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Observation of an Efimov spectrum in an atomic system Matteo Zaccanti LENS, University of Florence
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Observation of an Efimov spectrum in an atomic system
Matteo ZaccantiLENS, University of Florence
Universality in 2- 3-(4)-body systems
Low energy and |a|>>r0 : universal regime
V(r) 0
2r
p
Point-like particles
01/aE=0
a >0
If |a|r0
Scattering doesn’t depend on the microscopic V(r)
Universality in 2- 3-(4)-body systems
?
Log(1/a)1/a*
21/a
*
11/a
_
2 ...1/a_
1
Log
(|Ene
rgy|
)
Log(1/|a|) ...
)/( 01
*1
*
snExpa
a
a
a nn
3.21*
n
n
a
a
See B&H review 2006
r0 |a|
2
20
2
4/1
R
s
|a|∞Efimov effect
00624.1
)/2(
0
20
2
1
01
s
MrE
sExpE
E
n
n
Each Efimov state accompanied by two 4B states
Universality in 2- 3-(4)-body systems
von Stecher et al, arXiv:0810.3876
nn
nn
aa
aa
90.0
43.0
,2,4
,1,4
No additional 4-body parameter
Tunable a over wide range o Low collision energy
To summarize…
Ultracold atomic gases + Feshbach resonances
Ideal test bed for Efimov physics!!!Kraemer et al., Nature 440, 315 (2006)Knoop et al., Nat. Phys. 5, 227 (2009)Ottenstein et al., PRL 101, 203202 (2008)Ferlaino et al. PRL 102, 140401 (2009)Barontini et al., arXiv:0901.4584
To explore Efimov scenario:
To take in mind…
Efimov scenarior0 =0
Real worldr0 ~lVdW
Expected deviations for the deepest states of the ladder
Coincidence only as |a|>>r0
Additional 3-body parameter is needed!!!
(Unknown location of the first Efimov feature)
4.5
a+
2
22.7
a+
1
a)
b)
a-1
a-2
E (ar
b. u
nits
)
a*1 a-
1 /21.3 a*
2
22.7
22.7
Log
()
Log
(A
D )
Log(|a|)
Efimov scenario & 3-body observables
Universal relations between a-n a+
n and a*n
))]/ln(([sin /0
2/
4/
aasfa ))]/ln(([sin *
022 aasgaAD
Efimov physics in 39K
340 360 380 400 420 440-2000
-1500
-1000
-500
0
500
1000
1500
2000
K S
catter
ing
leng
th (a
0)
Magnetic field (G)
no narrow resonances
connection between a >0 and a <0 via universal region
Small background scattering length: abg =-29 a0 D’Errico et al. NJP 9, 223 (2007)
Broad Feshbach resonances: good control on a (B)Roati et al, PRL 99, 010403 (2007)
Fattori et al, PRL 101, 190405 (2008)
5.402)(
1.5210.29)(
GBBa
Efimov physics in 39K
3-body recombination vs a
3/bE
3/2 bE
(usually)3
)()()( 33
3
l
l
n
tnKtnntn
Explored: [-300; -8000] a0
[+20;+25000] a0
Efimov physics in 39K
4.5
a+
2
E1
E2 (22.7)-2 E
1 22.7
a+
1
a)
b)
a-
1a-
2
E (ar
b. u
nits
)
a*1 a-
1 /21.3
E=0
a*2
22.7
22.7
Log
()
Log(|a|)
Presence of two trimer states very close to Efimov scenario!!!
104 103 102
10-29
10-27
10-25
10-23
10-21
4.5
a+
2
E1
E2 (22.7)-2 E
1 22.7
a+
1
c)
a)
b)
a-
1a-
2
E (ar
b. u
nits
)
a*1 a-
1 /21.3
E=0
a*2
22.7
22.7
++
Log
()
K3(
cm6 s-1
)
Log(|a|)
+
-103 -104
10-23
10-21
10-19
Scattering length a (a0)
Efimov physics in 39K
a > 0 Two recombination minima
)11(043.0
)7(224 01
aa
)17(030.0
)900(5650 02
aa
104 102
10-28
10-26
10-24
10-22
10-20
K3
(cm
6 /s)
Scattering length (a0)
Efimov physics in 39K
a < 0One Efimov resonance…
)2(14.0
)90(1500 01
aa
-103 -104
10-23
10-21
10-19
Scattering length a(a0 )
K3 (c
m6 /s
)
… and associated 4-body feature(s)01,4 )50(660 aa
)4(44.0/ 11,4 aa
Efimov physics in 39K
104 103 102
10-29
10-27
10-25
10-23
10-21
Ene
rgy
(arb
. uni
ts)
Log
( A
D)
K3 (c
m6 /s
)
Log
()
a > 0 Two recombination maxima…
… close to the AD resonances (unexpected)
What’s going on?
Efimov physics in 39K
enhanced AD scattering cross section
(usually)3
)()()( 33
3
l
l
n
tnKtnntn
Where are the dimers? Three body processes!
K3 enhancement due to nl enhancement
Usually, the three-body process causes nl =3
Efimov physics in 39K
…but if AD cross section is large it can be nl >>3!!!
Efimov physics in 39K
Similar in BEC of 87Rb above critical opacity
Efimov physics in 39K
Idea confirmed from a simple model for AD collisions…
60 50 40 30 200.0
0.5
1.0
1.5
2.0
3
6
9
12
15
Scattering length (a0)
K
3(cm
6 /s)
AD/30
AD/2
n l
AD0
*1 )5(4.30 aa
Efimov physics in 39K
Similar mechanism at the other feature
1600 1400 1200 1000 800 600 400
0.1
1
10
K3(
10-2
4 cm
6 /s)
Scattering length (a0)
0*2 )40(930 aa
Ideal vs real world: a comparison
Ratio Exp Theo (Exp-Theo)/Theo (%)
a>0 a+1/a
*1 7.4(3) 4.5 +64(7)
a*2/a
*1 30.6(14) 22.7 +35(6)
a+2/a
*2 6(1) 4.5 +33(22)
a+2/a
+1 25(4) 22.7 +10(18)
104 103 102
10-29
10-27
10-25
10-23
10-21
Log
( A
D)
K3 (c
m6 /s
)
Log
()
Simultaneous fit gives: s0=0.956(18) Larger spacing
Ideal vs real world: a comparison
104 103 102
10-29
10-27
10-25
10-23
10-21
Log
()
Log
(A
D )
K3(
cm6 s-1
)
Log(|a|)
-103 -104
10-23
10-21
10-19
Scattering length a (a0)
Ratio Exp. UT (Exp.-UT.)/UT (%)
a<0
a*2/a
-1 -0.62(5) -1.06 -40(5)
a+2/a
-1 -3.8(6) -4.79 -21(12)
Consistent with recent theo. predictionsHammer et al., PRA 75 032715 (2007)Thøgersten et al., PRA 78 020501 (2008)Platter et al., PRA 79 022702 (2009)D’incao et al., J. Phys. B 42 044016 (2009)
Outlook & Conclusions
Universality vs short-range effects
Further investigations: AD scattering, 4-body features, association of Efimov trimers, low dimensions…
Different FRs:other Efimov spectra on the same atomic system.
Thank you for your attention!
Efimov scenario & 3-body observables
3-body recombination
atom-dimer scattering
Efimov scenario & 3-body observables
3-body recombination
particle-dimer scattering
0if)(sinh))ln((sin
)2sinh(4590
)]1(84.16
0if))(sinh))ln(((sin12.67[
)(
20
2
4
4
20
224
3
a
aa
sM
an
e
aa
ase
M
an
aK
l
l
22*2
2*2
2*2
)(sinh))/ln((sin
)(sinh)97.0)/ln((sin9.84
)(sinh))/ln((sin
)2sinh(3.20
0
0
0
aaas
aas
M
a
aas
Note that inelastic AD scattering causes an increase of only nl =3->nl =4
Efimov physics in 39K: experiment & analysis
pe= prob. for D of elastic AD collision before leaving the trap
Pi(k)= prob. for D of inelastic AD collision before the k-th el. coll
Simple considerations allow to estimate them:
pe = 1- exp(-n lAD)
pi(0) = 1- exp(-AD/(v0 AD))
))5
9(
v(1)(
0
2/
0
k
j
j
AD
ADi Expkp
After kM ED<ETrap
kS such that pi(kS)=1
Analytic exp. available in the low-energy limit for AD and AD
Efimov physics in 39K: experiment & analysis
You can include all possible events, extract nl
})()1()()1()1))((1({3100
S
M
k
kkiM
ke
Kc
ki
ke
Kc
kei
kel kpkpkpkppkpkpn
Efimov physics in 39K: experiment & analysis
UT/30…and compare exp. with K3=nl
20 30 40 50 60
0.0
0.4
0.8
1.2
1.6
2.0
2.4
K3(
10-2
9 cm6 /s
)
Scattering length (a0)
UT
Outline
Universality in few-body systems
Efimov’s physics in 39K
Outlook & perspectives
Ideal vs real world: a comparison
Efimov’s scenario & 3-body observables
