Penning Traps and Strong Correlations

Penning Traps and Strong Correlations

John BollingerNIST-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

Outline:

Ref: Dubin and O’Neil, Rev. Mod. Physics 71, 87 (1999)

Please ask questions!!

● Review of single particle motion● T=0 and → constant density spheroidal equilibria characterized by N

and rotation frequency ● Some experimental examples (NIST, Imperial College)● Connection with strongly coupled OCP’s● Types of correlations:

Shell structure3-d periodic crystals2-d crystal arraysExperimental examples

Outline:

● Review of single particle motion● T=0 and → constant density spheroidal equilibria characterized by N

and rotation frequency ● Some experimental examples (NIST, Imperial College)● Connection with strongly coupled OCP’s● Types of correlations:

Shell structure3-d periodic crystals2-d crystal arraysExperimental examples

4 cm

12 c

mSingle particle motion in a Penning trap

221),(

,dimensions trap,For 2

22 rzmzr

zr

ztrap

)sin(

)(sin)()sin()(

mmm

cmcc

zzo

trtrtr

tztz

222,

22zcc

mc mqB

cyclotron frequency single particle magnetron frequency

B=4.5 T

T

trapz R

MV characteristic trap

dimension

9Be+

Ωc/(2π) ~ 7.6 MHzωz/(2π) ~ 800 kHzωm/(2π) ~ 40 kHz

Single particle motion in a Penning trap

)sin(

)(sin)()sin()(

mmm

cmcc

zzo

trtrtr

tztz

222,

22zcc

mc mqB

stability “diagram”

2c

z

B=4.5 T

4 cm

Single particle motion in a Penning trap

9Be+

Ωc/(2π) ~ 7.6 MHzωz/(2π) ~ 800 kHzωm/(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?

Vtrap= 1 kV

B=4.5 T

Outline:

● Review of single particle motion● T=0 and → constant density spheroidal equilibria characterized by N

and rotation frequency ● Some experimental examples (NIST, Imperial College)● Connection with strongly coupled OCP’s● Types of correlations:

Shell structure3-d periodic crystals2-d crystal arraysExperimental examples

Non-neutral plasmas in traps evolve into bounded thermal equilibrium states

thermal equilibrium, Hamiltonian and total canonical angular momentum conserved ⇒

22

2 and )(2/ where

]/)(exp[),(

rqBrmvprqvmh

Tkphvrf Br

plasma rotates rigidly at frequency r

}density distribution

)]2/()ˆ(exp[),(),( 2 Tkrvmzrnvrf Br

plasma potential

trappotential

Lorentz force

potential

centrifugal potential

Lorentz-force potential gives radial confinement !!

c = eB/m = cyclotron frequency

2)(),(),(1exp),(

2rmzrqzrqTk

zrn rcrTpB

2)(),(),(1exp),(

2rmzrqzrqTk

zrn rcrTpB

02

)(),(),(0 interior, plasma,For 2

rmzrqzrqTzr rcrTp

)(2),(),(

2)(),(),(

22

0

2

rcro

po

rcrTp

mq

zrq

zrn

rmzrqzrq

→ constant density plasma; density determined by rotation frequency

T→0 limit

Quadratic trap potential (and T→0)

rcro

pp

rcrz

rcrTp

mnqzbra

m

rmrzm

rmzrqzrq

2,6

2)(

22

2)(),(),(

02

2222

222

2

2

plasma shape is a spheroid

r

z

2ro

2zo

r

aspect ratio zo/ro determined by r

associated Legendre function

B

T=0 plasma state characterized by: ωz , Ωc , and ωr , NBounded equilibrium requires: ωm< ωr < Ωc -ω

c m

Dens

ity n

o

m c / 2Rotation frequency r

nB

Plasma aspect ratio determined by r

Rotation frequency→

Plas

ma

radi

us→ Simple equilibrium

theory describes the plasma shapes

Bollinger et al., PRA 48, 525 (1993)

z2 /

2r (

c -

r)

= z0/r0

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)

rotating wall

Outline:

● Review of single particle motion● T=0 and → constant density spheroidal equilibria characterized by N

and rotation frequency ● Some experimental examples (NIST, Imperial College)● Connection with strongly coupled OCP’s● Types of correlations:

Shell structure3-d periodic crystals2-d crystal arraysExperimental examples

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))

2)(),(),(1exp),(

2rmzrqzrqTk

zrn rcrTpB

looks like neutralizing background

qm

rmq

zr

rcrobkgnd

o

bkgndrcrT

)(22

)(1),(2

2

134 , 3

2

naTka

qWS

BWS

energy lion therma

ions gneighborinbetween energy potential

● thermodynamic state of an OCP determined by:

> 1 ⇒ strongly coupled OCP

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

bodycenteredcubic

facecentered cubic

hexagonalclose packed

● with trapped ions and laser cooling, no~ 109 cm-3

T < 5 mK ⇒ > 500

Laser-cooled ion crystals

White DwarfInteriors

Neutron StarCrusts

= 1

75 = 2

Increasing Correlation

Plasmas vs strongly coupled plasmas

Correlations with small plasmas –effect of the boundary shell structure

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)

Observations of shell structure

● 1987 – Coulomb clusters in Paul trapsMPI Garching (Walther)NIST (Wineland)

● 1988 – shell structures in Penning trapsNIST group

● 1992 – 1-D periodic crystals in linear Paul trapsMPI Garching

● 1998 – 1-D periodic crystals with plasma diameter > 30 aWS

Aarhus group

Nature 357, 310 (92)

PRL 81, 2878 (98)

PRL 60, 2022 (1988)

See Drewsen presentation/Jan. 16

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 behaviorN > 105 in a spherical plasma bcc lattice

2001 – Totsji, simulations, spherical plasmas, N120 k

PRL 88, 125002 (2002)result: N>15 k in a spherical plasma

bcc lattice

NIST Penning trap – designed to look for “large” bcc crystals

4 cm

12 c

m

B=4.5 T

S1/2

P3/2 9Be+

(neglecting hyperfine structure)

+3/2

+1/2

-1/2

-3/2

+1/2

-1/2

no l313 nm)

Tmin(9Be+) ~ 0.5 mK

Tmeasured < 1 mK

Doppler laser cooling

J.N. Tan, et al., Phys. Rev. Lett. 72, 4198 (1995)W.M. Itano, et al., Science 279, 686 (1998)

Bragg scattering set-up

B0

-V0

axialcooling beam

Bragg diffractionCCD camera

side-viewcamera

deflector

9Be+

yx

z

frotation = 240 kHzn = 7.2 x 108 /cm3

compensation electrodes

(660o)

B0

-V0

axialcooling beam

Bragg diffractionCCD camera

side-viewcamera

deflector

9Be+

yx

z

compensation electrodes (660o)

Bragg scattering

Bragg scattering from spherical plasmas with N~ 270 k ions

Evidence for bcc crystals

scattering angle

B0

w

rotating quadrupole field (top-view)

B0

-V0

axialcooling beam

strobing

Bragg diffractionCCD camera

side-viewcamera

deflector

9Be+

yx

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 (660o)

- See Wednesday 10:15 AM talk -

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

1.2 mm

1.2

mm

compensation electrodes (660o)B

0

-V0

axialcooling beam

side-viewcamera

deflector

9Be+

yx

z

f/2 objective

relay lens

gateable camerastrobe signal

Mitchell. et al., Science 282, 1290 (98)

Real space imaging

compensation electrodes (660o)B

0

-V0

axialcooling beam

deflector

9Be+

yx

z

f/2 objective

relay lens

gateable camerastrobe signal

r = 2120 kHz

bcc (100) planepredicted spacing: 12.5 mmmeasured: 12.8 ± 0.3 mm

0.5 mm

Top-view images in a spherical plasma of 180,000 ions

bcc (111) planepredicted spacing: 14.4 mmmeasured: 14.6 ± 0.3 mm

Summary of correlation observations in approx. spherical plasmas

● 105 observe bcc crystal structure

● > N> 2observe other crystal structures (fcc, hcp?, ..) in addition to bcc

● shell structure dominates

c m

Dens

ity n

o

m c / 2Rotation frequency r

nB

Real-space images with planar plasmas

with planar plasmas all the ions can reside within the depth of focus

side-views

top-views

1 lattice plane, hexagonal order 2 planes, cubic order

65.70 kHz66.50 kHz

Planar structural phases can be ‘tuned’ by changing r

a b

c

Top- (a,b) and side-view (c) images of crystallized 9Be+ 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.

1 2 3 4 5

-2

-1

0

1

2

z / a

0 (pl

ane

axia

l pos

ition

)

a02 (areal charge density)

rhombic hexagonal

y

z

x

z1z2

z3

increasing rotation frequency

1 2 3 4 5

-2

-1

0

1

2

Theoretical curve from Dan Dubin, UCSDMitchell, et al., Science 282, 1290 (98)

Summary of Penning traps and strong correlations● Single particle orbits characterized by 3 motional frequencies:

T

trapz

zccmc R

MV

mqB

,

222,

22

● T=0 and constant density spheroidal equilibria characterized by N and rotation frequency

r

2ro

2zo

r

● Ions in a trap one-component plasma

● Large spheroidal plasmas (N > 105) stable bcc crystals

● 2-d triangular-lattice crystals obtained at low ωr (weak radial confinement)

metastable 3-d periodic crystals observed with 103 ionsDrewsen – PRL 96 (2006)

Not yet observed – liquid-solid phase transition at 172

Present effort: quantum simulation/control experimentsBiercuk et al., Nature 458, 996 (2009)Biercuk, et al., Quantum Info. and Comp. 9, 920-949 (2009)

4 cm

Single particle motion in a Penning trap

9Be+

Ωc/(2π) ~ 7.6 MHzωz/(2π) ~ 800 kHzωm/(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?

Vtrap= 1 kV

B=4.5 T