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 Walther Max-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 2002Cvitas and Siber, PRB 2003 Carbon nanotubes: Xe atoms trapped in a tube

plasma = electrons + ions What is a dusty plasma? Debye shielding small particle of solid matter becomes negatively charged absorbs electrons and ions & neutral gas

polymer microspheres 8.05  m diameter Q  - 6   10 3 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 Equipotential contours electrode positive potential electrode With gravity, particles sediment to high-field region  monolayer possible Without gravity, particles fill 3-D volume QE mg

Chamber top-view camera laser illumination side-view camera vacuum chamber

Comparison of dusty 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 10 5  charge no inherent rotation gravity effects 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 Horizontal: sheath conforms to shape of groove in lower electrode

Setup Argon laser pushes particles in the monolayer

Radiation Pressure Force transparent microsphere momentum imparted to microsphere Force = 0.97 I  r p 2 incident laser intensity I

Ar laser mirror scanning mirror chops the beam beam dump Chopping chopped beam

scanning mirror Scanning mirror Ar laser beam

scanning mirror partially blocks the beam sinusoidally-modulated beam Sinusoidal modulation beam dump

Two-axis scanning mirrors For steering the laser beam

Experiments with a 1-D Chain

Image of chain in experiment

Confinement is parabolic in all three directions Measured values of single-particle resonance frequency

Modes in a 1-D chain: Longitudinal restoring forceinterparticle repulsion experimentHomann et al theoryMelands  “dust lattice wave DLW” 1997

Modes in a 1-D chain: Transverse Vertical motion: restoring forcegravity + sheath experimentMisawa et al theoryVladimirov et al oscillation.gif Horizontal motion: restoring forcecurved sheath experimentTHIS TALK theoryIvlev 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  k backward  k “backward” = “negative dispersion”

Natural motion of a 1-D chain Central portion of a 28-particle chain 1 mm

Spectrum of natural motion Calculate: particle velocities v x v y cross-correlation functions  v x v x  longitudinal  v y v y  transverse 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 Ar laser beam 1

Radiation pressure excites a wave Wave propagates to two ends of chain modulated beam -I 0 ( 1 + sin  t ) continuous beam I0I0 Net force: I 0 sin  t 1 mm

Measure real part of k from phase vs x fit to straight line yields k r

Measure imaginary part of k from amplitude vs x fit to exponential yields k i transverse mode

CMCM Experimental dispersion relation (real part of k) Wave is: backward i.e., negative dispersion smaller N  largera larger 

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 = 6200e 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: backward i.e., negative dispersion RR LL I II III  CM LL  (s -1 ) Evanescent Theoretical dispersion relation of optical mode (without damping)  CM = frequency of sloshing-mode

 CM  L I II III small 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  10 3 e  =0.88 a=0.73mm  CM =18.84 s -1 real part of k

Damping: theory & simulation assume E =4 s -1 imaginary part of k Results: experiment, theory & simulation

Why is the wave backward? k = 0 Particles all move together Center-of-mass oscillation in confining potential at  cm Compare two cases: 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

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) wavenumber ka/  Frequency   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, Avinash PRL 2002

Phonon spectrum & sinusoidally-excited waves S. Nunomura, Goree, Hu, Wang, Bhattacharjee, Avinash PRL 2002

Phonon spectrum & theory S. Nunomura, J. Goree, S. Hu, X. Wang, A. Bhattacharjee and K. Avinash PRL 2002

Damping With dissipation (e.g. gas drag) method of excitation  k naturalcomplexreal external realcomplex (from localized source) later this talk earlier this talk

incident laser intensity I Radiation Pressure Force transparent microsphere momentum imparted to microsphere Force = 0.97 I  r p 2

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:v y vs.t Fourier transform:v y (t)  v y (  ) Calculate k phase anglevsx  k r amplitudevsx  k i

Analogy with optical mode in ionic crystal negativepositive + negative external confining potential attraction to opposite ions 1D Yukawa chain i onic crystal charges restoring force M m m M >> m

Electrostatic modes (restoring force) longitudinal acoustictransverse acoustic transverse optical (inter-particle) (inter- particle) (confining potential) v x v y v z v y v z 1D    2D    3D  

groove on electrode x y z Confinement of 1D Yukawa chain 28-particle chain UxUx x UyUy y

Confinement is parabolic in all three directions method of measurementverified: xlaserpurely harmonic ylaserpurely harmonic zRF modulation Single-particle resonance frequency