1. Penning Traps and Strong Correlations John Bollinger NIST-Boulder Ion Storage group Wayne Itano, David Wineland, Joseph Tan, Pei Huang, Brana Jelenkovic, Travis Mitchell, Brad King, Jason Kriesel, Marie Jensen, Taro Hasegawa, Nobuysau Shiga, Michael Biercuk, Hermann Uys, Joe Britton, Brian Sawyer

2. Outline: Ref: Dubin and O’Neil, Rev. Mod. Physics 71, 87 (1999) Please ask questions!!

3. Outline:

4. 4 cm 12 cm Single particle motion in a Penning trap cyclotron frequencysingle particle magnetron frequency B=4.5 T characteristic trap dimension

5. 9 Be + Ω c /(2π) ~ 7.6 MHz ω z /(2π) ~ 800 kHz ω m /(2π) ~ 40 kHz Single particle motion in a Penning trap stability “diagram” B=4.5 T

6. 4 cm Single particle motion in a Penning trap 9 Be + Ω c /(2π) ~ 7.6 MHz ω z /(2π) ~ 800 kHz ω m /(2π) ~ 40 kHz Tutorial problem: background gas collisions produce BeH + and BeOH + ions -What critical trap voltage will dump BeOH + -What critical trap voltage will dump BeH + -How much energy must be imparted to the BeH + cyclotron motion to drive it out of the trap? V trap = 1 kV B=4.5 T

7. Outline:

8. Non-neutral plasmas in traps evolve into bounded thermal equilibrium states thermal equilibrium, Hamiltonian and total canonical angular momentum conserved ⇒ plasma rotates rigidly at frequency  r } density distribution plasma potential trap potential Lorentz force potential centrifugal potential Lorentz-force potential gives radial confinement !!  c = eB/m = cyclotron frequency

9. → constant density plasma; density determined by rotation frequency T → 0 limit

10. Quadratic trap potential (and T→0) plasma shape is a spheroid r z 2r o 2z o rr aspect ratio   z o /r o determined by  r associated Legendre function B T=0 plasma state characterized by: ω z, Ω c, and ω r, N Bounded equilibrium requires: ω m < ω r < Ω c -ω

11.  c  m Density n o mm cc Rotation frequency  r nBnB Plasma aspect ratio determined by  r Rotation frequency→ Plasma radius→ Simple equilibrium theory describes the plasma shapes Bollinger et al., PRA 48, 525 (1993)

12.  z 2 /2  r (  c -  r )  = z 0 /r 0 experimental measurements of plasma aspect ratio vs trap parameters Measurements by X.-P. Huang, NIST Simple equilibrium theory describes spheroidal plasma shapes From Bharadia, Vogel, Segal, and Thompson, Applied Physics B (to be published)

13. Outline:

14. Ions in a Penning trap are an example of a one-component plasma ● one component plasma (OCP) – consists of a single species of charged particles immersed in a neutralizing background charge ● ions in a trap are an example of an OCP (Malmberg and O’Neil PRL 39, (77)) looks like neutralizing background ● thermodynamic state of an OCP determined by:   > 1 ⇒ strongly coupled OCP

15. Why are strongly coupled OCP’s interesting? ● For an infinite OCP,  >2  liquid behavior  ~173  liquid-solid phase transition to bcc lattice Brush, Salin, Teller (1966)  ~125 Hansen (1973)  ~155 Slatterly, Doolen, DeWitt(1980)  ~168 Ichimaru; DeWitt; Dubin (87-93)  ~172-174 ● Strongly coupled OCP’s are models of dense astrophysical matter – example: outer crust of a neutron star ● Coulomb energies/ion of bulk bcc, fcc, and hcp lattices differ by < 10 -4 body centered cubic face centered cubic hexagonal close packed ● with trapped ions and laser cooling, n o ~ 10 9 cm -3 T < 5 mK ⇒  > 500

16. Laser-cooled ion crystals White Dwarf Interiors Neutron Star Crusts  = 175  = 2 Increasing Correlation  Plasmas vs strongly coupled plasmas

17. Rahman and Schiffer, Structure of a One-Component Plasma in an External Field: A Molecular-Dynamics Study of Particle Arrangement in a Heavy-Ion Storage Ring, PRL 57, 1133 (1986) Dubin and O’Neil, Computer Simulation of Ion Clouds in a Penning Trap, PRL 60, 511 (1988)

18. Observations of shell structure ● 1987 – Coulomb clusters in Paul traps MPI Garching (Walther) NIST (Wineland) ● 1988 – shell structures in Penning traps NIST group ● 1992 – 1-D periodic crystals in linear Paul traps MPI Garching ● 1998 – 1-D periodic crystals with plasma diameter > 30 a WS Aarhus group Nature 357, 310 (92) PRL 81, 2878 (98) PRL 60, 2022 (1988) See Drewsen presentation/Jan. 16

19. How large must a plasma be to exhibit a bcc lattice? 1989 - Dubin, planar model PRA 40, 1140 (89) result: plasma dimensions  60 interparticle spacings required for bulk behavior N > 10 5 in a spherical plasma  bcc lattice 2001 – Totsji, simulations, spherical plasmas, N  120 k PRL 88, 125002 (2002) result: N>15 k in a spherical plasma  bcc lattice

20. NIST Penning trap – designed to look for “large” bcc crystals 4 cm 12 cm B=4.5 T S 1/2 P 3/2 9 Be + (neglecting hyperfine structure) +3/2 +1/2 -1/2 -3/2 +1/2 -1/2    nm) T min ( 9 Be + ) ~ 0.5 mK T measured < 1 mK Doppler laser cooling

21. J.N. Tan, et al., Phys. Rev. Lett. 72, 4198 (1995) W.M. Itano, et al., Science 279, 686 (1998) Bragg scattering set-up B0B0 -V 0 axial cooling beam Bragg diffraction CCD camera side-view camera deflector 9 Be + y x z f rotation = 240 kHz n = 7.2 x 10 8 /cm 3 compensation electrodes (6  60 o )

22. B0B0 -V 0 axial cooling beam Bragg diffraction CCD camera side-view camera deflector 9 Be + y x z compensation electrodes (6  60 o ) Bragg scattering

23. Bragg scattering from spherical plasmas with N~ 270 k ions Evidence for bcc crystals scattering angle

24. B0B0 ww rotating quadrupole field (top- view)       B0B0 -V 0 axial cooling beam strobing Bragg diffraction CCD camera side-view camera deflector 9 Be + y x z Rotating wall control of the plasma rotation frequency Huang, et al. (UCSD), PRL 78, 875 (97) Huang, et al. (NIST), PRL 80, 73 (98) compensation electrodes (6  60 o ) - See Wednesday 10:15 AM talk -

25. Phase-locked control of the plasma rotation frequency time averaged Bragg scattering camera strobed by the rotating wall Huang, et al., Phys. Rev. Lett. 80, 73 (98) ● determine if crystal pattern due to 1 or multiple crystals ● enables real space imaging of ion crystals

26. 1.2 mm compensation electrodes (6  60 o ) B0B0 -V 0 axial cooling beam side-view camera deflector 9 Be + y x z f/2 objective relay lens gateable camera strobe signal Mitchell. et al., Science 282, 1290 (98) Real space imaging

27. compensation electrodes (6  60 o ) B0B0 -V 0 axial cooling beam deflector 9 Be + y x z f/2 objective relay lens gateable camera strobe signal bcc (100) plane predicted spacing: 12.5  m measured: 12.8 ± 0.3  m 0.5 mm bcc (111) plane predicted spacing: 14.4  m measured: 14.6 ± 0.3  m

28. Summary of correlation observations in approx. spherical plasmas

29.  c  m Density n o mm cc Rotation frequency  r nBnB Real-space images with planar plasmas with planar plasmas all the ions can reside within the depth of focus

30. side-views top-views 1 lattice plane, hexagonal order2 planes, cubic order 65.70 kHz 66.50 kHz Planar structural phases can be ‘tuned’ by changing  r

31. a b c Top- (a,b) and side-view (c) images of crystallized 9 Be + ions contained in a Penning trap. The energetically favored phase structure can be selected by changing the density or shape of the ion plasma. Examples of the (a) staggered rhombic and (b) hexagonal close packed phases are shown.

32. y z x z1z1 z2z2 z3z3 increasing rotation frequency  Theoretical curve from Dan Dubin, UCSD Mitchell, et al., Science 282, 1290 (98) Theoretical curve from Dan Dubin, UCSD Mitchell, et al., Science 282, 1290 (98)

33. Summary of Penning traps and strong correlations ● Single particle orbits characterized by 3 motional frequencies: r 2r o 2z o rr ● 2-d triangular-lattice crystals obtained at low ω r (weak radial confinement) Present effort: quantum simulation/control experiments Biercuk et al., Nature 458, 996 (2009) Biercuk, et al., Quantum Info. and Comp. 9, 920-949 (2009)

34. 4 cm Single particle motion in a Penning trap 9 Be + Ω c /(2π) ~ 7.6 MHz ω z /(2π) ~ 800 kHz ω m /(2π) ~ 40 kHz Tutorial problem: background gas collisions produce BeH + and BeOH + ions -What critical trap voltage will dump BeOH + -What critical trap voltage will dump BeH + -How much energy must be imparted to the BeH + cyclotron motion to drive it out of the trap? V trap = 1 kV B=4.5 T