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ATOM-ION COLLISIONS

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

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ATOM-ION COLLISIONS

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

  2. Outline • 1.Analytical model of ultracold atom-ion collisions • Exact solutions for 1/r4 potential – single channel QDT • Multichannel quantum-defect theory • Frame transformation 2. Results for Ca+-Na system 3. Controlled collisions of atom and ions in movable trapping potentials

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

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

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

  6. Quantum defect parameter Exchange interaction, higher order dispersion terms: C6/r6, C8/r8, ... Separation of length scales  short-range phase is independent of energy and angular momentum R*– polarization forces Quantum-defect parameter Boundary condition imposed by  represents short-range part of potential Short range-wave function fulfills Schrödinger equation at E=0 and l=0 Behavior at large distances r Relation to the s-wave scattering length

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

  8. Quantum-defect theory of ultracold collisions Reference potentials: Solutions with WKB-like normalization at small distances Analytic across threshold! Solutions with energy-like normalization at r R* Rmin Non-analytic across threshold!

  9. QDT functions connect f,ĝ with f,g, 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) Quantum defect matrix Y(E) Y very weakly depends on energy: Expressing the wave function in terms of another pair of solutions R matrix strongly depends on energy and is nonanalytic across threshold

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

  11. Quantum-defect theory of ultracold collisions QDT functions relate Y(E)to observable quantities, e.g. scattering matrices 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 All the channels are closed  bound states

  12. Ultracold atom-ion collisions Example: 23Na and 40Ca+ • Both individual species are widely used in experiments • ab-initio calculations of interaction potentials and dipole moments are available Radial transition dipole matrix elements for transition between A1+ and X1+ states Born-Oppenheimer potential-energy curves for the (Na-Ca)+ molecular complex O.P. Makarov, R. Côté, H. Michels, and W.W. Smith, Phys.Rev.A 67, 042705, (2005).

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

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

  15. Frame transformation Frame transformation: unitary transformation between (asymptotic) and  (is) basis Clebsch-Gordan coefficients Transformation between(f1f2) and (is) basis

  16. Frame transformation r0~ exchange interaction Separation of length scales  polarization forces ~ R* At distances we can neglect • exchange interaction • hyperfine splittings • centrifugal barrier Then Quantum defect matrix in short-range(is) basis WKB-like normalized solutions as, at – singlet and triplet scattering lengths Unitary transformation between  (asymptotic) and  (short-range) basis

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

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

  19. Collisional rates for 23Na and 40Ca+ Rates of elastic collisions in the singlet channel A1+ Threshold behavior for C4 potential Rates of the radiative charge transfer in the singlet channel A1+ Maxima due to the shape resonances

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

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

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

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

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

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

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

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

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

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

  30. Controlled collisions between atoms and ions 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)

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