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

polymer microspheres 8.05  m diameter Q  - 6   10 3 e Particles Interparticle interaction is repulsive

Confinement of 1-D chain Vertical: gravity + vertical E Horizontal: sheath conforms to shape of groove in lower electrode

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  1997

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

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

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  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 Wave damping is higher for: smaller N larger 

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

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

Example of 1D chain: trapped ions Applications: Quantum computing Atomic clock Ion chain: trapped in a linear ion trap would form a register of quantum computer Walther, laser physics division, Max-Planck-Institut

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