Kaiser, R ., Vansteenkiste , N ., and CohenTannoudji , C., 1988, Laser coolin g below the one-photo n recoil energy by velocity-selectiv e coherent populatio n trapping, Phys. Rev. L ett., 61, 826±829. Barenco, A., 1996, Quantum physics and computers, Contemp. Phys., 37, 375±389. BascheÂ , Th., Kummer, S., and BraÈ uchle, C., 1995, Direct spectroscop y observatio n of quantu m jumps of a single molecule, N ature, 373, 132±134.


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Gabriel H orvat h studied Physics at the Ecole Polytechniqu e F eÂ deÂ rale de Lausanne. H e then obtained a M aster s D egree at Imperia l College, London , where he also obtained a PhD degree under the supervisio n of D r R . C. Thompson in the ® eld of ion traps. H e is currently workin g in Alain Aspect ’s group in Orsay on atom optics and atomic mirrors. R ichard T hompson is a R eader in Physics at Imperia l College. H e originally studied Physics at the U niversit y of Oxfor d and took a D Phil in atomic spectroscop y there. F ollowin g a postdoctora l appointmen t at the Kernforschungszentru m Karlsruh e workin g on laser spectroscop y of unstable isotopes, he moved in

1983 to the Nationa l Physica l Laboratory, Teddington , to work in the area of ion traps. In 1986 he was appointed a Lecturer at Imperial College, where he has continued to pursue his interest in laser coolin g of trapped ions. H e is Book R eviews Editor of Contemporary Physics. Peter Knight is Professo r of Quantum Optics and H ead of the Laser Optics and Spectroscopy G roup in the Physics D epartmen t of Imperia l College, London . H e has worked on the propertie s of atoms in intense laser ® elds and the quantu m properties of ligh t for more year s than he (or anyon e else) cares to remember and becam e interested in the physics of trapped ions a decad e ago with the obser vatio n of quantu m jumps in such systems. H e is the Editor of Contemporary Physics (as well as the Journal of M odern Optics) and past Chairma n of the Quantu m Electronics D ivision of the Europea n Physica l Society.