12 Ion traps 12.1 The force on ions in an electric field 12.2 Earnsha’s theorem 12.3 The Paul trap 12.4 Buffer gas cooling 12.5 Laser cooling of trapped ions 12.6 Quantum jumps 12.7 The Penning trap and the Paul trap 12.8 Electron beam ion trap 12.9 Resolved sideband cooling

12.1 The force on ions in an electric field An Ion with a single charge: An electric field : Experiences a force: A Neutral Atom with a magnetic moment of one Bohr magneton in a magnetic field gradient of Experiences a force

Ions feel a force 10 8 times greater than magneticlly-trapped neutrals. In a trap operating with a voltage of 500v, single-charged ions have a maximum ‘ binding ’ energy at a temperature of 6  10 6 K. In contrast, a magnetic trap for neutral has a maximum depth of only 0.07K. These estimates show that neutral atoms must be cooled before trapping but ion trapping requires only moderate electric fields to capture the charged particles directly.

12.2 Earnsha’s theorem Earnshaw proved that: A charge acted on by electrostatic forces cannot rest in stable equilibrium in an electric field. 带电粒子不能束缚在静电场中，也就是 说静电场中没有最大值和最小值，不能形成 一个束缚的势阱。

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

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

The gravitational potential energy of the ball on the surface has the same form as the potential energy of an ion close to a saddle point of the electrostatic potential. we assume here a symmetric saddle whose curvature has the same magnitude, but opposite sign, along the principal axes: (12.4) Where and are coordinates in a frame rotating with respect to the laboratory frame of reference The Paul trap

In the mechanical analogue, the rotation of the saddle at a suitable speed causes the ball to undergo a wobbling motion; the ball’s mean position only changes by a small amount during each rotation. the mean position of the object moves as if it is in an effective potential that keeps the ball near the centre of the saddle. We will find that an ion jiggling about in an a.c. field has a similar behaviour: a fast oscillation at a frequency close to that of the applied field and a slower change of its mean position Eqilibrium of a ball on a rotating saddle

A a.c. electric field An ion of charge e and mass M feels a force Newton ’ s second law (12.5) Two successive integrations give : The velocity (12.6) The displacement It has been assumed that the initial velocity is zero and r 0 is a constant of integration The effective potential in an a.c. field

V ac (a) (b) Fig A linear Paul trap used to store a string of ions. (a) A view looking along the four rods with the end-cap electrode and ions in the centre. Each of the rods is connected to the one diagonally opposite so that a voltage between the pairs gives a quadrupole field. (b) A side view of the rod and end-rap electrodes which have a.c. and positive d.c. voltages, respectively. A string of trapped ions is indicated. + + ….. y x

From the gradient of potential we find the electric field According Any place potential The equation of motion in the x-direction is If

This is a simplified form of the Mathieu equation : At the same way,we can This question only and take some numerical,it have stability equations.

Trapping of ions requires a vacuum, but the presence of a small background of helium gas at a pressure (10-4mbar) gives very effective cooling of hot ions. The ions dissipate their kinetic energy through collisions with the buffer gas atoms and this quickly brings the ions into thermal equilibrium with the gas at room temperature. For ions that start off above room temperature the buffer gas cooling actually reduces the perturbations on the ions. Any slight broadening and shift of the ions’ micromotion-the ions stay closer to the trap centre where they see smaller a.c. fields. Buffer gas cooling can be compared to having a vacuum flask cools down because the low pressure of gas between the walls provides much less thermal insulation than a good vacuum Buffer gas cooling

12.5 Laser cooling of trapped ions The cooling of ions uses the same scattering force as the laser cooling of neutral atoms A. This Doppler cooling works in much the same way as in optical molasses but there is no need for a counter -propagating laser beam because the velocity reverses direction in a bound system. B. Each trapped ion behaves as a three-dimensional simple harmonic oscillator but a single laser beam damps the motion in all directions. Differences Laser cooling of trapped ions and the laser cooling of neutral atoms Same

Fig A string of calcium ions in a linear Paul trap.The ion have an average separation of 10μm and the strong fluorescence enables each ion to be detected individually. The minimum size of the image for each ion is determined by the spatial resolution of the imaging system.

Time (S) Conuts per20 ms Fig The fluorescence signal from a single calcium ion undergoing quantum jumps. The ion gives a strong signal when it is in the ground state and it is ‘dark’ while the ion is shelved in the long-lived metastable state.

12.6 Quantum jumps In addition to the strong resonance transition of natural width used for laser cooling, ions have many other transitions and we now consider excitation of a weak optical transition with a natural width, where. Figure shows both of the transitions and the relevant energy levels. The first application described here simply uses a narrow transition to measure the temperature accurately.

︱2〉︱2〉 ︱3〉︱3〉 ︱1〉︱1〉 Fig Three energy levels of an ion. The allowed transition between levels 1 and 2 gives a strong fluorescence signal when excited by laser light. The weak transition between 1 and 3 means that level 3 has a long lifetime and (a metastable state).

Fig The electrode configuration of (a) the Paul trap and (b) the Penning trap, shown in cross-section.The lines between the end caps and ring electrode indicate the electric field lines; the Paul trap has an oscillating electric field but the Penning trap has static electric and magnetic fields. The electrodes shown have a hyberbolic shape, but for a small cloud of ions confined near to the centre any reasonable shape with cylindrical symmetry will do. Small ion traps with dimensions ~ 1mm generally have simple electrodes with cylindrical or spherical surfaces (a)(b) 12.7 The Penning trap and the Paul trap

The Penning trap The Penning trap has the same electrode shape as the Paul trap, but uses static fields. In a Penning trap for positive ions, both end caps have the same positive voltage to repel the ions and prevent them escaping along the axis. With only a d.c electric field the ions fly off in the radial direction, as expected from Earnshaw’s theorem, but a strong magnetic field along the z-axis confines the ions.

E x y (a) x y (b) Fig (a) A uniform electric field along the x-direction accelerates the ion in that direction. (b) A uniform magnetic flux density B along the z-direction leads to a circular motion in the plane perpendicular to B, at the cyclotron frequency

(c) In a region of crossed electric and magnetic fields the motion described by eqn12.25 is drift at velocity E/B perpendicular to the uniform electric field in addition to the cyclotron orbits. (d) In a Penning trap the combination of a radial electric field and axial magnetic field causes the ion to move around in a circle at the magnetron frequency. x y (c) (d)

This assumes the simplest case with two species of equal charge, but the ratio of the charges is always known exactly Mass spectroscopy of ions

12.8 Electron beam ion trap The electron beam ion trap ( EBIT ) was developed to confine ions that have lost many electrons and which have energies much higher than those in typical experiments with Paul and Penning traps.

Fig.12.9 A cross-section of an electron beam ion trap that has cylindrical symmetry. The high-energy electron beam along the axis of the trap attracts positive ions to give radial confinement and ionizes them further. The electrodes give confinement along the axis, i.e. the top and bottom drift tubes act like end caps as in a Penning trap but with much higher positive voltage. To the right is an enlarged view of the ions in the electrostatic potentials along the radial and axial directions. middle drift tube Top drift tube bottom drift tube Electron beam Radial potential Electron beam Axial potential Fig.12.9

12.9 Resolved sideband cooling The vibrational energy levels have the same spacing in both the ground and excited states of the ion.The trapped ion absorbs light at the angular frequency ωo of the narrow transition for a free ion,and also at the frequencies ωo±ωv, ωo±2ωv,etc.that correspond to transitions in which the vibrational motion of the ion charges.

ωUωU (a) Excitation by light of frequency, followed by spontaneous emission, leads to a decrease in the vibrational quantum number, until the ion reaches the lowest level with U=0, from whence there is no transition, as indicated by the dotted line at this frequency. U ′ =0 2 ωL=ωO-ωVωL=ωO-ωV U= Fig.12.10

SLSL SUSU Frequency of laser radiation, ω L′ Signal (arbitrary units) ω 0-ωVω 0-ωV ω 0+ ω V ( b) The spectrum of a trapped ion shows sidebands on either side of the main transition. this asymmetry indicates that the ion is mainly in the lowest vibrational state. The vertical axis gives the transition probability, or the probability of a quantum jump during the excitation of the narrow transition. Fig.12.10

Summary of ion traps This chapter explored some of the diverse physics of ion trapping, ranging from the cooling of ions to temperatures of only 10-3 K in small ion traps to the production of highly -charged ions in the EBIT. In the future it will be possible to do anti-atomic physics,This high-energy trapping work has developed from accelerator-based experiments and probes similar physics. At the opposite pole lies the work on the laser cooling of ions to extremely low energies. We have seen that the fundamental limit to the cooling of a bound system is quite different to the laser cooling of free atoms. Experimenters have developed powerful techniques to manipulate single ions and make frequency standards of extreme precision. Such experimental techniques give exquisite control over the state of the whole quantum system in a way that the founders of quantum mechanics could only dream about.