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