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 !