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
Outline:
● Review of single particle motion
● T=0 and 𝑧 2 − 𝑟 2 /2 → constant density spheroidal equilibria characterized by N
and rotation frequency 𝜔𝑟
● Some experimental examples (NIST, Imperial College)
● Connection with strongly coupled OCP’s
● Types of correlations:
Shell structure
3-d periodic crystals
2-d crystal arrays
Experimental examples
Please ask questions!!
Ref: Dubin and O’Neil, Rev. Mod. Physics 71, 87 (1999)
Outline:
● Review of single particle motion
● T=0 and 𝑧 2 − 𝑟 2 /2 → constant density spheroidal equilibria characterized by N
and rotation frequency 𝜔𝑟
● Some experimental examples (NIST, Imperial College)
● Connection with strongly coupled OCP’s
● Types of correlations:
Shell structure
3-d periodic crystals
2-d crystal arrays
Experimental examples
Single particle motion in a Penning trap
For r , z  trap dimensions ,
B=4.5 T
2


1
r
2
2
trap (r , z )  m z  z  
2
2

z 
Vtrap M
RT
characteristic trap
dimension
z (t )  zo sin( z t   z )
12 cm
r (t )  rc sin ( c  m )t  c  
rm sin(mt  m )
2
c
qB
 c  z
c 
, m 
 
 
m
2
2
 2 
2
cyclotron frequency
4 cm
single particle
magnetron frequency
Single particle motion in a Penning trap
z (t )  zo sin(  z t   z )
r (t )  rc sin ( c  m )t  c  
B=4.5 T
rm sin( mt  m )
2
c
qB
 c   z
c 
, m 
 
 
m
2
2
 2 
2
9Be+
Ωc/(2π) ~ 7.6 MHz
ωz/(2π) ~ 800 kHz
ωm/(2π) ~ 40 kHz
stability “diagram”
c
z 
2
Single particle motion in a Penning trap
B=4.5 T
9Be+
Ωc/(2π) ~ 7.6 MHz
ωz/(2π) ~ 800 kHz
ωm/(2π) ~ 40 kHz
Vtrap= 1 kV
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?
4 cm
Outline:
● Review of single particle motion
● T=0 and 𝑧 2 − 𝑟 2 /2 → constant density spheroidal equilibria characterized by N
and rotation frequency 𝜔𝑟
● Some experimental examples (NIST, Imperial College)
● Connection with strongly coupled OCP’s
● Types of correlations:
Shell structure
3-d periodic crystals
2-d crystal arrays
Experimental examples
Non-neutral plasmas in traps evolve into
bounded thermal equilibrium states
thermal equilibrium, Hamiltonian and total canonical angular momentum conserved
 
f (r , v )  exp[ (h  r p ) / k BT ]
⇒
2

qB 2
where h  mv / 2  q (r ) and p  mv r 
r
2
}
 

f (r , v )  n(r , z ) exp[ m(v  r rˆ) 2 /( 2k BT )]
plasma rotates rigidly at frequency r
density distribution
 1
n(r , z )  exp  
 k BT
c = eB/m =
cyclotron frequency

r2 
q p (r , z )  qT (r , z )  mr ( c  r )  
2 

plasma
potential
trap
potential
Lorentz
force
potential
Lorentz-force potential gives radial confinement !!
centrifugal
potential
T→0 limit
 1
n(r , z )  exp 
 k BT

r2 
q p (r , z )  qT (r , z )  mr ( c  r )  
2 

r2
For r , z  plasma interior, T  0  q p (r , z )  qT (r , z )  mr ( c  r )  0
2
r2
q p (r , z )  qT (r , z )  mr ( c  r )
2
 o 2
o
n0 (r , z ) 
  p (r , z )  2 2mr ( c  r )
q
q
→ constant density plasma; density determined by rotation frequency
Quadratic trap potential (and T→0)
r2
q p (r , z )  qT (r , z )  mr ( c  r )
2
m z2  2 r 2 
r2
 z    mr ( c  r )

2 
2
2

m p2
6


q 2 n0
ar  b z ,  
 2r  c  r 
 om
2
2
2
p
plasma shape is a spheroid
z
aspect ratio   zo/ro
determined by r
B
2zo
r
associated Legendre function
r
2ro
T=0 plasma state characterized by: ωz , Ωc , and ωr , N
Bounded equilibrium requires: ωm< ωr < Ωc -ω
Density no
Plasma aspect ratio determined by r
nB
m
c / 2
c  m
Rotation frequency r
Plasma radius→
Simple equilibrium
theory describes the
plasma shapes
Bollinger et al., PRA 48, 525 (1993)
Rotation frequency→
Simple equilibrium theory describes spheroidal plasma shapes
From Bharadia, Vogel, Segal, and Thompson,
Applied Physics B (to be published)
𝑚𝜃 = 1 rotating wall
z2/2r (c - r)
experimental measurements of plasma aspect ratio vs trap parameters
Measurements by X.-P. Huang, NIST
 = z0/r0
Outline:
● Review of single particle motion
● T=0 and 𝑧 2 − 𝑟 2 /2 → constant density spheroidal equilibria characterized by N
and rotation frequency 𝜔𝑟
● Some experimental examples (NIST, Imperial College)
● Connection with strongly coupled OCP’s
● Types of correlations:
Shell structure
3-d periodic crystals
2-d crystal arrays
Experimental 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))
 1
n(r , z )  exp 
 k BT

r2 
q p (r , z )  qT (r , z )  mr ( c  r )  
2 

looks like neutralizing background
 bkgnd

1
r2 
 T (r , z )  mr ( c  r )   
q
2
o

2 m (  r )
 bkgnd   o r c
q
2
● thermodynamic state of an OCP determined by:
q2
4

,  aWS 3n  1
aWS kB T 3
 > 1 ⇒ strongly coupled OCP
 
potential energy between neighboring ions
ion thermal energy
Why are strongly coupled OCP’s interesting?
● Strongly coupled OCP’s are models of dense astrophysical matter –
example: outer crust of a neutron star
>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
● For an infinite OCP,
● 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, no~ 109 cm-3
T < 5 mK ⇒  > 500
Plasmas vs strongly coupled plasmas
White
Dwarf
Interiors
Neutron
Star
Crusts
Increasing
Correlation 
Laser-cooled
ion crystals
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 traps
MPI Garching (Walther)
NIST (Wineland)
● 1988 – shell structures in Penning traps
NIST group
PRL 60, 2022 (1988)
● 1992 – 1-D periodic crystals in linear Paul traps
MPI Garching
Nature 357, 310 (92)
● 1998 – 1-D periodic crystals with plasma diameter
> 30 aWS
Aarhus group
PRL 81, 2878 (98)
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 behavior
N > 105 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
NIST Penning trap – designed to look for “large” bcc crystals
B=4.5 T
Doppler laser cooling
+3/2
+1/2
P3/2
(neglecting hyperfine
structure)
12 cm
9Be+
-1/2
-3/2
no l313 nm)
+1/2
S1/2
-1/2
Tmin(9Be+) ~ 0.5 mK
4 cm
Tmeasured < 1 mK
Bragg scattering set-up
Bragg diffraction
CCD camera
frotation = 240 kHz
n = 7.2 x 108 /cm3
deflector
side-view
camera
9Be+
J.N. Tan, et al., Phys. Rev. Lett. 72, 4198 (1995)
W.M. Itano, et al., Science 279, 686 (1998)
B0
-V0
compensation
electrodes
(660o)
z
axial
cooling beam
x
y
Bragg scattering
Bragg diffraction
CCD camera
deflector
side-view
camera
9Be+
B0
-V0
compensation
electrodes
(660o)
z
axial
cooling beam
x
y
Bragg scattering from spherical plasmas with N~ 270 k ions
Evidence for bcc
crystals
scattering angle
Rotating wall control of the plasma rotation frequency
- See Wednesday 10:15 AM talk Bragg diffraction
CCD camera
strobing
Huang, et al. (UCSD), PRL 78, 875 (97)
Huang, et al. (NIST), PRL 80, 73 (98)
deflector
rotating quadrupole field (topview)

side-view
camera
w

9Be+
B
compensation
electrodes
(660o)
0
z
axial
cooling beam

-V0
x
y



B0
Phase-locked control of the plasma rotation frequency
Huang, et al., Phys. Rev. Lett. 80, 73 (98)
time averaged Bragg scattering
camera strobed by the rotating wall
● determine if crystal pattern due to 1 or multiple crystals
● enables real space imaging of ion crystals
Real space imaging
gateable camera
Mitchell. et al., Science 282, 1290 (98)
strobe signal
relay lens
f/2 objective
1.2 mm
1.2 mm
deflector
side-view
camera
9Be+
B
-V0
compensation
electrodes
(660o)
0
axial
cooling beam
z
x
y
Top-view images in a spherical plasma of ~180,000 ions
gateable camera
strobe signal
relay lens
f/2 objective
deflector
9Be+
B
0
axial
cooling beam
-V0
0.5 mm
compensation
bcc (100) plane
bcc
(111) plane
electrodes
predicted
spacing: 12.5 mm
predicted
spacing:
o
(660 )
measured: 12.814.4
± 0.3mm
mm
measured: 14.6 ± 0.3 mm
z
x
y
Summary of correlation observations in approx. spherical plasmas
● 𝑵 ≳ 𝟐 × 105 observe bcc crystal structure
● 𝟏 × 𝟏𝟎𝟓 > N> 2× 𝟏𝟎𝟒 observe other crystal structures (fcc, hcp?, ..) in addition to bcc
● 𝟐 × 𝟏𝟎𝟒 > 𝑵 shell structure dominates
Density no
Real-space images with planar plasmas
nB
m
c / 2
c  m
Rotation frequency r
with planar plasmas all the ions can reside within the depth of focus
Planar structural phases can be ‘tuned’ by changing r
65.70 kHz
66.50 kHz
top-views
side-views
1 lattice plane, hexagonal order
2 planes, cubic order
a
c
b
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.
Mitchell, et al., Science 282, 1290 (98)
Theoretical curve from Dan Dubin, UCSD
z / a0 (plane axial position)
2
z
z
z2 3
z1
x
y
1
0
-1
rhombic
hexagonal
-2
1
2
3
2
4
a0 (areal charge density)
increasing rotation frequency 
5
Summary of Penning traps and strong correlations
● Single particle orbits characterized by 3 motional frequencies:
2
2
Vtrap M
c
qB
 c  z
c 
, m 
 

,



z
m
2
2
RT
 2 
● T=0 and 𝝋𝒕𝒓𝒂𝒑 ∝ 𝒛𝟐 −𝒓𝟐 /𝟐 ⟹ constant density spheroidal equilibria 2zo
characterized by N and rotation frequency 𝝎𝒓
● Ions in a trap ⇔ one-component plasma
● Large spheroidal plasmas (N > 105) ⇒ stable bcc crystals
metastable 3-d periodic crystals observed with ~ 103 ions
Drewsen – PRL 96 (2006)
Not yet observed – liquid-solid phase transition at Γ~172
● 2-d triangular-lattice crystals obtained at low ωr
(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)
r
r
2ro
Single particle motion in a Penning trap
B=4.5 T
9Be+
Ωc/(2π) ~ 7.6 MHz
ωz/(2π) ~ 800 kHz
ωm/(2π) ~ 40 kHz
Vtrap= 1 kV
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?
4 cm