1. INTRODUCTION TO PHYSICS OF ULTRACOLD COLLISIONS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 14 February 2008 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. J. Weiner, V.S. Bagnato, S. Zilio, and P.S. Julienne, Rev. Mod. Phys. 71, 1 (1999) (Ultra)cold atomic collisions cold collisions ultracold collisions

4. Typical interaction potential 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 moments dipole-dipole interaction, n = 3 V(r) r short-range part: chemical binding forces centrifugal barrier:

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: Energy scale: Typical range of the potential 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)  consequences for collisions in traps R* for atom-atom << size of the typical trapping potentials E* for atom-ion is 10 3 lower than for atom-atom  higher partial waves ( l > 0) not negligible for ultracold atom-ion collisions (~  K), whereas negligible for atom-atom collisions R* for atom-ion ~ size of the trapping potentials (rf + optical traps)

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

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

9. 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 Threshold laws for elastic collisions Special case n=3

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

11. Higher partial waves l -wave Scattering length For p-wave - scattering volume In the Wigner threshold regime Each time new bound state enters the potential a diverges and changes sign V(r) r R0R0 V0V0 V0V0 a(V0)a(V0)

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

13. Fermi pseudopotential E. Fermi, La Ricerca Scientifica, Serie II 7, 13 (1936) regularization operator (removes divergences of the 3D wave function at r  0) Pseudopotential supports single bound state for a>0 Correct for a weakly bound state with E<<E* Pseudopotentials r V(r) R0R0 Pseudopotential Asymptotic solution

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

15. Test: square-well potential + harmonic confinement V(r) r R0R0 V0V0 Energy spectrum for R 0 =0.01d Scattering volume Energy spectrum for R 0 =0.2 d Pseudopotential method valid for Pseudopotentials

16. Quantum-defect theory of ultracold collisions R*R* R min 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 (~R min ) Independent of energy for a wide range of kinetic energies Scattering phases (r~  )  quantum defect parameters (r~R min ) Knowledge of the scattering phases at a single value of energy allows to determine the scattering properties + position of bound states at different energies 3) Quantum-defect functions Can be found analytically for inverse power-law potentials Deep potential, wave function weakly depends on E Shallow potential, wave function strongly depends on E r>>R *

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

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

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

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