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
