1. ATOM-ION COLLISIONS ZBIGNIEW IDZIASZEK Institute for Quantum Information, University of Ulm, 20 February 2008 Institute for Theoretical Physics, University of Warsaw and Center for Theoretical Physics, Polish Academy of Science

2. Outline 2. Results for Ca + -Na system 1. Analytical model of ultracold atom-ion collisions - Exact solutions for 1/r 4 potential – single channel QDT - Multichannel quantum-defect theory - Frame transformation 3. Controlled collisions of atom and ions in movable trapping potentials

3. Atom-ion interaction state quadrupole moment: - atomic polarizability Large distances, atom in S state induced dipole ATOMION Large distances, atom in P state (or other with a quadrupole moment) graph from: F.H. Mies, PRA (1973)

4. Radial Schrödinger equation for partial wave l Transformation: Mathieu’s equation of imaginary argument To solve one can use the ansatz: Three-terms recurrence relation Solution in terms of continued fractions - characteristic exponent Analytical solution for polarization potential E. Vogt and G. Wannier, Phys. Rev. 95, 1190 (1954)

5. Analytical solution for polarization potential Short-range phase: Behavior of the solution at large distances Positive energies (scattering state) : s = s (, k, l ) – expressed in terms of continuous fractions Behavior of the solution at short distances scattering phase : Negative energies (bound state) :

6. Quantum defect parameter Short range-wave function fulfills Schrödinger equation at E=0 and l=0 Relation to the s-wave scattering length Behavior at large distances r  Exchange interaction, higher order dispersion terms: C 6 /r 6, C 8 /r 8,... R * – polarization forces Separation of length scales  short-range phase is independent of energy and angular momentum Boundary condition imposed by  represents short- range part of potential Quantum-defect parameter

7. Multichannel formalism - interaction potential Open channel: Closed channel: Classification into open and closed channels - matrix of N independent radial solutions Asymptotic behavior of the solution Interaction at large distances In the single channel case N – number of channels Radial coupled-channel Schrödinger equation

8. 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! Reference potentials:

9. Quantum-defect theory of ultracold collisions QDT functions connect f,ĝ with f,g,   Seaton, Proc. Phys. Soc. London 88, 801 (1966) Green, Rau and Fano, PRA 26, 2441 (1986) Mies, J. Chem. Phys. 80, 2514 (1984) Y very weakly depends on energy: Quantum defect matrix Y(E) Expressing the wave function in terms of another pair of solutions R matrix strongly depends on energy and is nonanalytic across threshold

10. For large energies semiclassical description is valid at all distances, and the two sets of solutions are equivalent Semiclassical approximation is valid when Quantum-defect theory of ultracold collisions For E 

11. Quantum-defect theory of ultracold collisions QDT functions relate Y(E) to observable quantities, e.g. scattering matrices All the channels are closed  bound states For a single channel scattering Renormalization of Y(E) in the presence of the closed channels This assures that only exponentially decaying (physical) solutions are present in the closed channels Scattering matrices are obtained from

12. Both individual species are widely used in experiments ab-initio calculations of interaction potentials and dipole moments are available O.P. Makarov, R. Côté, H. Michels, and W.W. Smith, Phys.Rev.A 67, 042705, (2005). Ultracold atom-ion collisions Born-Oppenheimer potential-energy curves for the (Na-Ca) + molecular complex Example: 23 Na and 40 Ca + Radial transition dipole matrix elements for transition between A 1  + and X 1  + states

13. Hyperfine structure Zeeman levels of the 23 Na atom versus magnetic field Zeeman levels of the 40 Ca ion versus magnetic field 23 Na: s=1/2 i=3/2 23 Ca + : s=1/2 i=0

14. Scattering channels Ca Na + Ca + Na Conserved quantities: m f, l, m l (neglecting small spin dipole-dipole interaction) Asymptotic channels states Channel states in (is) representation (short-range basis) m f =1/2 and l=0 Na Ca +

15. Frame transformation Frame transformation: unitary transformation between  (asymptotic) and  (is) basis Clebsch-Gordan coefficients Transformation between  (f 1 f 2 ) and (is) basis

16. Frame transformation polarization forces ~ R * Separation of length scales  r 0 ~ exchange interaction At distances we can neglect - exchange interaction - hyperfine splittings - centrifugal barrier Then Quantum defect matrix in short-range (is) basis a s, a t – singlet and triplet scattering lengths WKB-like normalized solutions Unitary transformation between  (asymptotic) and  (short-range) basis

17. Frame transformation Applying unitary transformation between  (asymptotic) and  (short-range) basis Example 23 Na and 40 Ca + - determines strength of coupling between channels Additional transformation necessary in the presence of a magnetic field B Quantum defect matrix for B  0 U

18. Example: energies of the atom-ion molecular complex Solid lines: quantum-defect theory for Y 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 Assumption of angular-momentum-insensitive Y becomes less accurate for higher partial waves

19. Collisional rates for 23 Na and 40 Ca + Rates of elastic collisions in the singlet channel A 1  + Rates of the radiative charge transfer in the singlet channel A 1  + Threshold behavior for C 4 potential Maxima due to the shape resonances

20. Scattering length versus magnetic field Energies of bound states versus magnetic field Feshbach resonances for 23 Na and 40 Ca + a s,  a t  weak resonances

21. s-wave scattering length Feshbach resonances for 23 Na and 40 Ca + Energies of bound states Charge transfer rate a s,  - a t  strong resonances

22. Feshbach resonances for 23 Na and 40 Ca + s-wave scattering length versus B, singlet and triplet scattering lengths MQDT model only

23. Shape resonances The resonance appear when the kinetic energy matches energy of a quasi-bound state Resonance in the total cross section Breit-Wigner formula  - lifetime of the quasi-bound state Due to the centrifugal barrier Due to the external trapping potental V(r) r R. Stock et al., Phys. Rev. Lett. 91, 183201 (2003) V(r)

24. R. Stock et al., Phys. Rev. Lett. 91, 183201 (2003) V(r) Trap-induced shape resonances Two particles in separate traps Relative and center-of-mass motions are decoupled Energy spectrum versus trap separation a<0 a>0

25. ATOM JON Controlled collisions between atoms and ions Atom and ion in separate traps + short-range phase   single channel model trap size  range of potential particles follow the external potential Controlled collisions Applications Spectroscopy/creation of atom-ion molecular complexes Quantum state engineering Quantum information processing: quantum gates

26. Identical trap frequencies:  i =  a =  Energy spectrum versus distance between traps Relative motion: harmonic oscillator states Bound state of r -4 potential (+correction due to trap) + short-range phase  Avoided crossings (position depend on energies of bound states Controlled collisions between atoms and ions

27. Identical trap frequencies:  i =  a =  + quasi-1D system Energy spectrum versus distance d Selected wave functions + potential  e,  o : short-range phases (even + odd states) Controlled collisions between atoms and ions

28. Avoided crossings: vibrational states in the trap  molecular states Dynamics in the vicinity of avoided crossings: (Landau-Zener theory) Probability of adiabatic transition Controlled collisions between atoms and ions

29. Energy gap  E at avoided crossing versus distance d Depends on the symmetry of the molecular state Decays exponentially with the trap separation Semiclassical approximation (instanton method) : 40 Ca + - 87 Rb  i =  a =2  100 kHz Controlled collisions between atoms and ions

30. Different trap frequencies:  i  a  Center of mass and relative motion are coupled Energy spectrum versus trap separation in quasi 1D system States of two separated harmonic oscillators Molecular states + center-of- mass excitations  e,  o : short-range phases (even + odd states) Controlled collisions between atoms and ions