Hyperfine-Changing Collisions of Cold Molecules J. Aldegunde, Piotr Żuchowski and Jeremy M. Hutson University of Durham EuroQUAM meeting Durham 18th April

Hyperfine-Changing Collisions of Cold Molecules

J. Aldegunde, Piotr Żuchowski and Jeremy M. HutsonUniversity of Durham

EuroQUAM meetingDurham18th April 2009

Contents

1. Hyperfine molecular levels (QUDIPMOL).

2. Hyperfine changing collisions (CoPoMol).

Fine

and

hyp

erfin

e st

ruct

ure

Hyperfine molecular levels

Atomic physics: •Gross spectra: the spectra predicted by considering non-relativistic 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.

This classification can be extended into the molecular realm.

Molecular fine and hyperfine levels

• Stability.

• Bose-Einstein condensate formation.

Gross structure >> Fine structure >> Hyperfine structure

Atom

ic h

yper

fine

stru

ctur

e

Hyperfine splitting ≈ GHz ≈ 10-1 K

S → Electronic spinL → Orbital angular momentumI → 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

1 Σ d

iato

mic

mol

ecul

es

1Σ molecules

7Li133Cs ( M. Weidemüller (Freiburg))

133Cs2 (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.


1 Σ d

iato

mic

mol

ecul

es

H = Hr +Hhf +HS +HZ

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1

Vi : Q i +c4I 1 ¢I 2+other terms

HS = ¡ ¹ ¢E

HZ = ¡2X

i=1

gi ¹ N I i ¢B ¡ gr¹ NN ¢B

N ! Rotational angular momentum

I 1; I 2 ! Nuclear spins


1 Σ d

iato

mic

mol

ecul

es

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1


HS = ¡ ¹ ¢E

HZ = ¡2X

i=1





H = Hr +Hhf +HS +HZ

1 Σ d

iato

mic

mol

ecul

es

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1


HS = ¡ ¹ ¢E

HZ = ¡2X

i=1





H = Hr +Hhf +HS +HZ

1 Σ d

iato

mic

mol

ecul

es

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1


HS = ¡ ¹ ¢E

HZ = ¡2X

i=1





H = Hr +Hhf +HS +HZ

1 Σ d

iato

mic

mol

ecul

es

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1


HS = ¡ ¹ ¢E

HZ = ¡2X

i=1





H = Hr +Hhf +HS +HZ

1 Σ d

iato

mic

mol

ecul

es

Hr = BÀN 2 ¡ DÀN 2 ¢N 2

Hhf =2X

i=1


HS = ¡ ¹ ¢E

HZ = ¡2X

i=1





H = Hr +Hhf +HS +HZ

1 Σ(N

=0) d

iato

mic

mol

ecul

es.

Zero

fiel

d sp

litting

s.

Zero field splittings dominated by the scalar spin-spin interaction (c4I1·I2).

c4(133Cs2) ≈ 13 kHzc4(40K87Rb) ≈ -2 kHz

Hyperfine splitting ≈ tens to hundreds of kHz ≈ 1 to 10 μK


1 Σ (N

≠0) d

iato

mic

mol

ecul

es.

Zero

fiel

d sp

litting

s.

The ratio |c4/(eQq)| ratio determines the zero field splitting partner:• Large |c4/(eQq)| values → the splitting is determined by the scalar spin-spin Interaction and coincides with that for N=0.• Small |c4/(eQq)| values → the splitting is determined by the electric quadrupole interaction.

eQq(85Rb2) ≈ 2 MHz

85Rb2 (N=1)

Hyperfine splitting ≈ hundreds to thousands of kHZ ≈ 10 to 100 μK


1 Σ d

iato

mic

mol

ecul

es.

Zeem

an s

plitti

ng.

(2I+1) components (N=0).Each level splits into (2F+1) components (N≠0).

The slope of the energy levels and the corresponding splittings are determined by the nuclear g-factors.


Hyperfine molecular levels1 Σ

dia

tom

ic m

olec

ules

.Ze

eman

spl

itting

.

• 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.

Hyperfine molecular levels1 Σ

dia

tom

ic m

olec

ules

.Ze

eman

spl

itting

.

• 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.

1 Σ d

iato

mic

mol

ecul

es.

Star

k sp

litting

.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.

1 Σ d

iato

mic

mol

ecul

es.

Star

k sp

litting

.Hyperfine molecular levels

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 collisionsRb

+ O

H(2 Π

3/2)

colli

sion

s

Rb + OH(2Π3/2)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 trappedin their absolute ground states.

Hyperfine changing collisionsCs

+ C

s co

llisi

ons

Cs(2S)+Cs(2S)Interact through a singlet and

a triplet potential

H =¹h2

2¹

"

¡ R¡ 1 d2

dR2R +L2

R2

#

+h1+h2+ V c(R) + Vd(R)

central dipolar

h1; h2 ! Monomer Hamiltonians

V c(R) = V0(R) P (0) + V1(R) P (1) (isotropic)

Vd(R) ! Small anisotropic spin-dependent interactionssinglet triplet

Cs +

Cs

colli

sion

sHyperfine changing collisions

H =¹h2

2¹

"

¡ R¡ 1 d2

dR2R +L2

R2

#

+h1+h2+ V c(R) + Vd(R)

central dipolar

Inelasticity due to Vd(R)

M S1 ! M S1 § 1M S2 ! M S2 ¨ 1V0(R) 6= V1(R) ! V c(R) drives spin-exchange collisions:

V c(R) diagonal in S;M S ;M S1 and M S2 . Its matrixrepresentation is proportional to the unit matrix.

V0(R) = V1(R) !

Slow inelastic collisions.

Rb +

CO

(1 Σ)

colli

sion

s

H =¹h2

2¹

"

¡ R¡ 1 d2

dR2R +L2

R2

#

+hRb +hCO +V(R;µ)

I (12C) = I (16O) = 0! no hyper¯nestructure

hCO = BÀN2 ¡ gr¹ NB ¢N

hRb = AI Rb ¢SRb +gs¹ BB ¢SRb ¡ gN ¹ RbB ¢I Rb

Rb and 12C16O interact through a doublet potential

No spin-relaxation collisions will takeplace.

H =¹h2

2¹

"

¡ R¡ 1 d2

dR2R +L2

R2

#

+hCO +V(R;µ) + hRb

jLMLNMN ijSMS I M I i

Hyperfine changing collisions

1. The dipolar interaction between the Rb electronic magnetic momentand the rotational magnetic moment of the CO molecule is given by

U = ¡¹ 04¼

gs¹ Bgr ¹ NR3

(S ¢N ¡ 3(S ¢R )(N ¢R ))

U \ connects" the jLM LNMN i , jSM S I M I i spaces and drives transitionswhereM S, MN and M L change by up to one. Matrix elements of Ubetween L = 0 or N = 0 states are zero.

2. The atomic hyper¯ne coupling constant (A) is R dependent. This isequivalent to include and extra-term in the Hamiltonian

H =¹h2

2¹

·¡ R ¡ 1 d

dR2R +

L 2

R2

¸+V (R;µ)+(A(R;µ) ¡ A(1 ))I Rb ¢SRb+hCO+hRb

that causes transitions

M S = 1=2$ M S = ¡ 1=2 M I $ M I + 1

orM S = ¡ 1=2$ M S = 1=2 M I $ M I ¡ 1

whenever the electric ¯eld is di®erent from zero.

Rb +

CO

(1 Σ)

colli

sion

sHyperfine changing collisions

Rb +

CO

(1 Σ)

colli

sion

s

Collisions of alkali atomswith singlet moleculeswill not cause fast atomic inelasticity


Both mechanisms will drive slow Rb inelastic collisions.

1. U / gsgr wheregs À gr.

2. The term (A(R;µ) ¡ A(1 ))I Rb ¢SRb is short-range.

(A(R;µ) ¡ A(1 )) ¡¡ ¡ ¡ ¡ ¡ ¡! 0R ! 1

Rb +

ND

3 co

llisi

ons

TheND3 molecule in low-¯eld-seeking states correlatingwith the jJ ;K i=j1;1ui statecan beStark-decelerated.

Cross sections for thej1;1ui ! j1;1ui (| )j1;1ui ! j1;1li (| )processes in theabsenceof ¯eld

ND3 molecules in low-¯eld-seeking states can probably not becooledto sub-mK temperatures by collisions with ultracold Rb atoms


Rb +

ND

3 co

llisi

ons

What about sympathetic cooling of ND3 high-¯eld-seeking states?

Rb and ND3 interactthrough a doublet potential

jSMSI M I i (Rb monomer)

jLML i jJ K IN ID i

The same analysis applied to Rb+CO collisions suggests that sympatheticcolling of ND3 (or NH3) molecules in high-¯eld-seeking statesby magnetically trapped Rb atoms is likely to be feasible.

1. Dipole interactions between magnetic moments ofthemonomers.

2. R dependenceof theRb hyper¯necoupling constant.


gs À gH > gD > gN > grRb +

ND

3 co

llisi

ons


² The interaction involving theammonia nuclear spinspredominates. Nevertheless both interactions aremuchsmaller than theelectron-electron dipole interaction.

² ND3 may bea better candidate to sympathetic coolingthan NH3.

Dipoleinteractions

² Interaction between theelectronic magneticmoment of Rb and the rotational magneticmoment of ND3.U / gsgr

² Interaction between theelectronic magneticmoment of Rb and thenuclear magneticmoments of ND3.U / gsgN ;H ;D

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 atomicspin 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 ishighly 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.


+ O

H(2 Π

3/2)

colli

sion

s

Thereare ¯vepotential energy surfaces corresponding to theRb(2S) and OH(2¦ 3=2) interaction.

Themonomers Hilbert spaces are coupled by the V operator.Atomic inelastic collisions will be fast.

M. Lara et al, Phys. Rev. A 75, 012704 (2007)


+ O

H(2 Π

3/2)

colli

sion

s

² Prospects for sympathetic cooling from the low-¯eld-seeking thresholds correlating with C1 are bleak.

² Atomic inelastic 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 doubletor 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.

Documents

Hyperfine-Changing Collisions of Cold Molecules J. Aldegunde, Piotr Żuchowski and Jeremy M. Hutson University of Durham EuroQUAM meeting Durham 18th April