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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 Cvitas and Siber, PRB 2003
Carbon nanotubes:
Xe atoms trapped in a tube
polymer microspheres
8.05 m diameter
Q - 6 103 e
Particles
Interparticle interaction is repulsive
Confinement of 1-D chain
Vertical: gravity + vertical E
lowerelectrode
groove mg
QE
Horizontal:sheath conforms to shape of groove in lower electrode
Image of chain in experiment
Confinement is parabolicin all three directions
lowerelectrode
x 0.1 Hz
groove y 3 Hz
z 15 Hz
Measured values of single-particle resonance frequency
Modes in a 1-D chain: Longitudinal
restoring force interparticle repulsion
experiment Homannet al. 1997
theory Melands 1997
Modes in a 1-D chain: Longitudinal
restoring force interparticle repulsion
experiment Homannet al. 1997
theory Melands “dust lattice wave DLW”1997
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
oscillation.gif
Horizontal motion:
restoring force curved sheath
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
not optical
k
k
optical
k
Optical mode in an ionic crystal
Terminology:“Backward” mode
forward
kbackward
k
“backward” = “negative dispersion”
Natural motion of a 1-D chain
Central portionof a 28-particle chain
1 mm
Spectrum of natural motion
Calculate:
• particle velocities
vx
vy
• cross-correlation functions
vx vxlongitudinal
vy vytransverse
• Fourier transform power spectrum
Longitudinal power spectrum
Power spectrum
negative slope
wave is backward
Transverse power spectrum
No wave at = 0, k = 0
wave is optical
Next: Waves excited by external force
Setup
Argon laser pushes only one particle
video camera(top view)
lower electrodeRF
Ar laser beam 2 Ar lase beam1
microsphere scanningmirror
Ar laser beam 1
Radiation pressure excites a wave
Wave propagatesto two ends of chain
modulated beam-I0 ( 1 + sint )
continuous beamI0
Net force: I0 sint
1 mm
Measure real part of k from phase vs x
fit to straight lineyields kr
0 5 100.00
0.01
0.02
0.03
0.04
0.05
0.06
exponential fitting
Am
plit
ud
e (
mm
/s)
position (mm)
Measure imaginary part of k from amplitude vs x
fit to exponentialyields ki
transverse mode
0 1 2 30
10
20
30
N = 10 N = 19 N = 28
(s-1)
kr a
CM
Experimental dispersion relation (real part of k)
Wave is:backwardi.e., negative dispersion
smaller N larger a
larger
0 1 2 30
10
20
30
N = 10 N = 19 N = 28
(
s-1 )
ki a
Experimental dispersion relation (imaginary part of k) for three different chain lengths
Wave damping is weakest in the frequency band
Wave damping is higher for:smaller Nlarger
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
• Different from experiment:
• Infinite 1-D chain
• Uniform interparticle distance
• Interact with nearest two neighbors only
Assumptions:
• Probably same as in experiment:
• Parabolic confining potential
• Yukawa interaction
• Epstein damping
• No coupling between L & T modes
Wave is allowed in a frequency band
Wave is:backwardi.e., negative dispersion
R
L
0 1 2 30
10
20
(s
-1)
k a
I
II
III
CM
L
(
s-1)
Evanescent
Evanescent
Theoretical dispersion relation of optical mode (without damping)
CM = frequency of sloshing-mode
0 1 2 30
10
20
30
ki
kr
(s
-1)
k a
C
M
L
I
II
IIIsmall damping
high damping
Theoretical dispersion relation (with damping)
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
Q = 6 103e = 0.88a = 0.73 mmCM = 18.84 s-1
real part of k
Damping:theory & simulation assume E = 4 s-1
0 1 2 30
10
20
30 experiment MDsimulation theory 3
(s-1)
ki a
imaginary part of k
Results: experiment, theory & simulation
Why is the wave backward?
k = 0Particles all move togetherCenter-of-mass oscillation in confining
potential at cm
Compare two cases:
k > 0Particle repulsion acts oppositely to
restoring force of the confining potentialreduces 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 k
natural complex real
external real complex
(from localized source)
laterthis talk
earlier this talk
incident laser intensity I
Radiation Pressure Force
transparent microsphere
momentum imparted to microsphere
Force = 0.97 I rp2
Example of 1D chain: trapped ions
Applications:
• Quantum computing • Atomic clock
Ion chain:
trapped in a linear ion trapwould form a register of quantum computer
Walther, laser physics division, Max-Planck-Institut
How to measure wave number
• Excite wavelocal in xsinusoidal with timetransverse to chain
• Measure the particles’ position:x vs. t, y vs. tvelocity: vy vs. t
• Fourier transform: vy(t) vy()
• Calculate k
phase angle vs x kr
amplitude vs x ki
Analogy with optical mode in ionic crystal
negative positive + negative
external confining potential
attraction to opposite ions
1D Yukawa chain ionic crystal
charges
restoring force
M m
+ -- + -- + ---- -- --
m mM >> m
Electrostatic modes(restoring force)
longitudinal acoustic transverse acoustic transverse optical (inter-particle) (inter-particle) (confining potential)
vx vy vz vy
vz
1D
2D
3D
groove on electrode
x
y
z
Confinement of 1D Yukawa chain
28-particle chain
Ux
x
Uy
y
Confinement is parabolicin all three directions
method of measurement verified:
x laser purely harmonic
y laser purely harmonic
z RF modulation
lowerelectrode
x 0.1 Hz
groove y 3 Hz
z 15 Hz
Single-particleresonance frequency