ATOM-ION COLLISIONS
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Transcript of 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
Quantum-defect theory of ultracold collisionsQuantum-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
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 collisionsQuantum-defect theory of ultracold collisions
Ergrg
rfrf
ii
ii
)(ˆ)(
)(ˆ)(
For E
Quantum-defect theory of ultracold collisionsQuantum-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 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
Frame transformationFrame transformation
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
Frame transformationFrame transformation
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).
Quantum-defect theory of ultracold collisionsQuantum-defect theory of ultracold collisions
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
Feshbach resonances for 23Na and 40Ca+Feshbach resonances for 23Na and 40Ca+
Energies of bound states
Charge transfer rate
as, - at strong resonances
tsc aaa
111
Feshbach resonances for 23Na and 40Ca+Feshbach resonances for 23Na and 40Ca+
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
Controlled collisions between atoms and ionsControlled collisions between atoms and ions
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)
Controlled collisions between atoms and ionsControlled 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 ionsControlled collisions between atoms and ions
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
Controlled collisions between atoms and ionsControlled collisions between atoms and ions
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)
Controlled collisions between atoms and ionsControlled collisions between atoms and ions