Download Two ultracold atoms in harmonic trap

Institute for Quantum Information,
University of Ulm, 18 February 2008
ULTRACOLD COLLISIONS IN THE
PRESENCE OF TRAPPING POTENTIALS
ZBIGNIEW IDZIASZEK
Institute for Theoretical Physics, University of Warsaw
and
Center for Theoretical Physics, Polish Academy of Science
Outline
1. Binary collisions in harmonic traps
- collisions in s-wave
- collisions in higher partial waves
2. Energy dependent scattering length
3. Scattering in quasi-1D and and quasi-2D traps
- confinement –induced resonances
4. Feshbach resonances
System
1. Ultracold atoms in the trapping potential
magnetic traps, optical dipole traps, electro-magnetic traps for charged particles, ...
Typical trapping potentials are harmonic close to the center
VT (r ) 

1
m  x2 x 2   y2 y 2   z2 z 2
2

2. Characteristic range of interaction R* << length scale of the trapping potential
trap size
R*
Interactions can be modeled via contact pseudopotential
- Very accurate for neutral atoms
- Not applicable for charged particles, e.g. for atom-ion collisions
Two ultracold atoms in harmonic trap
Hamiltonian (harmonic-oscillator units)
length unit:
energy unit:
Axially symmetric trap:
Contact pseudopotential for s-wave scattering (low energies):
CM and relative motions can be separated in harmonic potential
Two ultracold atoms in harmonic trap
Relative motion
We expand
into basis of harmonic oscillator wave functions
radial:
axial:
Contact pseudopotential affects only states with mz=0 and k even
(non vanishing at r=0 )
For mz0 or k odd trivial solution:
Two ultracold atoms in harmonic trap
Substituting expansion into Schrödinger equation and projecting on
Eigenfunctions:
Eigenenergies:
Integral representation can be obtained from:
Two ultracold atoms in harmonic trap
Energy spectrum in cigar-shape traps ( > 1)
Energy spectrum in pancake-shape traps ( < 1)
Energy spectrum for  = 5
For
Energy spectrum for  = 1/5
For
Z.I., T. Calarco, PRA 71, 050701 (2005)
Two ultracold atoms in harmonic trap
Comparison of theory vs. experiment: atoms in optical lattice
8
Energy [E/]
6
4
2
0
-2
-10
-5
0
5
10
a/aHO
• solid line – theory (spherically symmetric trap)
T. Bush et al., Found. Phys. 28, 549 (1998)
d ( E  12 )

a ( E  32 )
d   m
• points – experimental data
T. Stöferle et al., Phys. Rev. Lett. 96, 030401 (2006)
Bound state for positive and negative energies due to the trap
Two ultracold atoms in harmonic trap
Energy spectrum and wave functions for
very elongated cigar-shape trap
First excited state
Elongated in the direction of weak trapping
Dip in the center due to the strong interaction
Trap-induced bound state (a < 0)
Energy spectrum for  = 100
exact energies
1D model + g1D
Size determined by the strong confinement
Wave function is nearly isotropic
Two ultracold atoms in harmonic trap
Identical fermions can only interact in odd partial waves (l = 2n+1)
No interactions in higher partial waves at E0 (Wigner threshold law)
tan  l ~ k 2l 1
Scattering for l > 0 can be enhanced in the presence of resonances  Feshbach resonances
Two ultracold fermions in harmonic trap
Hamitonian of the relative motion:
Energy spectrum for  = 1/10
Energy spectrum for  = 1/10
Energy-dependent scattering length
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)
E.L Bolda et al., PRA 66, 013403 (2002)
D. Blume and C.H. Greene, PRA 65, 043613 (2002)
Energy-dependent scattering length
At small energies (k  0): aeff(E)  a
Schrödinger equation is solved in a self-consistent way
H0  V (E) 
E
Applicable only when CM and relative motions can be separated.
Energy-dependent scattering length
TEST: two interacting atoms in harmonic trap, s-wave states
Scattering
length
Model potential: square well
V(r)
R0
r
Energy
spectrum
V0
Parameters:
exact energies
pseudopotential approximation
pseudopotential with aeff(E)
Energy-dependent scattering length
TEST: two interacting atoms in harmonic trap, p-wave states
V(r)
R0
r
 3 2
 2a p (E) 
EDP: V p (r) 
 (r )r 3 r

r
V0
Scattering volume:
Energy spectrum for R0=0.05d
Energy-dependent pseudopotential
applicable even for R0 /d not very small
Energy spectrum for R0=0.2 d
Atomic collisions in quasi-1D traps
optical lattice
Quasi-1D traps
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
Hamiltonian of relative motion:
Asymptotic solution at small energies
f+ - even scattering wave
f- - odd scattering wave
After collision atoms remain in ground-state of transverse motion
Collisions of bosons in quasi-1D traps
Even scattered wave for bosons
M. Olshanii PRL 81, 938 (1998)
Confinement induced resonance (CIR)
occurs for
For k  0 ( E   )
a  0.68d
Transmission coefficient T
T  1  fe
2
Collisions of bosons in 1D system
Interactions of bosons in 1D can be modeled with:
Contact pseudopotential
Interaction strength for quasi-1D trap obtained from 3D solution
M. Olshanii PRL 81, 938 (1998)
Confinement induced resonance at
T. Bergeman et al. PRL 91, 163201(2003)
Gas of strongly interacting bosons in 1D: Tonks-Girardeau gas
Collisions of fermions in quasi-1D traps
Odd scattered wave for fermions
B. Granger, D. Blume, PRL (2004)
CIR
Resonance in p-wave for
For k  0 ( E   )
Scattering amplitude f-
Feshbach resonances
H1  1  W  2  E  1
W 1  H2  2  E  2
(2)
Em(B)
(1)
2
H1  
  V1 (r )
2
– entrance channel
2
H2  
  V2 (r )
2
– closed channel
W (r ) – coupling between channels
Inverting 1st equation with the help of Green’s functions
 1     G1W  2
W 1  H2  2  E  2
(1)
(2)
G1 
Substituting (1) into (2) and solving for 2
1
W 

E  H 2  WG1 W
1
    G1W
W 

E  H 2  WG1 W
2 
1
1
E  H1  i 0
H1    E  
r

r
 e

ik r
eikr
 f ( ,  )
r
Feshbach resonances
2 
1
W 

E  H 2  WG1 W
 1     G1W
Em(B)
2)
1)
1
W 

E  H 2  WG1 W
Close to a resonance only single bound-state from a closed channel contributes
res res
1

E  H 2  WG1W E  Em ( B)   m  i 2
Em ( B)  res H 2 res  m B  Bres 
Em ( Bres )  0
 m  Re res WG1W res
res
- resonant bound state in the closed channel
- energy of bound state
Bres – magnetic field when the bound state crosses the threshold
- energy shift due to the couppling

  Im res WG1W res - resonance width
2
Feshbach resonances
Phase shift


2

 E  Em ( B)   m 
 0   bg  arctan 
bg – background phase shift (in the absence
of coupling between channels)
B0  Bres   m 
Energy dependent scattering length
 2k
k 0 a 
bg m


B(1  E Eb )
aeff ( E, B)  abg 1 

 B  B0  E   B E Eb 
Background scattering length:
B  lim
 tan  bg (k ) 

abg  lim  
k 0
k


2
Eb 
2
2mabg
B0

B 

a( B)  abg 1 
 B  B0 
a (scattering length)
Typically for ultracold collisions
B
abg
B (magnetic field)
Trapped atoms + Feshbach resonances
Example: Energy spectrum of two
87Rb
atoms in a tight trap
Quasi-1D trap
Parameters of resonance
energy spectrum
resonance position
Lippmann-Schwinger equation and Green’s functions
H0  V 
 E
Green’s operator G  
     G V  2
Solution for V=0
1
E  H 0  i0
r   eik r
Lippmann Schwinger equation
2
Green’s function in position representation in free space H 0  

2m
G r, r  r E  H 0  i0
1

0
 ik r r
2m e
r   2
 4 r  r
Lippmann-Schwinger equation in position representation
  (r )  eik r   d 3r G  (r, r)V (r)  (r)
Behavior of (r) at large distances
 (r ) r
 e


Scattering amplitude
ik r
eikr
 f ( ,  )
r
f  ,   
1 2m 3 ik r
d r e V (r)  (r)
2 
4 
+ outgoing spherical wave
Atomic collisions in quasi-1D and quasi-2D traps
1D and 2D effective interactions in comparison to full 3D treatment
Energy spectrum in cigar-shape trap
Energy spectrum in pancake-shape trap
exact energies (3D)
exact energies (3D)
1D trap + g1D
2D trap + g2D
ZI, T. Calarco, PRA 74, 022712 (2006)
Realization of 1D and 2D regimes does not require very large anisotropy of the trap
H
H   1
W
W

H 2 
 
   1 
 2 
Then E  (kinetic energy at r = 0)
Z. Idziaszek, T. Calarco,
PRA 74, 022712 (2006)
QUASI-2D SYSTEMS
Scattering of spin-polarized fermions in quasi-2D
Asymptotic solution for kinetic energies
Atoms remain in the ground state in z direction
Solving the scattering problem ...
m=1 scattering wave for p-wave interacting fermions
Scattering in quasi-2D traps
Similar scattering confinement-induced resonaces as in quasi-1D traps
Example: two fermions, p-wave interactions
Scattering amplitude
CIR
Scattering amplitude in forward
direction for different values of
energy
2D scattering amplitude:
Zderzenia atomów w pułapkach kwazi-1D i kwazi-2D
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:
Rozwiązanie problemu rozpraszania:
fala m=1 dla fermionów oddziałujących w fali p
Zderzenia atomów w fali p w układzie kwazi-2D
Rezonans indukowany ściśnięciem gdy
położenie rezonansu:
ZI, and T. Calarco, PRL (2006)
Dla niskich energii (
):
Amplituda rozpraszania do przodu dla różnych energii kinetycznych
CIR
Rezonans nie widoczny powyżej
energii