Transcript No Slide Title

Transverse optical mode in a 1-D chain
J. Goree, B. Liu & K. Avinash
Motivation: 1-D chains in condensed matter
Colloids:
Polymer microspheres
trapped by laser beams
Tatarkova, et al., PRL 2002
Carbon nanotubes:
Xe atoms trapped in a tube
Cvitas and Siber, PRB 2003
Particles
polymer microspheres
8.05 mm diameter
Q - 6  10 e
3
Interparticle interaction is repulsive
Confinement of 1-D chain
Horizontal:
sheath conforms to shape
of groove in lower electrode
Vertical:
gravity + vertical E
QE
mg
groove
lower
electrode
Image of chain in experiment
Confinement is parabolic
in all three directions
Measured values of
single-particle resonance frequency
x 0.1 Hz
groove
y 3 Hz
lower
electrode
z 15 Hz
Modes in a 1-D chain: Longitudinal
restoring force
interparticle repulsion
experiment
Homann
et al. 1997
theory
Melands
1997
Modes in a 1-D chain: Longitudinal
restoring force
interparticle repulsion
experiment
Homann
et al. 1997
theory
Melands
1997
“dust lattice wave DLW”
longitudinal mode
Modes in a 1-D chain: Transverse
Vertical motion:
restoring force
gravity + sheath
experiment
Misawa et al.
2001
theory
Vladimirov et al. 1997
Horizontal motion:
restoring force
curved sheath
oscillation.gif
experiment
THIS TALK
theory
Ivlev et al. 2000
Unusual properties of this wave:
The transverse mode in a 1-D chain is:
• optical
• backward
Terminology:
“Optical” mode
w
w
k
not optical
k
w
k
optical
Optical mode in
an ionic crystal
Terminology:
“Backward” mode
w
w
k
backward
“backward” =
“negative dispersion”
k
forward
Natural motion of a 1-D chain
1 mm
Central portion
of a 28-particle
chain
Spectrum of natural motion
Calculate:
• particle velocities
vx
vy
• cross-correlation functions
vx vx longitudinal
vy vy transverse
• Fourier transform  power spectrum
Longitudinal
power
spectrum
Power spectrum
Transverse
power
spectrum
No wave at
w = 0, k = 0
wave is optical
Next: Waves excited by external force
Setup
video camera
(top view)
scanning
mirror
microsphere
RF
lower electrode
Ar laser beam 2
lase beam1
Ar laser
beam 1
Argon laser pushes only one particle
Radiation pressure excites a wave
1 mm
modulated beam
-I0 ( 1 + sinwt )
continuous beam
I0
Net force: I0 sinwt
Wave propagates
to two ends of chain
Measure real part of k from phase vs x
fit to straight line
yields kr
Measure imaginary part of k from amplitude vs x
0.06
transverse mode
Amplitude (mm/s)
0.05
0.04
exponential
0.03
fitting
0.02
0.01
0.00
0
5
10
position (mm)
fit to exponential
yields ki
Experimental dispersion relation (real part of k)
30
wC
20
Wave is:
backward
i.e., negative dispersion
-1
w (s )
M
10
0
0
N = 10
N = 19
N = 28
1
2
kr a
3
smaller N 
larger a
larger 
Experimental dispersion relation (imaginary part of k)
for three different chain lengths
30
20
-1
w (s )
Wave damping is weakest
in the frequency band
10
N = 10
N = 19
N = 28
0
0
1
2
ki a
3
Wave damping is higher for:
smaller N
larger 
Experimental parameters
To determine Q and D from experiment:
We used equilibrium particle positions & force balance
 Q = 6200 e
D = 0.86 mm
Theory
Derivation:
• Eq. of motion for each particle, linearized & Fourier-transformed
Assumptions:
• Probably same as in experiment:
• Parabolic confining potential
• Yukawa interaction
• Epstein damping
• No coupling between L & T modes
• Different from experiment:
• Infinite 1-D chain
• Uniform interparticle distance
• Interact with nearest two neighbors only
Theoretical dispersion relation of optical mode
(without damping)
I
Evanescent
20
wCM
II
wL
-1
ww
(s-1
(s) )
wR
wL
Evanescent
III
10
0
0
1
2
ka
3
Wave is allowed in
a frequency band
Wave is:
backward
i.e., negative
dispersion
wCM =
frequency of
sloshing-mode
Theoretical dispersion relation
(with damping)
30
high damping
kr
20
-1
w (s )
I
wCM
II
10
wL
III
small
damping
ki
0
0
1
ka
2
3
Wave damping is weakest
in the frequency band
Molecular Dynamics Simulation
Solve equation of motion for N= 28 particles
Assumptions:
• Finite length chain
• Parabolic confining potential
• Yukawa interaction
• All particles interact
• Epstein damping
• External force to simulate laser
Results: experiment, theory & simulation
real part of k
Q

a
wCM
= 6  103 e
= 0.88
= 0.73 mm
= 18.84 s-1
Results: experiment, theory & simulation
imaginary part of k
-1
w (s )
30
experiment
MDsimulation
theory 3
Damping:
theory & simulation
assume E = 4 s-1
20
10
0
0
1
2
ki a
3
Why is the wave backward?
Compare two cases:
k=0
Particles all move together
Center-of-mass oscillation in confining
potential at wcm
k>0
Particle repulsion acts oppositely to
restoring force of the confining potential
 reduces the oscillation frequency
Conclusion
Transverse Optical Mode
• is due to confining potential & interparticle repulsion
• is a backward wave
• was observed in experiment
Real part of dispersion relation was measured:
experiment agrees with theory
Damping
With dissipation (e.g. gas drag)
method of excitation
w
earlier
this talk
natural
complex
later
this talk
external
real
(from localized source)
k
real
complex
Radiation Pressure Force
momentum
imparted to
microsphere
incident laser
intensity I
transparent
microsphere
Force = 0.97 I  rp2
Example of 1D chain: trapped ions
Walther, laser physics division, Max-Planck-Institut
Ion chain:
trapped in a linear ion trap
would form a register of quantum computer
Applications:
• Quantum computing
• Atomic clock
How to measure wave number
• Excite wave
local in x
sinusoidal with time
transverse to chain
• Measure the particles’
position: x vs. t, y vs. t
velocity: vy vs. t
• Fourier transform:
vy(t)  vy(w)
• Calculate k
phase angle vs x  kr
amplitude
vs x  ki
Analogy with optical mode in ionic crystal
1D Yukawa chain
ionic crystal
charges
negative
positive + negative
restoring
force
external confining
potential
attraction to opposite ions
-
-
-
m
m
-
-
-
+
M >> m
-
-
M m
+
-
-
+
-
-
Electrostatic modes
(restoring force)
longitudinal acoustic
(inter-particle)
vx
transverse acoustic
(inter-particle)
vy
vz
1D

2D


3D


transverse optical
(confining potential)
vy
vz




Confinement of 1D Yukawa chain
groove on electrode
28-particle chain
y
z
Uy
y
x
Ux
x
Confinement is parabolic
in all three directions
x 0.1 Hz
groove
y 3 Hz
lower
electrode
z 15 Hz
method of
measurement
x
y
z
laser
laser
RF modulation
verified:
purely harmonic
purely harmonic
Single-particle
resonance frequency