C o n tro llin g C old C o llisio n s o f P olar M o lecu les
w ith E x tern a l F ield sbyC hristop her Carl Ticknor
B.S., Bucknell Univeristy, 2001
M.S., U niversity o f Colorado, 2004

A thesis subm itted to the
Faculty o f the Graduate School o f the
U niversity o f Colorado in p a rtia l fu lfillm ent
o f the requirements for the degree ofD octor o f Philosophy
Departm ent o f Physics
2005

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UMI Number: 3188487

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T his thesis entitled:
C ontrolling Cold Collisions o f Polar Molecules w ith E xternal Fields
w ritte n by Christopher C arl Ticknor
has been approved for the Departm ent o f Physics

John L. Bohn

rWoisuiM u rray Holland

Date

U

3 .Q O S

The final copy o f this thesis has been examined by the signatories, and we fin d th a t
b o th the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

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iii
T icknor, C hristopher C arl (Ph.D ., Physics)
C o n trolling Cold Collisions of Polar Molecules w ith E xternal Fields
Thesis directed by Prof. John L. Bohn

In this thesis we explore how external fields can be used to control collisionso f ultracold polar molecules. F irs t we review the S tark and Zeeman effects for polar
molecules and two body m ulti-channel scattering theory. A general treatm ent o f the
Stark effect and d ipolar interactions is also presented. We consider cold collisions o f OH
molecules in the 2 n 3/ 2 ground state under the influence o f a magnetic field. We findth a t modest fields o f several thousand Gauss can act to suppress inelastic collisions of
weak-field-seeking states by two orders o f magnitude. We a ttrib u te this suppression to
two factors: (i) an indirect coupling o f the entrance and the e xit channel, in contrast
to the effect o f an applied electric field and (ii) the relative sh ift o f the entrance ande xit scattering thresholds. In view o f these results, magnetic tra p p in g o f O H may proveexperim entally feasible.
We also present first steps toward understanding the ultracold scattering prop­
erties o f polar molecules in strong electric field-seeking states. We have found th a t theelastic cross section displays a quasi-regular set of potential resonances as a function o f
the electric field, which p o tentia lly offers intim ate details about the interm olecular in ­teraction. We illu stra te these resonances using a “toy” model composed o f pure dipoles
and a more physically realistic system. To analyze these resonances, we use a simple
W K B approxim ation to the eigenphase, which proves both reasonably accurate andm eaningful.

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A ck n o w led g em en ts

F irs t, I w ill acknowledge and thank all those th a t helped me get through graduate
school by clim bing and road bikin g w ith me. Then I would like to thank the ones I love
for th e ir support, m y parents and Alison.
I would like to thank the members o f the Bohn group for th e ir various contribu­
tions: D. C. E. B o rto lo tti, A. Avdeenkov, A. Volpi, M . Lara, and E. Meyer.
I wish to acknowledge and thank the experimentalists I worked w ith , who tru ly
made graduate school a learning experience: T . Loftus, C. A. Regal, M . Griener, S.
Inouye, J. G oldw in, M . L. Olsen, and D. S. Jin.
F in a lly I would like to thank John L. Bohn for his patience, incredible willingness
to teach, and enthusiasm for science.

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V

C o n ten ts

C hapterj|

1 In tro d u c tio n

1

i

!

1.1

P roduction o f Cold Polar Molecules..............................................................

3

1.2

A Review U ltra cold C ollision P h ysics ..............................................................

4

1.3

T his W o r k .............................................................................................................

6

|

j

2 E xternal Fields and Molecules
2.1

2.2

|

8

General Form of the S tark E f f e c t ...................................................................

8

2.1.1

M olecular Examples: H und’s Case ( b ) ..............................................

12

2.1.2

H u n d ’s Case (a).....................................................................................

13

2.1.3

A sym m etric R o to rs .................................................................................

17

2.1.4

The Zeeman E f f e c t .................................................................................

21

E lectric versus M agnetic F ie ld s ......................................................................

24

2.2.1

S tark Effect in OH w ith H yperfine S t r u c t u r e .................................

24

2.2.2

Zeeman Effect in O H w ith H yperfine S tr u c tu r e ..............................

25

3 T w o-B ody M ultichannel Scattering Theory

29

3.1

M ultichannel F o rm a lis m ...................................................................................

30

3.2

Field D ressin g.......................................................................................................

35

3.3

Symmetrized W a v e fu n c tio n .............................................................................

39

3.4

D ipole-D ipole I n t e r a c t io n ................................................................................

40

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vi
3.5

The Born A p p r o x im a tio n ................................................................................

42

3.6

A diabatic Representation and the B orn A pproxim ation...........................

44

3.7

N um erical T e c h n iq u e s ......................................................................................

45

4 W eak-Field Seekers in the Presenceo f M agnetic and E lectric Fields
4.1

4.2

47

Inelastic Rates o f O H -O H Collisions in E xternal F ie ld s ..............................

50

4.1.1

E lectric Field C a s e ................................................................................

51

4.1.2

Magnetic Field C a s e .............................................................................

53

4.1.3

A diabatic Analysis of the M agnetic Field C a s e ...............................

56

C o n c lu s io n s ..........................................................................................................

62

5 Strong-Field Seekers and the Influence of an E lectric Field

63

5.1

D ip ola r S c a t t e r in g .............................................................................................

67

5.2

Strong-Field S e e k e rs ..........................................................................................

71

5.3

C ollisional S p e c tro sco p y...................................................................................

78

5.4

Conclusion

81..........................................................................................................

6 The Last and Shortest C hapter82

B ibliograph y83

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vii

T ables

Table
2.1

2.2

Correspondence o f principal axes to the molecular axes for different lim its
o f ac[ 1 0 1 ] ...................................................................................................................

18

Parameters F,G , and H th a t are p a rt o f H ( k ) [101].....................................

19

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viii

F igures

Figure
2.1

The S tarkeffectfor R b C s ................................................................................

14

2.2

The S tarkeffectfor O H ...................................................................................

16

2.3

The Starkeffectfor form aldehyd e...................................................................

20

2.4

The S tarkeffectfor OH w ith hyperfine s t r u c t u r e ......................................

26

2.5

The Zeeman effect for O H w ith hyperfine s tr u c tu r e ....................................

27

4.1

T h e rm a lly averaged rate constants for collisions o f weak-field-seeking
states of O H, as a function o f applied electric f i e l d ....................................

4.2

P a rtia l inelastic rate constants for O H -O H collisions as a function ofelectric f i e l d ..........................................................................................................

4.3

49

52

P a rtia l inelastic rate constants for O H -O H collisions as a function o f
m agnetic f i e l d .......................................................................................................

54

4.4

Inelastic rate constants for the three-channel model s y s t e m ...................

57

4.5

A diabatic potential curves for the O H -O H syste m .......................................

58

4.6

Inelastic rate constants as estimated by the adiabatic distorted wave Born
approxim ation for the three-channel s y s te m ................................................

59

4.7

Illu stra tio n s o f the origin of the zero in the p a rtia l r a t e s ..........................

61

5.1

A diabatic curves of the pure dipole syste m ....................................................

66

5.2

Eigenphase and adiabatic W K B phase for the dipole-scattering model

69

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.

ix
5.3 A diabatic curves for R b C s ..................................................................................

73

5.4 Cross section for RbCs in strong-field seeking states..................................

75

5.5 Cross section for S r O ...........................................................................................

77

5.6

AW P predictions o f potentia l resonances and comparison w ith the fu ll
calculations w ith

different in itia l boundary conditions...................

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79

C h a p ter 1

In tr o d u ctio n

Dipoles and th e ir interactions constitute a dom inant d riv in g force behind thestructure and physical properties o f m atter.

Aspects of dipolar driven structure canbe found in plasmas [1], biomolecular fluids [2, 3], magnetic materials [4, 5], and semi­
conductors [4, 6 , 7]. Such systems display fantastically rich collective behavior o rig i­nating from anisotropic d ipolar interactions. For example, the dipolar interaction can
induce phase transitions, in which the systems become more ordered such as when aferromagnet is cooled below its Curie tem perature, the dipoles begin to align and themacroscopic system gains macroscopic order [8 ]. D ipolar interactions can also have sub­tle, yet significant, effects, such as in high T q superconductors where magnetic order
plays a significant role in th e ir properties [9, 10, 11, 12, 13].
D ip ola r systems become even more intricate w ith the application o f an external
field th a t also interacts w ith the dipoles. T his field can be used to control the dipoles
and, consequently, the physical properties o f the system. Examples can be found insuch phenomena as colossal magneto-resistance [14], mesoscopic ferro-fluids [15, 16], orthe phases o f high T q superconductors [17, 18, 19]. Since the dipole-dipole and fielddipole interactions determine the properties o f many physical systems, it is desirable tounderstand them . Even simple models o f dipoles have rich properties. For example, theIsing model is an array o f fixed dipoles on a la ttice th a t only interact w ith th e ir nearest
neighbors. T his model displays phase transitions much like those found in ferromagnets

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2[8, 20],However, physical systems are complicated and have m any co n trib u tin g inter­actions th a t lead to th e ir physical properties; thus understanding th e ir properties isd iffic u lt. The desire to gain insight in to complex systems has led to the p ursuit o f rel­
atively simple physical systems th a t can be studied and explored more throughly. The
relatively new field o f ultracold 1 atoms is a wonderful example o f such systems [21]. A
notable success o f this field was the production o f Bose-Einstein Condensation (B E C ) ina gas composed o f d ilu te alkali atoms in 1995 [2 2 , 23, 24], Since then, ultracold atom ic
systems have allured researchers to explore the fundamental properties o f m atter [25].
The techniques to control ultracold atoms th a t led to BEC have been used to create
degenerate Fermi gases [26]. Furtherm ore, the a b ility to control atom ic interactions has
led to the study o f the BEC-BCS crossover problem [27, 28, 29]. Such rapid advances
have led to new insights in to the properties o f m atter because these systems can the
finely tuned, controlled and then explored.
W ith the tremendous success and advancements in ultracold atom ic physics, sci­
entists have begun to tu rn th e ir attention towards polar molecules, w ith hopes o f placing
them in the ultracold environment [30]. We are excited about the possibilities th a t dipo­la r interaction w ill lead to strongly correlated systems, which can be finely controlled.
Just as the study o f ultracold alkali atoms yielded the a b ility to carefully study manybody systems, scientists speculate the investigation o f ultracold polar molecules could
lead to advances in the understanding o f highly correlated, strongly interacting dipolarsystems and even insight in to d ipolar phase transitions.
E xcitin g predictions have begun to emerge for ultracold d ipolar systems; most
predict novel collective properties [31]. These predictions include the creation o f BECs
w ith novel physical properties, exotic collective excitations [32, 33, 34], super-solid phase
1 By convention, “cold” atomic gases have temperature equal to or less than 1 mK, and “ultracold”
gases have temperatures roughly equal to or less than jj,K.

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3transitions in dipolar BECs [35], and novel types of pairing physics for fermions [36,37, 38].

These possibilities are exciting because o f insights they m ight offer in thefundam ental properties o f m atter. Furthermore, they may have exciting applications,such as the use o f ultracold polar molecules for robust quantum com puting schemes [39].
A t the heart o f such predictions are the tw o-body dipolar collisions, which are
not well understood. There is a growing need to understand these collision properties
as the experiments begin to tra p polar molecules. We need to understand the molecularcollisional properties to b u ild accurate predictions about their collective behavior andexploit them. I t is the intention o f this work to explore the theory o f ultracold twobody collisions o f polar molecules and to understand how external fields can be used tocontrol the collisions.
In the next two sections we briefly review the current state o f research in to the
production and understanding o f ultracold molecules. We first present an overview ofexperim ental methods [30], followed by a review o f the state o f theory for cold collisionso f polar molecules.

1.1

P r o d u c tio n o f C old P o la r M o lecu les
One way to produce cold molecules is to sta rt w ith cold atoms and then associatethem. T his can be achieved w ith a magnetic Feshbach resonance, i.e., using an appliedmagnetic field to alter the molecular structure o f colliding pairs such th a t they can be
dynam ically converted in to molecules. The use of these resonances to produce ultracold
molecules has been experim entally demonstrated w ith several different alkali systems[40, 41, 42, 43, 44]. Recently Feshbach resonances between different alkali atoms havebeen observed [45, 46], leading to the possibility o f polar molecules.

The molecules

produced are vib ra tio n a lly excited. To get to absolute ground state molecules requires
another technique such as the techniques developed for photo-association (PA). PA uses
a laser to associate two colliding atoms in to a molecule th a t has an excited electronic

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4state. Then, w ith the application o f another laser, a stim ulated Raman process candrive the excited molecule to the molecular ground state.

PA has successfully beenused to produce RbCs molecules in th e ir absolute vib rational ground state [47, 48, 49].
K R b has also been photo-associated [50, 51], although not yet in its vib ra tio n a l groundstate. These ind ire ct methods are currently confined to producing hetero-nuclear alkali
molecules.
A nother set o f experim ental techniques used to produce cold polar molecules arecalled direct methods, because they sta rt w ith molecules and then cool them directly.
There are many ways to do this, the firs t being buffer gas cooling [52, 53, 54, 55], th a t
uses cold Helium to cool the molecules through collisions. Another successful means ofcooling molecules is by electric-field-pulsed slowing or Stark slowing [56, 57, 58, 59, 60].
O ther means o f slowing molecules are light-held slowing [61], laser cooling [62], counterrotating-nozzle slowing [63], single collision scattering [64], and beam skim m ing [65, 6 6 ].
The advantages o f these techniques is th a t they are general enough to be used on almost
any polar molecule or even non-polax molecules in some cases. These direct methods
have been used to slow CaH, NH, OH, N H 3 , and H 2 CO, among others. The disadvantage
is th a t direct methods have not achieved ultracold temperatures.
Regardless o f the means by which molecules are cooled, it is necessary to under­
stand the m olecular interactions and how external fields can be used to influence them.
W ith such an understanding, the d ipolar interaction could be controlled and therefore
the properties o f the gas could be chosen. We now review the state o f ultracold molecularcollision theory.

1.2

A R e v ie w U ltra co ld C o llisio n P h y sics
A n area o f rapid development in cold collision theory is in the understanding of

the vib ra tio n a l quenching o f molecules. T his research was m otivated by the experimen­
ta l p roduction o f Feshbach-associated molecules in highly excited vib ra tio n a l states;

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I

5such processes are a first step toward understanding and accurately m odeling ultracold
chemical reactions [67, 6 8 , 69, 70, 71, 72, 73, 74, 75, 76]. These studies focused on
im p o rta n t issues such as the life tim e o f v ib ra tio n a lly excited molecules in the presenceo f other molecules or atoms.
I t m ight also be possible to control the chemical properties of ultracold molecules
through various means such as external fields.

A n example would be the hydrogen

abstraction process in O H + H 2 CO —>H2 0 + H C 0 , which by using electric fields could
control in tim ate details o f the chemical reaction [77]. The idea o f sensitively controllinga chemical reaction could pave the way for detailed studies and significant advances in
the understanding and observation of real tim e chemical reactions.
There have been a few theoretical studies on ultracold chemical reactions [78, 79,80]. One p a rtic u la rly interesting study found th a t the chemical reaction H 2 + F —>H F + H w ill s till take place at significant rates even though the reaction has a barriero f 700 K [79]. Such studies show th a t studies o f chemical reactions should be possible at
u ltracold temperatures. Later it was found th a t such large rates were resonance assisted[80], fu rth e r bolstering the claim o f control.
A t the heart o f all these studies is the tw o-body interaction o f molecules th a teither make up the molecular gas or the reactants. Thus for an accurate theory o f such
systems, there m ust be a clear understanding o f the tw o-body collision physics.
Some o f the firs t studies e x p lic itly dealing w ith ultracold collisions o f molecules
were those considering buffer-gas cooling and molecule-helium collisions [81, 82, 83,
84, 85]. B o th were o f great im portance in understanding how the buffer gas system
would affect the molecules to be cooled. These studies also explored the influence o f theexternal magnetic field on the collisional properties o f an ultracold gas as a means tocontrol it. Since the success o f these studies and the experim ental achievement o f cold
trapped molecules, the attention o f theorists has turned to the collisions o f molecules.
There have been several studies o f polar molecule-molecule collisions [8 6 , 87, 8 8 ,Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

689, 90, 91, 92, 93] whose prim a ry focus was the influence o f external fields on thecollisions.

For a review, see Ref.

[94]. One in trig u in g discovery has been the novel

long-range scattering resonances dubbed “ field-linked” states [90]. Field-linked states
are produced when two weak-field seeking states collide and th e ir dipolar interactions
tem po ra rily create a long-range bound state. A nother proposed use of a magnetic field
is to break weakly bound complexes [95]. Such studies have shown possible means o fcontrolling collisions w ith external fields.

1.3

T h is W ork
The focus o f this thesis is to understand dipolar interactions o f polar molecules

in an ultracold environment. These studies are carried out through fu ll quantum me­
chanical calculations exploring the collisions o f two polar molecules and how external
fields influence the collisions. T h is work was m otivated by a desire to understand how
to control the interaction o f polar molecules.

Such an understanding is increasingly

param ount as experiments strive towards trapped ultracold polar molecules. C ontrol­
lin g the collisions o f polar molecules could lead to a means to increase trap-life tim e orprovide prescriptions for probing p a rticu la r aspects o f the d ipolar gas one is interestedin.
T his theory w ill also facilitate many other theoretical studies and advance thebroader understanding and applications o f d ipolar interactions. One avenue o f co n tri­b u tio n leads to the a ddition o f realistic molecular interactions in to m any-body theories.
In analogy w ith ultracold atoms, a clear understanding of the tw o-body interaction w ill
be essential for the m any-body theories to accurately describe and predict the collective
behavior.
T his w ork pursues an understanding o f the long-range scattering properties ofpolar molecules and largely ignores the effects o f short-range molecular interaction.
There are two reasons for this omission: first, the surface th a t would govern such a

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collision may not be well known and second, molecular scattering is largely determinedby the long-range d ip ola r properties o f the interaction. Thus m any aspects o f collisional
control w ill originate from long-range scattering. Then as the experiments achieve colder
temperatures, they w ill be able to system atically probe the short-range interaction w ith
control already achieved by an understanding o f the long-range scattering.
The order in which this thesis is presented is as follows: Chapter (2) reviews
how molecules behave in the presence o f external magnetic and electric fields, Chapter(3) reviews m ultichannel scattering theory and aspects o f field dressing, C hapter (4)discusses how a magnetic field could be used to suppress inelastic losses in collisions of
weak-field-seeking states of polar molecules [92], Chapter (5) discusses how an electricfield can be used to control collisions o f strong-field-seeking states o f polar molecules
and identifies an in trig u in g set of scattering resonances found in this system [93].

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C h a p ter 2

E x tern a l F ield s an d M o lecu les

T h is chapter describes how external fields influence the internal states o f molecules.
To begin, we look at the Stark and Zeeman effects. We offer examples of H und’s case (a)and (b) molecules. The theory o f how to treat asymm etric rig id rotors is also discussed.
Then we look specifically at OH in the hyperfine basis and contrast the effects o f electricand magnetic fields.

2.1

G en eral Form o f th e S tark E ffectSince the electric field is a true vector (as opposed to a pseudovector), it only

couples states o f opposite p a rity [96]. For molecules, these states are usually separated
in energy by an amount, which we w ill denote A . T his means the S tark energies varyquadratically w ith low electric field and linearly only at higher fields once the Stark
energy is greater than the sp littin g .

T his coupling o f states o f opposite p a rity is a

general feature o f the Stark effect, whether the electric field originates from an externalsource or from another molecule as in the dipolar interaction. In the following section
we develop the Stark effect in a general manner to allow its results to be used in thed ipolar interaction o f rig id ro to r molecules.
We assume th a t vib ra tio n a l degrees of freedom are frozen out and the geometryo f the molecule is constant. T his is the rig id -ro to r assumption th a t is ju stifie d at low
tem peratures because there is not enough energy to excite vib ra tio n a l modes th a t arewith permission of the copyright owner. Further reproduction prohibited without permission.

i9usually several hundred K e lvin away in energy [97]. A molecular state is described interms o f the state vector |J M j S l ) , where J is the molecule’s rotational plus electronic
angular momenta, M j is the projection o f J onto the lab axis, H is J ’s projection
onto the molecular axis. We have suppressed the electronic degrees o f freedom th a t areneeded to specify the molecular state because they are spectators in the S tark effect at
low fields and are constants o f m otion for our studies.
To describe the rig id ro to r molecular wave function, we use (a, /?,7 |J M j f l ) =y j™ #
andf j , 7 ), where a, (3,7 are the Euler angles defining the molecular frame
is a W igner D function (or ro ta tio n m a trix).

We use the short hand toto represent the Euler angles. For a complete discussion on the W igner D fu n ctio n ’s
properties see Refs. [97, 98, 99].
W ith this description o f the molecular state, we can explore the basic propertiesof the Stark effect. The Stark H am iltonian has the form

Hs = -$ ■ £ ,(2.1)

where j l is the electric dipole moment o f the molecule and £ is the electric field. We w ill
take the field to be in the z direction o f the fixed lab frame in subsequent chapters, b u t
for the current discussion we pursue a general approach and allow the field to p oint in an
a rb itra ry direction. The Stark interaction can be evaluated by decomposing the electricfield in to its spherical components and ro ta tin g the molecular dipole operator in to thelab frame. To rotate an operator from the molecular frame, we again use a W igner D
function. In our case, the two frames are the lab and molecular frame, which are relatedby the same Euler angles Cj defining the molecular orientation. A rank k operator in
the lab frame,can be expressed in terms o f the molecular frame operators w ith thetransform ation: T * =D q \ { & ) T \ i where A (q) is projection o f the operator onto themolecular (lab) axes [97]. The inverse, a molecular frame operator described in termsof lab frame operators, is= J2q Dq\(& )Tq.

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10
The dipole is n a tu ra lly defined in the molecular frame and most generally has theform j l = Ylk^k&ki b u t this is overkill. The dom inant axis of sym m etry (eo) usually
has the only significant dipole moment because of symmetry. For example, a diatom ic
molecule has only one dipole moment [97]. For simple polyatom ic molecules, there is
generally only one component o f the dipole moment th a t needs to be considered to geta physically realistic description. Thus the dipole moment has the form f l = /mo- The
S tark H am iltonian can be evaluated by ro ta tin g the dipole in to the lab frame. Then in
spherical coordinates, the S tark H am iltonian has the form

ffs = - £ £ , £ > > ( £ ) .
Q( 2 .2 )To get a feel for what the W igner D -function is, we note th a t D ^ ( a , /?, 7 ) is equal to( —l ) 2qCq(fl, a), whereis a reduced spherical harmonic.

A t the heart o f evaluating the Stark effect, we find the field operator D * q coupling
two molecular states, which themselves are described by D functions. Thus to evaluate
the m a trix element, we have an integral o f three W igner D -functions integrated overthe molecular coordinates. W ritte n out, the integral is(JMjSl\D$(u,)\J'M'Jn) = M^

jJ

1 J' ^

/

J

1

. \
J1
(2.3)

\

-M j

q
-i

‘M
- jj /

y~fl

0

O' j

Here (;;;) is a 3-J sym bol th a t describes the addition of two angular momenta. One ofthe properties o f the 3-J symbol is th a t the bo tto m row must sum to zero to yield anonzero result. T his pro pe rty tells us th a t q = M j —M j and f l = f l', which is a p a rtia lstatem ent o f angular m om entum conservation. We have also introduced the notation:b i>hi= v^(2 ji + 1 )(2 j 2 + 1) • ■■(2j/v + !)•W ith this m a trix element, we areable to construct the Stark Effect for any rig id ro to r molecule. For valuable discussions
on the Stark effect, see Refs. [97, 100].

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11
A molecule may have a nuclear spin th a t generates a hyperfine structure. Becauseo f this, it is more appropriate to present the m a trix elements in the hyperfine basis,
where F and M F define the state. Here F is the sum o f J and the nuclear spin I in theusual angular m om entum sense. Thus F can have the values ranging between J + 1 and|J —1\. The energy separation between hyperfine levels is usually much less than either
the ro tatio na l or A-doublet s p littin g between different p a rity states o f the molecules.
We use the W igner-Eckart theorem to compute the Stark m a trix elements in acompact form . The m a trix elements o f the Stark effect are{ a F M F \H s \a 'F 'M 'F ) = - f i £ ( a F M F \ D ^ \ a ’ F ' M'F ),(2.4)

which is w ritte n in terms o f a purely geometrical m a trix element.

T his geometricalm a trix element can be evaluated w ith the application of the W igner-Eckart Theorem( a F M F |Dgo |F'M 'pO i) = [ F ] ( - 1 ) 1 + f' +m^
(

,
F'

1

F

\

M' f

q

—M f

\
(a F || Dq* II * ' F ' ) .

(2.5)

Here ( a F || D q * || a 'F ' ) is the reduced m a trix element, and a represents all rem aining
quantum numbers needed to uniquely determine the quantum state, such as fl.

Weretained q as a place holder th a t takes on the appropriate value M F — M'F when being
evaluated. I f we exploit the fact th a t the electric field is in the z direction, then q = 0,
and we fin d th a t M F = M 'F . We use the angular momentum conventions and statemento f the W igner-Eckart theorem defined by B rin k and Satchler [98] th a t differ from otherconventions like those used by Ref. [97].
A standard means of obtaining reduced m a trix elements is by using the W ignerEckart Theorem on Eq. (2.3):{ a J M \ D \ Z \ a ' J 'M 'j ) = [J ] ( - l ) 1+J' +M'( a J || D$* || ot'J').
y M j

q

~M j jReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(2.6)

12
Then we set this result equal to the RHS o f Eq. (2.3). From this equality, one can thensolve for the reduced m a trix element, which yields
(

J

, \
J'

1

(2.7)

( a J || D p || a 'J ' ) = [ J ' ] ( - l )\-n o o' /We now look in to specific molecular examples and discuss the e xp licit basis sets
used to describe molecules. W hen doing this, we m aintain the use of hyperfine notation,b ut, at first, we offer sim plified examples w ith 7 = 0. Then after we have illustrated the
Stark effect, we closely study OH w ith hyperfine structure.

2 .1 .1

M o le c u la r E x a m p le s :H u n d ’ s C ase (b )F irs t, consider a H u nd’s case (b) molecule w ith L=

0.H und’s case (b) occurs

when the electronic spin is decoupled from the molecular axis, b u t the o rb ita l angu­
lar m om entum is coupled to the molecular axis. Falling into this category are many
molecules such as heteronuclear alkali dimers and SrO w ith 25+1S ground states. Herethe p a rity o f a state is identified by the quantum number J, p a r it y = ( —1)J [97]. The
S tark effect therefore d ire ctly mixes the ground state, J=

0,w ith the firs t rotation-a lly excited state, J = 1. Thus A , the energy s p littin g between the p a rity states, is
2B = B J ( J + 1), where B is the rotational constant th a t is inversely proportion al tothe molecule’s moment o f inertia. The state is described sim ply by \J M j ) .
A ccounting for the possibility o f nuclear spin and electronic spin, we produce gen­eral m a trix elements in the hyperfine basis. Including the nuclear spin does not change
the form o f the Stark effect although the Stark effect can couple different hyperfine
states together i f they have the same M p . In the hyperfine basis, we find the reduced
m a trix element to be( a F || D p || F 'a ' ) = [TV, TV', J, J ', F ] ( - 1 )
v

X

f F

F'

1

J'

.7

7

/

><

/

J

.7'

TV'

TV S

1

,
TV

1

TV

\

(2.8)

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13
where the index a represents N , S and {;;;} is a 6 -J symbol th a t describes the additiono f three angular momenta. We are able to relate reduced m a trix elements in the F basis
to reduced m a trix elements in the J basis through relations given in B rin k and Satchler[98]. The one used for the current discussion is o f the form:
F

F'

1

J'

J

I

{ a { J I ) F || I>o* || a ' ( J ' I ' ) F ' ) oc

{ a J || £>** ||(2.9)

Here operator Dq* does not act on / , and therefore it is left unchanged by the operator.
However, it does act on J, and by decomposing F into J and I w ith some angularmom entum algebra, which results in the 6 -J symbol, we are able to obtain the reduced
m a trix element in the J basis.
As an example o f this system, we pick RbCs, which has aground state. Since

it is a singlet state, J = N . We have plotted the Stark effect for RbCs in Fig. 2.1. In
this plot, we have also assumed th a t / is zero. A common feature of the S tark effect isth a t states w ith the same M j repel each other. For example, this can be seen by looking
at the |J M j ) = |00) and |10). These states are coupled and repel each other. For thisreason the |0 0 ) state, which goes down in energy, is deemed a strong-field seeker, andthe 110) is a weak-field seeker (at small field).

One other feature o f the S tark effectw o rth noting is the degeneracy between the values o f ± M . T his degeneracy explains
w hy J = 1 has only two curves, one for M j = 0 and the other for \ M j \ = 1.

2.1.2

H und’s Case (a)Now consider a H u n d ’s case (a) molecule w ith 0 ^ 0 .

H und’s case (a) occurs

when the electronic spin and o rb ita l angular m omentum are strongly coupled to themolecular axis. A good example o f this case is the OH radical, which has a ground stateo f 2n3/2. In a ddition to being a n state, this molecule has a pair o f nearly degeneratestates w ith A = ± 1 , where A is the projection o f the o rb ita l angular momentum onto
the molecular axis. The energy eigenstates o f such a molecule in zero electric field are

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14

0.20
0.15

* 0.10Nw'
1 0.05
C
111

0.00
-

0.05
0

1000

2000

3000

4000

5000

Electric Field (V/cm)

Figure 2.1: The Stark effect in RbCs, which has a * £ ground state. The Stark effectcouples different ro tatio na l levels b u t preserves \ M j \ . We have used R=0.0245 (K ) andf i = 1.3 (D ) [49],

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15
eigenstates of parity, |J M j S l t ) , where e = + ( —) represents the e ( / ) lower (upper)p a rity state o f the doublet and H = |0 |. The p a rity o f this molecule is e(—I ) '7-1/ 2; fordetails see Refs. [92, 100]. The p a rity basis is w ritte n\jM jU ± ) = ( W Q > + W- n > ) .

(2 .10)In the p a rity basis, there is no dipole moment, because this basis is a linear com binationo f electric dipole “ up” and “down.” T his fact has im p o rta n t im plications for the dipoledipole interaction. In the p a rity basis, the energy s p littin g between the different p a rity
states ( e / / ) is called the A-doublet. The A-doublet arises from a coupling to a nearby
E state. I t is the coupling of the E state to I I state o f the same p a rity th a t breaks thedegeneracy o f the two I I p a rity states [97].
In this discussion, we e x p lic itly take the field to be in the z direction. In the rig id
ro to r basis Q has a definite sign, and the m a trix elements are well known [ 1 0 2 ]:( J M j Q \ H s \ J M j n ) = Zj ^[ y(2-11)In the S tark effect, there is a degeneracy between states w ith the same sign o f Q M j ,meaning ± M j are degenerate in an electric field. We can recast the Stark H am iltonian
in to the J -p a rity basis set from Eq. (2.10). Doing so, we find

( J M M e \H s \J M jn e ') =•(2-12)In this expression, the factor (1 - eeO/2 e xp licitly represents the electric field coupling
between states of opposite parity, since it vanishes for e = e'.
We have p lotted the S tark effect for OH in Fig. 2.2, and we have ignored the next
ro ta tion al state because i t is over 80 (K ) away. We have assumed th a t J = f l = 3/2
and 1 = 0. Again we see the strong- ( / ) and weak- (e) field seeking states. In fact, thelabels e ( /) are reserved for the lower (upper) set o f states in a A-doublet. So in general,
e is usually a weak-field seeker. A weak-field seeker is a molecular state whose internalenergy increases as an external field increases.

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16|Mj |=3/2
0.150 . 10 1= 1 / 2

0.05U|Mj |=1/2

0 .00 -0.05-

-

1=3/2

0.10
0

1000

2000

3000

4000

5000

Electric Field (V/cm)

Figure 2.2: The Stark effect for OH excluding hyperfine structure, w ith its 2 n 3/ 2 groundstate. We have set the A-doublet energy equal to 0.08 (K ) and electric dipole of /x =1.668 (D) [87]. One key feature o f this p lo t is th a t at low field, the energies varyquadratically and then vary linearly at high field once the Stark energy is larger than
the sp littin g . T his tra n sitio n happens at roughly 1000 (V /c m ) for O H [87].

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17
To get a feel for this m a trix element, we need to understand in which direction
the dipole points or the orientation o f the molecular axis (from O to H ). We can first
determine M j , b u t this is not enough because J can be either aligned or anti-aligned
w ith the molecular axis ( + or - f i ) . Thus we also need to know the signed values Q.
W ith the com bination M j C l , we can determine the dipole orientation. T his is why there
is a degeneracy between ± M because for M jC l > 0, we find the dipole is oriented alongz and for M j f l < 0, we find the dipole is oriented against z.
Including the hyperfine structure we find the reduced m a trix element to be

{

J'

1

\

F

F'

1

J'

J

I

J + J * -|—f” 1(2.13)\-noHere the index a represents e, Q, and J.

2.1.3

A sym m etric R otors
To extend this form alism to include asym m etric rotors, there is the additional

step o f including the ro ta tio n a l H am iltonian th a t mixes Q in constructing the moleculareigenstates. For an asym m etric rotor, there are three d istinct moments o f in e rtia andtherefore three d istin ct ro ta tio n a l constants.

The rotational H am iltonian is H rot =AJ% + B 3 \ + CJ%, where the ro ta tio n a l constants are labeled so th a t A > B > C anda, b, and c are the axis labels in the molecular frame. This additional structure mixes
O such th a t it is no longer a good quantum number. T his implies th a t we need todiagonalize the ro ta tio n a l H am iltonian to get the molecular eigenstates in addition to
the Stark H am iltonian i f such a field is present.
To describe an asymm etric rotor, we firs t define Ray’s asym m etry parameter k ,Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

body fixed axis

I

X

y

b
c

z

a

II
c
a
b

III
a
b
c

Table 2.1: Correspondence o f principal axes to the molecular axes for different lim its o f/c[101].

which defines the shape of the asym metric ro to r [ 1 0 1 ]:2B - A - C
A - C

'

(2.14)

T his param eter describes the geometry of the molecule as it ranges between the sym­m etric top lim its. In the prolate lim it (cigar) we have B = C, and thus k = -1. Whereas
in the oblate lim it (pancake), we have A = B , and thus k = 1.
As the molecular geometry changes w ith varying k , s o does the way in which thero ta tio n a l H am iltonian is best described. To describe a given molecule, we must assign
axis labels (a,b ,c) to the molecular frame ( x , y , z ). For different lim its o f k , we define
different cases o f axes assignment. For k = - 1 , we use case I, which assigns the a axis
to the m olecular z axis. W hen k = 0, we use case II , which assigns b to z . F in a lly fork= 1, we use case I I I , which assigns c to z . The complete list o f axes correspondence

are in Table 2.1, which is adapted from Ref. [101].
There is a standard way to w rite down the ro ta tio n a l H am iltonian for an asymmet­
ric ro to r takin g advantage o f molecular symmetries. We now present this methodology.
A more complete discussion is given in Refs. [101, 99]. F irst we rew rite the rotational
H am iltonian as

Hrot =+ ^ ^ H ( k ),(2.15)

where

H ( k ) = J I + k J I ~ J c2.

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(2.16)

19
ParameterFGHI(k- l ) / 2

II

III

0

(k + 1 )/2

1

K

-1

- ( /c + l ) / 2

1(k - 1 ) / 2

Table 2.2: Parameters F,G , and H th a t are part o f H ( k ) [101].

The m a trix elements of H ( k ) are:

{ J M Q \ H { k ) \ J M Q ) = F ( J ( J + 1) - 0 2) + G f l 2(J M S l ± 2 \H ( k ) \J M Q ) = H f ( J , 0 ± l ) 1/ 2

/ ( j , q ± i) =j (j+ 1) - n(n ± i )} { j ( j + 1) - n(n ± i)fi(n ± 2)}.(2.i7)

The second equation is off-diagonal in f2; it is nonzero only if fF = ft ± 2 . The parameters
F , G, and H are case dependent and specified in Table 2.2.
The zero-field eigenvalues and molecular eigenstates are achieved by diagonalizing
Eq. (2.15). The eigenvalues have the formE(J tM ) =+ 1) +(2.18)

Here E( k ) are the eigenvalues o f H ( k ). We have used a standard labeling scheme forthe asym m etric ro to r’s eigenstates, \JTM ) , where r is the phenomenological quantum
number taking the place o f fi.

For a given J and M , r starts at —J for the lowest

energy state and ends w ith J for the highest ro ta tio n a l state, for a to ta l o f 2 J + 1 states
for a given value of J and M .
To get field molecular eigenstates in a field, one must diagonalize the rotational
H am iltonian described above in the rig id ro to r basis (Eq. 2.15) together w ith the Stark
H am iltonian in Eqs. (2.4) and (2.8).
To illu s tra te the ro ta tio n a lly excited spectrum of an asym m etric ro to r in an elec­
tric field, we picked formaldehyde (H 2 C O ) as an example.

T his example is in Fig.

2.3. Its rota tion al constants are A = 13.558, B = 1.867, and C = 1.675 (K ), yieldingk = —0.96, and id e ntifying this molecule as being in the prolate lim it. Its electric dipole

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20

16

-

1 2 ' |2 20 >Electric Field (kV/cm)

Figure 2.3: T his is the Stark effect in formaldehyde, H 2 CO. The states are labeled w ith
the usual asym m etric ro to r notation \ J t M j ).

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21
moment is 2.34 (D ) [101]. For ultracold collisions, the main deviation o f this example
from other sym m etric rig id rotors is th a t the ro ta tio n a lly excited states are ordereddifferently than a sym m etric rotor. Furthermore, some ro ta tio n a lly excited states havedifferent zero-field basis sets. Figure 2.3 shows some J = 2 states are lower in energy
tha n some J = 1 ro ta tio n a lly excited states.

2.1.4

T h e Z eem an Effect
The effect o f the magnetic field can be treated in much the same way as the effecto f an electric field. The Stark and Zeeman effects in molecules have a sim ilar form,
since b o th arise as the scalar product of a dipole moment w ith an external field. T heir
influence on the molecule is quite different, however, since they act on fundam entallydifferent degrees o f freedom. The electric field is concerned p rim a rily w ith where thecharges are in the molecule, whereas the magnetic field is concerned w ith where they
are going. T his is o f param ount im portance, since it implies th a t the electric field isa true vector (odd under the p a rity operation), whereas the magnetic field is a pseu­
dovector (even under p a rity) [96]. T his means th a t the magnetic field respects parity.
Furtherm ore, there are diagonal contributions th a t directly sh ift molecular energy levels.
The vector form o f the Zeeman effect is

H z = ~ P b ■B = no(geS + gL L ) ■B.

Here po is the Bohr magneton, ge is the electron’s g factor (ge ~ 2.002),(2.19)= 1, and

S and L are the spin and o rb ita l angular m om entum vectors o f the electron. We use
atom ic units, so h = 1. We assume th a t the magnetic field is in the laboratory-fixed z
direction. There are other co n trib u tin g terms, b u t they are smaller by roughly a factoro f 10- 3 [97]. The magnetic field’s interaction w ith the molecule is diagonal in parity,and it has the a b ility to shift and possibly separate energy eigenstates of the molecule.
The Zeeman H am iltonian can be treated in a parallel manner to the S tark H am il­

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22
tonian by using the W igner Eckart theorem to decompose it in to molecule and casedependent reduced m a trix elements. Then the m a trix elements of Eq. (2.19) have theform/
H z = - g BB [ F ] ( - l )l+ F '+ M p

F'

1

F
( a F || D l * || a 'F ' ) .

y Mp

0

(2.20)—M f

We have suppressed the electronic degrees o f freedom because they are constants of them otion, b u t now we need to know th e ir form . The different H und’s cases are the way
in which the electronic structure is categorized. We now discuss the Zeeman effect for
H u n d ’s cases (a) and (b).
In H u nd ’s case (b) w ith L — 0, the electronic spin degrees o f freedom are describedmost accurately in the lab frame. T his means th a t the spin angular momenta operator
is |S M s ), w hich is embedded in |J M j ) . T his implies th a t there is no ro ta tio n needed
to evaluate the H am iltonian. Thus the Zeeman H am iltonian becomes nogeSzB z, w ith
m a trix elements propo rtio n al to HqgeM s B .

Thus to understand the magnetic fie ld ’s

effect, we must decompose the quantum number F in to J + 7 and J in to S + N to evaluate
the reduced m a trix element. For the H und’s case (b) molecule, we find the reduced
m a trix element to be( a F || P b D ^ O I b ) || a 'F ' ) = g,0 [S, J, J r, f ,] ( - 1 )i + f + j + j ' + n + s(2.21)In H un d’s case (a), the electronic degrees o f freedom are accurately represented by
the state vector |5 E )|L A ), where E (A) is the projection of the electronic spin (o rb ita langular m om entum ) onto the molecular axis.

Thus to evaluate the m a trix element,

we need to rotate the dipole operators in to the lab frame. The Zeeman H am iltonianm a trix elements are therefore proportion al to fio B (g eT, + <7lA ) .

In the J basis, theReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

23
Zeeman H am iltonian takes the form [103]( jM jn \ H z \JMjn) = ^ B ( A + g e m M J(2 22)

T h is is quite sim ilar to the equivalent expression (2.11) for the Stark effect, except th a t
the electron’s g-factor plays a role. Interestingly, for a 2n state, the prefactor ( A + g e£ ) f i
is always greater than zero. We now recast the Zeeman interaction in to the J -p a rity
basis set (2.10). T his gives us( J M j f l e \ H z \ J M j f l e ') =^for H = A + £ states. Here we have use A to denote |A|, likewise for S.
I t is w orth th a t noting if O = 1/2 for a 2 n 3/ 2 molecule, such as OH, then the
Zeeman m a trix element is( J M j f l e \ H z \ J M j f l e ') =Here the o rb ita l^J ( J + ) )(2-24)and spin contributions to the molecular magnetic moment nearly cancel,

to w ith in the deviation o f g e / 2 from one.
The key feature o f the Zeeman m a trix element (2.23) is th a t it is diagonal in e,
in contrast to the Stark m a trix element. The fu ll m a trix element including hyperfine
structure is( a F ||X ^ ( A + S e £ ), 1 +|| F W ) = [ J , J ' , F ' } ( - i) i+ J + J '+ /+ F - n
F

F'

1

1

J

1

J' ^

J'

J

I

^

-U

0

ft j

(2.25)^ ^For the O H ground state where J = 3 /2 = f2, an im p o rta n t feature o f this interactionis 1 + ee'(—1)J+J>+2n reduces to 5(fj . For a complete discussion on the Zeeman effect inmolecules, see Ref. [97].

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24

2.2

E lectric versu s M a g n etic F ield s
We now dire ctly contrast the effect o f electric and magnetic fields in O H and focus

on two features th a t can be exploited in collisions. F irst, the electric field couples tostates o f opposite p a rity and polarizes the molecules as the electric field is increased.
Second, the magnetic field directly changes the energy levels o f molecular states andrespects parity.
Relevant molecular energy scales for O H in this scattering problem are the Adoublet s p littin g th a t is

A ~ 0.0797 K , and the hyperfine s p littin g is A

~ 0.0038 K.

O H also has an electric dipole moment th a t is (i ~ 1.668 D.

2.2.1

Stark Effect in OH w ith H yp erfine Structure
The distinguishing feature o f the Stark effect is th a t it mixes molecular stateso f opposite p a rity separated by the A doublet sp littin g . A consequence o f this is th a t
the S tark energies vary quadratically w ith electric field at low fields and linearly onlyat higher fields. The field where this tra n sitio n occurs is given roughly by equating
the fie ld ’s effect j l ■£ to the A doublet sp littin g .

In OH, this field is approxim ately£0 ~ A//i ~ 1000(F/cm ) [87].
F inally, using Eq. (2.5) and (2.13), we arrive at the working m a trix elements of
the Stark effect:( F M F e\Hs \ F ' M F e') = - » £(1+ ee'(—1 )J+J'+ 2fi+ i \+ ■^-----------------1

x ( - l y + J ' + F + F ' + i + M p - n ^ F / ) J J/]
1

J’

1

./ ^

(
1

F

'

\

F

F'

1

J'

J

I

i

X

\

m f

t

(2.26)

Figure 2.4 shows the energy levels o f O H in the presence of an electric field. B oth p a rity
states are shown, labeled e and / . A n essential p o in t o f Fig. 2.4 is th a t the e and /states repel as the electric field is increased. T his means th a t all o f the / (e) stateswith permission of the copyright owner. Further reproduction prohibited without permission.

25
increase (decrease) in energy as the field in increased, im plying th a t states of the samep a rity stay close together in energy as the field is increased. T his fact has a crucial
effect on inelastic scattering, as we w ill show.
The highest-energy state in Fig. 2.4 is the stretched state w ith quantum numbers|F M fc ) — |22—). I t is this state whose cold collisions we are most interested in, because(i) it is weak-field seeking and (ii) collisions at low tem perature result almost entirely
from long-range dipole-dipole interactions [87].

2.2.2

Zeeman Effect in OH w ith Hyperfine Structure
The Zeeman effect respects p a rity and this gives rise to a very different result

when an external field is applied.

Using Eq.

(2.20) and (2.25), we find the m a trix

elements are( F M p e | H z | F ' M f c ') =HqB (A + geT,) I+ Ce {

J ------------x ( - ^ J + J ' + f + f ' + i + M f - Q f /; ^ j ,j
/

\

J

K ~^

0

J

(2.27)
'

(
F'

1

F

F

F'

1

J'

J

I

<

Mp

0

-M f ^

Figure 2.5 shows the Zeeman energies in the hyperfine basis, for low [Fig. 2.5(a)] andhigh [Fig. 2.5(b)] fields. For OH in the 2 n 3/ 2 state, the p a rity factor^^

reduces sim ply to 6 (Ci . Because the magnetic field preserves parity, Fig. 2.5 (b) amounts
to two copies o f the same energy level diagram, separated in energy by the lam bda
doublet energy. For small magnetic fields, the molecular <7-factor is g ^ g oc ( F 2 + J 2 —I 2)
and is always positive for OH. T h is is in contrast to the low-field magnetic moment ofalkali atoms th a t is g ^ g h oc ( F 2 — J 2 — I 2) (and where J , o f course, refers to the sumo f o rb ita l and spin angular momenta). In Eq. (2.27) for H = 1/2, the factor A + geT,
goes to A - gef.1.

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0.0900.088/-N*

0.086-

P<D 0.084c
LU

0.0820.0800

200

400

600

800

1000

Electric Field (V/cm)

0.0040 .002 0 .0 0 0 -

^

>
0)
-

0 002
.

-

c

LU

-0.004-0.0060

200

400

600

800

1000

Electric Field (V/cm)

Figure 2.4: S tark effect for the ground state o f O H w ith the hyperfine structure ac­counted for. In zero field, the / and e states are separated by the A-doublet energy.
The gray line indicates the state of interest for our analysis, the |2 2 —) state. A n im ­p o rta n t feature o f this interaction is th a t the opposite p a rity states repel, and thus likep a rity states stay close together in energy.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

27

0 .02 -

0 01
.

-

*

0 .00 -

O)cm

-0 .0 1 --0 . 02 0

50

100

150

200

250

Magnetic Field (gauss)
0.200.150.10s

Y

>
P
<0
c
til

0.050.00-0.05-0.10-0150

250

500

750

1000 1250 1500

Magnetic Field (gauss)

Figure 2.5: Zeeman effect for the ground state o f OH, in low (a) and high (b) fields.
T h is p lo t is the same for b o th the e and / states for zero electric field because the
Zeeman interaction preserves parity. The |22—) state is indicated in gray.

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28
I t is im p o rta n t to dire ctly contrast the effects o f the external fields fo r the ex­amples given here. F irs t, the S tark effect polarized the molecule by m ixing the p a rity
states; therefore all o f the / states increase in energy while all o f the e states decrease as
the electric field is increased, as shown in Fig. 2.4. T his results in the weak-field seeking
molecular states rem aining relatively close together in energy as the field is increased.
T his is im p o rta n t for inelastic collisions, which w ill be discussed in C hapt. 4.
The behavior o f the molecular states in a magnetic field is dram atically different.
F irst, p a rity is a conserved. Second, the Zeeman energy depends on the M quantum
number, this as seen in the m a trix element given in Eq. (2.23). These two facts im p ly
states o f a given p a rity w ill separate in energy as the field is increased; this is seen in
Figs. 2.5 (a) and (b). T his result can be used to suppress inelastic collisions, it w ill also
be discussed in C hapt. 4.

1

I

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C h a p ter 3

T w o -B o d y M u ltich a n n el S ca tterin g T h eo ry

T his chapter outlines m ultichannel scattering theory, field dressing, the dipoledipole interaction, the B orn approxim ation, and adiabatic representation.
The theory o f tw o-body scattering is a m ature and established field; therefore weon ly provide an overview o f scattering theory as needed for our studies o f molecularcollisions. For a complete development of scattering theory see Refs. [104, 105]. To
study collisions o f two molecules, we w ill sta rt w ith the tim e-independent Schrodinger
equation (T IS E ):= £ |tf).

(3.1)

Here H is the to ta l H am iltonian describing the system, E is energy eigenvalue, and |T )is the quantum state o f the two molecules whose interaction we are interested in. To
solve this problem, we use coordinates describing the center o f mass (Rm) and relative
coordinate (R ). We solve Eq. (3.1) in free space, so the this coordinate system leads toan uncoupled set o f equations. For our purposes o f studying molecular collisions, theT IS E governing the relative m otion of the molecules has the form

(Z + Z ( R ) + 2 £ i„ t ) W= E |’F).

(3.2)

The terms in the H am iltonian are the kinetic energy (T ) and molecular interaction (F ).
H_mt describes the internal energy and the interaction w ith any external fields for eachmolecule. We tre a t the effect o f the external fields in p e rturbatio n theory and assume

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30
it is independent o f R. The underlined operators indicate they are matrices, im plying
Eq. (3.2) is actually a set of coupled equations. There is an equation for each uniquepair o f molecular states needed to describe the molecular dynamics. We now discuss theasym ptotic form o f Eq. (3.2) and how to extract physical inform ation from boundary
conditions.

3.1

M u ltich a n n el F orm alism
The need for m ultichannel form alism originates from the internal structure o fmolecules. For example, different molecular states have d istinct behavior in an external
field, as do th e ir pairwise interactions. T his makes it necessary to account for molecular
structure, to achieve an accurate model o f the molecular interaction. To understandhow these internal degrees o f freedom enrich scattering, we w rite the wavefunction interm s o f an incident plane wave and scattered spherical waves. For the m ultichannel
scattering problem , the wavefunction is( R \ * W ) = eik^ \ Xm) + £ik R-e ^ - f i m)(e,<t>)\Xn).

(3.3)

n

Here we have introduced Xm> which represents the internal state o f b o th molecules,
including all internal molecular degrees o f freedom. The scattering am plitude(f>)

depends on the incident (m ) and final state (n) o f the molecules. T his implies th a t ascattering event can change the internal states o f the molecules. I f the molecular statechanges, i.e., m ^ n, then the collision is deemed inelastic. I f the molecular state isunchanged by the collisions, it is called elastic.
To account for the relative angular degrees of freedom we expand the R dependent
wavefunction in terms o f spherical harmonics, this is the p a rtia l wave expansion. The
spherical harmonics account for the o rb ita l m otion of the colliding molecular pair.
We form unique combinations o f spherical harmonics and molecular pairs called
channels to expand the wavefunction in. For example, i f we have molecules w ith two

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internal states, say | f ) and | j ) , then the molecular scattering states |xn) could be
I T1 T2 ), | I 1 T2 ), | T1 I 2 ), and | J,1 J,2 )- To construct channels, we also need to factor in thep a rtia l wave component. T his results in a channel, | / ) , being defined by the quantum
states, \Xnf ) \ l f m i f ), where {9(p\l f m i f ) = Yifm^ (9, (p) is a spherical harmonic. In thisexample, a few possible channels are | TiT 2 ) |0 0 ) , | T1 T2 )| 2 0 ) , or | XiT 2 ) |42). Anothernotation we use when denoting the scattering basis is |12 /m ;), where the 11 2 ) = \x n)or | 1 ) ( 12 )) denotes all o f the quantum numbers needed to define the first (second)
molecule.
We expand the wavefunction in channels, i.e.

(R \*® ) =

Here

(3.4)

is a radial wavefunction for the f th channel, o f the i th solution to Eq. (3.2). In

this expansion, we have included a factor o f R l to elim inate the firs t derivative in theradial coordinate o f the radial T IS E . The channel index, / , denotes all of the quantum
numbers, including the p a rtia l wave number, I f and n iif and quantum numbers for thetwo molecular states, r i f .
Furtherm ore, we pick the molecular states (or linear combinations o f them ) suchth a t they diagonalize H i nt, which can lead to different thresholds. T his implies channels
can have different wave numbers, i.e., k f = ^ 2 m r { E — E j ), where H intx nf = ^ j X n f
and m r is the reduced mass. Here Ej defines the threshold for a given set o f molecularstates. T his choice o f states is called “ field dressed” i f there is an external field, which
w ill be discussed later. We adopt a shorthand, Yijmi^{R )X ns — </>/, th a t describes all of
the channel’s quantum numbers except the radial coordinate.
B y expanding the wavefunction, Eq. (3.4), and by m u ltip ly in g byon the left

and integrating over these degrees of freedom, we can reduce Eq. (3.2) in to a set of1i

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32
coupled equations describing the interaction o f the molecules. These equations are£ { { £ m

~ E + E ’ ) 5’ >+ v '<<*>)

+

=a

<3-5)

The interm olecular potential V fi(R ) is (<j)f \ V (R)\<j)i), which has averaged over all coj

ordinates other than R. Thus it contains all o f the inform ation about the molecularinteraction. Sfi is a Kronecker delta which has the value of 1 if i = / or zero otherwise.
m r is the reduced mass o f the system. We make the association th a t the i th solution
has the i th channel as the incident channel.
In principle, this in fin ite set o f equations must be simultaneously solved. However
this is a few too many equations to actually solve numerically. So to solve the system,
we need to converge any given calculation w ith respect to a number o f p a rtia l waves
and a num ber of molecular states included in a calculation.

For ultracold collisions,few p a rtia l waves are actually needed because at low energy, the centrifugal barrier, the+ 1) term in Eq. 3.5, suppresses the effect o f scattering w ith large I. However, cold
collisions o f d ipolar objects are subtle and require careful attention.
We need to apply boundary conditions to get a physically m eaningful solution
to Eq. (3.5). The firs t boundary condition is to demand th a t all solutions be zero at
R = 0:^ W (0) = 0.

(3.6)

T his boundary demands th a t the wavefunction be fin ite at R = 0. The next boundary
condition is applied at large R, where V ji —» 0.

I t is this condition th a t allows us

to extract the scattering inform ation from the solutions to Eq. (3.5). T h is boundary
condition can be w ritte n in terms o f the S m a trix, which is defined by [106]...

_( e-i(kfR-lfn/2)s;) « E ( f -----------------^_ i(kf R-lfir/2) a\-------------- — j -S fi is the m ultichannel scattering m atrix.

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M33
W ith this boundary condition in Eq.

(3.7) and the solution to Eq.

(3.5), we

can invert this relation and obtain the S fi. Another way to encapsulate the scattering
inform atio n is in the T m a trix, T j t . T his is defined as [104]T fi = i ( S fi -Sf i ).

(3.8)

The S and T matrices have significant meaning in scattering theory [104]. TheS

m a trix is the operator th a t evolves the quantum state from its incident state (IV’o))to its fina l state including scattering, SjV’o) = iVh) + iV’sc)) where \ipsc) is the scatteredfraction o f the fin al state. Thus, the S m a trix has inform ation about b o th scattered andunscattered contributions to the final wavefunction. I f we compare |^o) + IV-’sc) to theform o f the solution given in Eq. (3.3), we find th a t \ipo) is the incident plane wave and\4>sc)

is the spherically sym m etric scattered wave. The T m a trix, or transfer m a trix,

is the operator th a t accounts for ju s t the scattered p o rtio n of the final wavefunction:
T\lpo) = - # s c ) Physical observables, such as the p a rtia l cross section, can be extracted from theT

m a trix , [104]:= ^ \ Tfi\2-(3-9)

T his q u a n tity describes how much incident flu x in state i is scattered in to state / . I t
has units o f scattering flu x per incident flu x per u n it area or area.Specifically a f i istheq u a n tity

th a t describes how likely it is th a t molecules entering in channeli areconverted in to channel / by th e ir interaction. I f the molecules in channel i are identical,
then there is an additional factor of 2 required because o f sym m etry requirements [106].
A nother q u a n tity com m only used to describe collisions is the rate constant K tf = ViOf,
where u, is the incident speed o f the colliding particles.
One can rew rite the boundary condition in Eq. (3.7) in terms o f real-valued spatial
functions. In doing so, we must introduce the solutions to Eq. (3.5) when the V fi = 0.

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34
They are p roportion al to spherical Bessel functions: { R \ip ^ ) = y
and y / : h\j k

/ R j i f (k f R )kf R n if ( k fR ). Here we have picked the prefactor so th a t these functions areenergy norm alized .2

Spherical Bessel functions have the asym ptotic formk R j i ( k R —> oo) ~ sin(fcf? — ln / 2 )k R n i( k R —> oo) ~ cos(kR — ln /2 ) .

(3.10)j i (n i ) is referred to as the regular (irregular) solution.
Using these solutions to Eq. (3.5), we can w rite the large R boundary conditionin terms of the K m a trix. T his is

^ { R

-

oo) oc ' £ kf R { j i f (kf R )Sf i ~ n l f (kf R ) K f i ) .
f(3.11)

I t is from this representation th a t the scattering phaseshift (Si)emerges.The scattering
phase sh ift is found by using the asym ptotic form o f the Besselfunctions in Eq.

(3.10)

w ith trigonom etric identities to w rite the diagonal p o rtio n of the wavefunction as
ip\z^

oc cos(Si) sin(ki — l n / 2 ) + s in (^ ) cos(ki — l n /2 )oc cos(5j){sin(A;j — l n / 2 ) + tan(5*) cos(ki — l n / 2 ) }oc sin(ki — l n / 2 + 5{).

(3.12)

I t is from the second equation here th a t we see the diagonal K u is related to ta n (^ ).
U ltim a te ly a p a rtia l cross section is related to the phase shift an =sin 2 (5j).

There is a possibility of a channel being energetically forbidden at large R. Thisimplies E j —E > 0 , which means kj is purely imaginary, and the asym ptotic solutionsof Eq. (3.5) are growing and decaying exponentials. To get a physical wavefunction, we
demand an exponentially decaying solution in this channel. To do so, we enforce theboundary condition such th a t
i p f ( R ^ o o ) o c e - k} R.

(3.13)

2 I f we denote the wavefunction simply as (/?|<?t ) = y / ^ Tk k R j i ( k R ), where k = y /2 m r t. Then thenormalization is {ge\ge') = <5(e — (.')■

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35
T h is is achieved by taking the appropriate linear com bination of mathem atical solutions
such th a t the wavefunction at large R decays in the f th channel.

3.2

F ield D ressin g
In the previous chapter, we explored the effects of external fields on ind ivid u a lmolecules. Now it is tim e to incorporate the effects o f external fields in to the collisiono f polar molecules. The effect of an external field is felt in two places in Eq. (3.2), firstin H int and second in V. We w ill discuss how the field enters into H int and V_, andhow they are influenced by external fields. F irst, we discuss the notation th a t we use
to explain field dressing.
To illu stra te ket notation and its u tility , we show its equivalent in m a trix notation.
A state vector, | ^ ) , can be represented in an appropriate basis, say a. T his representa­
tio n could represent anything from spatial coordinates to spin states. W ritin g the statein the a representation we find: X )0 la a)(c*a|'-'/), where (a a| ^ ) is the projection o f | ^ )onto the ath component o f the a basis. I f there is another equivalent representation,say (3, we could w rite | ^ ) = Y.b \Pb){Pb\'~')- A n im p o rta n t concept is the relationshipbetween the two representations. Since a and (3 are equivalent, one must be able totransform a state in itia lly given in the a representation into the (3 representation. The
relationship between the two representations is a u n ita ry transform ation.

Changing

representation a to f3 is
l° > = £ l - W & l « a ) K R .

(3.14)

ba

Here we ide ntify Y l b a \ f i b ) { f l b \ a a ) ( a a \ as the u n ita ry transform ation operator, labeled
C/4. A n in d iv id u a l element o f this m a trix is=(ftb \n a), which describes how the b t hcomponent o f (3 basis projects onto the a th component o f the a basis.
In Eq. (3.2), we have H i nt; in Eq. (3.5), we have E fS f i. Because we have useda basis which diagonalized H i nt, we say we are using the field-dressed basis. T h is iswith permission of the copyright owner. Further reproduction prohibited without permission.

i36
necessary num erically because if H i nt were not diagonal, then it could have nonzero
channel couplings at large R th a t prevents the numerical convergence o f a solution. B u t
more im p ortan tly, the basis th a t diagonalizes Hint is the set of physical states of thetwo molecules when they are in fin ite ly separated.
To illu stra te how to do field dressing, we need to know the form o f H i nt. Using
the e xp licit | 12 /m ) basis,

can be w ritte n as: H_i ® I 2+ 11 ® K 2 ® h m t • HereH i and H 2 contain all o f the Ham iltonians describing the molecular degrees o f freedom.
T his means these H am iltonians include the molecules’ Stark and Zeeman interactions,as well as all other necessary H am iltonians such as ro ta tio n and vibration. Thus using
a basis th a t diagonalizes H i nt implies th a t the molecular states composing the channels,|1) and 12), diagonalize th e ir respective Ham iltonians, H i and I / 2 .
We focus on one o f the molecular Ham iltonians, H i , to fu rth e r explain field dress­
ing. As stated above, field dressing requires using the states th a t diagonalize the molec­u la r H am iltonian to compose the scattering basis. For example, if we consider a H u n d ’s
case (b) polar molecule, such as RbCs, then H i could include the Stark H am iltonian
and the ro ta tio n H am iltonian. Then the undressed field basis would then be the rig id
ro to r basis, and w ith a nonzero electric field, the dressed basis would be the eigenstateso f H i . These eigenstates are appropriate linear combinations o f ro ta tio n a l states. In
general, the undressed basis is the zero-field basis.
To illu stra te

the effect o f field dressing on the molecular interaction,

we need toe x p lic itly considerthe dressed (|d)) and undressed (|u)) molecular state bases.

We canw rite the operator H i in the undressed basis:(3-15)Hiuu'

B y diagonalizing the H i operator, we obtain the eigenstates, or the find the field-dressedbasis, |d), i.e.,
H i\d ) = \ d\d).

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(3.16)

37
The eigenvectors form the columns o f a m a trix U_, th a t can be used to transform between
dressed and undressed basis sets.
To give a concrete molecular example, consider the two e and / states o f OH
w ith o u t hyperfine structure, where J = M j = 0 = 3/2. The e and / states w ill be
our undressed basis, where \u\) = \J M p & e ) and \u2 ) = |J M / T l/ ) . We have alreadycomposed the Stark H am iltonian in the undressed basis in Section (2.1.2). The Stark
H am iltonian and the energy s p littin g between the e and / states, A , is:

(

0

q

q

A

\

H, =

(3.17)

where q = f i £ M Q / J ( J + 1), as given in Eq. (2.23). This equation can be diagonalized analytically, yielding the energy eigenvalues: A i = 5 ( A — ^ A 2 + 4q2^j and A2 =^ ^ A + y / A 2 + 4q2^. The eigenstates or field-dressed states are |d i) = c ^ | u i ) + d p \ u 2)
S2)
and \d2) = ci^ |1i i) ~^'c'
2 >1^ 2 )- For this example, these field-dependent coefficients are ana ly tic and are

k { > + -v . / i W(2)

1 and Ci

2 ''y/A 2+4q2 J

However in general, the c ^ s are num erically determined.
A molecular operator w ritte n in the undressed basis can be transform ed in to thedressed basis by using a linear transform ation:A= U ^H^U=Y,\d){d\u){u \H i\u'){u'\d'){d'\dud'u'(3.18)= Y *d \d )(d \.
Between the second and th ird line, we have used Eq. (3.16) and (d\d') = S^d’ ■In the OH example, we can transform the operator H 1 from the undressed basis/ ,(i)
to the dressed basis by constructing U_ =J 2) ^-1Ji) J2)

1 ( u i\ d i)

(u i\d 2) N
. Then(u 2 \di)(u 2 \d2)

H \ in the dressed basis isA= U ^ H iU

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(3.19)

There is a lo t o f algebra to get between the second and th ird lines b u t it does work, andit had better, because we picked the dressed basis such th a t H \ was diagonal.
Now th a t we have diagonalized the molecular Ham iltonians, we can construct thefield-dressed scattering basis, i.e., for a given channel | 1 2 /m (), | 1 ) and |2 ) are the energy
eigenstates o f the field. We can see where the scattering thresholds originate from whenacting on H i nt. The result is( 1 2 l m i |H in t|l ' 2 'l'm \)= { \ 2 l m i \ I i \ ® 1 2 ® Jjm, + l i ® i £ 2 ®

= (Ai + A2)<5n/#22

mjmJ(3.22)The last line uses the notation in Eq.

(3.5) and corresponds / (i) to the unprim ed(prim ed) states.
Field dressing the molecular interaction accounts for the effect o f the field on themolecules, and consequently using the field-dressed molecular interaction accounts forthe fie ld ’s influence on the interaction. To compute the interaction, we first construct
the molecular interaction in the undressed basis (we only need to do this once), andthen for a p a rticu la r value o f the field, we need to construct the transform ation between
the dressed and undressed basis. Then the field-dressed molecular interaction is(3.23)( 1 2 l m i \u\U 2 l m 1)
U\U2Ull U,2
x {u\U 2 l m i \ V {R)\u\iU 2i l ' m 'i)(3.24)

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39
Here we use |it*) to denote the undressed basis o f the i th molecule. T h is expression
shows us th a t the external field can change the way in which the colliding moleculesinteract to rearrange the couplings. The field dressing can also be w ritte n in matrices.
The transform ation m a trix is U_\ <8 >U_\ <8 >h mi ■ A n ind ivid u a l element o f this m a trixis { \ 2 im 1\u\U 2 l ' m 'j), where an element is ( l | u i ) ( 2 |u 2 )<S;;'5 m(m/, which is the product ofcoefficients from the in d ivid u a l molecular field dressing matrices: U_\ and U}2-

3.3

S y m m etrized W a v efu n ctio n
I f the molecules are identical particles, then th e ir wavefunction must be prop­

erly symmetrized. For bosons (fermions) the wavefunction must be sym m etric (anti­
sym m etric) upon particle exchange. There are two operations to ensure this symme­
try : one must b o th swap internal coordinates o f the molecules and spatially invert
the coordinate system. T his implies th a t the angular and molecular state components,
Ylf ml f {0,<t>)Xnf , carry the sym m etry o f the wavefunction. Under spatial inversion, I ,the spherical harmonics have the property

iY lm(0,</>) - Ylm(7T - 0 , 4, + 7T
) = ( - 1 )lYlm(0, 4,).

(3.25)

To complete the sym m etrization, we construct a sym m etric basis under particle ex­change. The symmetrized basis is

Here s denotes the exchange sym m etry of the molecular basis set and 812 im plies adelta function for all molecular quantum numbers.

For bosons (fermions), we needs (—1)( to be positive (negative). Therefore i f we have a pair of indistinguishable bosons(fermions), s must be 1, then we must restrict I to even (odd) values. For a complete
discussion o f sym m etry properties in cold collisions see Ref. [106].

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40

3.4

D ip o le -D ip o le In tera ctio n
The molecular interaction has many contributions. However, for the purposes o f

our studies, we focus solely on the dipole-dipole interaction. The ju stifica tio n for thisis th a t all other long range forces are much smaller, such as the quadrupole-quadrupoleand dispersion interactions [91]. W ith regards to the short-range interaction, there are
generally no accurate potential energy surfaces (PESs). Even w ith o u t PESs, the colli­
sions o f ultracold polar molecules are in trig u in g because at near zero collision energies,
the d ipolar interaction strongly influences the molecular scattering. For example, in the
scattering o f weak-field seekers, it has been found th a t the scattering is nearly purely
long range in nature [87]; this prevents the molecules from reaching short range. How­ever, for strong-field seekers, there is no way to avoid short-range interactions. This
makes it necessary to approach scattering and its result w ith the intention o f eventuallyadding the PES or a param eterization accounting for short-range molecular interaction.
Even w ith o u t short-range interaction, much can be learned from studying the dipolarinteraction. The p rim a ry result w ill be an understanding o f the long-range scattering,which can lead the way for a clear comprehension o f the fu ll molecular interaction. We
w ill now discuss the general form o f the dipole-dipole interaction.
The dipole-dipole interaction is long range and anisotropic. The interaction invector form is [96]3(R ■fa)(fi ■fa) - fa- fa
R3(3.27)

where fa is the dipole moment o f molecule i, R is the interm olecular separation, andR is the u n it vector defining the interm olecular axis. This interaction is conveniently
re w ritte n in terms o f tensorial operators as [106]:q
Here C^_q{ 6 , <f>) is a reduced spherical harm onic th a t acts on the relative angular coordi­
nate of the molecules, w hile (mi ® M2 )q is the second rank tensor formed from two rankwith permission of the copyright owner. Further reproduction prohibited without permission.

41
one operators found in the Stark H am iltonian. For this reason, m a trix elements o f theinteraction are given conveniently in terms o f the m a trix elements in Eq. (2.5).
The physical significance o f this fact originates from the dipole-dipole interaction
describing how a molecule’s dipole field influences another molecule’s dipole when theyscatter. Thus to understand the form o f the interaction, we need to understand how
an electric field influences a polar molecule, which we have done w ith the S tark effect.
Therefore the m a trix elements of the d ipolar interaction are:( l 2 l m i \ H Ftl\l'2 'l'm 'i) =x ( a 2F 2M F2\ D 1*q0\a'2F ' M ,F2)

\

/
1

1

2

M Fl - M 'Fi

M F2 - M ',2mi - m[5(3.29)

J

where

0 0

0

\
(3.30)—m i

m i — m\m\

Here we have used q\ = M f x - M'Fi and q% = M p 2 — M F . Equation (3.29) shows th a t
once the Stark H am iltonian has been constructed for a particular molecule by obtaining
the m a trix elements {a F M F \D ]£ \a !F 'M 'p ), then the H am iltonian describing the dipolarinteraction can be achieved w ith little extra effort. T his general form applies to all rigid
ro to r molecules.
Equations (3.27) and (3.28) are the general form of the dipole-dipole interaction.
They can be electric dipoles or magnetic dipoles. For polar molecules the interaction ofelectric dipoles is much larger than magnetic dipoles. This can be seen by comparing
the size o f the dipoles th a t appear in Eq.

(3.27).

For polar molecules, the dipole

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42
originates from a charge d istrib u tio n , and this can lead to electric dipoles o f order 1(a.u.). For magnetic dipoles, whose origin is the intrinsic spin of the electron, have amagnetic dipole on the order o f a ~ 1/137 (a.u.), the fine structure constant. Since
the interaction goes as yu2, one can see th a t the electric dipole-dipole interaction can bemuch larger.

3.5

T h e B o rn A p p ro x im a tio n
The behavior of the dipolar interaction is d istin ct from previously observed non­polar collisions in u ltracold gases for two reasons: ( 1 ) it is anisotropic and ( 2 ) i t has
d is tin c t threshold behavior. There is a large body o f work on the threshold behavioro f collisions,or scattering behavior, as thethreshold w ork see Ref.[107]energy goes to zero.

For a review o f theand references therein. Generally,threshold lawsdependon b o th the p a rtia l wave and the interm olecular potential, ~ 1 /R S. The scattering
phase shift can be w ritte n as

S a a k 2 l+ 1 + p k s~2.

(3.31)

The coefficients a and /? depend on the details o f the potential. For the dipolar interac­
tio n where s = 3, the second term dominates for I > 0 where 6 ~ k. T his shows us th a t
the po ten tia l has a significant effect on the scattering properties as the collision energy
approaches zero.
To firs t explore the threshold behavior, we use the Born approxim ation [104, 105].
We consider the d ipo lar interaction to be small. Then we can expand the wavefunctionin terms o f spherical Bessel functions th a t comprise the unperturbed incident planewave. We can then use the Born approxim ation to study the threshold behavior. Here
we follow the discussion laid out by Refs. [8 8 , 105].
To first order, the B orn approxim ation gives the T m a trix as( f \ T \ i) cc (f\V \i) .

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(3.32)

43
Here ( f \ T \ i) is a more e xplicit way of w ritin g T /j, which relates the molecular interaction
to changes in the molecular state from |z) to the final state |/ ) . Using the Bessel functions
and the potentia l in Eq. (3.28), we find the Born approxim ation gives us the following
approxim ation to the T m a trix [105]< /|T |i) = 2^ k f J™

(3.33)

Here we have used C % {fi) to represent the coupling between the channels i and / , which
is given by Eq. (3.29).
There are two cases we wish to discuss: (1) the elastic case (k i = k f) and (2) the
inelastic, exotherm ic case (ki < k f) as ki —> 0. The elastic case be solved for l{ + I f > 0
and gives the result( f \ T \ i) = 4 m r cr 23 ( / i ) fc.
\j ,kt / _________________________________
r
r

(3
34)
(3.34)

where T is a gamma function. The im p o rta n t feature o f this result is T / j ~ fc,, whichmeans the cross section for any incident p a rtia l wave does n o t vanish in the zero-energy
lim it.
The inelastic case can also be solved for li + l f > 0.

v4klj 1/2r ( li+^ +3) r ( l i + 3/ 2 ) jli H- I f

1%— I f

—1

.

/
\ 2
j k■

j,

(3.35)

where F is a hypergeometric function. The im portance o f this result is if we have k f »k i, which is the case for u ltracold exotherm ic collisions, then we find the leading orderterm o f hyper geometric function is a constant. Thus Eq. (3.35) gives us T f i ~ fc-!+1^2.
2/ —1/• —1/2
T his leads to cry* ~ ki 1 ~ E?
. T his result shows us th a t for an incident channel
w ith .s-wave character, the cross section diverges as the collision energy goes to zero.

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44

3.6

A d ia b a tic R ep resen ta tio n an d th e B o rn A p p ro x im a tio n
The o rdinary B orn approxim ation is useful for getting a feel for the threshold

behavior, b u t the form given above does not help interpret our results when the dipolarinteraction is strong. To improve upon the Born approxim ation for our applications,
we use the adiabatic representation [91, 105]. T his representation allows us to moreaccurately represent the wavefunctions and channel coupling for the molecular systems.
The adiabatic representation assumes th a t R is a “slow” coordinate. A t every R, wediagonalize the H am iltonian in all rem aining degrees o f freedom. Since it is not rig ­orously true th a t R varies in fin ite ly slowly, the residual nonadiabatic couplings can be
accounted for in the kinetic energy operator. In a more form al treatm ent, we diagonalize<3'36>
where the term s are, respectively, the centrifugal barrier, potential m a trix including
dipole-dipole interaction, and the molecular H am iltonian in the diabatic representation.
Diagonalizing the m a trix in Eq. (3.36), we get W |a (I? )) = Ua (R )\a (R )}, where Ua (R)are the eigenvalues and |o (A )) are the eigenvectors. W ith the eigenvectors we are able toform a linear transform ation X (R ) th a t transform s between the diabatic and adiabatic
representations, i.e., X T W (R ) X — U_(R). The eigenvalues and eigenvectors have radial
dependence b u t for notational sim plicity (R ) w ill be suppressed hereafter.
To distinguish between adiabatic and diabatic representations, we use Greek let­ters ( a , /?,...) to denote the adiabatic channels and Roman letters (i,j,...)

to denotediabatic channels. W hen considering specific inelastic processes in the diabatic basis,
we denote in itia l and fin al channels as i and / and the adiabatic channels as l and tp.
In the lim it R —> oo, the two sets of channels coincide. The transform ation between
the representations is R dependent, im plying th a t the channel couplings shift from thepo te ntia l to the kinetic energy operator. Using the adiabatic representation changes

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45
Eq. (3.5) to( x T— ~ x + u \ my

2 m r dR 2h2 ( *+ 2p 4 e + Q \ + U \ m
2m r ydR?
dR

J== E m -

(3-37)

Here £ = X T ip, w ith ip ^ w ritte n in vector notation.
To get the channel couplings in the adiabatic picture, we need m a trix elementso f the derivative operators, defined as P ap = (o| ^ |/3) and Q af) = (a| jp p |/3). We eval­uate the Pa /3 m a trix, the dom inant off-diagonal channel coupling, using the Hellm annFeynman theorem [105]pJ2ki X l kXVktX i0
a0 _ — u ^ u p — '(}In general, the co n trib u tio n o f Q is much smaller than F , so consequently we ignore Q
for this analysis.
Scattering am plitudes are then easily estimated in the adiabatic-distorted-wave
B orn approxim ation (A D W B A ). Namely, we construct incident and fin a l radial wave
functions (phV th a t propagate according to the adiabatic potentials U ljp. In terms ofthese adiabatic wavefunctions, the scattering T -m a trix is given by an overlap integral
analogous to Eq. (3.33) [105]n h 2 , , 'd /d R P + P d /d R ,W ------------ 7=mry/ 2Here d /d R (d /d R ) is the radial derivative operator acting to the left (right). The crosssection for identical bosons is a ljp = ^ | T tv,|2. From here we are able to num ericallycalculate a rate constant for inelastic loss K ™61 = v,oup, where vL is the asym ptotic
velocity given by ^/2U, (R —> o o )/rn r .

3 .7

N u m erica l T ech n iq u es
M odeling molecular collisions is a large numerical undertaking. A typical OH

scattering problem in a field involves several hundred channels to be simultaneously

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46
solved. To do these calculations, we have used the Johnson log-derivative propagator[108], which num erically propagates the Y m a trix in R. The Y m a trix is Y_ = ip'ip~l ,where ip' =We also used the R -m a trix method w ith fin ite element basis numer­

ical techniques to produce wavefunctions [106].
Once we num erically calculated the Y m a trix, we could apply boundary condi­
tions and solve for the K m a trix. The wavefunction is ip oc / — qK , where / (g) isdiagonal m a trix w ith the regular (irregular) solution as its entries, these functions are
the energy-normalized spherical Bessel functions. Since we have num erically solved for
the Y m a trix, we can solve for the K m a trix in terms o f / , g, and Y_. Thus the Km a trix is K_ = (Yjg -- / ') . W ith the K m atrix, all physical observables canbe determined. For more details on such m atters, see Ref. [106].

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C h a p ter 4

W eak -F ield Seek ers in th e P r e se n c e o f M a g n etic and E lectric F ield s

As the experim ental reality o f trapping ultracold polar molecules approaches, aclear understanding is needed o f how the molecules interact in the tra p environment.
O n the most straightforw ard level, collisions are essential for cooling the gas by either
evaporative or sym pathetic cooling methods. A high rate o f elastic collisions is desirable,
w hile a low rate o f exothermic, state-changing collisions is essential i f the cold gas is tosurvive at all.
One prom ising strategy for trapping ultracold molecules m ight be to follow up on
successes in tra pp in g cold atoms and to construct electrostatic [56,109] or magnetostatic[52] traps th a t can hold molecules in a weak-field-seeking state. Cold collisions o f polar
molecules in this environment have been analyzed in the past, w ith the finding th a tinelastic collision rates were unacceptably high in the presence of the electric field,
lim itin g the possibilities for stable trapping [87]. Reference [87] found th a t the large
inelastic rates were due to the strong dipole-dipole interaction between the molecules.
One im p o rta n t feature o f the dipole-dipole interaction is its com paratively long range.
Even w ith o u t know ing the details o f the short-range molecule-molecule interactions,the dipole forces alone were sufficient to change the internal molecular states. Indeed,a significant finding was th a t for weak-field seekers, the molecules are prevented fromapproaching close to one another due to a set o f long-range avoided crossings. Therefore,a reasonably accurate description o f molecular scattering may be made using the dipolar

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48
forces alone [90].
A com plim entary set o f theoretical analysis have considered the problem o f collisional s ta b ility o f paramagnetic molecules in a magnetostatic trap. For example, theweak-field-seeking states o f molecules are expected to survive collisions w ith He buffer
gas atoms quite well [110, 111]. Collisions of molecules w ith each other are also expected
to preserve th e ir spin orientation fa irly well, and hence remain trapped [87]. However,this effect is m itigated in the presence of a magnetic field [85, 112].
So far, no one appears to have considered the influence o f magnetic fields on
cold molecule-molecule collisions where b o th species have electric dipole moments. The
effects o f magnetic and electric fields were contrasted in Section 2.2, and in this chapter,
we approach th e ir use to control collisions; we consider cold O H ( 2 n 3/ 2 )-O H ( 2 n 3/ 2 )collisions in an electric or a magnetic field. To the extent th a t the applied electric fieldis zero, one m ight expect th a t dipole forces average to zero and thus do not contributeto destabilizing the spin orientation. I t turns out th a t this is not quite correct and th a t
dipole-dipole forces s till dom inate long-range scattering. However, applying a suitably
strong magnetic field m itigates this effect significantly. Interestingly, even in this case,the residual second-order dipole interactions are sufficiently strong to restrict scattering
to large interm olecular separation.
The m ain result o f the chapter is summarized in Fig. 4.1, which contrasts theinfluence o f electric and magnetic fields. Figure 4.1(a) plots the elastic (solid curves) andinelastic (dashed curves) collision rate constants for OH molecules in th e ir |F M f c ) =|22—) weak-field-seeking hyperfine state (for details on quantum numbers, see Chapt.
2). As the electric field is increased, the inelastic rate constant grows to alarm ingly
large values, m aking the gas collisionally unstable, as was shown in Ref. [87]. Figure
4.1(b) shows the analogous rate constants in a magnetic field (in b o th cases the field
is assumed to lie along the positive z axis o f the laboratory reference frame). In thiscase, the magnetic field has the effect o f suppressing collisions all the way down to a

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49

10

T=1mK

-

T=10mK

.—
s
</>

co"

E

o'
'w

200

400

600

800

1000

Electric Field (V/cm)

T=1mK
T=10nK
(/)
CO

E
o

,-12

.-1
3
0

1000

2000

3000

4000

Magnetic Field (gauss)

Figure 4.1: T he rm ally averaged rate constants for collisions of weak-field-seeking stateso f OH as a function o f applied electric field (a) and magnetic field (b). In b o th cases,
solid lines denote elastic scattering rates, while dashed lines denote inelastic scattering
rates. T w o temperatures are considered. A p p lyin g an electric fields drives the inelastic
collisions rates up, w hile an applied magnetic field drives inelastic collision rates down.

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rate constant o f 2 x 10~ 13 cm3/sec at fields o f B — 3000 Gauss. These results are fa irlyrobust against raising the tem perature to the merely cold (not ultracold) temperatures,~ I m K , attainable in buffer-gas loading or Stark slowing experiments. T his is goodnews for experiments, i t implies th a t cooling strategies th a t rely on collisions may befeasible, provided a suitably large bias magnetic field is applied.
O ur m ain goal here is to analyze the suppression of rates in a magnetic field. We
present an analysis of the system in a magnetic field using a reduced channel modelth a t encapsulates the essential collision physics.

Finally, the model is q u a litative lyunderstood using the adiabatic representation.

4.1

In ela stic R a te s o f O H -O H C o llisio n s in E x tern a l F ield s
We now study the consequences o f the dipolar interaction, Eq. (3.29), on scatter­ing. Scattering calculations are done using the log-derivative propagator m ethod [108].
To ensure convergence at all collision energies and applied fields, it is necessary to in ­clude p a rtia l waves up to I = 6 and to carry the propagation out to an interm oleculardistance of R = 104 (a.u.) before m atching to long-range wave functions. Cross sections
and rate constants are computed in the standard way for anisotropic potentials [ 1 1 0 ].
We rem ind the reader th a t throughout this chapter we consider collisions ofmolecules in itia lly in th e ir \F M y e ) = |2 2 —) states, w hich are weak-field seeking for bothelectric and magnetic fields. Thus for a scattering process incident on an s- p a rtia l wave,
the incident channel w ill be w ritte n |i) = | F i e i ) | F2 ^ < 2 e2 ) |) = |2 2 , —) |2 2 , —)| 0 0 ).

In this section, we w ill make frequent reference to “energy gap suppression” ofcollision rates. T h is notion arises from a perturbative view o f inelastic collisions, inwhich case the tra n sitio n p ro b a b ility am plitude is proportional to the overlap integral

j d R ^ { R ) V i } {R )i,s {R )(4.1)

where i p i j denotes the incident and final channel radial wave functions and V )/ is thewith permission of the copyright owner. Further reproduction prohibited without permission.

51
coupling m a trix element between them. In our case, ipi w ill have a long de Broglie
wavelength corresponding to its essentially zero collision energy. The de Broglie wave­
length o f ipf w ill instead grow smaller as the energy gap— E j between incident andfinal thresholds grows. Thus the integral in Eq. (4.1), and correspondingly the collision
rates, w ill dim inish. For this reason, the collisions we consider tend to favor changing
the hyperfine states o f the molecules over changing the p a rity states, since the hyperfines p littin g o f OH is smaller than the A-doubling.

4.1.1

E lectric F ield C ase
To calculate scattering in the presence of an electric field, we only need to include

p a rtia l waves / =0 , 2 for num erical accuracy of the inelastic rate constant, K ™ el, forthe field range th a t we consider, E < 1000(F/cm ), and at a collision energy o f 10 ~5 K .
Here we are only interested in the trend and identification of the loss mechanism. To
num erically converge the inelastic rates at higher field values, where the induced dipoles
are large, n a tu ra lly requires more p a rtia l waves.
Figure 4.2 (a) shows the to ta l (black) and p a rtia l (color) inelastic rate constant
R in d as a function o f the electric field [compare to Fig. 4.1 (a)]. Even in zero field,
where the d ipolar forces nom inally average out, the rate constant is large, comparable
to the elastic rate constant. T h is fact attests to the strength o f dipolar forces in OH,
even in second order.
The green line in Fig.

4.2(a) represents losses to the dom inant zero-field loss

channel 110—) |22—) |22). The blue curve in Fig. 4.2 (a) represents instead the dom inant
loss process at higher electric field values in channel [21 —)[22—)|21). Whereas the form er
rate remains relatively insensitive to field, the la tte r rises dram atically.
T h is behavior arises from two competing tendencies in an electric field. The firstis the increasing m ixin g o f different p a rity states as the field is turned on, leading to anincreasing strength o f the direct dipole-dipole coupling th a t affects b o th e xit channels.

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0

200

400

600

800

1000

i

S

Electric Field (V/cm)

o.ooo-

0 002 -

- .

<D

£

-0.004-p0

x: -0.006w<D

jE -0.008-

0.010

Electric Field (V/cm)

Figure 4.2: (a) T o ta l (black) and selected p a rtia l (color) inelastic rates for O H -O H
collisions as a function of electric field. The green curve is the dom inant zero field
inelastic loss process to channel 110—)|2 2 -)|2 2 ) (green). In the presence of the field,a different channel, |21—)|22—)|21), becomes dom inant (blue), (b) The thresholds forthese e xit channels, relative to the incident threshold.

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53
T his addition al coupling would, in general, cause inelastic rates to rise. I t is, however,offset by the com peting tendency for inelastic rates to become less likely when thechange in relative kinetic energy o f the collision partners is larger. Figure 4.2(b) shows
the threshold energies for the two e xit channels in Fig. 4.2(a), versus field, w ith zero
representing the energy o f the incident threshold. Here it is evident th a t loss to thechannel |2 2 —) 11 0 —)| 2 2 ) (green line) is accompanied by a large gain in kinetic energy,
whereas loss to channel |2 1 —)| 2 2 - ) | 2 1 ) (blue line) gains com paratively little kinetic
energy, and thus the later channel more strongly affected by the increased coupling
generated by the field.

4.1.2

M agnetic Field Case
To gain insight in to the suppression o f the inelastic rates in a magnetic field [Fig.

4.3 (a)], calculations were at a representative collision energy E = 10~ 5 K . To converge
the calculations in high field (B > 1500 Gauss) required p a rtia l waves I = 0 ,2 ,4 , 6 . We
have only considered collisions w ith incident p a rtia l wave I = 0 , since higher p a rtia l wave
contributions, w hile they exist, only contribute to rates at the fraction o f a percent level.
Because the electric field remains zero, p a rity is s till a rigorously good quantum
number. Therefore states o f the same p a rity are not dire ctly coupled. Nevertheless,the dom inant loss channels in a magnetic field share the p a rity o f the incident chan­nel wave function, |i) = |22, —)|22, —) |00). Figure 4.3 (a) illustrates this by showing
the to ta l (black) and p a rtia l (color) inelastic rates as a function o f the magnetic field.
The loss rates shown correspond to the e xit channels | / ) = |1 0 -)|2 2 -)|2 2 )(g re e n ),111 —) |22—) |41)(blue), and 11 0 - ) 11 0 - ) |44)(red).
Since direct coupling to the final channels is forbidden to the dipolar interaction,
all coupling must occur through some interm ediate channel |in t). Moreover, owing to
the p a rity selection rules in the m a trix element [Eq. (3.29)], this interm ediate channelmust have p a rity quantum numbers ei = €2 = + . Since this coupling is second order,

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54

10'

w
n'

E

o

10'14
250

500

750

1000

Magnetic Field (gauss)-0.05>

B

0)

c

-

0.10

LU
p
0

-0.15

(0

a)
L.
■C
h

-

0 20
.

-

-0.25
250

500

750

1000

Magnetic Field (gauss)

Figure 4.3: (a) The to ta l (black) and p a rtia l (color) inelastic rates for O H -O H collisions
as a function o f magnetic field. The colors are explained in the te xt, (b) The corre­
sponding thresholds, referred to the incident channel’s threshold (E i = 0). The lowest
curve is one possible interm ediate channel.

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55
the dom inant e xit channels can consist of b o th d-wave (If — 2) and g-wave (If = 4)
contributions, in contrast to the electric field case.
The prim a ry feature o f the inelastic rates in Fig. 4.3 (a) is th a t they decreasesignificantly at large field.

T his decline is the m ain reason for optim ism regarding

evaporative cooling strategies in OH; an applied bias field of 3000 Gauss can reducethe inelastic rate constant to below 2 x 10-1 3 cm3/sec (see Fig. 4.1). The cause ofthis decrease can be traced dire ctly to the relative separation o f the incident and final
channel thresholds, along w ith the indirect nature o f the coupling.
To see this, we reduce the model to its essential ingredients: (1) a strong dipoledipole interaction, ( 2 ) the relative sh ift o f the thresholds as the magnetic field is tuned,(3) an extrem ely exotherm ic interm ediate channel, and (4) the centrifugal barrier inthe fina l and interm ediate channels. The H am iltonian for a reduced model is HModel =To + VModeU where To is the kinetic energy operator and VModel in m a trix form is/^M o d e l

—

,
E i

0

a /R

0

Ef + Cf/R2

b /R 3/ R3b /R 3E in t+ C in t/R 2

\
(4.2)

Here Cj is a centrifugal repulsion w ith Cj = h 2 l j ( l j + l) /2 m r , a and 6 are dipole-dipole
coupling strengths, and E j is the threshold energies for the j th channel, w hich m im ic
the essential feature o f the Zeeman effect in this system. The channels { i, / , in t } havequantum numbers ( e i^ ) * = ( e i^ K = ( — ) an<i (e i ^ ) * ” 4 = (+ + )• The incident channel
has a p a rtia l wave /j = 0 , w hile dipole-coupling selection rules allow l i nt = 2 and I f = 2
or I f = 4.
The model H am iltonian [Eq.

(4.2] e x p lic itly excludes direct coupling between

incident and fina l channels, whereas coupling is mediated through the in t channel.
Parameters characteristic o f the physical problem are a = 0.12 (a.u.), b = 0.10 (a.u.),

E i — 0, E f = —0.003 —» —0.1(K ), and E int = —0.17(K), li = 0, I f = 2 or I f = 4 andhnt — 2. The energy gap separation losses to the interm ediate channels are negligible.

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We find, in addition, th a t moving E int has little effect on the rate constants for loss tochannel / .
In this model, we use the threshold separation instead o f magnetic field because
th a t is the essential effect o f the field, to shift incident and final thresholds relative toeach other. We have chosen E f to vary over the range sim ilar to the effect o f changing
the field from 0 to 1000 Gauss for the dom inant zero-field loss channel, 110—) |22—).
Figure 4.4 shows the inelastic rates computed w ith in this model.

T his three-channel model does a reasonable jo b o f m im icking the prom inent features o f the fu ll
calculation, including the eventual and lasting decrease in rates as the states are sepa­
rated in energy. In addition, the g-wave rates decay more slowly as a function o f field
than do the d-wave rates, which is consistent w ith the fu ll calculation (compare Fig.
4.3).

The declining values o f the rate constant cannot, however, be a ttrib u te d to asimple overlap integral o f the form o f Eq. (4.1), since the incident and final channels
are not d ire c tly coupled. We therefore present a more refined adiabatic analysis o f thisprocess in the next subsection.

4.1.3

Adiabatic A nalysis o f the M agnetic Field CaseTo understand the system’s magnetic field behavior we analyze the reduced chan­nel model (4.2) in the adiabatic representation discussed in Sec. 3.6.
A p a rtia l set o f adiabatic potential curves generated in th is way is shown in Fig.
4.5, and exhibits an avoided crossing at R = 150.

Thus molecules incident on theupperm ost channel scatter p rim a rily at large values o f R. T h is p o in t has been made in
the past when an electric field is applied [87]; here we note th a t it is s till true in zero
electric field and th a t scattering calculations can proceed w ith o u t reference to shortrange dynamics.
The result o f the A D W B A is shown in Fig. 4.6. The two curves are for d (black)and ^-wave (red) inelastic channels. Several key features are present th a t also occur in

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

57

r 10

CO

E
o
0

£ eg
¥

,-12

0.00

0.02

0.04

0.06

0.08

0.10

E-Ef (K)

Figure 4.4: Inelastic rate constants for the three-channel model system, Eq. 4.2, as func­
tio n o f in itia l and final threshold separation. The two curves are for d and <7-wave exit
channels (black and red, respectively). In the g-wave channel, K ^ nel evolves more slowly
as the thresholds are separated. The threshold separation shown corresponds to varying
a magnetic field from 0 to 1000 Gauss for the dom inant loss channel, j 10—)|22—)|22).

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58

0 .0 4 !

2

0 .02 -/*-N

' q

0 .00 -

>

-

0.02
0

100

200

300

400

500

400

500

R (a.u.)0.04

0 . 02 £0 .00 -

>

-

0.02
100

200

300

R(a.u.)

Figure 4.5: The relevant adiabatic potentia l curves for the O H -O H system. Shown are
two different values o f the final threshold energy E i — E f = 5 rn K (a) and E{ — E f =15m K (b)t|Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

59

.-10

<A
CO

E
o
©

£ CM
*
.-13

.-14

0.00

0.02

0.04

0.06

0.08

0.10

E-E,(K)

Figure 4.6: Inelastic rate constants as estimated by the adiabatic distorted wave Born
approxim ation for the three-channel system. The black curve is for a d-wave e xit chan­nel, and the red for a g-wave exit channel. The threshold separation shown corre­
sponds to varying a magnetic field from 0 to 1000 Gauss for the dom inant loss channel,
110 ) |22 ).

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60
the fu ll calculation, namely: ( 1 ) the inelastic rate goes down w ith increasing thresholdseparation, (2) there is a zero in the rates, as seen in Fig. 4.3, and (3) the g-wave inelastic
rate goes more slowly than the d-wave, as seen in the model and the fu ll calculation.
The A D W B A accounts for all o f these. The first feature, dim inishing rates, s till arises
from an energy gap suppression, since the de Broglie wavelengths o f incident and final
channel s till do not match well. In the A D W B A , this process is fu rth e r helped alongby the fact th a t the residual channel coupling, represented by P , is localized near the
avoided crossings o f the adiabatic potential curves.
The A D W B A helps to visualize this suppression, as shown by the sample wave
functions in Fig. 4.7. T his figure shows ipl:

an(^f° r various values o f E f.

V arying E f mimics the sh ift o f the thresholds in an applied magnetic field. The values o f

E f o f Fig. 4.7 are (a) E f = - QmK , (b) E f = - 2 2 m K , and (c) E f = —Q2mK. The effectof the different E f s leave tpL m ostly unchanged. However, ipip becomes more exotherm ic
and therefore more oscillatory (Xdb clearly shortens). Moreover, the dom inant couplingregion where Ptp peaks moves to shorter R as E f increases. T his m otion is obvious
from the avoided crossing in Fig. 4.5.
The tra n sitio n am plitude in the A D W B A given in Eq. (3.39) is proportion al tothe integral o f the product o f the three quantities in Fig. 4.7. Because o f the shorteningo f the de Broglie wavelength in the e xit channel, this integral w ill eventually vanish,accounting for the zero in the inelastic rates. The total rate w ill, in general, not vanish,
since there are many e xit channels, and they w ill experience destructive interference at
different values o f the threshold, hence at different fields.
Finally, the g-wave inelastic rates are not so strongly affected by the separationo f E i and E f because the g-wave centrifugal barrier is larger, meaning a greater energy
is required to change the wave function at short range such th a t a node in the velocity
can pass through the coupling region. The zero in this rate constant w ill thus occur atlarger threshold separations.

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61

2000

1500
1000

c

x100

500

3

ro

-500
-1000

-1500
-2000

100

200

300

400

500

400

500

400

500

R(a.u.)
2000

1000 -

x100

c
3
A
(0
'w'

- 2000 -

100

200

300

R(a.u.)
3000
2000 -x100(01000 -c3- 1000 -- 2000 -

-3000
100

200

300

R(a.u.)

Figure 4.7: Illu stra tio n s o f the origin o f the zero in the p a rtia l rates. Each panel
shows curves
x 100 and PLtp x lO 9. The plots are for different fin a l energies,
and the inelastic rate from the Born approxim ation are (a) E f — —6 rn K and K™ el —5 x 10_ 11 (cra 3 /.s), (b) E f = —22m K , K™ el = 5 x 10- 14 (cm 3 /.s), and (c) E f — —Q2mK,K™ el = 6 x 1 0 “ 13 (cm 3 /.s), see te x t for details.

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62

4.2

C on clu sion s
We have explored the influence of a magnetic field on the cold collision dynamicso f polar molecules. The dipole-dipole interactions rem ain significant even in the absenceo f an electric field th a t polarizes the molecules. In general, this implies th a t molecularorientations are unstable in collisions, m aking magnetic tra p p in g infeasible. We havefound, however, th a t a suitably strong magnetic field can m itigate this instability.
Beyond this result, we note th a t laboratory strength fields can exert comparable
influence on cold collisions, if applied separately. A useful rule o f thum b in this regard
is th a t an electric field o f 300 V I cm acting on a 1 D dipole moment causes roughly thesame energy sh ift as a 100 Gauss field acting on a 1 Bohr magneton magnetic moment.
T h is raises the interesting question o f how the two fields can be applied simultaneously
to exert even finer control over collision dynamics. T his w ill be the subject o f future
investigations.

i1

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C h a p ter 5

S tro n g -F ield Seek ers and th e In flu en ce o f an E lectric F ield

Conventional spectroscopy o f atoms and molecules begins w ith the premise th a t
the energy levels o f the species being probed are fixed even though they may shift slightly
in electromagnetic fields. These levels are then interrogated by energy-dependent probes
such as photons or charged or neutral particles.

Inform ation on the energetics andstructure of the molecules is extracted from absorption energies, oscillator strengths,
selection rules, etc. In these investigations, the study of resonances has played a centralrole.
W ith the advent o f ultracold environments for atoms and molecules, this generalview o f spectroscopy can be inverted. Cold collisions provide a nearly monochrom atic
probe o f near-threshold interm olecular interactions, w ith resolution set by the m illiK e lv in or m icro-K elvin tem perature o f the gas. In this case, the energy levels o f nearby
resonant states can be tuned in to resonance w ith the zero-energy collisions. For coldgases o f alkali atoms, this strategy is already in widespread use. I t exploits the fact
th a t the Zeeman effect can shift the internal energies o f the atoms over ranges orderso f m agnitude larger than the collision energy itself. In this way, atoms can be madeto resonate when they would not n a tu ra lly do so (i.e., in zero field). Measurement ofthese “ Feshbach” resonances (more properly, Fano-Feshbach resonances) has yielded themost accurate determ ination o f alkali-alkali potentia l energy surfaces for near-thresholdprocesses [113].

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64
The current experim ental and theoretical push to create and study ultracold
molecules [30] w ill lead to many more opportunities for this novel kin d o f spectroscopy,
since molecules possess many more internal degrees o f freedom than do atoms. Therew ill be, for example, numerous Fano-Feshbach resonances in which one or both o f thecollision partners becomes vib ra tio n a lly or ro ta tio n a lly excited [71, 114, 115]; statisticalarguments suggest th a t these resonances w ill be quite narrow in energy, a fact related
to th e ir abundance [114]. A second class o f resonant states w ill occur when the con­stituents are excited in to higher-lying fine structure or hyperfine structure states, morereminiscent o f the resonances observed in alkali atoms. In molecules, these resonances
are n a tu ra lly also tunable in position using magnetic fields [116].
In this chapter, we are p rim a rily interested in a th ird class o f resonance: potential
resonances th a t are engendered by altering the interm olecular potentia l energy surface
itself. T his capability becomes especially prom inent in cold collisions o f heteronuclear
polar molecules, whose d ipolar interactions are quite strong on the scale o f the lowtranslational tem peratures o f the gas. A t the same tim e, the dipole moments o f thein d ivid u a l dipoles can be strengthened or weakened as well as aligned by an applied
electric field.
T his kin d o f resonance includes shape resonances, discussed in the context o f cold
atoms polarized by strong electric fields [117, 118]. A second set dubbed “ field-linked”
resonances has been studied in some detail in Refs [90, 87, 91]. These states appear
in P ES’s th a t correlate to weak electric field-seeking states o f free molecules. They are
weakly bound, long-range in character, and indeed do not appear to exist w ith o u t anelectric field present. S im ilar resonances are predicted to occur in metastable states of
the alkaline earth elements at low temperatures [119, 120].
A rich set o f potential resonances emerges among strong-field seeking states, andth is is the subject o f the present paper. Strong-field seekers are o f increasing im portanceexperim entally, since they enable molecules to be trapped in th e ir absolute ground

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65
states where no tw o-body inelastic collision processes are available to harm the gas. In
this case, the colliding molecules are free to approach to w ith in a small internuclear
distance o f one another; the resulting potential resonances therefore can probe detailed
interm olecular dynamics near threshold. The resulting data, consisting of scatteringpeaks as a function o f electric field, can be thought o f as a kind o f “ Stark spectroscopy.”
Just such a to ol has been applied previously in precision measurements o f alkali Rydberg
spectra [1 2 1 , 1 2 2 ],In this chapter, we explore such spectra in ultracold molecules, finding th a t the
spectrum is dom inated by a quasi-regular series in the electric field values.

Such aseries is the fundam ental b u ild in g block o f molecular Stark spectroscopy and plays arole analogous to the Rydberg series in atom ic spectroscopy. In b o th cases, the series
lays out the fundam ental structure of the unperturbed, long-range physics between
interacting entities. In the case o f atoms, the deviation from an unperturbed Rydberg
series, encapsulated in the quantum defects, yields inform ation on the electron-coreinteraction [123, 124]. Sim ilarly, it is expected th a t differences in observed S tark spectrafrom those presented here w ill probe the short-range interm olecular interactions.
Using this form alism , introduced in section 2, we study the potential resonances
using a sim plified version o f the molecular gas in which all dipoles are assumed to be
perfectly aligned and where molecular fine structure plays no role. In Sec. IV , we con­sider the case o f more realistic molecules, where fine structure does intervene. We show
th a t the structure of the potential resonances is unchanged, b u t th a t additional narrow
Fano-Feshbach resonances do appear. Throughout this chapter, we emphasize how thevarious types o f resonance can be classified and organized by simple considerations in ­volving the W K B approxim ation. To introduce the ideas, we firs t illu stra te them w ith
a simple “to y ” model o f pure polarized dipoles.

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66

0.501
0.25-

0 . 00 2025-

> -0 .5 0 -0.75-

1.00

40

60

80

100 120 140

R (a.u.)0.0

0.5 2 1.0

1.5

2.0

|i (a.u.)
Figure 5.1: (a) A diabatic curves of the pure dipole system for two values o f the dipole
moment fj,: fi = 0.1 (gray) and fj, = 1 (black) (b) The cross section o f the polarized
dipole model versus fi, at a collision energy E = 10~ 12 K . The brackets denote predictedresonant positions using the adiabatic W K B phase (AW P) approxim ation. The black
bracket represents the phase co n trib u tio n from the lowest adiabatic curve, and the grayis from all rem aining contributions.

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67

5.1

D ip o la r S ca tterin g
O ur p rim a ry interest in polar molecule scattering is how the strong anisotropicinteraction affects the system. As a firs t step illu s tra tin g the influence o f d ipolar in ­teractions, we present a simple model composed o f polarized dipoles w ith no internal
structure. S tric tly speaking, this system is created by an in fin ite ly strong electric field
th a t com pletely polarizes the molecules and raises all other internal states to experi­
m entally unattainable energies. Thus the only label required for a channel is its p a rtia lwave, I = { 0 ,2 ,4 ,...} in the numerical example given here. The m a trix elements o f thedipole-dipole interaction are taken to be (12Z0|//^|12Z'0) = —0.32/z2(Z0|Cq |Z'0), which
is typ ica l for molecules like RbCs or SrO. We then a rtific ia lly vary the dipole moment fi
to see the effect o f an increasingly strong dipolar interaction. P ragm atically speaking,varying /j parallels changing the electric field. The intention o f this model is to focus on
the effect o f direct anisotropic couplings between the degenerate channels, as measured
by th e ir effect on the p a rtia l wave channels.
For this model, we use a reduced mass o f m r = 104 a.u., typical o f very lig h tmolecules. We moreover assume th a t the molecules approach one another along thelaboratory z-axis, so th a t only the projection m i = 0 o f o rb ita l angular m omentumis relevant. To set a concrete boundary condition at small R , we apply “ hard w all”
boundary conditions, ip(R in) = 0 at a characteristic radius i?jn = 20 a.u. We pick thecollision energy to be nearly zero, namely 10~ 12 K . To converge the calculation for thismodel requires inclusion o f p a rtia l waves up to Z = 14 and numerical integration o f the
Schrodinger equation out to R = R 00 = 1 x 105 a.u. using the log-derivative propagatorm ethod o f Johnson [108].
To get a sense o f the influence of increasing the dipole moment, we firs t lookat adiabatic curves of the system. Figure 5.1 (a) shows two different sets o f adiabatic
curves: a gray set w ith /z = 0.1 (a.u.) and a black set /z = 1.0 (a.u.). In each set, the four

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68
lowest adiabatic curves are shown. Looking at these curves, we can see two characteristic
effects o f increasing fi.

F irst, the lowest curve becomes much deeper.

Second, thehigher adiabatic curves, o riginating from nonzero p a rtia l waves, may support boundstates at short distance states, i.e., w ith in the centrifugal barrier. B oth these effect may
generate bound states, leading to d istin ct classes o f scattering resonances as fj, is varied.
The deepening o f the lowest adiabatic curves induces potential resonances, whereas thehigher-lying curves lead to narrow shape resonances wherein the molecules must tunnel
through the centrifugal barrier.
The different classes o f resonances can clearly be seen in cross section, in Fig. 5.1(b). The broad quasi-regular set o f resonances seen in the cross section are the potentia lresonances o riginating from the lowest adiabatic curve. The narrow shape resonances
appearing sporadically in the spectrum originate from the higher-lying curves.

Forthe purpose o f this chapter, we focus on the wide potential resonances and sim ply
acknowledge the existence o f the narrow shape resonances.
To show th a t the broad resonances p rim a rily belong to the lowest potentia l andthe narrow shape resonances belong to the higher-lying potentials, we use an eigenphaseanalysis. The eigenphase can be thought o f as the sum o f the phase shifts for all o f thechannels; thus it tracks the behavior o f all the channels simultaneously. The eigenphaseis defined as(P e ig e n= ^ t a n _ 1 ( A f) .

(5.1)

i

Here \ f are the eigenvalues o f the K m a trix from the full-scattering calculation [104].
The K m a trix is related to the more fa m ilia r scattering m a trix by S — ( l + i K ) / ( l —iK ) .
W hen the system gains a bound state, it appears as a jt ju m p in eigenphase.

Theeigenphase o f the system is shown in Fig. 4.2 (a) as the solid line w ith many abrupt
steps.
To analyze this situation fu rth e r, we construct an approxim ate eigenphase as

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69

CD
V)

CD
JC

a

0.0

0.5

1.0

1.5

2.0

^ (a.u.)

WKB

0
WKB

0)

CD
-C
o .

WKB

2.0

0.5

j/(a .u .)Figure 5.2: (a) Eigenphase (solid) and to ta l adiabatic W K B phase (AWP, dashed) for
the dipole-scattering model, (b) A W P contributions for the lowest adiabatic curves{ 4> w kb)

the black bracket indicating when 4> w k b P^ses through a m u ltip le o f n.
(2 3 }The two rem aining curves are contributions w ith nonzero p a rtia l waves
Thegray bracket indicates where the sum o f these contributions passes through a m u ltiple
o f 7r.

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70
follows. F irs t, we assess the to ta l phase accumulated in each adiabatic channel using a
W K B prescription:$ w k b (A1) = /\Z - 2 m r F4£,(/x, R ) /h 2.

(5.2)

R in

Here (i) stands for the i th adiabatic curve. For the lowest adiabatic curve, w hich isalways a ttractive, the range o f integration is from R tn to Rout = oo. For high-lying
channels

th a t possess a b arrier to scattering at zero collision energy,the lim its

o f inte­

gration are from R in to the inner classical tu rn in g p oint o f the barrier. T h is w ill yieldsome info rm a tion on shape resonances trapped behind the barrier, b u t we w ill not makemuch o f this in the analysis to follow. Finally, we add together the in d ivid u a l W K Bphases to produce an approxim ate eigenphase shift, which we dub the “ adiabatic W K Bphase” (A W P ):<I>WKb { ^ ) =4>W K r { ^ ) -

(5-3)

i
Since we are not concerned w ith properties o f the phase associated w ith the higher-lying
adiabatic curves, we do not consider the connection form ula now.
The to ta l AW P for th is system is shown as a dashed line in Fig.

5.2 (a).

It

tracks the eigenphase well b u t offers more inform ation if we decompose the A W P into
its contributions. Figure 5.2 (b) shows the in d ivid u a l contributions o f the sum. The
largest co n trib u tio n is the phase from the lowest adiabatic curve, which can be associatedw ith potential resonances. A black bracket appears above this phase co n trib u tio n w ith
vertical marks indicatin g when it passes through an integer m u ltip le of 7r, i.e., when we
expect to see a potential resonance in the cross section. T his same bracket is plotted inFig. 5.1 and shows good agreement between the locations o f the potential resonances
and the A W P predictions. We conclude from this th a t the m ain resonance features in
the S tark spectrum arise p rim a rily from this single potential curve.
In Fig. 5.2 (b), the two rem aining phase contributions originate from nonzero
p a rtia l wave channels th a t possess centrifugal barriers; see Fig. 5.1(b). The gray bracket

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indicates where the sum o f these two contributions passes through an integer m u ltip le of
7r, and thus represents a guess for where the shape resonances lie. T h is gray bracket isalso shown in Fig. 5.1. The agreement w ith the position o f the narrow resonance featuresin the cross section is not nearly as good as it is w ith the broad potential resonances.
T his indicates a more involved crite rio n for shape resonances. Nevertheless, the AW P
predicts 15 shape resonances, and there are 13 in the range o f /i shown. T he AW P
appears to offer a means to roughly predict the number o f shape resonances in thissystem, even though it does not predict the locations exactly.
A m ain p o in t o f this analysis is th a t the A W P in the lowest adiabatic channel
th a t alone is sufficient to locate the potential (as opposed to shape) resonances w ith o u t
fu rth e r m odification. For the rest o f th is chapter, we focus on the potentia l resonances
in more realistic molecules w ith internal molecular structure. The general analysis interm s o f a single-channel AW P w ill s till hold, b u t an additional phase sh ift w ill be
required to describe the spectrum.

5.2

S tro n g -F ield Seekers
Strong-field-seeking molecules are approxim ately described as polarized in thesense o f the last section, because th e ir dipole moments are aligned w ith the field. Theyw ill, however, contain a richer resonance structure owing to the presence o f low -lying
excited states th a t can alter the dipolar potential energy surface at sm all R.
For concreteness, we focus here on molecules w ith aground state. Heteronul-

cear alkalis fit in to this category and are ra p id ly approaching ground state production
w ith various species [125]. As examples, we pick RbCs and SrO in th e ir ground states.
G round state RbCs has been produced experim entally [49]. As for SrO, prom ising newtechniques should lead to experim ental results soon [126]. For sim p licity we include only
the J = 0 and J — 1 ro ta tio n a l states and freeze the projection o f molecular angularm om entum to M j = 0. T his restricts the number of scattering thresholds to three,Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

72
identified by the p a rity quantum number o f the molecules in zero field.

The p a rity

quantum numbers for the three thresholds are ( — , — h, + + ) . T h is model is sim ilar tothe one presented in Ref. [87] th a t can be easily constructed for any rig id ro to r when
only including two molecular states. One im m ediate consequence o f m u ltiple thresholds
is the presence o f rota tion al Fano-Feshbach resonances in the collisional spectrum.
The firs t example is RbCs, whose physical parameters are /i = 1.3 D, m r = 110
a.m.u., and B e = 0.0245( K ) [49]. As before, we apply a vanishing boundary condition
at R in = 20 a.u. To converge this calculation over the field range considered, we requirep a rtia l waves up to / = 30.

We firs t look at the adiabatic curves o f the system toget an understanding o f how the real system deviates from the simple model presented
above. In Fig. 5.3 (a), we p lo t the six lowest adiabatic curves for the RbCs system w ith
o nly four p a rtia l waves, so the figure is more easily interpreted. The sets o f adiabatic
curves shown are for two different field values: the gray set has £ = 0 and the black set
has £ = 5000 (V /c m ).

There are two im p o rta n t features th a t differ from the dipole

example. F irs t, there are two higher thresholds, and the electric field shifts these apartin energy as the field is increased. Second,the electric field dram atically changes theradial dependence o f the Ham iltonian.
The difference in thresholds can be seen clearly in Fig. 5.3 (a) where the lowest
excited threshold moves from 0.05(R ) at £ = 0 to 0.18(R ) at £=5000 (V /c m ).

The

difference in radial dependences for the two cases is seen more clearly in a log-log p lo t
o f the two lowest adiabatic curves for b o th fields, as shown in Fig. 5.3 (b). The gray set
corresponding to zero field has two distin ct asym ptotic radial powers. A t large R, thelowest adiabatic curve has a 1 /R 6 behavior asym ptotically because o f couplings w ith
channels far away in energy [~ 0.05(A')]. However as R approaches zero, the dipolarinteraction has overwhelmed the rotational energy separation, and the radial dependence
becomes 1 /R 3 in character at about R =100 (a.u.). For reference, the dashed line is
propo rtio n a l to 1 /R 3. W ith £ = 5000 (V /c m ), the two black curves show the radial

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73-

0 . 1-

-

0.2

100

200

300

R (a.u.)

>

E
>
<D
c

LU

100

500

R (a.u.)Figure 5.3: (a) A diabatic curves for the RbCs system at two different field values,£ = 0(V /c m ) (gray) and £ = 5000{V /c m ) (black), (b) Log-log p lo t o f the absolutevalue o f the two lowest adiabatic curves. The blue dashed line is proportion al to l / R 3.

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74
dependence o f the adiabatic curves. The lowest curve now has nearly 1 / i ? 3 behavior
over the whole range shown. A sym p to tica lly when the centrifugal barrier is larger, theradial dependence w ill change to 1 / i ?4 [87]. The second lowest adiabatic curve is alsosignificantly altered by the strong dipolar interaction as can be seen in Fig. 5.3 (b).
In Fig.

5.4 (a), we p lo t the cross section for the model RbCs system.

This

spectrum is riddled w ith narrow Fano-Feshbach resonances b u t is s till dom inated by aseries o f po te ntia l resonances sim ilar to the one in Fig. 5.1 (b). There are tw o sets of
A W P predictions shown as over-brackets. To understand th e ir difference, we look to
Fig. 5.4 (b). The AW P for the lowest adiabatic curve is shown in Fig. 5.4 (b) for two
different zero-field phase values. The gray curve is the AW P th a t is dire ctly computedfrom the m ethod described in Eq.

(5.3).

The locations where it passes through an

integer m u ltip le o f 7r are indicated by the gray triangles. Referring back to Fig. 5.4(a), where the same gray triangles appear, we see th a t this simple estimate does not
reproduce the resonance position.
We can, however, introduce an additional overall phase sh ift to account for thedifference in short-range interactions from the pure polarized case. The shifted AW P
reads
& W K b (£ ) =$ W K B (£ )^ d e fe c t■

(5-4)

B y tre a tin g S^efect as a fittin g parameter, we can obtain the resonance positions in ­dicated by the black bracket in Fig.

5.4, which agree quite well w ith the resonancepositions in the close-coupled calculation. To do so requires, in this case, a phase shift<5d e fec t= 0.14. In analogy w ith R ydberg spectroscopy, we consider the sh ift we haveadded to be a “quantum defect” th a t accounts for the effect o f the short-range inter­action. T he addition al phase sh ift reflects the influence o f short-range physics on the
scattering such as curve crossings w ith curves from higher thresholds.
The A W P also saturates w ith field, as can be seen in Fig. 5.4 (b). T his saturation

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75

,-12

1000

2000

3000

4000 ' 5000

Electric Field (V/cm)

10

8
6
Q.

4
2
0

0

1000 2000 3000 4000 5000
Electric Field (V/cm)

Figure 5.4: (a) Cross section for RbCs in strong-field seeking states, including the 10
po te ntia l resonances th a t the AW P predicts (solid bracket). The AW P phases are shownin (b). The black bracket corresponds to the A W P predicted resonances for the lowest
adiabatic curve after a defect, Sdefect — 0.14, has been added according to Eq. (5.4).
The gray triangles are the A W P predicted resonances w ith o u t the defect.

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76
occurs because the electric field eventually fu lly polarizes the molecules, so the dipole
moment cannot increase further.

The effect can also be seen in the spacing o f thepoten tia l resonances. A t low fields, the potentia l resonances occur frequently in field.
Then, as the field is fu rth e r increased, the resonances occur less often in field, which
is a signature o f dipole moment saturating and therefore an increasing field having lesseffect on the molecular interaction.
As a second example, we consider SrO, which has the physical parameters m r = 52
a.m.u, f i = 8.9 D, and B e = 0.5 K [39]. We choose this molecule for its com paratively large
mass and dipole moment, which guarantee a large number o f resonances. In Fig. 5.5 wehave p lotted the cross section for SrO, which is dom inated by the quasi-regular potential
resonance series. As before, the black bracket indicates where the phase shifted, A W P
predicts the po ten tia l resonances, and we see the agreement is good.

Furtherm ore,

the series has not term inated since we have not completely polarized the dipole. The
series o f potential resonances saturates at 17.5 (k V /c m ) after a to ta l o f 43 potential
resonances have been induced. To line up the A W P ’s predictions and the actual crosssection requires a defect of Sfe/ect = 0.215.
We have picked two examples to illu stra te how the potential resonances w ill ap­
pear in the context o f collisional spectroscopy. These resonances w ill occur to varying
degrees in the strong-field seeking collisions o f all polar molecules. For example, we canalso make sim ilar predictions for an asymm etric ro to r molecule such as formaldehyde(H 2 C O ). We fin d th a t this molecule should possess six potential resonances in the fieldinterval from 0 to 50 k V /c m .
I t is w orth noting th a t portions o f sim ilar resonance series were anticipated incold atom ic gases subjected to electric fields [118]. However very few such resonances
are likely to be observed, owing to the enormous fields ~ M V /c m required to generatethem . In polar molecules, by contrast, the entire series should be readily observable.

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77

10*
10^
V 0-7:o.
bi1io"113
10'131
0

1000

2000

3000

4000

5000

Electric Field (V/cm)

Figure 5.5: Cross section for SrO w ith a bracket indicating resonance positions predictedby the phase shifted A W P for the lowest adiabatic curve. There are 33 P R shown inthis range o f electric field. Sdefect = 0.215.

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78

5.3

C o llision al S p ectro sco p y
T hrough the course of this work, we have shown th a t the zero-energy cross sectiono f strong-field-seeking molecules is dom inated by a set of broad potential resonances.
Even though these resonances are themselves intriguing, th e ir properties can be ex­ploited to learn much more about the system. The general structure o f the potential
resonances is governed by the long-range d ipolar interaction, which has a predictable andcommon form . W ith a clear understanding o f this interaction and how it induces reso­
nances, it could be exploited to learn about the short-range interaction o f the molecules.
T his is because details o f where the lines appear must also depend on the boundary con­d itio n experienced by the wave function at small values o f R. Therefore the spectrum
contains inform atio n on the sm all R interm olecular dynamics. Thus by studying thepotentia l resonance series, we can extract inform ation about the short-range dynamics.
T his idea is sim ilar to quantum defect theory, which has been very successful inthe spectroscopy o f R ydberg atoms. The short-range physics o f the electron interacting
w ith the nucleus is complicated and not easily solved. However, once the electron is out
o f the small R region, it enters in to a pure coulomb potential where its m otion is wellunderstood. The effect o f the short range must be merged w ith long-range physics toform a complete solution. To account for the short-range interaction, the energy can beparameterized by replacing the principal quantum number w ith an effective quantum
num ber n * = n — /i. T his procedure is tantam ount to identifying an additional phase
sh ift due to the interaction o f the electron w ith the atom ic core. The idea o f merging
standard long-range physics w ith complicated short-range behavior has been applied
successfully not only in Rydberg states o f atoms [124, 127] and molecules [128], b u t also
in atom ic collisions [129], cold collisions [131, 132, 130], and dipole-dipole interactionso f the type we envision here [118].
As a simple expression o f this idea, we can alter the boundary condition applied

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79

0 .&

0 . 2-

0.0
0

1000

2000

3000

4000

5000

Electric Field (V/cm)
Figure 5.6: The vertical curves represent electric field values where the A W P predicts
po te ntia l resonances for a given defect, £wKB($defect)- The points represent resonant
field values, £^b\ in the fu ll calculations w ith different in itia l boundary conditions. The
values o f b are given by cotan( 7rj0 ). The values o f f3 for the different fu ll num ericalcalculations are 0 (filled circle), 0.11 (filled square), 0.22 (filled triangle), 0.56 (hollow
circle), 0.78 (hollow square), and 0.89 (hollow triangle). Each set o f values £ ^ is plotted
at a height corresponding to its best-fit value o f 5defect-

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80
at R in when perform ing the fu ll scattering calculation and note its influence on the fielddependent spectrum. For the above calculations, we have imposed the standard van­ishing boundary condition, tp(R in) = 0, for all channels. We now replace this condition
w ith a unifo rm logarithm ic derivative, b = (ggV,)V,’~1> at R in. Thus previously we set
b = oo, b u t now we allow b to vary. The log-derivative is conveniently represented as aphase:
b = cotan( 7r/?),(5.5)

where (3 can lie between zero and one, covering all values o f 6 from —oo to +oo. For(3 = 0, the boundary condition is the one employed above, ip{R in ) = 0, whereas for
13 = 0.5, the boundary condition is ^ tp ( R in ) — 0.
We have recomputed the collisional spectrum o f RbCs for several different in itia lconditions, and p lotted the field values o f the potential resonances, £^b\ in Fig. 5.6 assets o f points. The values of f3 for the different calculations are 0 (filled circle), 0.11
(filled square), 0.22 (filled triangle), 0.56 (hollow circle), 0.78 (hollow square), and 0.89
(hollow triangle). The filled circles are resonant locations for cross section in Fig. 5.4(a).
We next wish to demonstrate th a t each such spectrum can be identified by a single
quantum defect parameter, as was done in the previous section. T his entails picking
a value o f Sdef ect and then setting the phase 4> w k b (^ ) equal to an integer m u ltiple o f
7r. T his yields a set o f resonant field values, SwKB^defect)- Each Sdefect corresponds

to a p a rticu la r approxim ate spectrum.

The set o f curves, £wKB{Sdefect)i generated

by continuously varying Sdefect are shown are shown in Fig.

5.6 as solid lines. The

bracket in Fig. 5.4 (a) corresponds to the set o f points where a vertical line intersects£\V K B { S d e f e c t ) w ith S d e fe c t = 0.14.
We can compare the resonant field values predicted by the AW P, £wKB(Sdefect)>and resonant field values given by the fu ll calculation w ith different in itia l conditions,Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

81
5 ® . To p lo t <?W, we have varied the height at which the set o f points

is plottedu n til it aligns w ith £ w k b {Sd efect)- Doing this, we are able to to see how b and Sdefect are
related. Thus in Fig. 5.6, we can see th a t even w ith different boundary conditions, thesingle A W P curve in Fig. 5.4 (b) can be used to predict the spacing between the potentia lresonances by varying a single parameter, Sdefect- T his shows the AW P represents thelong-range scattering physics well and th a t em pirically extracted parameters like Sdefectw ill carry inform atio n about the short-range physics such as th a t embodied in b.

5.4

C o n clu sio n
A number o f resonant processes may occur when two polar molecules meet in an

u ltracold gas. We have focused here on the dom inant, quasi-regular series o f potential
resonances between strong-field seeking states. These potential resonances originate in
the direct deform ation o f the potential energy surface upon which the molecules scatter.
Observation o f these resonances may offer a direct means for probing the short range
interaction between molecules. We have provided a means o f analyzing this system w ith
an adiabatic W K B phase integral. T his m ethod shows how the system evolves w ith theapplication o f an electric field.

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C h a p ter 6

T h e L ast an d S h o rtest C h ap ter

Throughout the course of this work, we have studied the effects o f external fields
on collisions o f polar molecules.

F irst, we explored how the use of a magnetic field

could suppress inelastic loss rates for weak-field seeking states o f OH. We used a simple
num erical m ethod w ith w hich the suppression could be understood in terms o f energy
gap suppression. Second, we began the exploration o f collisions between strong-fieldseeking states in the presence o f an electric field.

We found a quasi-regular set o fpotentia l resonances as a function o f the electric field. To analyze these resonances, weused a simple W K B approxim ation to the eigenphase and explored the nature o f thepotential resonances.
There are m any possible future directions for this work. One is to fu rth e r theunderstanding of how external fields can control long-range scattering such th a t themolecules approach short range a in determined manner. T his would lead to the a b ility
to system atically extract inform ation about the short-range interaction o f the moleculesand even the possibility o f controlling chemical reactions. A nother possible direction is
to explore the param eterization of the molecular scattering such th a t fa ith fu l represen­tations o f the tw o-body interaction may be inserted in to many body theories.

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B ib lio g ra p h y[1] K . Takahashi et al., Phys. Rev. E 58, 7805 (1998).
[2] N. T ja n d ra and A . Bax, Science, 278, 1015 (1997).
[3] J. H. Prestegard et al., Chem. Rev. 104, 3519 (2004).
[4] P. M ohn, Magnetic M aterials in the Solid State, (Springer, New York, 2002).
[5] K . BeBell et al., Rev. M od. Phys. 72, 225 (2000);
Zunger, Phys. Rev. L e tt. 69, 3766 (1992).

D. B. Laks, S. H.Wei, and A.

[6 ] A . M ichal and R. Tycko, Phys. Rev. L e tt. 81, 3988 (1998).
[7] L. L i and A . P. Alivisatos, Phys. Rev. L e tt. 90, 97402 (2003).
[8 ] G. F. Mazenko, E qu ilibrium Statistical Mechanics, (John W hiley and Sons, New
York, 2000).
[9] V . J. Em ery et al., Phys Rev. E. 56 6120 (1997).
[10] B. P. Stojkovic et al., Phys. Rev. L e tt. 82 4679 (1999).
[11] C. J. Olson Reichhardt et al., Phys. Rev. L e tt. 92 16801 (2004).
[12] H. J. K ang et al., N ature 423, 522 (2003).
[13] J. W . L ynn et al., Phys. Rev. L e tt. 60, 2781 (1988).
[14] M . Uehara et al., N ature 399 560 (1999).
[15] J. J. Weis, J. Phys. Cond. M at. 15 S1471 (2003).
[16] S. H. L. K la pp , J. Phys. Cond. M at. 17 R525 (2005).
[17] S. A . Kivelson et al., Proc. N atl. Acad. Sci. 98, 11903 (2001).
[18] S. M . Choi and J. W . Lynn, Phys. Rev. L e tt. 87, 107001 (2001).
[19] M . A . Kastner et al., Rev. M od. Phys. 70, 897 (1998).
[20] S. Brush, Rev. M od. Phys. 39, 883 (1967).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

84[21] J. Weiner e t al. Rev. M od. Phys. 71, 1 (1999).
[22] M . H. Anderson e t a l, Science 269, 198 (1995).
[23] K . B. Davis e t al. Phys. Rev. L e tt. 75, 3969 (1995).
[24] C. C. Bradely e t al., Phys. Rev. L e tt. 75, 1687 (1995).
[25] F. Dalfovo e t al., Rev. M od. Phys. 71, 463 (1999).
[26] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).
[27] C. A . Regal, M . Greiner, and D. S. Jin, Phys. Rev. Le tt. 92, 040403 (2004).
[28] M . W . Zw ierlein e t al., Phys. Rev. L e tt. 92, 120403 (2004).
[29] C. C hin e t al., Science 305, 1128 (2004).
[30] For a recent review, see J. Doyle, B. Friedrich, R.V. Krems, and F. MasnouSeeuws, European Physical Journal D 31, 149 (2004).
[31] M . A . Baranov e t al., Physica Scripta, T 102, 764 (2002).
[32] L. Santos e t al., Phys. Rev. Le tt. 85, 1791 (2000).
[33] S. Y i and L. You, Phys. Rev. A 61, 041604(R) (2000).
[34] S. Y i and L. You, Phys. Rev. A 6 6 , 013607 (2002).
[35] S. Giovanazzi, D. O ’D rell, and G. K u riz k i, Phys. Rev. L e tt. 8 8 , 130402 (2002).
[36] K . Goral, L. Santos, and M . Lewenstein, Phys. Rev. L e tt. 8 8 , 170406 (2002).
[37] M . A. Baranov, M . S. M a r’enko, Val. S. Rychkov, and G. V. Shlyapnikov, Phys.
Rev. A 6 6 , 013606 (2002).
[38] B. Dam ski e t al., Phys. Rev. Le tt. 90, 110401 (2002).
[39] D. Demille, Phys. Rev. L e tt. 92, 067901 (2002).
[40] E. A . Donley, N. R. Claussen, S. T . Thompson, and C. E. W ieman, Nature (Lon­don) 417, 529 (2002).
[41] C. A . Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature 424, 47 (2003).
[42] K . E. Strecker, G. B. Partridge, and R. G. H ulet, Phys. Rev. L e tt. 91, 080406(2003).
[43] J. Cubizolles e t al., Phys. Rev. L e tt. 91, 240401 (2003).
[44] S. Jochim e t al., Phys. Rev. L e tt. 91, 240402 (2003).
[45] C. A. Stan, M . W . Zwierlein, C. H. Schunck, S. M . F. Raupach, and W . Ketterle,
Phys. Rev. L e tt. 93 143001 (2004).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

85[46] S. Inouye, J. G oldw in, M . L. Olsen, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys.
Rev. Le tt. 93 183201 (2004).
[47] A . J. K erm an e t al., Phys. Rev. L e tt. 92, 033004 (2004).
[48] A. J. K erm an e t al., Phys. Rev. L e tt. 92, 153001 (2004).
[49] J. Sage e t al., Phys. Rev. Le tt. 94, 203001 (2005).
[50] M . W . M ancini, G. D. Telles, A. R. L. Caires, V . S. Bagnato, and L. G. Marcassa,
Phys. Rev. L e tt. 92, 133203 (2004) .
[51] D. Wang e t a l, Phys. Rev. L e tt. 93, 243005 (2004).
[52] J. W einstein, R. deCarvalho, T . G uillet, B. Friedrich, and J.M . Doyle, Nature
395, 148 (1998).
[53] C.L. Cesar, D .M . Silveira, M .G . Veloso, F. Zappa, and C.C. Rodegheri, Hydrogen
A to m I I I , The international conference on Precision Physics o f Simple Atom ic
Systems, edited by C .L. Cesar, S.G. Karshenboim , V .A . Shelyuto (2004).
[54] A . Peters, Q uantum Gases Internation al Workshop, p. 19 (2003).
[55] R. deCarvalho, N. Brahms, B. Newman, C. Johnson, L. W illm an, J. M . Doyle, T .
J. G reytak, and D. Kleppner, Hydrogen A to m I I I , The Internation al conference
on Precision Physics of Simple A tom ic Systems, edited by C. L. Cesar, S. G.
Karshenboim , V . A. Shelyuto (2004).
[56] H .L. Bethlem , G. M eijer, In t. Rev. Phys. Chem. 22, 73 (2003).
[57] J. R. Bochinski, E. R. Hudson, H. J. Lewandowski, and J. Ye, Phys. Rev. A(subm itted).
[58] E. A . Hinds, Phys. Scripta T70, 34 (1997).
[59] M . R. T a rb u tt, H. L. Bethlem, J. J. Hudson, V . L. Ryabov, V . A . Ryzhov, B. E.
Sauer, G. M eijer, and E. A . Hinds, Phys. Rev. L e tt. 92, 173002 (2004).
[60] H. Gould, Departm ent o f Energy, Basic Energy Science C ontractor’s Meeting,
2003.
[61] P. F. Barker, A . I. Bishop, R. Fulton, and M . N. Shneider, in ACS 2004 N ational
M eeting Technical Program (2004).
[62] M . D i Rosa, IT A M P -C U A Workshop on U ltracold Polar Molecules, Cambridge,(2004).
[63] M . G up ta and D. Herschbach, J. Phys. Chem. A 103, 10670 (1999).
[64] M . S. E lioff, J. J. V alentini and D. W . Chandler, Science 302, 1940 (2003).
[65] S. A . Rangwala, T . Junglen, T . Rieger, P. W . H. Pinkse, and G. Rempe, Phys.
Rev. A 67, 043406 (2003).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

86[6 6 ] E. N ik itin , E. Dashevskaya, J. A lnis, M . Auzinsh, E.R .I. Abraham , B.R. Furneaux,M . K eil, C. McRaven, N. Shafer-Ray, and R. Waskowsky, Phys. Rev. A 6 8 , 023403(2003).
[67] A. V olpi and J. L. Bohn, Phys. Rev. A 65, 064702 (2002).
[6 8 ] P. Soldan et al., Phys. Rev. L e tt. 89, 153201 (2002).
[69] N. Balakrishnan, R. C. Forrey, and A . Dalgarno, Phys. Rev. L e tt. 80, 3224 (1998).
[70] N. Balakrishnan et al., Astrophys. J. 524, 1122 (1999).
[71] R. C. Forrey, V . Kharchenko, N. Balakrishnan, and A . Dalgarno, Phys. Rev. A
59, 2146 (1999).
[72] N. Balakrishnan, A. Dalgarno, and R. C. Forrey, J. Chem. Phys. 113, 621 (2000).
[73] N. Balakrishnan and A . Dalgarno, J. Phys. Chem. A 105, 2348 (2001).
[74] E. Bodo, F. A. G ianturco, and A. Dalgaxno, Chem. Phys. Le tt. 353, 127 (2002).
[75] J. C. Flasher and R. C. Forrey, Phys. Rev. A 65, 032710 (2002).
[76] T . Stoecklin, A. Voronin, and J. C. Rayez, Phys. Rev. A 6 6 , 042703 (2002).
[77] E. Hudson et al., in preparation, (2005).
[78] P. F. Week and N. Balakrishnan, J. Chem. Phys. 122, 154309 (2005)[79] N. Balakrishnan and A . Dalgarno, Chem. Phys. Le tt. 341, 652 (2001).
[80] N. Balakrishnan and A. Dalgarno, J. Phys. Chem. A 107, 7101 (2003).
[81] J. L. Bohn, Phys. Rev. A 62, 032701 (2000).
[82] R. V . Krem s et al., Phys. Rev. A 67, 060703 (R) (2003).
[83] H. C ybulski et a l, J. Chem. Phys. 122, 094307 (2005).
[84] K . Maussang et al., Phys. Rev. L e tt. 94, 123002 (2005).
[85] A . V olpi and J. L. Bohn, Phys. Rev. A 65, 052712 (2002).
[8 6 ] J. L. Bohn, Phys. Rev. A 63, 052714 (2001).
[87]

A . V . Avdeenkov and J. L. Bohn, Phys. Rev. A 6 6 , 052718 (2002).

[8 8 ]A . V . Avdeenkov and J. L. Bohn, Phys. Rev. A 71, 022706 (2005).

[89] R. V . Krems and A . Dalgarno, J. Chem. Phys. 120, 2296 (2004).
[90]

A . V . Avdeenkov and J. L. Bohn, Phys. Rev. L e tt. 90, 043006 (2003).

[91]

A . V . Avdeenkov and J. L. Bohn, Phys. Rev. A 69, 012710 (2004).

[92] C. T ickn or and J. L. Bohn, Phys. Rev. A 71, 022709 (2005).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

87[93] C. T ickn o r and J. L. Bohn, subm itted to Phys. Rev. A (2005).
[94] R. V . Krems, In t. Rev. Phys. Chem., in press. (2005).
[95] R. V . Krems, Phys. Rev. L e tt. 93, 013201 (2004).
[96] J. D. Jackson, Classical Electrodynamics (2nd E dition, W iley, New York, 1975).
[97] J. Brow n and A . C arrington, R otational Spectroscopy o f D iatom ic Molecules,(Cambridge, Cambridge, 2003).
[98] D. M . B rin k and G. R. Satchler, A ngular Momentum, (Clardeon Press, O xford,
1993).
[99] M . T inkham , Group Theory and Quantum Mechanics, (Dover, New York, 2003).
[100] K . Schreel and J. J. te r Muelen, J. Phys. Chem. A 101, (1997).
[101] T . Hain, R. Moision, and T . Curtiss, J. Chem. Phys. I l l , 6797 (1999).
[102] C. H. Townes and A . L. Schawlow Microwave Spectroscopy, (Dover, New York,1975).
[103] G. Herzberg M olecular Spectra and M olecular Structure: vol. 1 Spectra o f D i­
atom ic molecules. (Kreiger, Florida, 1950).
[104] J. Taylor, Scattering Theory, (Krieger, Florida, 1983).
[105] M . S. C hild , M olecular C ollision Theory, (Dover, New York, 1974).
[106] James Burke, P h.D Thesis. U niversity o f Colorado (1999).
[107] H. R. Sadeghpour et al., J. Phys. B: A t. M ol. O pt. Phys. 33 R93 (2000).
[108] B. R. Johnson, J. C om put. Phys. 13, 445 (1973).
[109] S. Y . T . Meerakker et. al, physics/0407116 (2004).
[110] J. L. Bohn, Phys. Rev. A 62, 032701 (2000).
[111] R. V . Krems, A . Dalgarno, N. Balakrishnan, and G. C. Groenenboom, Phys. Rev.
A , 67 060703(R) (2003).
[112] R. V . Krem s and A. Dalgarno, J. Chem. Phys., 120 2296 (2004); R. V . Krems,
H.R. Sadeghpour, A . Dalgarno, D. Zgid, J. Klos, and G. Chalasinski, Phys Rev.
A , 6 8 051401 (R) (2004).
[113] N. R. Claussen et al., Phys. Rev. A 67, 060701 (2003); E. G. M . van Kem pen etal., Phys. Rev. L e tt. 8 8 , 093201 (2002); K . M . O ’Hara et al. Phys. Rev. A 6 6 ,041401(R) (2002); C. C hin et al. Phys. Rev. Le tt. 85, 2717 (2000); J. P. Leo etal., Phys. Rev. L e tt. 85, 2721 (2000).
[114] J. L. Bohn, A. V . Avdeenkov, and M . P. Deskevich, Phys. Rev. L e tt. 89, 203202(2002).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[115] N. Balakrishnan, R. C. Forrey, and A . Dalgarno, Phys. Rev. L e tt. 80, 3224 (1998).
[116] R. V . Krems, Phys Rev. L e tt. 93, 13201 (2004).
[117] M . Marinescu and L. You, Phys Rev. Le tt. 81, 4596 (1998); S. Y i and L. You,
Phys. Rev. A 63, 053607 (2001).
[118] B. Deb and L. You, Phys. Rev. A 64, 022717 (2001).
[119] A. Derevianko, S. G. Porsev, S. Kotochigova, E. Tiesinga, and P. S. Julienne,
Phys. Rev. L e tt. 90, 063002 (2003).
[120] R. Santra et al., Phys. Rev. A 67, 062713 (2003).
[121] G. D. Stevens et a l, Phys Rev. A 53, 1349 (1996).
[122] J. F. Baugh et al., Phys Rev. A 58, 1585 (1998).
[123] M . J. Seaton, Rep. Prog. Phys. 46 167 (1983).
[124] M . A ym a r et al., Rev. M od. Phys. 6 8 , 1015 (1996).
[125] C. Haimberger et a l, Phys. Rev. A 70, 021402 (R) (2004); A . J. Kerm an et a l,
Phys. Rev. L e tt. 92, 153001 (2004); D. Wang et al., Phys. Rev. L e tt. 93, 243005(2004); M . W . M ancini et a l, Phys. Rev. L e tt. 92, 133203 (2004).
[126] D. D eM ille, D. R. Glenn, and J. Petricka, Eur. Phys. J. D 31, 375-384 (2004).
[127] U. Fano and A. R. P. Rau, A tom ic Collisions and Spectra (Academic Press, Lon­don, 1986).
[128] C. H. Greene and Ch. Jungen, A dv. A t. M ol. Phys. 21, 51 (1985).
[129] F. H. Mies, J. Chem. Phys. 80, 2514 (1984).
[130] B. Gao, Phys. Rev. A 58, 4222 (1998); ibid. 62, 050702 (2000).
[131] J. P. Burke, Jr., C. H. Greene, and J. L. Bohn, Phys. Rev. L e tt. 81, 3355 (1998).
[132] F. H. Meis and M . R aoult, Phys. Rev. A 62, 012708 (2000).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.