1. ULTRACOLD COLLISIONS IN THE PRESENCE OF TRAPPING POTENTIALS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 18 February 2008 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 3. Scattering in quasi-1D and and quasi-2D traps - confinement –induced resonances 2. Energy dependent scattering length 4. Feshbach resonances

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

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

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

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

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

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

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

10. Identical fermions can only interact in odd partial waves ( l = 2n+1 ) Hamitonian of the relative motion: Two ultracold atoms in harmonic trap Energy spectrum for  = 1/10 Two ultracold fermions in harmonic trap 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

11. 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 ) Energy-dependent scattering length At small energies (k  0): a eff (E)  a Energy-dependent scattering length E.L Bolda et al., PRA 66, 013403 (2002) D. Blume and C.H. Greene, PRA 65, 043613 (2002) Schrödinger equation is solved in a self-consistent way Applicable only when CM and relative motions can be separated.

12. V(r) r R0R0 V0V0 Model potential: square well exact energies pseudopotential approximation pseudopotential with a eff ( E ) Scattering length Energy spectrum Parameters: Energy-dependent scattering length TEST: two interacting atoms in harmonic trap, s-wave states

13. V(r) r R0R0 V0V0 Energy spectrum for R 0 =0.05d Energy-dependent pseudopotential applicable even for R 0 /d not very small Energy spectrum for R 0 =0.2 d TEST: two interacting atoms in harmonic trap, p-wave states Energy-dependent scattering length Scattering volume: EDP:

14. Hamiltonian of relative motion: Quasi-1D traps Asymptotic solution at small energies 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 After collision atoms remain in ground-state of transverse motion Atomic collisions in quasi-1D traps f + - even scattering wave f - - odd scattering wave optical lattice

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

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

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

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

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

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

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

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

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

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

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

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

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

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