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Hyperfine-Changing Collisions of
Cold Molecules
J. Aldegunde, Piotr Żuchowski and Jeremy M. Hutson
University of Durham
EuroQUAM meeting
Durham
18th April 2009
Contents
1. Hyperfine molecular levels (QUDIPMOL).
2. Hyperfine changing collisions (CoPoMol).
Fine and hyperfine structure
Hyperfine molecular levels
Atomic physics:
•Gross spectra: the spectra predicted by considering nonrelativistic electrons and neglecting the effect of the spin.
• Fine structure: energy shifts and spectral lines splittings due
to relativistic corrections (including the interaction of the
electronic spin with the orbital angular momentum).
• Hyperfine structure: energy shifts and splittings due to the
interaction of the nuclear spin with the rest of the system.
Gross structure >> Fine structure >> Hyperfine structure
This classification can be extended into the molecular realm.
• Stability.
Molecular fine and
hyperfine levels
• Bose-Einstein condensate
formation.
Atomic hyperfine structure
Hyperfine molecular levels
S → Electronic spin
L → Orbital angular momentum
I → Nuclear spin
Alkali atoms → L=0, S=1/2
F=S+I
Ĥhf= A IRb ∙ SRb
Ĥz= gs μB B∙SRb - gRb μN B∙IRb
Hyperfine splitting ≈ GHz ≈ 10-1 K
Hyperfine molecular levels
1Σ
diatomic molecules
7Li133Cs
1Σ
molecules
133Cs
2
( M. Weidemüller (Freiburg))
(Hanns-Christoph Nägerl (Innsbruck))
40K87Rb
(Jun Ye, D. Jin (JILA))
S=0 (no fine structure)
Two sources of angular momentum:
• N → Rotational angular momentum (L in atom-atom collisions).
• I1, I2 → Nuclear spins of nucleus 1 and 2.
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
1Σ
diatomic molecules
N !
I 1; I 2 !
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
1Σ
diatomic molecules
N !
I 1; I 2 !
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
1Σ
diatomic molecules
N !
I 1; I 2 !
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
1Σ
diatomic molecules
N !
I 1; I 2 !
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
1Σ
diatomic molecules
N !
I 1; I 2 !
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
Rotational angular momentum
Nuclear spins
H^ = H^ r + H^ hf + H^ S + H^ Z
H^ r
=
H^ hf
=
B À N 2 ¡ D À N 2 ¢N 2
X2
Vi : Q i + c4 I 1 ¢I 2 + ot her t erms
i= 1
1Σ
diatomic molecules
N !
I 1; I 2 !
H^ S
=
H^ Z
=
¡ ¹ ¢E
X2
¡
gi ¹ N I i ¢B ¡ gr ¹ N N ¢B
i= 1
Hyperfine molecular levels
1Σ(N=0)
diatomic molecules.
Zero field splittings.
Zero field splittings dominated by the scalar spin-spin interaction (c4I1·I2).
c4(40K87Rb) ≈ -2 kHz
c4(133Cs2) ≈ 13 kHz
Hyperfine splitting ≈ tens to hundreds of kHz ≈ 1 to 10 μK
The ratio |c4/(eQq)| ratio determines the zero field splitting partner:
• Large |c4/(eQq)| values → the splitting is determined by the scalar spinspin Interaction and coincides with that for N=0.
• Small |c4/(eQq)| values → the splitting is determined by the electric
quadrupole interaction.
85Rb
2 (N=1)
Hyperfine splitting ≈ hundreds to
thousands of kHZ ≈ 10 to 100 μK
1Σ
(N≠0) diatomic molecules.
Zero field splittings.
Hyperfine molecular levels
eQq(85Rb2) ≈ 2 MHz
Hyperfine molecular levels
(2I+1) components (N=0).
Each level splits into
1Σ
diatomic molecules.
Zeeman splitting.
(2F+1) components (N≠0).
The slope of the energy levels and the corresponding splittings are
determined by the nuclear g-factors.
1Σ
diatomic molecules.
Zeeman splitting.
Hyperfine molecular levels
• Energy levels with the same value of MI display avoided crossings (the
red lines correspond to MI =-3)
• I remains a good quantum number for values of the magnetic field
below those for which the avoided crossings appear.
• For large values of the magnetic field the individual projections of the
nuclear spins become good quantum numbers.
1Σ
diatomic molecules.
Zeeman splitting.
Hyperfine molecular levels
• Energy levels with the same value of MI display avoided crossings (the
red lines correspond to MI =-3)
• I remains a good quantum number for values of the magnetic field
below those for which the avoided crossings appear.
• For large values of the magnetic field the individual projections of the
nuclear spins become good quantum numbers.
1Σ
diatomic molecules.
Stark splitting.
Hyperfine molecular levels
• Mixing between rotational levels is very important and increases with
the electric field.
• The number of rotational levels required for convergence becomes
larger with field.
• For the levels correlating with N=0, the Stark effect is quadratic at low
fields and becomes linear at high fields.
Hyperfine molecular levels
1Σ
diatomic molecules.
Stark splitting.
Energy levels correlating with N=0 referred to their field-dependent
average value:
• Each level splits into I+1 components labelled by |MI|.
• At large fields the splitting approach a limiting value
and the individual projections of the nuclear spins become
well defined.
Hyperfine changing collisions
Rb + OH(2Π3/2)
Rb + OH(2Π3/2) collisions
M.Lara et al studied these collisions (Phys. Rev. A 75, 012704 (2007)).
• Rb, OH or both of them undergo fast collisions into high-field-seeking
states.
• Sympathetic cooling is not going to work unless both species are trapped
in their absolute ground states.
Cs + Cs collisions
Hyperfine changing collisions
Cs( 2 S)+ Cs( 2 S)
Int eract t hrough a singlet and
a t riplet pot ent ial
2
¹
h
^
H =
2¹
"
¡ R
¡ 1
2
^2
#
d
L
^1 + h
^2 + V
^ c (R) + V
^ d (R)
R
+
+
h
dR 2
R2
cent ral
dipolar
^ 1; h
^ 2 ! Monomer Hamiltonians
h
^ c (R) = V0 (R) P^ (0) + V1(R) P^ (1)
V
singlet
(isotropic)
t riplet
^ d (R) ! Small anisotropic spin-dependent interactions
V
Hyperfine changing collisions
2
Cs + Cs collisions
¹
h
^
H =
2¹
"
¡ R
¡ 1
2
^2
#
d
L
^1 + h
^2 + V
^ c (R) + V
^ d (R)
R
+
+
h
dR 2
R2
cent ral
dipolar
^ c (R) drives spin-exchange collisions: M S1 ! M S1 § 1
V0 (R) 6
= V1 (R) ! V
M S2 ! M S2 ¨ 1
V0 (R) = V1 (R) !
^ c (R) diagonal in S; M S ; M S1 and M S2 . It s mat rix
V
represent at ion is proport ional t o t he unit mat rix.
^ d (R)
Inelasticity due to V
Slow inelastic collisions.
Hyperfine changing collisions
Rb + CO(1Σ) collisions
Rb and 12C16O interact through a doublet potential
I ( 12 C) = I ( 16 O) = 0 ! no hyper¯ne structure
"
2
¹
h
^
H =
2¹
#
¡ R
¡ 1
d2
L^ 2
^ Rb + h
^ CO + V
^ (R; µ)
R
+
+
h
2
2
dR
R
^ CO = B À N 2 ¡ gr ¹ N B ¢N
h
^ Rb = AI Rb ¢SRb + gs¹ B B ¢SRb ¡ gN ¹ Rb B ¢I Rb
h
No spin-relaxation collisions will take place.
2
¹
h
^
H =
2¹
"
¡ R
¡ 1
2
^2
#
d
L
^ CO + V
^ Rb
^ (R; µ) + h
R
+
+
h
dR 2
R2
jLM L N M N i
jSM S I M I i
Rb + CO(1Σ) collisions
Hyperfine changing collisions
1. T he dipolar int eract ion bet ween t he Rb elect ronic magnet ic moment
and t he rot at ional magnet ic moment of t he CO molecule is given by
^ = ¡ ¹ 0 gs ¹ B gr ¹ N (S ¢N ¡ 3(S ¢R^ )(N ¢R^ ))
U
4¼
R3
^ \ connect s" t he jL M L N M N i , jSM S I M I i spaces and drives t ransit ions
U
^
where M S , M N and M L change by up t o one. M at rix element s of U
bet ween L = 0 or N = 0 st at es are zero.
2. T he at omic hyper¯ne coupling const ant (A ) is R dependent . T his is
equivalent t o include and ext ra-t erm in t he Hamilt onian
2
h
¹
H^ =
2¹
·
¸
L^ 2
¡ 1 d
^CO + h
^Rb
^ (R; µ) + (A (R; µ) ¡ A (1 ))I R b ¢S R b + h
¡ R
R+ 2 +V
2
dR
R
t hat causes t ransit ions
M S = 1=2 $ M S = ¡ 1=2
MI $ MI + 1
M S = ¡ 1=2 $ M S = 1=2
MI $ MI ¡ 1
or
whenever t he elect ric ¯eld is di®erent from zero.
Hyperfine changing collisions
Rb + CO(1Σ) collisions
Bot h mechanisms will drive slow Rb inelast ic collisions.
^ / gsgr where gs À gr .
1. U
2. T he t erm (A(R; µ) ¡ A(1 ))I Rb ¢SR b is short -range.
(A(R; µ) ¡ A(1 )) ¡¡R
¡ ¡!¡ ¡ 1
¡! 0
Collisions of alkali at oms wit h singlet molecules
will not cause fast at omic inelast icity
Rb + ND3 collisions
Hyperfine changing collisions
The ND3 molecule in low-¯eld-seeking st at es correlat ing
wit h t he jJ; K i = j1; 1ui st at e can be St ark-decelerat ed.
Cross sect ions for t he
j1; 1ui ! j1; 1ui (| )
j1; 1ui ! j1; 1li (| )
processes in t he
absence of ¯eld
ND3 molecules in low-¯eld-seeking st at es can probably not be cooled
t o sub-mK t emperat ures by collisions wit h ult racold Rb at oms
Hyperfine changing collisions
Rb + ND3 collisions
What about sympathetic cooling of ND3 high-¯eld-seeking states?
T he same analysis applied t o Rb+ CO collisions suggest s t hat sympat het ic
colling of ND 3 (or NH 3 ) molecules in high-¯eld-seeking st at es
by magnet ically t rapped Rb at oms is likely t o be feasible.
jSM SI M I i (Rb monomer)
Rb and ND3 int eract
t hrough a doublet pot ential
jLM L i jJK I N I D i
1. Dipole int eract ions between magnet ic moment s of
t he monomers.
2. R dependence of t he Rb hyper¯ne coupling const ant .
Rb + ND3 collisions
Hyperfine changing collisions
² Int eract ion between t he elect ronic magnet ic
moment of Rb and t he rot at ional magnet ic
moment of ND 3 .
^ / gsgr
U
Dipole
int eract ions ² Int eract ion between t he elect ronic magnet ic
moment of Rb and t he nuclear magnet ic
moment s of ND 3 .
^ / gsgN ; H ; D
U
gs À gH > gD > gN > gr
² T he int eract ion involving t he ammonia nuclear spins
predominat es. Nevert heless bot h int eract ions are much
smaller t han t he elect ron-elect ron dipole int eract ion.
² ND 3 may be a bet t er candidat e t o sympat het ic cooling
t han NH3 .
Conclusions
• The rotational levels of 1Σ alkali metal dimers split into many hyperfine
components.
• For nonrotating states, the zero-field splitting is due to the scalar
spin-spin interaction and amounts to a few μK.
• For N≠1 dimers, the zero-field splitting is dominated by the electric
quadrupole interaction and amounts to a few tens of μK.
• External fields cause additional splittings and can produce avoided
crossings.
• For molecules in closed shell single states colliding with alkali atoms, the atomic
spin degrees of freedom are almost independent of the molecular degrees of
freedom and the collisions will not change the atomic state even if the potential is
highly anisotropic.
• Prospects for sympathetic cooling of ND3/NH3 molecules with cold Rb atoms:
1. Poor for ND3/NH3 low-field-seeking states.
2. Good for ND3/NH3 high-field-seeking states. ND3 better than NH3.
• Quantitative calculations are necessary.
Hyperfine changing collisions
Rb + OH(2Π3/2) collisions
There are ¯ve pot ent ial energy surfaces corresponding t o t he
Rb( 2 S) and OH( 2 ¦ 3=2 ) int eract ion.
^ operat or.
The monomers Hilbert spaces are coupled by t he V
At omic inelast ic collisions will be fast .
M. Lara et al, Phys. Rev. A 75, 012704 (2007)
Rb + OH(2Π3/2) collisions
Hyperfine changing collisions
² Prospect s for sympat het ic cooling from t he low-¯eldseeking t hresholds correlat ing wit h C1 are bleak.
² At omic inelast ic collisions will be fast .
Conclusions
•
Molecular energy levels split into many fine and hyperfine components.
• 1Σ alkali dimers only display hyperfine splittings.
• For nonrotating states, the zero-field splitting is due to the scalar
spin-spin interaction and amounts to a few μK.
• For N≠1 dimers, the zero-field splitting is dominated by the
electric quadrupole interaction and amounts to a few tens of μK.
• Except for short range terms, the system Hamiltonian for collisions between
2s atoms and singlet molecules can be factorised. The collisions will not cause
fast atomic inelasticity.
• This factorization will not be possible when the 2s atoms collides with doublet
or triplet molecules. In this case, the potential operator will drive fast atomic
Inelastic collisions.
• Prospects for sympathetic cooling of ND3/NH3 molecules with cold Rb atoms:
1. Poor for ND3/NH3 low-field-seeking states.
2. Good for ND3/NH3 high-field-seeking states. ND3 better than NH3.
• Quantitative calculations are necessary.