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ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS

Institute for Quantum Information, University of Ulm, 18 February 2008. ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS. ZBIGNIEW IDZIASZEK. Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science. Outline.

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ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS

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  1. Institute for Quantum Information,University of Ulm, 18 February 2008 ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS ZBIGNIEW IDZIASZEK Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science

  2. Outline • 1. Binary collisions in harmonic traps • collisions in s-wave • collisions in higher partial waves 2. Energy dependent scattering length 3. Scattering in quasi-1D and and quasi-2D traps- confinement –induced resonances 4. Feshbach resonances

  3. System 1. Ultracold atoms in the trapping potential magnetic traps, optical dipole traps, electro-magnetic traps for charged particles, ... Typical trapping potentials are harmonic close to the center 2. Characteristic range of interaction R* << length scale of the trapping potential trap size R* Interactions can be modeled via contact pseudopotential - Very accurate for neutral atoms - Not applicable for charged particles, e.g. for atom-ion collisions

  4. Two ultracold atoms in harmonic trap Hamiltonian (harmonic-oscillator units) length unit: energy unit: Axially symmetric trap: Contact pseudopotential for s-wave scattering (low energies): CM and relative motions can be separated in harmonic potential

  5. We expand into basis of harmonic oscillatorwave functions Two ultracold atoms in harmonic trap Relative motion radial: axial: Contact pseudopotential affects only states with mz=0 and k even(non vanishing at r=0 ) For mz0 or k odd trivial solution:

  6. Two ultracold atoms in harmonic trap Substituting expansion into Schrödinger equation and projecting on Eigenfunctions: Eigenenergies: Integral representation can be obtained from:

  7. Two ultracold atoms in harmonic trap Energy spectrum in pancake-shape traps ( < 1) Energy spectrum in cigar-shape traps ( > 1) Energy spectrum for  = 5 Energy spectrum for  = 1/5 For For Z.I., T. Calarco, PRA 71, 050701 (2005)

  8. Two ultracold atoms in harmonic trap Comparison of theory vs. experiment: atoms in optical lattice • solid line – theory (spherically symmetric trap) T. Bush et al., Found. Phys. 28, 549 (1998) • points – experimental data T. Stöferle et al., Phys. Rev. Lett. 96, 030401 (2006) Bound state for positive and negative energies due to the trap

  9. exact energies 1D model + g1D Two ultracold atoms in harmonic trap First excited state Energy spectrum and wave functions for very elongated cigar-shape trap Elongated in the direction of weak trapping Dip in the center due to the strong interaction Trap-induced bound state (a < 0) Energy spectrum for  = 100 Size determined by the strong confinement Wave function is nearly isotropic

  10. Two ultracold atoms in harmonic trap Identical fermions can only interact in oddpartial waves (l = 2n+1) No interactions in higher partial waves at E0 (Wigner threshold law) Scattering for l > 0 can be enhanced in the presence of resonances  Feshbach resonances Two ultracold fermions in harmonic trap Hamitonian of the relative motion: Energy spectrum for  = 1/10 Energy spectrum for  = 1/10

  11. Energy-dependent scattering length Fermi pseudopotential -applicable for: k R* 1, k a  1/ k R* s-wave scattering lenght: In the tight traps (large k) or close to resonances (large a) E.L Bolda et al., PRA 66, 013403 (2002)D. Blume and C.H. Greene, PRA 65, 043613 (2002) Energy-dependent scattering length At small energies (k  0): aeff(E)  a Schrödinger equation is solved in a self-consistent way Applicable only when CM and relative motions can be separated.

  12. V(r) R0 r V0 exact energies pseudopotential approximation pseudopotential with aeff(E) Energy-dependent scattering length TEST: two interactingatoms in harmonic trap, s-wave states Scattering length Model potential: square well Energy spectrum Parameters:

  13. V(r) R0 r V0 Energy-dependent scattering length TEST: two interactingatoms in harmonic trap, p-wave states EDP: Scattering volume: Energy spectrum for R0=0.05d Energy-dependent pseudopotential applicable even for R0 /d not very small Energy spectrum for R0=0.2 d

  14. Atomic collisions in quasi-1D traps optical lattice Quasi-1D traps Weak confinement along z Strong confinement along x,y Effective motion like in 1D system In the harmonic confinement CM and relative motions are not coupled Hamiltonian of relative motion: Asymptotic solution at small energies f+- even scattering wave f-- odd scattering wave After collision atoms remain in ground-state of transverse motion

  15. Collisions of bosons in quasi-1D traps Even scattered wavefor bosons M. Olshanii PRL 81, 938 (1998) Confinement induced resonance (CIR) occurs for Transmission coefficient T

  16. Collisions of bosons in 1D system Interactions of bosons in 1D can be modeled with: Contact pseudopotential Interaction strength for quasi-1D trap obtained from 3D solution M. Olshanii PRL 81, 938 (1998) Confinement induced resonanceat T. Bergeman et al. PRL 91, 163201(2003) Gas of strongly interacting bosons in 1D:Tonks-Girardeau gas

  17. CIR Collisions of fermions in quasi-1D traps Odd scattered wavefor fermions B. Granger, D. Blume, PRL (2004) Resonance in p-wave for Scattering amplitude f-

  18. (2) Em(B) (1) Feshbach resonances – entrance channel – closed channel – coupling between channels Inverting 1st equation with the help of Green’s functions Substituting (1) into (2) and solving for 2

  19. 2) Em(B) 1) Feshbach resonances Close to a resonance only single bound-state from a closed channel contributes - resonant bound state in the closed channel - energy of bound state Bres – magnetic field when the bound state crosses the threshold - energy shift due to the couppling - resonance width

  20. B Feshbach resonances Phase shift bg – background phase shift (in the absence of coupling between channels) Energy dependent scattering length Background scattering length: Typically for ultracold collisions

  21. energy spectrum resonance position Trapped atoms + Feshbach resonances Example: Energy spectrum of two 87Rb atoms in a tight trap Quasi-1D trap Parameters of resonance

  22. Lippmann-Schwinger equation and Green’s functions Green’s operator Solution for V=0 Lippmann Schwinger equation Green’s function in position representation in free space + outgoing spherical wave Lippmann-Schwinger equation in position representation Behavior of (r) at large distances Scattering amplitude

  23. exact energies (3D) 1D trap + g1D exact energies (3D) 2D trap + g2D Atomic collisions in quasi-1D and quasi-2D traps 1D and 2D effective interactions in comparison to full 3D treatment Energy spectrum in pancake-shape trap Energy spectrum in cigar-shape trap ZI, T. Calarco, PRA 74, 022712 (2006) Realization of 1D and 2D regimes does not require very large anisotropy of the trap

  24. Then E  (kinetic energy at r = 0) Z. Idziaszek, T. Calarco, PRA 74, 022712 (2006)

  25. Asymptotic solutionfor kinetic energies QUASI-2D SYSTEMS Scattering of spin-polarized fermions in quasi-2D Atoms remain in theground state in z direction Solving the scattering problem ... m=1 scattering wavefor p-wave interacting fermions

  26. Scattering in quasi-2D traps Similar scattering confinement-induced resonaces as in quasi-1D traps Example: two fermions, p-wave interactions Scattering amplitude CIR Scattering amplitude in forward direction for different values of energy 2D scattering amplitude:

  27. Zderzenia atomów w pułapkach kwazi-1D i kwazi-2D Rozpraszanie fermionów w fali p w układzie kwazi-2D Zachowanie asymptotycznedla energii kinetycznych Atomy pozostają w stanie podstawowym w kierunku z Amplituda rozpraszania w 2D: Rozwiązanie problemu rozpraszania: fala m=1 dla fermionów oddziałujących w fali p

  28. Zderzenia atomów w fali p w układzie kwazi-2D Rezonans indukowany ściśnięciem gdy położenie rezonansu: ZI, and T. Calarco, PRL (2006) Dla niskich energii ( ): Amplituda rozpraszaniado przodudla różnych energii kinetycznych CIR Rezonans nie widoczny powyżej energii

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