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INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS

Institute for Quantum Information, University of Ulm, 14 February 2008. INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS. ZBIGNIEW IDZIASZEK. Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science. Outline.

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INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS

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  1. Institute for Quantum Information,University of Ulm, 14 February 2008 INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS ZBIGNIEW IDZIASZEK Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science

  2. Outline 1. Characteristic scales associated with ultracold collisions 2. Wigner threshold laws 3. Scattering lengths and pseudopotentials 4. Quantum defect theory • 5. Resonance phenomena: • shape resonances • Feshbach resonances

  3. ultracold collisions (Ultra)cold atomic collisions cold collisions J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71, 1 (1999)

  4. Typical interaction potential V(r) centrifugal barrier: r long-range part: dispersion forces • neutral atoms, both in S state:van der Waals interaction, n = 6 • atom in S state-charged particle (ion): polarization forces, n = 4 • neutral atoms with dipole momentsdipole-dipole interaction, n = 3 short-range part: chemical binding forces

  5. Long-range dispersion forces At E0 (close to the threshold) scattering properties are determined by the part of the potential with the slowest decay at r Characteristic scales Length scale: Typical range of the potential Energy scale: Height of the centrifugal barrier, determines contribution of higher partial waves For EE* only s-wave (l = 0) collisions

  6. Characteristic scales Example values of R* and E* for different kinds of interactions Neutral atoms in S states (alkali) Atom(S)-ion (alkali atom-alkali earth ion) • R* for atom-atom << size of the typical trapping potentials • R* for atom-ion ~ size of the trapping potentials (rf + optical traps)  consequences for collisions in traps • E* for atom-ion is 103 lower than for atom-atom  higher partial waves (l > 0) not negligible for ultracold atom-ion collisions (~K), whereas negligible for atom-atom collisions

  7. Partial-wave expansion and phase shifts V(r) Partial wave expansion r At large distances without potential: l=0 attractive potential: l > 0 repulsive potential: l < 0

  8. Cross section for partial wave l Threshold laws for elastic collisions Behavior of cross-sections at E0 Wigner threshold laws for short-range potentials E. Wigner Phys. Rev. 73, 1002 (1948) decays faster than 1/rn Example: Yukawa potential Smooth and continuous matching

  9. Threshold laws for elastic collisions Long-range dispersion potentials First-order Born approximation (Landau-Lifshitz, QM) For 2l < n-3 Wigner threshold law is preserved For 2l >n-3 long-range contribution dominates Exact treatment Analytical solution at E=0 Special case n=3

  10. For l=0 Wigner threshold law: Scattering length Scattering length Physical interpretation: repulsive attractive Potential without bound states

  11. V(r) R0 r V0 For p-wave - scattering volume Scattering length Each time new bound state enters the potential adiverges and changes sign a(V0) V0 Higher partial waves In the Wigner threshold regime l-wave Scattering length

  12. Pseudopotentials At very low energies only s-wave scattering is present Total cross-section: - depends on a single parameter de Broglie wavelength range of the potential particles do not resolve details of the potential shape independent approximation J. Weiner et al. RMP 71 (1999)

  13. Pseudopotentials Fermi pseudopotential regularization operator (removes divergences of the 3D wave function at r0) E. Fermi, La Ricerca Scientifica, Serie II 7, 13 (1936) V(r) Asymptotic solution r R0 Pseudopotential Pseudopotential supports single bound state for a>0 Correct for a weakly bound state with E<<E*

  14. Pseudopotentials Generalized pseudopotential for all partial waves K. Huang & C. N. Yang, Phys. Rev. 105, 767 (1957) Correct version of Huang & Yang potential: l-wave scattering length R. Stock et al, PRL 94, 023202 (2005)A. Derevianko, PRA 72, 044701 (2005)ZI & TC, PRL 96, 013201 (2006) For particular partial waves it can be simplified ... Pseudopotential for p-wave scattering Pseudopotential for d-wave scattering

  15. V(r) R0 r V0 Pseudopotentials Test: square-well potential + harmonic confinement Pseudopotential method valid for Scattering volume Energy spectrum for R0=0.01d Energy spectrum for R0=0.2 d

  16. Quantum-defect theory of ultracold collisions Seaton, Proc. Phys. Soc. London 88, 801 (1966)Green, Rau and Fano, PRA 26, 2441 (1986)Mies, J. Chem. Phys. 80, 2514 (1984). 1) Reference potential(s) Asymptotic behavior, the same as for the real physical potential Arbitrary at small r (model potential) 2) Quantum-defect parameters Characterize the behavior of the wave function at small distances (~Rmin) R* r>>R* Shallow potential, wave function strongly depends on E Independent of energy for a wide range of kinetic energies Deep potential, wave function weakly depends on E Rmin 3) Quantum-defect functions Scattering phases (r~)  quantum defect parameters (r~Rmin) Can be found analytically for inverse power-law potentials Knowledge of the scattering phases at a single value of energy allows to determine the scattering properties + position of bound states at different energies

  17. Quantum-defect theory of ultracold collisions Linearly independent solutions of the radial Schrödinger equation Solutions with WKB-like normalization at small distances Analytic across threshold! R* Solutions with energy-like normalization at r Rmin For large energies when semiclassical description becomes applicable at all distances, two sets of solutions are the same Non-analytic across threshold!

  18. QDT functions connect f,ĝ with f,g, Quantum-defect theory of ultracold collisions In WKB approximation, small distances (r~Rmin) Physical interpretation of C(E), tan (E) and tan (E): C(E) - rescaling (E) and (E) – shift of the WKB phase For E, semiclassical description is valid at all distances For E0, analytic behavior requires

  19. Expressing the wave function in terms of f,ĝ functions Quantum-defect theory of ultracold collisions  - QDT parameter (short-range phase)  very weakly depends on energy: QDT functions relates  to observable quantities, e.g. scattering matrices The same parameter predicts positions of the bound states

  20. Quantum-defect theory of ultracold collisions Example: energies of the atom-ion molecular complex Solid lines:quantum-defect theory for independent ofE i l Points:numerical calculations for ab-initio potentials for 40Ca+ - 23Na Ab-initio potentials: O.P. Makarov, R. Côté, H. Michels, and W.W. Smith, Phys.Rev.A 67, 042705 (2005).

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