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Transverse optical mode in a 1-D chain

Transverse optical mode in a 1-D chain. J. Goree, B. Liu & K. Avinash . Colloids: Polymer microspheres trapped by laser beams. Carbon nanotubes: Xe atoms trapped in a tube. Tatarkova, et al., PRL 2002. Cvitas and Siber, PRB 2003. Motivation: 1-D chains in condensed matter. Particles .

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Transverse optical mode in a 1-D chain

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  1. Transverse optical mode in a 1-D chain J. Goree, B. Liu & K. Avinash

  2. Colloids: • Polymer microspheres trapped by laser beams • Carbon nanotubes: • Xe atoms trapped in a tube Tatarkova, et al., PRL 2002 Cvitas and Siber, PRB 2003 Motivation: 1-D chains in condensed matter

  3. Particles polymer microspheres 8.05 mm diameter Q » - 6 ´103e Interparticle interaction is repulsive

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

  5. Image of chain in experiment

  6. Confinement is parabolicin all three directions Measured values of single-particle resonance frequency

  7. Modes in a 1-D chain: Longitudinal restoring force interparticle repulsion experimentHomann et al. 1997 theoryMelands 1997

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

  9. oscillation.gif • Horizontal motion: • restoring force curved sheath • experimentTHIS TALK • theoryIvlev et al. 2000 Modes in a 1-D chain: Transverse • Vertical motion: • restoring force gravity + sheath • experimentMisawa et al. • 2001 • theoryVladimirov et al. 1997

  10. Unusual properties of this wave: • The transverse mode in a 1-D chain is: • optical • backward

  11. w w k Optical mode in an ionic crystal k w not optical optical k Terminology: “Optical” mode

  12. w w k forward k backward Terminology:“Backward” mode “backward” = “negative dispersion”

  13. 1 mm Natural motion of a 1-D chain Central portion of a 28-particle chain

  14. Spectrum of natural motion • Calculate: • particle velocities • vx • vy • cross-correlation functions • ávx vxñ longitudinal • ávy vyñ transverse • Fourier transform Þ power spectrum

  15. Longitudinal power spectrum Power spectrum

  16. Transverse power spectrum negative slope Þ wave is backward No wave at w = 0, k = 0 Þ wave is optical

  17. Next: Waves excited by external force

  18. Ar laser beam 1 Setup Argon laser pushes only one particle

  19. modulated beam 1 mm -I0 ( 1 + sint ) continuous beam I0 Radiation pressure excites a wave Net force: I0 sint Wave propagates to two ends of chain

  20. Measure real part of k from phase vs x fit to straight line yields kr

  21. Measure imaginary part of k from amplitude vs x fit to exponential yields ki transverse mode

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

  23. Wave damping is weakest in the frequency band Experimental dispersion relation (imaginary part of k) for three different chain lengths Wave damping is higher for: smaller N larger 

  24. Experimental parameters To determine Q and D from experiment: We used equilibrium particle positions & force balance Þ Q = 6200 e D = 0.86 mm

  25. Theory • Derivation: • Eq. of motion for each particle, linearized & Fourier-transformed • Assumptions: • Probably same as in experiment: • Parabolic confining potential • Yukawa interaction • Epstein damping • No coupling between L & T modes • Different from experiment: • Infinite 1-D chain • Uniform interparticle distance • Interact with nearest two neighbors only

  26. Evanescent I CM Wave is allowed in a frequency band Wave is: backward i.e., negative dispersion II R L  (s-1) Evanescent L III Theoretical dispersion relation of optical mode (without damping) CM = frequency of sloshing-mode

  27. Wave damping is weakest in the frequency band Theoretical dispersion relation (with damping) high damping I CM II L III small damping

  28. 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

  29. Results: experiment, theory & simulation real part of k Q = 6 ´ 103 e  = 0.88 a = 0.73 mm CM = 18.84 s-1

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

  31. Why is the wave backward? Compare two cases: • k = 0 • Particles all move together • Center-of-mass oscillation in confining potential at wcm • k > 0 • Particle repulsion acts oppositely to restoring force of the confining potential • reduces theoscillation frequency

  32. 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

  33. dusty.physics.uiowa.edu

  34. later this talk earlier this talk Damping With dissipation (e.g. gas drag) • method of excitation k • natural complex real • external real complex • (from localized source)

  35. momentum imparted to microsphere Radiation Pressure Force incident laser intensity I transparent microsphere Force =0.97I rp2

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

  37. 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: vy vs. t • Fourier transform: vy(t) ® vy() • Calculate k • phase angle vs x Þkr • amplitude vs x Þki

  38. - - - - - - - - - - - - + + + m m M m Analogy with optical mode in ionic crystal 1D Yukawa chainionic crystal negative positive + negative charges restoring force external confining potential attraction to opposite ions M >> m

  39. Electrostatic modes(restoring force) • longitudinal acoustic transverse acoustic transverse optical • (inter-particle)(inter-particle)(confining potential) • vxvy vzvy vz • 1D   • 2D    • 3D   

  40. y Uy z y Ux x x Confinement of 1D Yukawa chain groove on electrode 28-particle chain

  41. Confinement is parabolicin all three directions Single-particle resonance frequency method of measurement verified: x laser purely harmonic y laser purely harmonic zRF modulation

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