Physics of and in Ion Traps Proposed Topics: TRIUMF, Vancouver June 01 Basics of Paul- and Penning-traps (equ. of motion, trap geometries, influence of trap imperfections,) Ion detection and cooling (Buffer gas cooling, resistive cooling, Laser Doppler- and sideband-cooling, sympathetic cooling, ion crystallization) Zeeman spectroscopy (g factor determinations) Hyperfine spectroscopy Atomic clocks Mass spectrometry in Paul- and Penning-traps Quantum computing with trapped ions
Why particle trapping?
Why particle trapping? Hans Dehmelt: A single particle at rest floating forever in free space would be the ideal object
Why particle trapping? Hans Dehmelt: A single particle at rest floating forever in free space would be the ideal object Approximation: A single particle at very low momentum confined for long times by well known forces in a small volume in space would be a desirable object
Key words for trap spectroscopy Sensitivity Precision i Control
Pioneers of ion trapping Wolfgang Paul Hans Dehmelt Nobelprize 1989
Basics of Ion Traps
Trapping of charged particles by electromagnetic fields Required: 3-dimensional force towards center F = - e grad du Convenience: harmonic force F x,y,z U = ax +by + cz Laplace equ.: Δ(eU) = 0 a,b,c can not be all positive Convenience: rotational symmetry U = (U 0 /r 0 )(x +y -z ) Quadrupole potential Equipotentials: Hyperboloids of revolution
Problem: No 3-dimensional potential minimum because of different sign of the coefficients in the quadrupole potential Solutions: Application of r.f. voltage: dynamical trapping Paul trap d.c. voltage + magnetic field in z-direction: i Penning trap
The ideal Paul trap Time-averaged potential minimum
Equation of motion for a single particle Potential: U=(U 0 +V 0 cosωt)(r -z )/r 0 Using: a q u z z = = 8 eu mr = r, z 0 τ = Ω t / o Ω 4 ev 0 mr Ω 0 = = a q r r We obtain the normalized Mathieu differential equation d u dt + ( a q cos τ ) u = 0 The solutions are well known and depend on the size of the parameters a and q: When u remains finite in time:stable solutions When u goes to infinity: unstable solutions
Stable solutions of the Mathieu equation
First stability area For a=0:
solution of the equation of motion: u( t) = A n= 0 β c cos( + n)( Ωt / ) n β = β ( a, q ) c n = f ( a, q ) Approximate solution for a,q<<1: u( t) = A[1 ( q / ) cos ωt] cos β =a+q / Ωt This is a harmonic oscillation at frequency Ω (micromotion) modulated by an oscillation at frequency ω(macromotion)
Ion trajectories at different operating conditions
Trajectory of a single microscopic particle (Wuerker 1959)
Time averaged potential depth: D i m 8 = Ω r 0 Numerical example: m=50 Ω/π=1 MHz r 0 =1 cm ß=0.3 D = 5 ev Maximum ion density, when space charge potential equals trapping potential depth β n max 1o 6 cm -3 i Mean kinetic ion energy (no cooling) 1/10 D
Density distribution of an ion cloud in a Paul trap Experimentally measured distribution for uncooled ion cloud Calculated density distribution for different temperatures
The ideal Penning trap Axial harmonic potential Radial confinement by magnetic field
Equations of motion
Solutions: 3 harmonic oscillations ω z = 4 eu 0 md 1 ω = ( ω + ω 1/ ω 1 ) axial + c perturbed cyclotron = 1 ( ω ω 1 c magnetron ) ω 1 = ω c ω z
Ion Motion in a Penning Trap
Stability limit: z 8 ω c ω e M B U d
Quantum mechanical energy levels of a particle in the Penning trap: E = + + z ( n + 1/) h ω ( n + 1/) hω + ( n + 1/) hω 7 Negative sign for magnetronenergy indicates metastability of motion
Rotating wall compression of ion clouds Mg + Electrons
Comparison of Paul- and Penning traps Paul traps Penning traps Simple set up (no big magnet required) Three dimensional potential well Simultaneous trapping of both signs of charge Limited range for different charge states Rf heating All frequencies subject to space charge shift No rf heating effects High mass resolution Better stability for higher charge states cyclotron frequency insensitive to space charge Expensive (big magnet required) Trapping for only one sign of charge
References W. Paul, Electromagnetic Traps for Charged Particles, Rev. Mod. Phys. 6, 531 (1990) P.K. Ghosh, Ion Traps, Clarendon, Oxford (1995) H. Dehmelt, in: Advances in: Atom. Molec. Phys. Vol 3 (1967) F.G. Major, V. Gheorghe, G. Werth Charged Particle traps, Springer (005) G. Werth, F.G. Major, V. Gheorghe, Charged Particle traps II, Springer (009) G. Werth, Trapped Ions, Contemporary Physics 6, 41 (1985) G. Savard and G. Werth, Precision Nuclear Measurements with Ion Traps, Ann. Rev. Nucl. Part. Science 50, 119 (00)
Real Traps Deviations from ideal harmonic potential caused by: Truncation of trap electrode Imperfect electrode shape Misalignements Space charge from simultaneously trapped ions
Dealing with imperfections: Expansion of the trapping potential in spherical harmonics: ) (cos ) ( ϑ ϑ n P r c r Φ = Φ ) (cos ), ( 0 ϑ ϑ n n n P d r c = Φ = Φ Coordinate dependence of some higher order contributions: : z r n + = 4 4 8 4 3 : 4 3 : 3 : z z r r n r z z r n z r n + = + = + 6 4 4 6 16 10 90 5 : 6 z z r z r r n + + = n=: quadrupole n=3: hexapole n=4 octupole n=6: dodekapole n=: quadrupole, n=3: hexapole, n=4. octupole, n=6: dodekapole
Effects of trap imperfections: Coupling of oscillation modes Shift of eigenfrequencies Asymmetry of resonances Collective and noncollective oscillations Instabilities of ion trajectories
Coupling of ion oscillation modes of a stored ion cloud in a Paul Trap Ion number 1000 ω z + ω r ω z + ω r Ω -ω r /3 800 4 ω z 600 ω z / 3 ω r 4 ω r Ω -3ω z 3 ω z 400 ω r /3 ω z +ω r 00 0 ω r ω r ω z ω z Ω -ω z Ω - ω ω z Ω - ω r Ω -ω r 0 500 1000 1500 000 500 3000 Frequency [khz]
Motional spectrum of ions in a Paul trap measured by laser induced dfluorescence
Minimizing trap imperfections Additional electrodes between ring and endcap Cyclotron resonance at different values of the correction voltage
Asymmetry of axial resonance (taken at different excitation amplitudes)
Individual and center of mass oscillation of axial motion Detected Ion Number [a.u.] 1600 Center of Mass oscillation 100 800 400 0 Individual ion oscillation Noncollective Collective Resonance Resonance 1040 1050 1060 1070 Excitation Frequency [khz]
Space chargeshiftof axial resonance 1060 Axial oscillation Frequeny [khz] 1058 1056 1054 105 1050 1048 1046 0 10000 0000 30000 40000 Ion number
Threshold behaviour of center of mass resonance 10mV 0mV 3mV N d (a arb. units) 30mV 90mV 1045 1050 1055 1060
Dependence of thresghold amplitude on ion number 100 10 100 1000 10000 N d V th ~ N /3
Bistability in the excitation of motional resonances 14000 1000 N d (arb. units) 10000 8000 6000 4000 000 0 1054 1055 1056 1057 1058
Instabilities of the ion motion in a Paul trap occur when the ion oscillation freqencies ω r r, ω z are linear dependend on the traps driving frequency Ω n r ω r + n z ω z =k Ω n r, n z, k integer n r + n z = N N = order of perturbation
Instabilities of ion motion in a Paul trap
Observed Instabilities on electrons in a Penning trap ω z +ω =ω + ω z =ω + +3ω ω z + =ω + ω z +ω - =ω + Limit i of the stability region 3ω z =ω + ω z +ω =ω + Num mber of ele ectrons (a a.u.) Ring voltage (% of V max )
Instabilities of electrons stored in a Penning trap for different storage times Partic cle numb ber (a.u.) Stor rage tim me [ms] Trapping Voltage [V]