Physics of and in Ion Traps

Transcription

1 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

2 Why particle trapping?

3 Why particle trapping? Hans Dehmelt: A single particle at rest floating forever in free space would be the ideal object

4 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

5 Key words for trap spectroscopy Sensitivity Precision i Control

6 Pioneers of ion trapping Wolfgang Paul Hans Dehmelt Nobelprize 1989

7 Basics of Ion Traps

8 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

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10 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

11 The ideal Paul trap Time-averaged potential minimum

12 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

13 Stable solutions of the Mathieu equation

14 First stability area For a=0:

15 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)

16 Ion trajectories at different operating conditions

17 Trajectory of a single microscopic particle (Wuerker 1959)

18 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

19 Density distribution of an ion cloud in a Paul trap Experimentally measured distribution for uncooled ion cloud Calculated density distribution for different temperatures

20 The ideal Penning trap Axial harmonic potential Radial confinement by magnetic field

21 Equations of motion

22 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

23 Ion Motion in a Penning Trap

24 Stability limit: z 8 ω c ω e M B U d

25 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

26 Rotating wall compression of ion clouds Mg + Electrons

27 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

28 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)

29 Real Traps Deviations from ideal harmonic potential caused by: Truncation of trap electrode Imperfect electrode shape Misalignements Space charge from simultaneously trapped ions

30 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 3 : 3 : z z r r n r z z r n z r n + = + = : 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

31 Effects of trap imperfections: Coupling of oscillation modes Shift of eigenfrequencies Asymmetry of resonances Collective and noncollective oscillations Instabilities of ion trajectories

32 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 / 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 Frequency [khz]

33 Motional spectrum of ions in a Paul trap measured by laser induced dfluorescence

34 Minimizing trap imperfections Additional electrodes between ring and endcap Cyclotron resonance at different values of the correction voltage

35 Asymmetry of axial resonance (taken at different excitation amplitudes)

36 Individual and center of mass oscillation of axial motion Detected Ion Number [a.u.] 1600 Center of Mass oscillation Individual ion oscillation Noncollective Collective Resonance Resonance Excitation Frequency [khz]

37 Space chargeshiftof axial resonance 1060 Axial oscillation Frequeny [khz] Ion number

38 Threshold behaviour of center of mass resonance 10mV 0mV 3mV N d (a arb. units) 30mV 90mV

39 Dependence of thresghold amplitude on ion number N d V th ~ N /3

40 Bistability in the excitation of motional resonances N d (arb. units)

41 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

42 Instabilities of ion motion in a Paul trap

43 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 )

44 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]