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
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
Atom-ion interaction
Large distances, atom in S state
V (r ) ~ 
 e2
induced dipole
2r 4
 - atomic polarizability
ATOM
ION
Large distances, atom in P state (or other with a quadrupole moment)

state
2
2
quadrupole moment: Q  3 z  r
2Qe 2  e 2
V (r ) ~ 3  4
r
2r
 state
Qe 2  e 2
V (r ) ~  3  4
r
2r
graph from:
F.H. Mies, PRA (1973)
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
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):
Quantum defect parameter
Separation of length scales 
short-range phase is independent of energy
and angular momentum
Quantum-defect parameter
Exchange interaction, higher order
dispersion terms: C6/r6, C8/r8, ...
R*
– polarization forces
l ( E )  const
Boundary condition imposed by  represents shortrange part of potential
Short range-wave function fulfills Schrödinger equation at
E=0 and l=0
 
2
 2 2 
R* 
 4  0 r   0
 2

r
r

r
r 

 0 (r )  A sin R * r   
Behavior at large distances r
r 0 (r )  AR* cos   r sin 
Relation to the s-wave scattering length a R*   cot 
V (r ) ~ 
 e2
2r 4
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
K oo   tan  l
Foo (r ) 
1
k
sin( kr  l 2   l ) (r  )
Quantum-defect theory of ultracold collisions
Reference potentials:
Solutions with WKB-like normalization
at small distances
 2 2

 2l (l  1)


V
(
r
)


E
i

 i r   0
2
2
2


r
2

r


fˆ (r , E )
gˆ (r , E )
f (r , E )
g (r , E )
 (r , E )
Analytic across threshold!
Solutions with energy-like normalization at r
R*
Rmin
Non-analytic across threshold!
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)
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
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
f i (r )  fˆi (r ) 

g i (r )  gˆ i (r )
For E
E 
Quantum-defect theory of ultracold collisions
QDT functions relate Y(E) to observable quantities, e.g. scattering matrices
Renormalization of Y(E) in the presence of the closed channels
For a single channel scattering
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
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).
Hyperfine structure
23Na:
s=1/2 i=3/2
Zeeman levels of the 23Na atom versus magnetic field
23Ca+:
s=1/2 i=0
Zeeman levels of the 40Ca ion versus magnetic field
Scattering channels
Conserved quantities: mf, l, ml
(neglecting small spin dipole-dipole interaction)
Asymptotic channels states
mf =1/2 and l=0
Ca+
Na
Ca+ Na
Channel states in (is) representation (short-range basis)
Ca Na+
mf =1/2 and l=0
Frame transformation
Frame transformation: unitary transformation between (asymptotic) and  (is) basis
Clebsch-Gordan coefficients
Transformation between (f1f2) and (is) basis
Frame transformation
r0 ~ exchange interaction
Separation of length scales 
At distances
polarization forces ~
we can neglect
- exchange interaction
- hyperfine splittings
- centrifugal barrier
Wij (r ) 
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
C4
 ij
r4
R*
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
Quantum-defect theory of ultracold collisions
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).
cot as R* 
Assumption of angular-momentum-insensitive Y becomes less accurate for higher partial waves
Collisional rates for 23Na and 40Ca+
Rates of elastic collisions in
the singlet channel A1+
Threshold behavior for C4 potential
tan  l ~ k 2
 l ~ E, for l  0
Rates of the radiative charge
transfer in the singlet channel A1+
Maxima due to the shape resonances
Feshbach resonances for 23Na and 40Ca+
Scattering length versus
magnetic field
as, at  weak resonances
1
1 1
 
ac as at
Energies of bound states
versus magnetic field
Feshbach resonances for 23Na and 40Ca+
s-wave scattering length
as, - at  strong resonances
1
1 1
 
ac as at
Charge transfer rate
Energies of bound states
Feshbach resonances for 23Na and 40Ca+
s-wave scattering length versus B, singlet and triplet scattering lengths
MQDT model only
as
  cot  s
R*
at
  cot t
R*
Shape resonances
Due to the centrifugal barrier
V(r)
Due to the external trapping potental
2
l (l  1)
2r 2
V(r)
1
2
 2 ( z  z ) 2  V (r )
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
2

 / 2
~
2
( E  Eqb ) 2   / 2
 - lifetime of the quasi-bound state
2
2

dE  qb V  free ( E )  E  Eqb 
 
Resonance in the total cross section
Trap-induced shape resonances
Two particles in separate traps
2
2
1
1
2
2
H 
1 
 2  m 2 r1  d1   m 2 r2  d 2   V (r1  r2 )
2m
2m
2
2
z
Relative and center-of-mass motions are decoupled
2
1
2
H rel  
   2 z  z   V (r )
2
2
z  d1  d 2
Vtotal (r )  12  2 ( z  z ) 2  V (r )
d1  d1zˆ
d 2  d 2 zˆ
Energy spectrum versus trap separation
V(r)
a<0
z0   
a>0
R. Stock et al., Phys. Rev. Lett. 91, 183201 (2003)
Controlled collisions between atoms and ions
Atom and ion in separate traps
H 
2
+ short-range phase   single channel model
Controlled collisions
JON
ATOM


1
1
e
2
2
2
2
i 
  mii ri  d i   maa ra  d a   4
2mi
2mi
2
2
2r
2
2
a
d
(a)
• trap size  range of potential
• particles follow the external potential
(b)
Applications
(c)
• Spectroscopy/creation of atom-ion molecular complexes
• Quantum state engineering
• Quantum information processing: quantum gates
(d)
i
Controlled collisions between atoms and ions
Identical trap frequencies: i=a=
2
 e2 1 2
2
  4   r  d 
Relative motion: H rel  
2
2r
2
+ short-range phase 
d  di  d a
Energy spectrum versus distance between traps
harmonic
oscillator
states
Bound state of
r-4 potential
(+correction due
to trap)
Avoided crossings
(position depend on
energies of bound
states
Controlled collisions between atoms and ions
Identical trap frequencies: i=a= + quasi-1D system
H rel
2
 e2 1 2
2

  4   z  d 
2
2r
2
Selected wave functions + potential
Energy spectrum versus distance d
e, o : short-range phases (even + odd states)
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
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) :
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-ofmass
excitations
e, o : short-range
phases (even + odd
states)