ATOM-ION COLLISIONS

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

OutlineOutline

2. Results for Ca+-Na system

1. Analytical model of ultracold atom-ion collisions

- Exact solutions for 1/r4 potential – single channel QDT

- Multichannel quantum-defect theory

- Frame transformation

3. Controlled collisions of atom and ions in movable trapping potentials

Atom-ion interactionAtom-ion interaction

4

2

3

2

2

2~)(

r

e

r

QerV

4

2

3

2

2~)(

r

e

r

QerV

223 rzQ state

state

quadrupole moment:

4

2

2~)(

r

erV

- atomic polarizability

Large distances, atom in S state

induced dipole

ATOM ION

Large distances, atom in P state (or other with a quadrupole moment)

graph from:F.H. Mies, PRA (1973)

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 potentialAnalytical solution for polarization potential

E. Vogt and G. Wannier, Phys. Rev. 95, 1190 (1954)

Analytical solution for polarization potentialAnalytical 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):

Quantum defect parameterQuantum 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: C6/r6, C8/r8, ...

4

2

2~)(

r

erV

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 constEl )(

rRAr *0 sin)(

02

04

2*

2

2

rr

R

rrr

cot* Ra

sincos)( *0 rARrr

Multichannel formalismMultichannel 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

looK tan

)()sin()( 21 rlkrrF lkoo

N – number of channels

Radial coupled-channel Schrödinger equation

Quantum-defect theory of ultracold collisionsQuantum-defect theory of ultracold collisions

),(ˆ

),(ˆ

Erg

Erf

),(

),(

),(

Er

Erg

Erf

R*

Rmin

Solutions with WKB-like normalization

at small distances

Solutions with energy-like normalization at r

Analytic across threshold!

Non-analytic across threshold!

Reference potentials:

02

)1()(

2 2

2

2

22

rEr

llrV

r ii


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

For large energies semiclassical description is valid at all distances, and the two sets of

solutions are equivalent

Semiclassical approximation is valid when


Ergrg

rfrf

ii

ii

)(ˆ)(

)(ˆ)(

For E


QDT functions relate Y(E) to observable quantities, e.g. scattering matrices

All the channels are closed bound states

For a single channel scatteringRenormalization 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

• 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 collisionsUltracold atom-ion collisions

Born-Oppenheimer potential-energy curves for the (Na-Ca)+ molecular complex

Example: 23Na and 40Ca+

Radial transition dipole matrix elements for transition between A1+ and X1+ states

Hyperfine structureHyperfine structure

Zeeman levels of the 23Na atom versus magnetic field Zeeman levels of the 40Ca ion versus magnetic field

23Na: s=1/2 i=3/2 23Ca+: s=1/2 i=0

Scattering channelsScattering channels

Ca Na+

Ca+ Na

Conserved quantities: mf, l, ml

(neglecting small spin dipole-dipole interaction)

Asymptotic channels states

Channel states in (is) representation (short-range basis)

mf =1/2 and l=0

mf =1/2 and l=0

NaCa+

Frame transformationFrame transformation

Frame transformation: unitary transformation between (asymptotic) and (is) basis

Clebsch-Gordan coefficients

Transformation between(f1f2) and (is) basis


ijij r

CrW

44)(

polarization forces ~ R*

Separation of length scales r0 ~ exchange interaction

At distances we can neglect

- exchange interaction

- hyperfine splittings

- centrifugal barrier

Then

Quantum defect matrix in short-range (is) basis

as, at – singlet and triplet scattering lengths

WKB-like normalized solutions

Unitary transformation between (asymptotic) and (short-range) basis


Applying unitary transformation between (asymptotic) and (short-range) basis

Example 23Na and 40Ca+

- determines strength of coupling between channels

Additional transformation necessary in the presence of a magnetic field B

Quantum defect matrix for B 0

U

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 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 Y becomes less accurate for higher partial waves

*cot Ras

Collisional rates for 23Na and 40Ca+Collisional rates for 23Na and 40Ca+

Rates of elastic collisions in the singlet channel A1+

Rates of the radiative charge transfer in the singlet channel A1+

0for, ~ lEl

Threshold behavior for C4 potential2~tan kl

Maxima due to the shape resonances

Scattering length versus magnetic field

Energies of bound states versus magnetic field

Feshbach resonances for 23Na and 40Ca+Feshbach resonances for 23Na and 40Ca+

tsc aaa

111

as, at weak resonances

s-wave scattering length


Energies of bound states

Charge transfer rate

as, - at strong resonances

tsc aaa

111


s-wave scattering length versus B, singlet and triplet scattering lengths

MQDT model only

ss

R

a cot*

tt

R

a cot*

Shape resonancesShape resonances

The resonance appear when the kinetic energy matches energy of a quasi-bound state

Resonance in the total cross section

22

qb

2

2/)(

2/~

EEBreit-Wigner formula

- lifetime of the quasi-bound state

qb

2

freeqb )(2

EEEVdE

Due to the centrifugal barrier Due to the external trapping potental

)()( 2221 rVzz

V(r)

r

)1(2 2

2

llr

R. Stock et al., Phys. Rev. Lett. 91, 183201 (2003)

V(r)

z

)(2

1

222

2

rel rVzzH

)(2

1

2

1

22 212

2222

112

2

2

1

2

rrdrdr Vmmmm

H

)()()( 2221 rVzzVtotal r

R. Stock et al., Phys. Rev. Lett. 91, 183201 (2003)

V(r)

Trap-induced shape resonancesTrap-induced shape resonances

Two particles in separate traps

Relative and center-of-mass motions are decoupled

21 ddz

Energy spectrum versus trap separation

a<0 a>0

zd ˆ11 d

zd ˆ22 d

0z

d i

ATOM JON

a

Controlled collisions between atoms and ionsControlled 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

(a)

(b)

(c)

(d)

4

22222

22

22

1

2

1

22 r

emm

mmH aaaaiiii

ii

i

drdr

Identical trap frequencies: i=a=

Energy spectrum versus distance between traps

224

22

rel 2

1

22dr

r

eH

Relative motion:

harmonic oscillatorstates

Bound state of r-4 potential (+correction due to trap)

ai ddd + short-range phase

Avoided crossings (position depend on energies of bound states


Identical trap frequencies: i=a= + quasi-1D system

Energy spectrum versus distance d

Selected wave functions + potential

224

22

rel 2

1

22dz

r

eH

e, o : short-range phases (even + odd states)


Avoided crossings: vibrational states in the trap molecular states

Dynamics in the vicinity of avoided

crossings:

(Landau-Zener theory) Probability of adiabatic transition


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) :

40Ca+ - 87Rb

i=a=2100 kHz


Different trap frequencies: ia

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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ATOM-ION COLLISIONS