ultracold collisions in the presence of trapping potentials zbigniew idziaszek institute for quantum...
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ULTRACOLD COLLISIONS IN THE
PRESENCE OF TRAPPING POTENTIALS
ZBIGNIEW IDZIASZEK
Institute for Quantum Information,University of Ulm, 18 February 2008
Institute for Theoretical Physics, University of Warsaw
and
Center for Theoretical Physics, Polish Academy of Science
OutlineOutline
1. Binary collisions in harmonic traps
- collisions in s-wave
- collisions in higher partial waves
3. Scattering in quasi-1D and and quasi-2D traps
- confinement –induced resonances
2. Energy dependent scattering length
4. Feshbach resonances
SystemSystem
1. Ultracold atoms in the trapping potential
Typical trapping potentials are harmonic close to the center
magnetic traps, optical dipole traps, electro-magnetic traps for charged particles, ...
Interactions can be modeled via contact pseudopotential
2. Characteristic range of interaction R* << length scale of the trapping potential
- Very accurate for neutral atoms
- Not applicable for charged particles, e.g. for atom-ion collisions
222222
2
1)( zyxmV zyxT r
R*
trap size
CM and relative motions can be separated in harmonic potential
Axially symmetric trap:
Contact pseudopotential for s-wave scattering (low energies):
Hamiltonian (harmonic-oscillator units)
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
length unit: energy unit:
Relative motion
We expand into basis of harmonic oscillator wave functions
Contact pseudopotential affects only states with mz=0 and k even
(non vanishing at r=0 )
radial:
axial:
For mz0 or k odd trivial solution:
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Integral representation can be obtained from:
Eigenenergies:
Eigenfunctions:
Substituting expansion into Schrödinger equation and projecting on
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum for = 5 Energy spectrum for = 1/5
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum in cigar-shape traps ( > 1)
Energy spectrum in pancake-shape traps ( < 1)
For
Z.I., T. Calarco, PRA 71, 050701 (2005)
For
T. Stöferle et al., Phys. Rev. Lett. 96, 030401 (2006)
Bound state for positive and negative energies due to the trap
Comparison of theory vs. experiment: atoms in optical lattice
T. Bush et al., Found. Phys. 28, 549 (1998)
• solid line – theory (spherically symmetric trap)
• points – experimental data
-10 -5 0 5 10-2
0
2
4
6
8
En
erg
y [E
/]
a/aHO
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
)(
)(
23
21
E
E
a
dmd
Energy spectrum and wave functions for
very elongated cigar-shape trap
Energy spectrum for = 100
exact energies
1D model + g1D
First excited state
Elongated in the direction of weak trapping
Size determined by the strong confinement
Wave function is nearly isotropic
Trap-induced bound state (a < 0)
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Dip in the center due to the strong interaction
Identical fermions can only interact in odd partial waves (l = 2n+1)
Hamitonian of the relative motion:
Two ultracold atoms in harmonic trapTwo ultracold atoms in harmonic trap
Energy spectrum for = 1/10Energy spectrum for = 1/10
Two ultracold fermions in harmonic trap
No interactions in higher partial waves at E0 (Wigner threshold law)12~tan l
l k
Scattering for l > 0 can be enhanced in the presence of resonances Feshbach resonances
Fermi pseudopotential - applicable for: k R* 1, k a 1/ k R*
s-wave scattering lenght:
In the tight traps (large k) or close to resonances (large a)
Energy-dependent scattering length
At small energies (k 0): aeff(E) a
Energy-dependent scattering lengthEnergy-dependent scattering length
E.L Bolda et al., PRA 66, 013403 (2002)D. Blume and C.H. Greene, PRA 65, 043613 (2002)
Schrödinger equation is solved in a self-consistent way
EEVH )(0
Applicable only when CM and relative motions can be separated.
V(r)
r
R0
V0
Model potential: square well
exact energies
pseudopotential approximation
pseudopotential with aeff(E)
Scattering length
Energy spectrum
Parameters:
Energy-dependent scattering lengthEnergy-dependent scattering length
TEST: two interacting atoms in harmonic trap, s-wave states
V(r)
r
R0
V0
Energy spectrum for R0=0.05d
Energy-dependent pseudopotential
applicable even for R0 /d not very small
Energy spectrum for R0=0.2 d
TEST: two interacting atoms in harmonic trap, p-wave states
Energy-dependent scattering lengthEnergy-dependent scattering length
Scattering volume:
EDP: 23
32
)()(
)( rr
rEa
V pp
�
rr
Hamiltonian of relative motion:
Quasi-1D traps
Asymptotic solution at small energies
Weak confinement along z
Strong confinement along x,y
Effective motion like in 1D system
In the harmonic confinement CM and relative motions are not coupled
After collision atoms remain in ground-state of transverse motion
Atomic collisions in quasi-1D trapsAtomic collisions in quasi-1D traps
f+ - even scattering wave
f- - odd scattering wave
optical lattice
Collisions of bosons in quasi-1D trapsCollisions of bosons in quasi-1D traps
Even scattered wave for bosons
Confinement induced resonance (CIR)
occurs for
Transmission coefficient T
M. Olshanii PRL 81, 938 (1998)
21 efT
daEk 68.0)(0For
Interactions of bosons in 1D can be modeled with:
Contact pseudopotential
Interaction strength for quasi-1D trap obtained from 3D solution
Confinement induced resonance at
T. Bergeman et al. PRL 91, 163201(2003)
Gas of strongly interacting bosons in 1D: Tonks-Girardeau gas
Collisions of bosons in 1D systemCollisions of bosons in 1D system
M. Olshanii PRL 81, 938 (1998)
Odd scattered wave for fermions
B. Granger, D. Blume, PRL (2004)
Scattering amplitude f-
CIR
Collisions of fermions in quasi-1D trapsCollisions of fermions in quasi-1D traps
Resonance in p-wave for
)(0For Ek
Feshbach resonancesFeshbach resonances
)(2 1
2
1 rVH
)(2 2
2
2 rVH
)(rW
2221
1211
EHW
EWH
EH1
– entrance channel
Em(B)
(1)
(2)
– closed channel
– coupling between channels
0
1
11 iHE
G
Inverting 1st equation with the help of Green’s functions
r
efe
ikri
r),(
rkr)2(
)1(
2221
211
EHW
WG
Substituting (1) into (2) and solving for 2
WWWGHE 12
2
1
W
WWGHEWG
1211
1
Feshbach resonancesFeshbach resonances
Close to a resonance only single bound-state from a closed channel contributes
2)(
1 resres
12
iBEEWWGHE mm
resres2res)( BBHBE mm
res1resRe WWGm
res1resIm2
WWG
0)( res BEm
res - resonant bound state in the closed channel
Bres – magnetic field when the bound state crosses the threshold
- energy shift due to the couppling
- resonance width
WWWGHE 12
2
1
W
WWGHEWG
1211
1
Em(B)
1)
2)
- energy of bound state
Feshbach resonancesFeshbach resonances
mmbg BEE )(
2arctan0
0
1)(BB
BaBa bg
2
2
0
0
2
2lim
bgb
mbgk
mres
maE
a
kB
BB
Phase shift
bg – background phase shift (in the absence of coupling between channels)
Typically for ultracold collisions
Background scattering length:
abg
a (s
catte
ring
leng
th)
B (magnetic field)
B0
B
b
bbg EEBEBB
EEBaBEa
0eff
)1(1),(
Energy dependent scattering length
k
ka bg
kbg
)(tanlim
0
Parameters of resonance
Example: Energy spectrum of two 87Rb atoms in a tight trap
Quasi-1D trap
energy spectrum
resonance position
Trapped atoms + Feshbach resonancesTrapped atoms + Feshbach resonances
Lippmann-Schwinger equation and Green’s functionsLippmann-Schwinger equation and Green’s functions
2 VG
EVH 0
rkr ie
0
1
0 iHEG
rr
rrrrrr
4
20,
2
100
ikemiHEG
m
H2
2
0
)()(),()( 3 rrrrr rk VGrdei
r
efe
ikri
r),()(
rkr
)()(2
4
1, 3
2rrrk
Verd
mf i
Lippmann Schwinger equation
Green’s operator
Solution for V=0
Green’s function in position representation in free space
Lippmann-Schwinger equation in position representation
Behavior of (r) at large distances
Scattering amplitude
+ outgoing spherical wave
ZI, T. Calarco, PRA 74, 022712 (2006)
1D and 2D effective interactions in comparison to full 3D treatment
exact energies (3D)
1D trap + g1D
exact energies (3D)
2D trap + g2D
Realization of 1D and 2D regimes does not require very large anisotropy of the trap
Atomic collisions in quasi-1D and quasi-2D trapsAtomic collisions in quasi-1D and quasi-2D traps
Energy spectrum in cigar-shape trap Energy spectrum in pancake-shape trap
2
1
HW
WHH
2
1
Then E (kinetic energy at r = 0)
Z. Idziaszek, T. Calarco, PRA 74, 022712 (2006)
Scattering of spin-polarized fermions in quasi-2D
Asymptotic solution for kinetic energies
Atoms remain in the ground state in z direction
m=1 scattering wave for p-wave interacting fermions
Solving the scattering problem ...
QUASI-2D SYSTEMSQUASI-2D SYSTEMS
Scattering amplitude in forward direction for different values of energy
CIR
Scattering amplitude
2D scattering amplitude:
Scattering in quasi-2D trapsScattering in quasi-2D traps
Similar scattering confinement-induced resonaces as in quasi-1D traps
Example: two fermions, p-wave interactions
Rozpraszanie fermionów w fali p w układzie kwazi-2D
Zachowanie asymptotyczne dla energii kinetycznych
Atomy pozostają w stanie podstawowym w kierunku z
Amplituda rozpraszania w 2D:
fala m=1 dla fermionów oddziałujących w fali p
Rozwiązanie problemu rozpraszania:
Zderzenia atomów w pułapkach kwazi-1D i kwazi-2DZderzenia atomów w pułapkach kwazi-1D i kwazi-2D
położenie rezonansu:
Dla niskich energii ( ):
Rezonans indukowany ściśnięciem gdy
Amplituda rozpraszania do przodu dla różnych energii kinetycznych
CIR
Rezonans nie widoczny powyżej energii
ZI, and T. Calarco, PRL (2006)
Zderzenia atomów w fali p w układzie kwazi-2D Zderzenia atomów w fali p w układzie kwazi-2D