Triatomic states in ultracold gasesTriatomic states in ultracold gases
Marcelo Takeshi Yamashita
Universidade Estadual Paulista - Brazil
Lauro Tomio – IFT / Unesp Tobias Frederico – ITA Francis Bringas - ITA Antonio Delfino - UFF
Collaborators
Work partially supported by
Guidelines
Summary
The Efimov statesBound statesVirtual statesResonances
Triatomic continuum resonances
Three-body recombination for virtual and bound two-body states in ultracold traps
The Efimov effect - Thomas-Efimov equivalence
Three-body bound state equation with zero-range interaction with momenta cutoff
x
xyxy
xxd
y
y
22
3
3
232
2 1
43
)(momenta
yq
xp
energies
32
3
22
2
E
E
ε2 0
)(33
N (N = 0, 1, 2, ...) Efimov statesEfimov states
1) E2 tends to zero with Λ fixed – Efimov effect
2) Λ tends to infinity with E2 fixed – Thomas collapse
Adhikari, Frederico, and Goldman PRL 74, 487 (1995).
Skorniakov and Ter-Martirosian equation (1956)
The Efimov states – bound, virtual and resonances
Three-body bound state equation with zero-range interaction with subtraction
xyzxydzdxx
y
yfL
L
223
1
10
2
232
1
43
/2)(
)(1
222)3(
xfxyzxy
Three-body resonances
Three-body energy is complex
x
y
ixe
iye
Contour deformation method
Three-body virtual states
The Efimov states – bound and virtual states
Lines – Bound states
crosses – ground
squares – first excited
diamonds – second excited
Symbols – Virtual states
circles - refers to the first excited state
triangles – refers to the second excited state
Appearance of the virtual state (dashed line)
The virtual state turns into an excited state (solid line)
23 3
4
23
ε2 bound
MTY, Frederico, Delfino, and Tomio PRA 66, 052702 (2002)
The Efimov states - resonances
ε2 virtual
Resonances
Bringas, MTY, and Frederico PRA 69, 040702(R) (2004)
The Efimov states – trajectory of Efimov states
Complete trajectory of Efimov states
E3 boundE2 virtual
E3 resonanceE2 virtual
E3 boundE2 bound
E3 virtualE2 bound
The Efimov states – triatomic continuum resonances
from http://www.uibk.ac.at/exphys/ultracold/
“Evidence of Efimov quantum states in an ultracold gas of cesium atoms” !
T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl & R. Grimm, Nature 440, 315 (2006)
23
21 0297.0
mB
Ba
0.00 0.02 0.04 0.06 0.08 0.10 0.12
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
(Re(
E3(N
) ) / B
3(N -
1) )1/
2
(B2 / B
3
(N - 1))1/2
Excited Efimov stateturns into a resonance
From the experimentT = 0 a = -898 a0
Real part
Imaginary part x 0.1
Triatomic continuum resonances in an ultracold gas of cesium atoms
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.20
200
400
600
800
1000
E3(n
K)
a (1000a0)
-0.8 -0.7 -0.6 -0.50
200
400
600
800
0
400
800
1200
1600
(b)
ER (
nK)
ar- (1000a0)
ER = 1800(ar-+0.898)
(a)
EI = - 7100(ar-+0.898)2| EI |
(nK
)
From calculations
Analytic approximations
The Efimov states – triatomic continuum resonances
The Efimov states – triatomic continuum resonances
Adding the effects of triatomic continuum resonances in the recombination rate L3 for T = 0
where
The resonance energy can be approximated by
We can easily find the solution of ar- for Er
After performing the thermal average of the recombination rate <L3> th we have the recombination length
For T = 0E. Braaten, and H.-W. Hammer, Phys. Rep. 428, 259 (2006)
-2.0 -1.6 -1.2 -0.8 -0.4 0.00
5
10
15
20
25
30
Rec
ombi
natio
n le
ngth
(10
00a 0)
Scattering length (1000a0)
Recombination length in a cesium trapped gas as a function of the scattering length and temperature. Solid curves from up to bottom 10, 100, 200, 300, 400 and 500 nK. Symbols are the experimental results for 10 nK (full circles), 200 nK (full triangles) and 250 nK (open diamonds) from T. Kraemer et al., Nature 440, 315 (2006).
0 100 200 300 400 500 600-0.90
-0.89
-0.88
-0.87
-0.86
-0.85
-0.84
-0.83
-0.82
-0.81
Posi
tion
of r
eson
ance
(10
00a 0)
Temperature (nK)
Position of the maximum of the recombination length as a function of the temperature. Experimental data from B. Engeser et al., in preparation.
The Efimov states – triatomic continuum resonances
arxiv:cond-mat/0608542
Weakly bound molecules
Recombination for positive scattering lengths (two-body bound states)
m
aL
4
3
3
2
E
E
1 triatomic bound state2 triatomic bound states3 triatomic bound states
[1]
[2]
[3]
[1] E. A. Burt et al. Phys. Rev. Lett. 79, 337 (1997).[2] D. M. Stamper-Kurn et al. Phys. Rev. Lett. 80, 2027 (1998).[3] N. R. Claussen, E. A. Donley, S. T. Thompson e C. E. Wieman. Phys. Rev. Lett. 87, 160407 (2001); J. L. Roberts, N. R. Claussen, S. L. Cornish e C. E. Wieman. ibid. 85, 728 (2000).
Dimensionless recombination parameter α as a function of the ratio between the binding energies of the diatomic and triatomic molecules.
MTY, Frederico, Delfino, and Tomio PRA 68, 033406 (2003)
Weakly bound molecules
[1] E. A. Burt et al. Phys. Rev. Lett. 79, 337 (1997).[2] D. M. Stamper-Kurn et al. Phys. Rev. Lett. 80, 2027 (1998).[3] N. R. Claussen, E. A. Donley, S. T. Thompson e C. E. Wieman. Phys. Rev. Lett. 87, 160407 (2001); J. L. Roberts, N. R. Claussen, S. L. Cornish e C. E. Wieman. ibid. 85, 728 (2000).[4] J. Söding et al. Appl. Phys. B69, 257 (1999).
AZ|F,mF> a (nm) ρa 3α exp E 2 (mK) S 3 (mK) S 3 ' (mK)
23Na|1,-1> 2.75 6x10-5 42 ± 12 [2] 2.85 4.9 0.21
87Rb|1,-1> 5.8 1x10-5 52 ± 22* [1] 0.17 0.39 0.005
87Rb|1,-1> 5.8 1x10-4 41 ± 17** [1] 0.17 0.30 0.013
87Rb|2,2> 5.8 4x10-5 130 ± 36 [4] 0.17 - -85Rb|2,-2> 211.6 0.5 7.84 ± 3.4 [3] 1.3x10-4 1.14x10-4 3.8x10-5
* Non-condensate atoms ** Condensed atoms
Prediction of trimer binding energies with respect to the threshold, S3=E3-E2 and S’3=E’3-E2, considering the central values of the experimental recombination parameter exp. It is also shown the respective two-body scattering length and the diluteness parameter a3.
Summary
Complete trajectory of Efimov states for 3 identical bosons
Prediction of trimer energies in atomic trapsScattering length
andRecombination coefficient
Inclusion of the triatomic continuum resonance effect in the recombination length
Recombination length at finite temperatures
Good description of the position of resonance as a function of the temperature
Thank you !