Transverse optical mode in a 1-D Yukawa chain

J. Goree, B. Liu & K. Avinash

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)

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

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

polymer microspheres

8.05 m diameter

Q - 6 103 e

Particles

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?

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

Chamber

top-viewcamera

laser illumination

side-viewcamera

vacuum chamber

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

Confinement of a monolayer

– Particles repel each other

– External confinement by bowl-shaped electric sheath above lower electrode

Confinement of 1-D chain

Vertical: gravity + vertical E

lowerelectrode

groove mg

QE

Horizontal:sheath conforms to shape of groove in lower electrode

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

Radiation Pressure Force

transparent microsphere

momentum imparted to microsphere

Force = 0.97 I rp2

incident laser intensity I

Ar laser

mirror

scanning mirrorchopsthe beam

beam dump

Choppingchopped beam

scanningmirror

Scanningmirror

Ar laser beam

scanning mirror partially blocksthe beam

sinusoidally-modulated beamSinusoidalmodulation

beam dump

Two-axis scanning mirrors

For steering the laser beam

Experiments with a 1-D Chain

lowerelectrode

groove mg

QE

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 “dust lattice wave DLW”1997

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

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

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

Possibilities for non-neutral plasma experiments

Ion chain(Walther, Max-Planck-Institut für Quantenoptik )

Dust chain

2-D Monolayer

triangular lattice with hexagonal symmetry

2-D lattice

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

Longitudinal wave

4mm

k Laser incident here

f = 1.8 Hz

Nunomura, Goree, Hu, Wang, Bhattacharjee Phys. Rev. E 2002

Random particle motion

No Laser!

= compression + shear

4mm

S. Nunomura, Goree, Hu, Wang, Bhattacharjee, AvinashPRL 2002

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

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

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

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