Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.
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Transcript of Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.
![Page 1: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/1.jpg)
Transverse optical mode in a 1-D Yukawa chain
J. Goree, B. Liu & K. Avinash
![Page 2: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/2.jpg)
Example of 1-D chain
Applications:
• Quantum computing • Atomic clock
WaltherMax-Planck-Institut für Quantenoptik
linear ion trap
image of ion chain
(trapped in the central part of the linear ion trap)
![Page 3: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/3.jpg)
Examples of 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
![Page 4: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/4.jpg)
plasma = electrons + ions Plasma
+
-
+
+
+
+
+
+
+
- -
-
-
--
-
+
-
What is a dusty plasma?
D
• Debye shielding
small particle of solid matter
• becomes negatively charged
• absorbs electrons and ions
& neutral gas
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polymer microspheres
8.05 m diameter
Q - 6 103 e
Particles
![Page 6: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/6.jpg)
Solar system• Rings of Saturn• Comet tails
Fundamental science• Coulomb crystals• Waves
Manufacturing• Particle contamination
(Si wafer processing)• Nanomaterial synthesis
Who cares about dusty plasmas?
![Page 7: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/7.jpg)
Electrostatic trapping of particles
Equipotentialcontours
electrode
electrode
positive
potential
electrode
electrode
With gravity, particles sediment to high-field region monolayer possible
Without gravity, particles fill 3-D volume
QE
mg
![Page 8: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/8.jpg)
Chamber
top-viewcamera
laser illumination
side-viewcamera
vacuum chamber
![Page 9: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/9.jpg)
Comparison ofdusty plasma & pure ion plasmas
Similar:
• repulsive particles
• lattice, i.e., periodic phase
• 3-D, 2-D or 1-D suspensions
• direct imaging
• laser-manipulation of
particles
Different - dusty plasma has:
• gaseous background
• 105 charge• no inherent rotation• gravity effects
D
a
D
r
r
QrU
exp
4)(
0
• Yukawa potential
![Page 10: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/10.jpg)
Confinement of a monolayer
– Particles repel each other
– External confinement by bowl-shaped electric sheath above lower electrode
![Page 11: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/11.jpg)
Confinement of 1-D chain
Vertical: gravity + vertical E
lowerelectrode
groove mg
QE
Horizontal:sheath conforms to shape of groove in lower electrode
![Page 12: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/12.jpg)
Setup
Argon laser pushes particles in the monolayer
H eN e lase rho riz o nta ls he e t
v ideo cam e ra(to p v iew )
lo wer e lec tro deR F
two -axiss te e ring
m ic ro sphe res
m o d ula tio n
A r lase rbe a m
x
y
f ram egra bb e r
![Page 13: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/13.jpg)
Radiation Pressure Force
transparent microsphere
momentum imparted to microsphere
Force = 0.97 I rp2
incident laser intensity I
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Ar laser
mirror
scanning mirrorchopsthe beam
beam dump
Choppingchopped beam
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scanningmirror
Scanningmirror
Ar laser beam
![Page 16: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/16.jpg)
scanning mirror partially blocksthe beam
sinusoidally-modulated beamSinusoidalmodulation
beam dump
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Two-axis scanning mirrors
For steering the laser beam
![Page 18: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/18.jpg)
Experiments with a 1-D Chain
lowerelectrode
groove mg
QE
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Image of chain in experiment
![Page 20: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/20.jpg)
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
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Modes in a 1-D chain: Longitudinal
restoring force interparticle repulsion
experiment Homannet al. 1997
theory Melands “dust lattice wave DLW”1997
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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
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Properties of this wave:
The transverse mode in a 1-D chain is:• optical• backward
![Page 24: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/24.jpg)
Terminology: “Optical” mode
not optical
k
k
optical
k
Optical mode in an ionic crystal
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Terminology:“Backward” mode
forward
kbackward
k
“backward” = “negative dispersion”
![Page 26: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/26.jpg)
Natural motion of a 1-D chain
Central portionof a 28-particle chain
1 mm
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Spectrum of natural motion
Calculate:
• particle velocities
vx
vy
• cross-correlation functions
vx vxlongitudinal
vy vytransverse
• Fourier transform power spectrum
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Longitudinal power spectrum
Power spectrum
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negative slope
wave is backward
Transverse power spectrum
No wave at = 0, k = 0
wave is optical
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Next: Waves excited by external force
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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
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Radiation pressure excites a wave
Wave propagatesto two ends of chain
modulated beam-I0 ( 1 + sint )
continuous beamI0
Net force: I0 sint
1 mm
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Measure real part of k from phase vs x
fit to straight lineyields kr
![Page 34: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/34.jpg)
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
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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
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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
![Page 37: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/37.jpg)
Experimental parameters
To determine Q and D from experiment:
We used equilibrium particle positions & force balance
Q = 6200 e
D = 0.86 mm
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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
![Page 39: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/39.jpg)
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
![Page 40: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/40.jpg)
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
![Page 41: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/41.jpg)
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
![Page 42: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/42.jpg)
Results: experiment, theory & simulation
Q = 6 103e = 0.88a = 0.73 mmCM = 18.84 s-1
real part of k
![Page 43: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/43.jpg)
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
![Page 44: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/44.jpg)
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
![Page 45: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/45.jpg)
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
![Page 46: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/46.jpg)
Possibilities for non-neutral plasma experiments
Ion chain(Walther, Max-Planck-Institut für Quantenoptik )
Dust chain
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2-D Monolayer
![Page 49: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/49.jpg)
triangular lattice with hexagonal symmetry
2-D lattice
![Page 50: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/50.jpg)
Dispersion relation (phonon spectrum)
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4
wavenumber ka/
Fre
quen
cy
Theory for a triangular lattice, = 0°Wang, Bhattacharjee, Hu , PRL (2000)
compressional
shear
acoustic limit
![Page 51: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/51.jpg)
Longitudinal wave
4mm
k Laser incident here
f = 1.8 Hz
Nunomura, Goree, Hu, Wang, Bhattacharjee Phys. Rev. E 2002
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Random particle motion
No Laser!
= compression + shear
4mm
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
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Phonon spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
En
erg y
den
s ity
/ k B
T (
10-
3m
m2s)
k
a
= 0°
Transverse mode
& sinusoidally-excited waves
S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002
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Phonon spectrum
-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
6.0
4.0
2.0
0.0
Longitudinal mode6.0
4.0
2.0
0.0
k (mm-1)
f (H
z)f (
Hz)
ka/-2.0 -1.5 -1.0 0.5 0.0 0.5 1.0 1.5 2.0
/
0
3.0
2.0
1.0
0.0
4.0
/
0
3.0
2.0
1.0
0.0
4.0
5
10
15
En
erg y
den
s ity
/ k B
T (
10-
3m
m2s)
k
a
= 0°
Transverse mode
& theory
S. Nunomura, J. Goree, S. Hu, X. Wang, A. Bhattacharjee and K. AvinashPRL 2002
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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
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incident laser intensity I
Radiation Pressure Force
transparent microsphere
momentum imparted to microsphere
Force = 0.97 I rp2
![Page 58: Transverse optical mode in a 1-D Yukawa chain J. Goree, B. Liu & K. Avinash.](https://reader038.fdocuments.in/reader038/viewer/2022110402/56649e355503460f94b24b16/html5/thumbnails/58.jpg)
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
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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
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Electrostatic modes(restoring force)
longitudinal acoustic transverse acoustic transverse optical (inter-particle) (inter-particle) (confining potential)
vx vy vz vy
vz
1D
2D
3D
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groove on electrode
x
y
z
Confinement of 1D Yukawa chain
28-particle chain
Ux
x
Uy
y
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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