Efimov Effect IN ULTRACOLD ATOMIC GAS...ENHANCING THE OBSERVABILITY OF THE Efimov Effect IN ULTRACOLD ATOMIC GAS MIXTURES Jose· P. D’Incao and Brett D. Esry Department of Physics,

ENHANCING THE OBSERVABILITY OF THE

Efimov Effect IN ULTRACOLD ATOMIC GAS

MIXTURES

Jose P. D’Incao and Brett D. Esry

Department of Physics, Kansas State University

National Science Foundation

WHAT IS THIS EFIMOV EFFECT?

EFIMOV EFFECT

Vitaly Efimov, “Energy levels arising from resonant two-body forces ina three-body system”, Phys. Lett. 33, 563 (1970)

r

"quasi-bound"

"weakly bound"

V( )r

V( )r

Resonant Interaction

EFIMOV EFFECT


r

a<0

a>0

Eba

12

~

"quasi-bound"

"weakly bound"

V( )r

V( )r


EFIMOV EFFECT


r

a<0

a>0

"quasi-bound"

"weakly bound"

V( )r

V( )r

| |a


Eba

12

~

EFIMOV EFFECT


r

a<0

a>0

"quasi-bound"

"weakly bound"

V( )r

V( )r

| |a


Eba

12

~

r0 | |a

W R( ) ~

R

R

W R( )

EFIMOV EFFECT


r

a<0

a>0

"quasi-bound"

"weakly bound"

V( )r

V( )r

| |a


Eba

12

~

r0 | |a

W R( ) ~

R

R

W R( )

EFIMOV EFFECT


r

a<0

a>0

"quasi-bound"

"weakly bound"

V( )r

V( )r

| |a


Eba

12

~

r0

Efimov States !

| |a

W R( ) ~

R

R

W R( )

EFIMOV EFFECT


r

a<0

a>0

"quasi-bound"

"weakly bound"

V( )r

V( )r

| |a


Eba

12

~

r0

Efimov States !

| |a

W R( ) ~

R

R

W R( )

Finite Number of2-Body bound states(short-range interaction)

Large Number of3-Body bound states

EFIMOV EFFECT

Search for Efimov States: Nuclear Physics, Atomic Physics

Limitations: needs to find a system with large a

(at least 3 Efimov states)→ He3: only 1 Efimov state: open question !

→ extremely weakly-bound states !

Ultracold Quantum Gases

Experimentally accessible !Clear signature of the Efimov Effect !

(3-body collisions)

ULTRACOLD QUANTUM GASES

... after evaporativecooling ...

B

K

Feshbach Resonances



B

K

Feshbach Resonances

Scattering Length



B

K

Feshbach Resonances

Scattering Length

Tunability of a !



B

K

Feshbach Resonances

Efimov

Tunability of a !

EffectScattering Length



B

K

Feshbach Resonances

Efimov

Tunability of a !

EffectScattering Length

2-Body Collisions→ Suppressed !3-Body Collisions→ Lifetime, Stability ... Efimov Effect !


B

K

Three-Body Recombination: K3 ∝ (k/µ)σ

++ +

a

+a

12

~( (

Vibrational Relaxation: Vrel ∝ (k/µ)σ

+ +1

2~( (+

a

Rate Equations:nX(t) = −[K3(a)]n

3X − [Vrel(a)]nXnX2

nX2(t) = −[Vrel(a)]nXnX2

nX , nX2: experimental observables !

SIGNATURES OF EFIMOV EFFECT?!

BUILDING INTUITIVE PICTURE

r0 | |a

R

W R( )

Efimov States !

r0(

(

lnN =

Phase shift jumps of Cross section: Efimov features !

Increasing a ...... we are creating new

New Efimov State:

SIGNATURES OF THE EFIMOV EFFECT

Vibrational Relaxation + +1

2~( (+

a

+

~



2~( (+

a

~

+



2~( (+

a

+

~

+



2~( (+

a

+

Resonant

Transmission

~

+



2~( (+

a

+

Resonant

Transmission

Efimov State

~

+


Vibrational Relaxation

Vrel = Aηsinh(2η)

sin2[

s0 ln(a

r0)+Φ

]

+sinh2(η)a

Aη , Φ, η: details !

+ +1

2~( (+

a

+

Resonant

Transmission

Efimov State

~

+



Vrel = Aηsinh(2η)

sin2[

s0 ln(a

r0)+Φ

]

+sinh2(η)a

Aη , Φ, η: details !

+ +1

2~( (+

a

+

Resonant

Transmission

Efimov State

~

+


Three-Body Recombination ++ +

a

+a

12

~( (

++

~



a

+a

12

~( (

++

+

~

~



a

+a

12

~( (

++

+

~

~



a

+a

12

~( (

++

+

~

~



a

+a

12

~( (

Interference

++

+

~

~



a

+a

12

~( (

~

+

Interference

++

+

~

Small



a

+a

12

~( (

~

+

Interference

++

+

~

Small

K3 = Bηsin2[

s0 ln(ar0) + Φ

]

a4

Bη , Φ: details !



a

+a

12

~( (

~

+

Interference

++

+

~

Small

K3 = Bηsin2[

s0 ln(ar0) + Φ

]

a4

Bη , Φ: details !

OBSERVATION OF THE EFIMOV EFFECT

Rate Equations:nX(t) = −[K3(a)]n

3X − [Vrel(a)]nXnX2

nX2(t) = −[Vrel(a)]nXnX2

-

| |

Three features needto be observed !

LIMITATIONS

At finite temperatures only a finte number of featurescan be expected to be observed.

amin ≈ r0[eπ/s0 ]N ⇒ Tmax . 1/ma2min

(spacing: eπ/s0 ≈ 22.7) Large a / Low T !

10-30

10-25

10-20

10-15

10 100 1000 10000 100000

K3

(cm

6 /s)

−a (a.u.)

1 mK

100 µK

10 µK

1 µK

100 nK

10 nK

0.1 nK − 1 nK

−aI−aII−aIII−aIV−aV−aVI−aVII

a4

Three-BodyShape Resonances

AnalyticalFormula

B+B+B Collisions

10 100 1000 10000 100000

a (a.u.)

1 mK

100 µK

10 µK

1 µK

100 nK

10 nK0.1 nK − 1 nK

aI aII aIII aIV aV aVI aVII

a4 scaling

MultichannelInterference Minima

AnalyticalFormula

CESIUM EXPERIMENT - INNSBRUCK

RUDI GRIMM’S GROUP

Analytical Formula

400 nK250 nK

10 nK

400 nK



Minimum [Esry PRL (1999)]et. al,

400 nK250 nK

10 nK

400 nK



Minimum [Esry PRL (1999)]et. al,

Used to optimize loss !

250 nK400 nK

10 nK

400 nK



Peak [Esry PRL (1999)]et. al,

400 nK250 nK

10 nK

400 nK



Different slopes [Esry PRL (1999)]et. al,

400 nK250 nK

10 nK

400 nK



Finite Energy Effects [D'Incao, Suno, Esry PRL (2004)],

400 nK250 nK

10 nK

400 nK

0+

2+

SINGLE SPECIE ATOMIC GASES

• Large spacing (eπ/s0 ≈ 22.7) between Efimov features

• amin ≈ r0[eπ/s0 ]N → Tmax . 1/ma2min: large a, low T

• 133Cs: amin ≈ 4.× 106 a.u.; Tmax ≈ 8.× 10

−5 nK (✗)

TWO SPECIES ATOMIC GASES

• Spacing (eπ/s0) can be made smaller (✓)

• Experimentally accessible amin and Tmax (✓)

• Competition different 3-body processes: important (?)

• Favorable conditions: Boson-Fermion mixtures (✓)


EXPERIMENTS: 87Rb-40K (JILA)23Na-6Li (MIT) Interspecies Feshbach resonances !

Two types of collisions are important: XXY and XY Y

Competition !

Recombination (no molecules !)

nX ≈ −[KX+X+Y3 (a)]nYn

2X − [K

X+Y+Y3 (a)]n2

YnX

nY ≈ −[KX+X+Y3 (a)]n2

XnY − [KX+Y+Y3 (a)]nXn

2Y,

Relaxation

nX ≈ −[V XY+Xrel

(a)]nXYnX

nY ≈ −[V XY+Yrel

(a)]nXYnY

nXY ≈ −[V XY+Xrel

(a)]nXnXY − [VXY+Yrel

(a)]nYnXY


Efimov Physics → a dependence in K3, Vrel.

[Gross scaling, valid for E . 1/2µa2]

Vrel K3 (D3)

Jπ E a > 0 a < 0 E a > 0 a < 0

BBX 0+* const ∗[Ps0 ]a const const(k4) ∗[Ms0 ]a4 ∗[Ps0 ]|a|

4

δ =mXmB

1− k2 a3−2p0 const k2(k6) a6 |a|6−2p0

2+δ<δc* k4 ∗[Ps0 ]a

5 const k4(k8) ∗[Ms0 ]a8 ∗[Ps0 ]|a|

8

2+δ>δck4 a5−2p0 const k4(k8) a8 |a|8−2p0

FFX 0+ const a1−2p0 const k4(k8) a8 |a|8−2p0

δ =mXmF

1−δ<δc* k2 ∗[Ps0 ]a

3 const k2(k6) ∗[Ms0 ]a6 ∗[Ps0 ]|a|

6

1−δ>δck2 a3−2p0 const k2(k6) a6 |a|6−2p0

2+ k4 a5−2p0 const k4(k8) a8 |a|8−2p0

D’Incao and Esry, Submitted to PRL.(*) Efimov Effect !

s0 depends on δ!


-6

-4

-2

0

2

4

6

δ = mX/mB

(a) BBX

Jπ= 0+

Jπ= 1- Jπ= 2+

-s0

p0 p0

-s0

0.01 1 100

δc

-6

-4

-2

0

2

4

6

δ = mX/mF

(b) FFX

0.01 1 100

Jπ= 0+

Jπ= 1-

Jπ= 2+p0

p0

p0

-s0

δc

* Here !


-8

-4

0

4

8

δ = mX/mB

BBX Systems

-s0

eπ/s0

0.01 1 100

mB=mX

We want mixtures with heavy Bosons (mB >> mX) !

EFIMOV EFFECT IN BOSON-BOSON MIXTURES

mB À mb BB b collisions: s0 large→ eπ/s0 small (✓)B b b collisions: s0 small→ eπ/s0 large (✗)



Three-Body Recombination

B+B+b

B+b+b

?

| |

B+B+b

B+b+b

?

| |

| |

Condition: nB À nb

nB = −KB+B+b3 nbnB

2

nb = −KB+B+b3 n2

Bnb

enough signal ? (✗)




B+B+b

B+b+b

?

| |

B+B+b

B+b+b

?

| |

| |

Condition: nB À nb

nB = −KB+B+b3 nbnB

2

nb = −KB+B+b3 n2

Bnb

enough signal ? (✗)


Bb+b

Bb+B

?

Condition: nb = 0

nB = −VBb+Brel

nBnBb

nBb = −VBb+Brel

nBbnB

Bb+Bb collisions can be important ! (✗)

EFIMOV EFFECT IN BOSON-FERMION MIXTURES

mB À mF BBF collisions: s0 large→ eπ/s0 small (✓)BFF collisions: no Efimov Effect (✗)




B+B+F

B+F+F

6k

| |

| |

| |k

B+B+F

B+F+F

Condition: T . Tmax (✓)nB = −K

B+B+F3 nFnB

2

nF = −KB+B+F3 n2

BnF




B+B+F

B+F+F

6k

| |

| |

| |k

B+B+F

B+F+F

Condition: T . Tmax (✓)nB = −K

B+B+F3 nFnB

2

nF = −KB+B+F3 n2

BnF


BF+B

BF+F

-3.33

Condition: T . Tmax (✓)nB = −V

BF+Brel

nBnBF

nBF = −VBF+Brel

nBFnB

BF +BF collisions suppressed (p-wave)

BOSON-FERMION MIXTURES

KB+B+F3 and V BF+B

rel

B − F eπ/s0 |amin|(a.u.) Tmax(nK)133Cs−6Li 4.877 1.6× 104 60.0 ✓

87Rb−6Li 6.856 5.6× 104 5.00 ✓

23Na−6Li 36.28 3.3× 108 ¿ 0.1 ✗ (MIT)7Li−6Li > 100 À 108 ¿ 0.1 ✗

133Cs−40K 47.02 9.2× 107 ¿ 0.1 ✗

87Rb−40K > 100 À 108 ¿ 0.1 ✗ (JILA)23Na−40K > 100 À 108 ¿ 0.1 ✗

7Li−40K > 100 À 108 ¿ 0.1 ✗

BOSON-FERMION MIXTURES

7.5

5

2.5

0 1000 10000

K3/

a4 (10

-39 c

m2 /s

)

a (a.u.)

a4

eπ/s0

s-wave CsLi*

133Cs + 133Cs + 6Li

p-wave CsLi

SUMMARY

• Ultracold Quantum Gases: clear signature of EFIMOV EFFECT

• Boson-Fermion mixtures (mB À mF ): favorable system

• Extremely long-lived BF molecules: EFIMOV PHYSICS

Documents

Efimov Effect IN ULTRACOLD ATOMIC GAS...ENHANCING THE OBSERVABILITY OF THE Efimov Effect IN ULTRACOLD ATOMIC GAS MIXTURES Jose· P. D’Incao and Brett D. Esry Department of Physics,