reactive collisions in confined geometries

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Reactive collisions in confined geometries

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2015 New J. Phys. 17 035007

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New J. Phys. 17 (2015) 035007 doi:10.1088/1367-2630/17/3/035007

PAPER

Reactive collisions in conned geometries

Zbigniew Idziaszek1, Krzysztof Jachymski1 and Paul S Julienne2

1 Faculty of Physics, University ofWarsaw, Pasteura 5, 02-093Warsaw, Poland2 JointQuantum Institute, University ofMaryland andNational Institute of Standards andTechnology, College Park,MD20742,USA

E-mail: [email protected]

Keywords: cold collisions, chemical reactions, reduced dimensions

AbstractWeconsider low energy threshold reactive collisions of particles interacting via a van derWaalspotential at long range in the presence of external connement and give analytic formulas for theconnementmodied scattering in such circumstances. The reaction process is described in terms ofthe short range reaction probability. Quantumdefect theory is used to express elastic and inelastic orreaction collision rates analytically in terms of two dimensionless parameters representing phase andreactivity.We discuss themodications toWigner threshold laws for quasi-one-dimensional andquasi-two-dimensional geometries. Connement-induced resonances are suppressed due to reac-tions and are completely absent in the universal limit where the short-range loss probabilityapproaches unity.

1. Introduction

Cold and ultracoldmolecular collisions are an important research topic for a number of reasons, as reviewed by(Carr et al 2009,Qumner and Julienne 2012). In particular, chemical reactions near zero collision energy(200 nK) can be studied experimentally (Ni et al 2010, Ospelkaus et al 2010) and explained by relatively simplequantum scatteringmodels based on the properties of the long range potential (Idziaszek and Julienne 2010,Idziaszek et al 2010). Since reactive collisions can result in the rapid loss of trappedmolecules, it is important tounderstand and control them asmuch as possible. Reaction rates can bemodied and even greatly reduced byaligning dipolarmolecules in optical lattice structures of reduced dimensions (Qumner andBohn 2010, 2011,Julienne et al 2011, Zhu et al 2013, Simoni et al 2015).

In this work, we present an analytic treatment of ultracold reactive collisions between particles interactingwith an isotropic potential, such as S-state atoms or rotationless polarmolecules in the absence of an externalelectric eld, conned in a trap that effectively reduces the dimensionality of the system.We parametrize thereaction at short range using a simple quantumdefect parameterization and extend the analytical resultsobtained in (Micheli et al 2010) based on a long range van derWaals potential. In the formerwork it wasassumed that the reaction happens at short rangewith unit probability, which gives the process several universalfeatures.Here we generalize this treatment to consider the case of non-universal collisions, where the short-range reaction probability is in general smaller than unity. This gives rise to the possibility of resonances in thereaction rates. Highly reactivemolecules in reduced dimensional lattice structures can also experience the Zenoeffect, where reaction rates can be suppressed throughmany-body correlations that develop (Syassen et al 2008,Drr et al 2009, Zhu et al 2014). However, wewill not treat such correlations or the Zeno effect here.

A number of workers have developed the idea of using a pseudopotential proportional to the s-wavescattering length to represent short range interactions in traps, including highly anisotropic traps that effectivelyreduce the dimensionality of the system (Bush et al 1998, Tiesinga et al 2000, Bolda et al 2002, Blume andGreene 2002, Bolda et al 2003, Idziaszek andCalarco 2005, 2006,Naidon et al 2007, Yurovsky andBand 2007,Idziaszek 2009).Quasi-1D systems are of large interest in the context of integrability and exactly solvablemodels(Yurovsky et al 2008). Such trapping potentials can result in connement-induced resonances (CIRs)(Olshanii 1998, Petrov et al 2000), which have been studied theoretically (Petrov and Shlyapnikov 2001, GrangerandBlume 2004, Kanjilal and Blume 2004,Melezhik and Schmelcher 2009, Peng et al 2010, 2011, Sala et al 2012)

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and observed experimentally (Moritz et al 2005,Haller et al 2010, Sala et al 2013).Our theory extends theseconventional treatments to give the proper energy-dependent complex scattering length that is needed tocalculate such resonances accurately (Naidon et al 2007), including also the effect of loss channels due tochemical reactions or inelastic collisions, if they be present.

This work is structured as follows. In section 2we introduce the general scattering problem in the presence ofan external trap and the complex scattering length that describes both elastic and inelastic processes. Section 3discusses the case of quasi-one-dimensional trap geometry, while section 4 is dedicated to the quasi-2D case. Ineach sectionwe introduce the effective scattering lengths and rate constants and discuss their behavior andpossible resonances. Section 5 summarizes the results.

2. Reactive scattering process in the presence of a trap

Let us consider two particles conned in an external harmonic trap, which can be described by the stationarySchrdinger equation (the center ofmassmotion has been separated out)

+ + =r r r rU V E2

( ) ( ) ( ) ( ). (1)2

2tr

Here is the reducedmass of the pair, rU ( ) is the interparticle interaction andVtr is the harmonic trap, whichcan conne the particles in two direction x and y (with free quasi-1Dmotion along the z axis) or one direction z(with free quasi-2Dmotion in the x y, plane),

= =V V z12

,1

2. (2)tr 1D 2 2 tr 2D 2 2

It is also possible to consider anisotropic quasi-1D connement, = +V x y( )1D 122 2 2 . These kinds of the

trapping potential can be realized in experiment using optical lattices. The harmonic potential is described bycharacteristic length =di i . For each harmonic oscillator state, the total energy =E EiD, i=1, 2, iscomposed of the oscillator energy and the free particle energy in the unconned direction(s)

= + + + = + +E n m p E q(1 2 )2

,1

2 2. (3)1D

2 2

2D

2 2

Herewe follow (Naidon and Julienne 2006), where the indicesn m, denote the state of the 2Dharmonicoscillator described by thewave functionnm, is the state of the 1Dharmonic oscillator, and p and q representthe respective quasi-1D and quasi-2Dmomenta of freemotion. The asymptotic stationary state solution of theSchrdinger equation is the conventional one representing the sumof an incident planewave and a scatteredwave. Thewave function at large distances is then

+ + + r f fz

z( ) ( )e e e (4)nmp

r

nmpz

n m nm n m n mp z

n m nm n m n mp z1D i

,i

,i

+

r z f zq

( ) ( )e ( )i

8e . (5)qq

r q2D i i

Wewill describe the reactive collisions using a simplemodel based on quantumdefect theory. In thistreatment the fullmultichannel interaction potential is replaced by an effective single-channelmodel withproper boundary conditions at short range (Idziaszek and Julienne 2010, Jachymski et al 2013).This singlechannel represents the s- or p-wave channel inwhich the initial ultracold reactant species have been prepared,where s and p respectively represent relative angularmomentumquantumnumbers= 0 and 1; the formerapplies to nonidentical species or identical bosons and the latter applies to identical fermions. Themainassumption of themodel is the separation of length and energy scales between the chemical reaction and longrange scattering processes. The former happens at distances r0much smaller than typical length scale associatedwith the long range interactions, which for van derWaals potentialC r6 6 is given by

=

( )a

C

2 2. (6)

1

4

2

6

2

1 4

In our treatment we parametrize thewave function at short range and connect it with the long range solutionof the van derWaals potential. Themodel is schematically presented ongure 1 and discussed in (Jachymskiet al 2013). Briey, if the colliding particles reach the short range, part of the uxwill be absorbed there due toreaction or inelastic scattering and part will come backwith an additional phase shift. This allows forparametrization of the scattering process using two quantumdefect parameters y and s, where the y parameter

2

New J. Phys. 17 (2015) 035007 Z Idziaszek et al

determines the short range reaction probability = +Py

yre4

(1 )2, and s represents the scattering length of the full

interaction potential in units of a for scattering in the absence of any loss from the entrance channel. Essentially,s parameterizes the phase of thewave function due to incoming ux back-scattered into the entrance channel,and y parameterizes any loss of incoming ux due to any inelastic or reactive collision event at short range(Idziaszek and Julienne 2010).

In this workwewill focus on the case of low energies, where only scattering in the lowest partial wave allowedby the symmetry is relevant. Provided that the van derWaals length scale ismuch smaller than the connementlength a d so that the trapping potential can be regarded as constant in the interaction range, the scatteringprocess can be described by the s or p-wave pseudopotential (Bolda et al 2002, Bolda et al 2003,Naidon et al 2007,Idziaszek andCalarco 2006, Idziaszek 2009)

=

r rUa k

rr( )

2 ( )( ) , (7)s

2

=

r rU

V k

rr( )

( )( ) , (8)p

2 3

33

where

= =a k k k( ) tan ( ) , (9)0

= =V k k k( ) tan ( ) (10)1 3

are the energy-dependent s-wave scattering length and p-wave scattering volume, respectively and are denedconventionally using the 3Dphase shift k( ). In the case of reactive collisions these quantities take complexvalues and can bewritten in terms of our y and s parameters. For van derWaals interactions (Jachymskiet al 2013)

+ + +

a k a s y

s

y s( )

1 (1 )

(1 ) i, (11)

k 0 2

+ +

V k V

y s

ys s( ) 2

i( 1)

i( 2), (12)

k 0

where =

( )( )VC

18

2 3 4

3

4

2

6

2 is themean p-wave scattering volume.

3.Quasi-1D case

It is possible to solve the Schrdinger equation (1) with the boundary condition (4) at any energy and nd therelation between the scattering amplitudes f and the 3D scattering length for the swave or volume for the pwave.At low enough energies

=

+ f p

p( )

1

1 icot ( ). (13)

Note that the 1Dwavenumber p is different from the 3Dwavenumber k, since = = + Ek p

2 2

2 2 2 2

. It is often

convenient to use 1D scattering lengths, which can be dened in variousways.Here we choose to dene an evenand odd scattering length in the form

= = +

a p p p

p S p

S p ( ) tan ( )

i

1 ( )

1 ( ). (14)

Note that the a p ( ) so dened has units of inverse length.Within this notation, the subsequent formulas,including the rate constants, have a form independent of parity. It is also possible to dene the 1D scatteringlengths which have the unit of length (Olshanii 1998, Girardeau et al 2004)

= = a pp p

a pp

p( )

1

tan ( ), ( )

tan ( ). (15)e

1D

eo1D o

Our quantities can be easily related to these scattering lengths. In the general case a p ( )will be complex due tothe reaction process, = a i .

From the experimental point of view, themost relevant quantities are the rate constants, in one dimensiondened as

= ( )p g p S p( )2

1 ( ) , (16)1D,re 2

= p g p S p( )2

1 ( ) , (17)1D,el2

where g is the statistical factor equal to 1 for distinguishable particles and 2 for identical particles in the sameinternal states. It can be convenient to rewrite these denitions using scattering lengths, obtaining

= p gp

p f p( )2

( ) ( ) (18)1D,re2

1D

= p gp

a p f p( )2

( ) ( ), (19)1D,el 2 1D

where

=

+ + f p

p a p p p( )

1

( ) 2 ( ). (20)1D

2 2

The loss rate constants determine the decay of one-dimensional particle density n1D of the homogenous gasaccording to

= n n . (21)e1D r 1D2Note that n1D has units of (length)

1 and 1D,re has units of (length)/(time), so that their product nre 1Drepresents a loss rate per particle that can be compared to the similar 3D loss rate per particle with theconventional 3D rate constant and density.We also note that in the presence ofmore than two particles, thedensity decay in one dimension can be greatly affected bymany-body correlations. For example, in the case ofstrong interactions the particles can form aTonksGirardeau gas even in the presence of dissipation, therebyslowing down the reaction rate (Syassen et al 2008,Drr et al 2009, Zhu et al 2014). Treating such a case isbeyond the scope of this paper.

3.1. Even andodd scattering lengthsSolving the Schrdinger equationwith the pseudopotentials in equations (7) and (8) and boundary conditions(4) yields

= ++a p

d

a k

d1

( ) 2 ( ) 2

1

2, (22)

2

= a p

d

p V k p d

1

( ) 6 ( )

2 1

2, (23)

2

2 2

4


quite similar to the zero-energy result for elastic scattering. These formulas are only valid in the limit p

2

2 2

,

where = Ek

2

2 2

so that ka a d 2 1. The latter condition is also required for the pseudopotentialapproximation do be valid in the presence of external trap.

Using the low k expansions (11) and (12), the formulas (22) and (23) can bewritten as

= + + + +a p

d d

a

y s

s y s

1

( ) 2

1

2

1 i ( 1)

i ( 2), (24)

= a p p d

d

p V

s sy

s y

1

( )

2 ( 1 2)

12

2 i

1 i. (25)

2

2

2

It is instructive to investigate both the y 0 and y 1 limits of the expressions above. The former correspondsto the case where reactions are absent and should reduce to thewell-known results, while the latter is theuniversal reactive case forwhich there should be no dependence on s parameter. Indeed, for y 0we recoverthe formulas of (Olshanii 1998, Granger andBlume 2004):

++

a

d d

sa

1

2(1 2)

, (26)

y 0

a p d

d

p V

s

s

1

2 ( 1 2)

12 2

1. (27)

y 0

2

2

2

In the universal reactive limit

++

a p

d d

a

1

( ) 2(1 2)

(1 i), (28)

y 1

a p p d

d

p V

1

( )

2 ( 1 2) (1 i)

12 . (29)

y 1

2

2

2

The 1D scattering lengths dene the one-dimensional coupling constants g which describe the effective one-

dimensional contact interactions =+ +U z g z( ) ( )1D for evenwaves and =

U z g z( ) ( )z z1D for oddwaves.

Using our denitions, we have =+ +g p a p( ) ( )2 and = g p a p p( ) ( ) ( )2 2 .

3.2. Rate constantsWecannow calculate the elastic and reactive rate constants using (16) and (17). In the limit of very low collisionenergies pd a 12 , the even scattering rates read

+ + +

g p d ya

s s

s y s

2 ( 2)

( 2), (30)

p1D,re 02 2

2 2 2

+ g p2 . (31)p1D,el 0

Wenote that the elastic rate in the low energy limit approaches a universal value, independent of the reactivity,scattering length and the transverse trap strength. In the case of odd scattering, the formulas are rathercomplicated even in the low energy limit pV d 12 , but we provide them for completeness

+ g p V

d

y s s12 2 (2 ( 2) ), (32)

p

D

1D,re 02

21

+

( )g p V

d

y s144 2 (1 ), (33)

p

D

1D,el 03 2

4

2 2

1

where = + + + + + + + s s sy s s y y s4 ( 4 ) 8 4 ( 3 ) 4 ( ( 1) )1D 2 2 2 2 2 and = V d12 ( 1 2) 3. Inthe universal case y 1, this yields

+ +

g p Vd

12 1

1 2 2, (34)

p1D,re 02

2 2

+ +

g p Vd2

144 1

1 2 2. (35)

p1D,el 03 2

4 2

5


The low energy behavior of the reactive rate constants is illustrated ongure 2. In both cases only thedimensionless part is plotted for convenience, so in the even case the rate has been divided by g p d a( )2 2 , andin the odd case by g p V d12 ( )2 2 . For the odd case, inwhich the denominator still depends onV d 3 via the function, a connement strength corresponding to =d a10 has been assumed.Wenote that the strongest lossesin the even case can be found for low values of y and s close to zero. This stems from the fact that at p 0

+ +f a p1 ( )1D 2. For the odd case, where at low energy f p1

1D 2, the reaction rate depends only on theimaginary part of the scattering length. As a result, s= 2 gives the largest reaction rates. If the connement is nottoo strong, exhibits amaximumat this particular value associatedwith a CIR, aswill be shown in the nextsection.

3.3. Impact of reactions onCIRThe effective 1D coupling constants can diverge if the 3D scattering length is tuned to a certain nite value,which is known asCIR (Olshanii 1998).Wewill nowdiscuss how the presence of reactionmodies theproperties of this resonance. In the absence of reactions, one can calculate at which point the one-dimensional

coupling constant approaches innity andnd the resonance at = s a d(1 2) 1 for even, and =

++

s 112

for

oddwaves, with dened as in equation (32). However, it is easy to verify that the coupling is not divergent ifreactions are present. The 1D scattering length, however, is still strongly varying close to the resonance position,and the imaginary part of a exhibits amaximum there. For higher values of the y parameter, this effect getssuppressed, and in the limit of y 1 the resonance disappears completely. This effect is intuitively clear, sincefor unit loss probability at short range there is noux reected back from short range and thus nothing toresonate, so no resonances can be present. The behavior of even and odd scattering lengths at different values ofy is illustrated ongures 3 and 4.We picked y=0 (purely elastic collision), y=0.03 (veryweakly reactive), y=0.1(intermediate case), y=0.3 (quite strongly reactive) and y= 1 (universal reactive) as examples. For y=0weobserve the conventional CIRwith imaginary part equal to zero, corresponding to no losses. As the reactivitygrows, the real parts of the scattering length do not diverge anymore and the resonance is washed out. In the

Figure 2.Reactive rate constants rescaled to dimensionless form for quasi-1D even (left) and odd (right) scattering, as given byequations (30), (32) (see themain text for details).

Figure 3.The even one-dimensional scattering length = +a i as a function of the 3D scattering length =s a a3D for differentloss parameter y . Left: real part , right: imaginary part . The transverse connement corresponds to =d a10.

6


universal y=1 case approaches a constant value. The imaginary parts showquite similar behavior. As soon asthe losses are switched on ( y 0), a sharp peak appears in at the resonance position. Increasing the value of ymakes it less pronounced. In the universal case does not depend on s anymore.

We note that it is straightforward to generalize the results from this section to the case of anisotropictransverse connement, x y. The resulting formulas are slightlymore complicated, but apart from thatanisotropy does not introduce any new effects. For example, the even scattering length is given by

= ++ =a p

d

a k

dC1

( ) 2 ( )

( )

2, (36)x

2

0

where = y x and C ( ) is a generalization of function, dened in (Peng et al 2011).

4.Quasi-2D case

Wewill now analyze the case of planar connement, again assuming that the transversemotion is frozen andweare in the quasi-two-dimensional case. The energy in quasi-2D consists of harmonic oscillator energy and the 2D

wavenumber q, so in this case = = + Ek q

2

1

2 2

2 2 2 2

. For q 1 the only relevant termof the 2D scatteringamplitude f , is the f00 term, further denoted as f. It can be decomposed into partial waves = =

f f em mmi .

The 2D Smatrix is related to the amplitude via = + S f1i

2(Naidon et al 2007) and to the 2Dphase shift via

=

+ f q

q( )

4i

1 icot ( ), (37)

where the index = m, which is 0 and 1 for the two lowest partial waves in this case.We choose the 2Dcomplex scattering length to be dened by

= = +

a q q

S q

S q ( ) tan ( )

1

i

1 ( )

1 ( ), (38)

which is a dimensionless, complex quantity = a q ( ) i . This choice is againmotivated by the simplicity ofthe following formulas.

4.1. Scattering lengthsBy solving the Schrdinger equation to connect the 2D and 3D scattering lengths, we obtain the following resultfor the lowest partial waves

= +

a q

B

q d

d

a k

1

( )

1ln

2

( ), (39)

02 2

=

a q

d

q V k q d

1

( )

2

3 ( )

2

3(0), (40)

12 2 2

where B 0.9049 and (0) 0.328 (Idziaszek andCalarco 2006). Applying formulas (11), and (12) to theseequations yields

Figure 4. Same as on gure 3, but for the odd scattering length a .Mind the logarithmic scale used for the imaginary part.

7


= +

a qa

d

s y y

y s s y y ( )

( i) 2

( 1) i ( ( i) 2 ), (41)0

= + + + +

a qVq

d

y s

y s ys s ( )

3 i( 1)

( i( 1)) i( 2), (42)1

2

where = ( )lna d Bq d 22 2 and = Vd2 (0) 3 .We note that when a d , we have 1, but the parameterdepends strongly on energy and cannot in general be neglected. In the universal limit the formulas reduce to

+ + +

a q

a

d ( )

1 2 i

1 2 2, (43)

y0

1

2

+ ++ +

a q

Vq

d ( )

3 1 i

2 2, (44)

y1

1 2

2

whereas in the nonreactive case

+

a q

a

d

s

s ( )

1, (45)

y0

0

+

a q

Vq

d

s

s s ( )

3 1

2 ( 1). (46)

y1

0 2

4.2. Rate constantsThe two-dimensional reaction rate constants are dened as

= ( )q g S q( ) 1 ( ) , (47)2D,re 2

= q g S q( ) 1 ( ) . (48)2D,el 2

The decay of two-dimensional densityn2D is governed by

= n n , (49)2D 2D,re 2D2wheren2D has units of (length)

2, 2D,re has units of (length)2 (time), and as for quasi-1D and 3D, their productrepresents the loss rate per particle.

As in the previous section, it is convenient to rewrite this in the form

= q g q f q( ) 4 ( ) ( ), (50)2D,re 2D

= q g a q f q( ) 4 ( ) ( ), (51)2D,el 2 2D

where

=

+ + f q

a q q( )

1

1 ( ) 2 ( ). (52)2D

2

When plugging in formulas (41) and (42) to obtain the rates, we notice that the rates form=0 haveadditional energy dependence via the parameter, which contains a term logarithmic in q 2. As a result, at

q 0 both rates go to zero logarithmically:

= += q g y s s( ) 4 ( 2) 2 (53)m 02D,re2D,0

= + = q g s s y( ) 4 ( 2) , (54)m 02D,el2 2 2 2

2D,0

where = + + + + + + +s s y s s y y s s s2 (2 ( 2) ) ( ( 2) ) ( 1 ( 2) ) (1 )2D,0 2 2 2 2 2 2 2 and = a d . In the = m 1case, at sufciently low energies q V d 12 , we get

+

q g Vq

d

y s s( )

3 ( ( 2) 2), (55)

q1

2D,re 02

2D, 1

8


+

q g V q

d

y s( )

9 ( 1), (56)

q1

2D,el 02 4

2

2 2

2D, 1

where = + + + + + + ( ) ( )y s s s sy s s y( 1) 2 2 3 ( 2)2D, 1 2 2 2 2 2 2 2 2. Figure 5 shows the behaviorof the rate constants for different s and y. As in the 1D case, only the dimensionless part is plotted. Them=0 ratewas thus divided by g4 and the = m 1 rate by g Vq d3 ( )2 2 .

Figure 5.Reactive rate constants rescaled to dimensionless form for quasi-2D =m 0 (left, given by equation (53)) and = m 1 (right,given by equation (55)) scattering. The harmonic oscillator length =d a10 and in the =m 0 casewe assumed =qa 0.05.

Figure 6.The two-dimensional scattering length = a i0 corresponding to =m 0 as a function of the 3D scattering length=s a a3D for different loss parameters y . Left: real part, right: imaginary part . The transverse connement corresponds to=d a10 and =qa 0.05. Note the logarithmic scale used for the imaginary part.

Figure 7. Same as on gure 6, but for = m 1 case. Real (left) and imaginary (right) parts of = a i1 are shown.

9


4.3. Two-dimensional CIRIn quasi-2D systems, the CIR for elastic interactions andm=0 can be found by solving the equation

+ = ( )ln 0dsa Bq d 22 2 . Its position strongly depends on energy due to the logarithmic term. In them= 1 case,the resonance occurs at =

++s

2

1with dened as in (42). Similarly to the quasi-1D case, adding chemical

reactions results innite coupling constant with a pronouncedmaximumof the imaginary part at the resonanceposition. Figures 6 and 7 show some examples for different values of y fromnonreactive to universal case. In thereal parts we observe the same kind of behavior as for the quasi-1D case, with a single resonance beingwashedout as y 1. For y=0we have = 0 as before. A sharp resonance appears in the imaginary part for small butnonzero y and is washed out formore reactive collisions. In them=0 case exhibits also a visibleminimumat sclose to zero. In this regime the real part is also very small. This corresponds to the limit of noninteractingparticles.

5. Conclusion

In conclusion, we have given the analytical formulas in the near-threshold limit for the scattering lengths(equations (24) and (25), and (41) and (42)) aswell as elastic and reactive rate constants (equations (30)(33)and (53)(56)) for atomic ormolecular species interacting by a long range van derWaals potential andundergoing inelastic loss or chemical reactions in the presence of strong connement of the initial reactantspecies. These formulas are based on a powerful quantumdefect treatment, which allows the separation of thecollision into a long range and a short range part, with the latter being characterized by two quantumdefectparameters. One, s, represents a dimensionless phase and the other, y, represents the loss probability ofux fromthe initially prepared incoming channel of the reactants. The quantumdefect framework allows the elastic andinelastic or reactive collision rates for quasi-1D or quasi-2D connement to be expressed in terms of the energy-dependent 3D complex scattering length. The theory gives analytic predictionswith clear intuitivemeaning.While our implementation has been analytic, a numerical implementation is also possible. This could be usefulto extend the theory to species with dipole or quadrupolemoments, wheremore than one inverse power lawlong range form contributes to the overall potential. It is also possible to generalize the results to amultimodecase when several transverse states are occupied.Our theory also shows howCIRs aremodied in the presence ofinelastic or reactive processes. Such resonances are also potential sources of collision control bymanipulatingthe connement strength.

Acknowledgments

Thisworkwas supported by the Foundation for Polish Science International PhDProject co-nanced by the EUEuropeanRegional Development Fund, byNational Center for ScienceGrantNos. DEC-2011/01/B/ST2/02030andDEC-2013/09/N/ST2/02188, and by anAFOSRMURI FA955009-10617.

References

BlumeD andGreeneCH2002Phys. Rev.A 65 043613Bolda E L, Tiesinga E and Julienne P S 2002Phys. Rev.A 66 013403Bolda E L, Tiesinga E and Julienne P S 2003Phys. Rev.A 68 032702BushT, Englert B, Rzaewski K andWilkensM1998 Found. Phys. 28 549Carr LD,DeMille D, KremsRV andYe J 2009New J. Phys. 11 055049Drr S, Garcia-Ripoll J, SyassenN, BauerD, LettnerM,Cirac J andRempeG 2009Phys. Rev.A 79 023614GirardeauM,NguyenHandOlshaniiM2004Opt. Commun. 243 322Granger BE andBlumeD2004Phys. Rev. Lett. 92 133202Haller E,MarkM J,Hart R,Danzl J G, Reichsllner L,Melezhik V, Schmelcher P andNgerlHC2010 Phys. Rev. Lett. 104 153203Idziaszek Z 2009Phys. Rev.A 79 062701Idziaszek Z andCalarco T 2005Phys. Rev.A 71 050701Idziaszek Z andCalarco T 2006Phys. Rev.A 74 022712Idziaszek Z and Julienne P S 2010Phys. Rev. Lett. 104 113202Idziaszek Z,QumnerG, Bohn J L and Julienne P S 2010Phys. Rev.A 82 020703Jachymski K, KrychM, Julienne P S and Idziaszek Z 2013Phys. Rev. Lett. 110 213202Julienne P S,HannaTMand Idziaszek Z 2011Phys. Chem. Chem. Phys. 13 1911424Kanjilal K andBlumeD2004Phys. Rev.A 70 042709MelezhikV S and Schmelcher P 2009New J. Phys. 11 073031Micheli A, Idziaszek Z, PupilloG, BaranovMA, Zoller P and Julienne P S 2010Phys. Rev. Lett. 105 073202MoritzH, Stferle T, Gnter K, KhlM and Esslinger T 2005Phys. Rev. Lett. 94 210401Naidon P and Julienne P S 2006Phys. Rev.A 74 062713Naidon P, Tiesinga E,MitchellWF and Julienne P S 2007New J. Phys. 9 19

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NiKK,Ospelkaus S,WangD,QumnerG,Neyenhuis B, deMirandaMHG, Bohn J L, Ye J and JinD S 2010Nature 464 13248OlshaniiM1998Phys. Rev. Lett. 81 93841Ospelkaus S,Ni KK,WangD, deMirandaMHG,Neyenhuis B,QumnerG, Julienne P S, Bohn J L, JinD S andYe J 2010 Science 327 8537Peng SG, Bohloul S S, LiuX J,HuH andDrummondPD2010Phys. Rev.A 82 063633Peng SG,HuH, LiuX J andDrummondPD2011Phys. Rev.A 84 043619PetrovDS,HolzmannMand ShlyapnikovGV2000Phys. Rev. Lett. 84 25515PetrovD and ShlyapnikovG2001Phys. Rev.A 64 012706QumnerG andBohn J L 2010Phys. Rev.A 81 060701(R)QumnerG andBohn J L 2011Phys. Rev.A 83 012705QumnerG and Julienne P S 2012Chem. Rev. 112 49495011Sala S, Schneider P I and SaenzA 2012Phys. Rev. Lett. 109 073201Sala S, ZrnG, LompeT,WenzAN,Murmann S, Serwane F, Jochim S and Saenz A 2013Phys. Rev. Lett. 110 203202Simoni A, Srinivasan S, Launay JM, Jachymski K, Idziaszek Z and Julienne P S 2015New J. Phys. 17 013020SyassenN, BauerD, LettnerM,Volz T, DietzeD, Garcia-Ripoll J, Cirac J, RempeG andDrr S 2008 Science 320 132931Tiesinga E,WilliamsC J,Mies FH and Julienne P S 2000Phys. Rev.A 61 063416YurovskyVA andBandYB 2007Phys. Rev.A 75 012717YurovskyVA,OlshaniiM andWeissD S 2008Adv. At.Mol. Opt. Phys. 55 61138ZhuB et al 2014Phys. Rev. Lett. 112 070404ZhuB,QumnerG, ReyAMandHollandM J 2013Phys. Rev.A 88 063405

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1. Introduction2. Reactive scattering process in the presence of a trap3. Quasi-1D case3.1. Even and odd scattering lengths3.2. Rate constants3.3. Impact of reactions on CIR

4. Quasi-2D case4.1. Scattering lengths4.2. Rate constants4.3. Two-dimensional CIR

5. ConclusionAcknowledgmentsReferences

reactive collisions in confined geometries

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