Chapter \thechapter Fewbody physics of ultracold atoms and molecules with longrange interactions
1 Introduction
The quantum mechanical fewbody problem at ultracold energies poses severe challenges to theoretical techniques, particularly when longrange interactions are present that decay only as a powerlaw potential. One familiar result is the modification of the elastic scattering Wigner threshold laws [sadeghpour2000jpb] in the presence of powerlaw interaction potentials. Analogously, the nearthreshold behaviors of all twobody scattering and bound state observables require generalizations in order to correctly describe the role of long range potentials. These generalizations are often termed modified effective range theories [omalley1961jmp, idziaszek2006pra, friedrich2009pra, friedrich2011pra] or generalized quantum defect theories [greene1979prab, greene1982prab, watanabe1980pra, mies1984jcpa, mies2000prab, burke1998prlb, gao1998pra, gao1998bpra, gao2001pra, gao2005pra, ruzic2013pra, idziaszek2011njp, gao2013pra, ruzic2013pra] in the literature.
For simple, isotropic longrange interaction potentials involving one or more powerlaw potentials, analytical solutions are often known at least at zero energy, and allow extensive development in terms of closedform solutions through the techniques of classical mathematics. [seaton1983rpp, watanabe1980pra, gao1998bpra]
With anisotropic longrange interactions however, such as the dipoledipole interaction characteristic for a gas composed of polar molecules or else of strong magnetic dipoles, often the theoretical heavy lifting relies on numerical solution techniques. [ni2010nature, quemener2012cr, blume2012rpp] For dipolar scattering, numerical methods provide critical detailed information beyond the insights gleaned from simpler analytical approaches such as the partialwave Born approximation. [kanjilal2008pra, bohn2009njp, cavagnero2009njp, wang2012pra, dincao2011pra, ticknor2011pra, giannakeas2013prl]
Moving from the problem of two interacting particles to three or more involves a tremendous leap in the complexity of the theoretical description, as well as a commensurate richness in the phenomena that can occur. For three interacting particles having no long range Coulomb interactions, the realm of universal Efimov physics has received tremendous theoretical interest [amado1972prd, amado1973prd, efimov1972jetpl, efimov1973npa, efimov1979sjnp, macek1986zpd, adhikari1988pra, esry1996jpb, esry1996pra, nielsen1999prlb, esry1999prl, bedaque2000prl, nielsen2001prep, macek2006pra, deltuva2010prab, deltuva2010pra, efremov2009pra, wang2010prl, wang2011njp, wang2012prl, naidon2014prl, wang2014natphys] during the several decades since its original prediction[efimov1970plb, efimov1971sjnp] in 1970. Moreover, it has received extensive experimental attention since the first clear observation of the Efimov effect in the 2006 measurement by the Innsbruck group of Grimm and coworkers.[kraemer2006nature] The community of theorists interested in the Efimov effect started initially in nuclear physics, but the recent studies have concentrated on ultracold atomic and molecular systems. This change in the relevant subfield occurred because the key parameter controlling Efimov states is the atomatom scattering length, which is nowadays controllable near a FanoFeshbach resonance,[fano1935nc, feshbach1958ap, fano1961pr, fano2005J.Res.Natl.Inst.Stand.Technol., feshbach1962ap] through the application of external fields.[stwalley1976prl, tiesinga1993pra, inouye1998NT, chin2010rmp]
One of the interesting aspects of the Efimov effect is that for sufficiently large twobody scattering lengths, the sum of pairwise particle interactions produces a net effective attraction, namely a hyperradial potential energy curve proportional to where is the threebody hyperadius [see Eq. (14)]. Such a potential is sometimes called a “dipole potential” because it also arises in various atomic and molecular twobody systems, in particular when a charge moves in the field of a permanent dipole, which is either a polar molecule[garrett2970cpl, engelking1984pra, lineberger1997pra, clark1980jpb] or else an excited, degenerate hydrogen atom [gailitis1963procphyssoc, seaton1961procphyssoc, greene1980jpb]. A peculiarity of such an attractive inverse square potential is that it is too singular to produce a unique solution near the origin, and that behavior must be regularized in any given physical system. For the chargedipole system, it occurs naturally because any dipole has finite extent and so the dipole potential does not hold once the charge moves inside the charge distribution that produces the dipole. In Efimov physics, the shortrange hyperradial phase as the system moves into the effective dipole potential (Efimov) region of the hyperradius is regulated by the socalled “threebody parameter”. [braaten2006prep, dincao2009jpb]
We discuss the details of the threebody parameter in Sec. 3.1. The main idea is the following: in the presence of zerorange twobody interactions, the energy of the lowest Efimov resonance would drop to negative infinity were it not for the fact that the potential energy does not hold all the way down to for any real system. Zerorange interactions are a convenient model of Efimov physics, but they require the introduction of a threebody parameter to truncate the ground state energy to a finite value. One can view the threebody parameter as setting either a characteristic length or energy related to the deepest Efimov trimer. When expressed as an energy, it is the lowest Efimov energy level at unitarity, and when expressed as a length it is the value of the negative scattering length at which the lowest Efimov state binding energy goes to zero (for ). In some models, such as effective field theory [braaten2006prep] or in a model analysis of the adiabatic hyperspherical potential curves, [wang2013amop, mehta2009prl] it is possible to derive an approximate relationship between these two different ways of defining the threebody parameter. In the first few decades of theoretical study of Efimov physics, it was taken as a matter of fact that the threebody parameter could not be predicted in general on the basis of twobody interactions alone, and that for different real systems that parameter would vary almost randomly. This view was expressed, for instance, in [dincao2009jpb]. Thus it was a major surprise when experiments on the system of three bosonic Cs atoms began to show[berninger2011prl] that several different Efimov resonances have almost the same value of the three body parameter expressed as . In particular, expressed in units of the van der Waals length , which is defined by [gribakin1993pra, chin2010rmp]
(1) 
for two atoms of reduced mass interacting via the van der Waals potential ( is the twobody distance), experiments showed that . Around one year later, theory was able to explain this quasiuniversality of systems having a longrange van der Waals interaction between each pair of particles, but this eventually was explained for a system of three identical bosonic atoms [wang2012prl] which was largely confirmed later by [naidon2014prl] and for a threeboson system with only two of the atoms identical by [wang2012prlb].
It remains of considerable interest to map out this recently identified quasiuniversality of the threebody parameter to see how it depends on the nature of the longrange twobody interactions. Evidence that there could be a universality for interacting dipoles was in fact suggested in two studies of the threedipole problem, first for three oriented bosonic dipoles where an Efimov effect was predicted for the first time in a 3D system where angular momentum is not conserved [wang2011prl]. That study was later extended to three oriented identical fermionic dipoles where there is no predicted Efimov effect, but nevertheless an interesting universal state could be predicted. [wang2011prlb]
The present review summarizes recent developments that have led to an improved understanding of longrange interactions, focusing on twobody and threebody systems with ultracold atoms or molecules. While there have been tremendous accomplishments in experimental fewbody physics in recent years, the present review concentrates on the theoretical understanding that has emerged from combined experimental and theoretical efforts. The theory has been developed to the point where we increasingly understand which aspects of fewbody collisions, bound states, and resonances are universal and controlled only by the longrange Hamiltonian, and which aspects differ and distinguish one species from another having identical longrange forces.
2 van der Waals physics for two atoms
2.1 van der Waals universality and Feshbach resonances with atoms
Let us consider two atoms interacting by a long range potential of the form , where is the distance between the atoms. While the physics associated with various values of has been widely studied, e. g., (the Coulomb potential), (two dipoles, see Sec. 5.1), or (an ion and an atom, see Sec. 6), in this section we consider the specific case of for the van der Waals (vdW) potential between two neutral state atoms. When the length and potential are scaled by their respective van der Waals units of length and energy , the long range potential between the two atoms including the centrifugal barrier for partial wave is [jones2006rmp]
(2) 
It is important to note that there are other conventions in the literature defining a length and corresponding energy associated with the van der Waals potential. Gao [gao1998bpra] uses and Gribakin and Flambaum [gribakin1993pra] introduced the mean scattering length [gribakin1993pra]. The latter is especially useful for giving a simpler form to theoretical expressions.
Gao [gao1998bpra] has worked out the analytic solutions for the bound and scattering states of this potential, which are especially relevant to two and fewbody physics with cold atoms, which to a large extent is governed by the states near the collision threshold, where represents energy. These analytic solutions are parameterized in terms of the wave scattering length [gao2004jpb], a threshold property of the scattering phase shift for the partial wave as collision momentum . Specifying specifies the bound () and scattering () solutions away from for all [gao1998pra, gao2009pra, gao2001pra]. For example, if , there is not only an wave bound state at , but also an bound state for partial waves ; similarly, an wave bound state exists if and a wave bound state if .
It is much more effective to use the analytic solutions to the van der Waals potential instead of the solutions to a zerorange pseudopotential to characterize nearthreshold cold atom physics, since the former are accurate over a much wider range of energy near threshold. This is a consequence of the fact that tends to be much larger than the range of strong chemical interactions that occur when . Thus, the van der Waals potential spans a wide range of between the chemical and asymptotic regions, and its solutions accurately span a range of bound state and collision energies large compared to and large compared to energy scales relevant to cold atom phenomena; see Refs. [jones2006rmp, chin2010rmp]. For example, Chin et al. [chin2010rmp] give examples showing the near threshold spectrum of bound states for waves () and other partial waves () on a scale spanning hundreds of . These bound state energies in units of are universal functions of , that is, speciesspecific parameters like the reduced mass and the magnitude of are scaled out by scaling by and , and the scaled captures the effect of all short range physics.
When the scattering length is large and positive, the energy of the last wave bound state of the potential is universally related to the scattering length by the simple relation,
(3) 
It is simple to make universal corrections to this expression due to the van der Waals potential. Using reduced units of for energy and length, Gribakin and Flambaum [gribakin1993pra] and Gao [gao2004jpb] have developed corrections to the universal binding energy , valid when :
(4) 
where and . Furthermore, the universal van der Waals effective range correction (defined in Section 2.2) to the scattering phase shift for is given by [gao1998bpra, flambaum1999pra]:
(5) 
Note that the effective range expansion breaks down near , or .
FanoFeshbach resonances are possible when a closed channel bound state is tuned in the vicinity of an open channel threshold. A closed spin channel is one with separated atom energy (or threshold energy) that, in the case of alkali metal species with Zeeman spin structure, has a bound state with a different magnetic moment than the separated atoms of the threshold open channel, the difference being . The “bare” closed channel bound state energy tunes with magnetic field as , whereas when the bare closed and open channels are mixed by interaction terms in the Hamiltonian, the scattering length takes on the following form near the resonance pole:
(6) 
where represents the “’background” or “bare” open channel scattering length in the absence of coupling to the closed channel bound state and is the resonance “width.” The field position of the pole at where is normally shifted from , as described in the next paragraph. It is the ability to tune the scattering length to any value using resonance tuning that makes cold atoms such good probes of fewbody physics.
It turns out to be very useful to characterize the properties of various Feshbach resonance in terms of universal van der Waals parameters. In fact, Chin et al [chin2010rmp] introduced a dimensionless parameter to classify the strength of a resonance. Broad, open channel dominated resonances are those with , and narrow, closed channel dominated ones are those with . The former tends to behave like single open channel states characterized by the scattering length alone. The latter tend to exhibit their mixed closed/open channel character, and can not be fully characterized by their scattering length alone. However, all isolated resonances take on the following simple threshold BreitWigner form for the phase shift [julienne2006intconfap, blackley2014pra], here given in its nearthreshold limiting form using the leading terms in an expansion in , valid when :
(7) 
where the reduced slope , and the pole is shifted from the “bare” crossing by . The universal van der Waals Eq. (7) depends on the background scattering length , the “pole strength” in the numerator of the resonance term, and the slope . When collision energy is large enough that the condition is not satisfied, then the simple threshold forms for the numerator and shift term in Eq. 7 need to be replaced by the more complex energydependent universal van der Waals functions of as explained in Refs. [julienne2006intconfap, blackley2014pra]
Julienne and Hutson [julienne2014pra] give examples with Li atoms illustrating the universal binding energy formulas in Eqs. (34). The experimentally measured and coupled channels calculated binding energies for the broad 832G resonance of Li with agree well with the Gao expression , but clear departures are seen from the conventional universal formula . On the other hand, the more closed channel dominant 738G resonance of Li with shows clear departures of calculated and measured binding energies from these simple formulas as the field is tuned away from the pole position. Blackley et al. [blackley2014pra] test the effectiveness of the effective range expansion for for finite energies near the resonance pole for several different cases. While this expansion tends to be quite good near the poles of broad resonances, it tends to fail for narrow resonances, whereas the universal van der Waals expression of Eq. (7), or its energydependent generalization, tends to be much more accurate.
Figure 1 illustrates some basic features of nearthreshold bound states and the scattering length as is tuned for the case of two Cs atoms in their lowest energy Zeeman sub level, based on the theoretical coupled channels model in Ref. [berninger2013pra]. There are three prominent broad wave resonances present with poles near 0 G, 550 G, and 800 G. The corresponding ramping states are evident; the displacement between the apparent crossing of the linear ramping level from the nearby resonance pole position is a manifestation of the shift discussed above. While the last wave bound state is too close to threshold to be seen, the flat and bound states for a scattering length are quite close to the actual calculated levels. Here the integer quantum number counts the bound states down from the threshold at . The agreement of calculated and universal van der Waals  and wave levels illustrates how the pure van der Waals levels are good indicators of the levels of the real system, even on an energy scale large compared to ( 2.9 MHz in this case).
It is also apparent that the different resonances in Fig. 1 overlap to some extent with one another. Reference [jachymski2013prl] shows how to extend the isolated resonance treatment above to a series of interacting resonances. While it is still a good approximation to treat each resonance in an overlapping series as an isolated resonance near its pole, with a uniquely defined parameter, its “local” background (in near a pole) can be a consequence of its interactions with other nearby resonances. While it is usually adequate to consider resonances as isolated ones, we will give an example below in Sec. 3.2 where the threebody Efimov physics of Cs atoms is affected by two overlapping resonances.
2.2 Feshbach resonances and multichannel quantum defect theory (MQDT)
Theories that are typically grouped under the name “quantum defect theory” can be viewed as having two main elements: (1) They systematically separate energy dependences arising from longrange interactions from those associated with shortrange physics; and (2) They connect and interrelate the physics of highlying bound states (spectroscopy) with the properties of lowenergy scattering states (collisions). Historically, the first study along these lines was the development of effective range theory of Schwinger, Bethe, Fermi and Marshall, and others, which expresses the cotangent of the lowenergy wave scattering phase shift as and relates to the energy of a highlying bound state when the scattering length is large and positive [see Eq. (3)]
The most systematic separation of shortrange and longrange influences on singlechannel collisions and spectroscopy in the context of atomic Rydberg states was developed by [seaton1983rpp], who described his theoretical framework using the term “quantum defect theory” (QDT). An earlier forerunner of these ideas was a derivation by [Hartree1928] of the single channel Rydberg formula for any system like an alkali atom with a single electron moving outside of a closed shell ion core. Generalizations exist for effective range theory, e.g. to include energy dependences associated with long range potentials due to polarization [omalley1961jmp] or repulsive Coulomb forces relevant in nuclear physics [Berger1965PR]. One key difference between these generalized effective range theories and quantum defect theories is that the former typically concentrate on developing lowenergy expansions of phase shift behavior, whereas quantum defect theory normally relies on exact solutions as a function of energy and has a wider range of applicability as a result. Quantum defect theory was also generalized to treat attractive or repulsive chargedipole potentials [greene1979prab, greene1982prab] and polarization potentials that are attractive [watanabe1980pra, idziaszek2011njp, gao2013pra] or repulsive [watanabe1982pra].
Typically quantum defect theories have a wider energy range of validity than the simpler effective range theories, because the latter rely on Taylor expansions of the longrange field parameters in powers of the threshold energy whereas the former utilize exact (often analytically known) properties. Subsequent generalizations to treat cold collisions began in the 1980s in the context of chemical physics [mies1984jcpa, mies1984jcpb]. Later when ultracold collisions became crucial to understand for the burgeoning field of dilute quantum gases, theories aiming more concretely at describing twobody interactions with long range van der Waals tails were developed in earnest [julienne1989josab, gao1998bpra, gao2001pra, burke1998prlb, mies2000prab, gao2005pra], and they all demonstrated great efficiency in characterizing the energy dependence of singlechannel and multichannel solutions to the coupled equations whose solutions govern the properties of FanoFeshbach resonances. One of the most recent generalizations and improvements to the theory by [ruzic2013pra] has been applied to the LiCs collision system [Pires2014pra] where multichannel QDT (MQDT) predictions are compared with other treatments such as the full closecoupling solutions as well as the simpler and more approximate “asymptotic bound state model” (ABM) [Tiecke2010pra].
The key idea of all these theories is that beyond some range , it is usually an excellent approximation to treat the motion of separating particles as though they are moving in a simple, onedimensional longrange potential . In a typical system involving cold collisions of ground state atoms, when particles separate beyond a distance of around , the potential energy can be accurately approximated by a van der Waals potential . The solutions to the corresponding radial Schrödinger equation have been worked out in fully analytical form by [gao1998bpra] but it is often simpler to rely on numerical QDT solutions to the radial differential equation. Two linearly independent solutions in this theory are denoted where the superscript “sr” is meant to indicate that these are characterized by “short range” boundary conditions that guarantee these solutions are smooth and analytic functions of energy at small distances .
In the case of long range attractive singlepowerlaw potentials with , all linearlyindependent solutions at small distance are oscillatory all the way down to even for nonzero centrifugal potentials with . Two energyanalytic solutions in the notation of [ruzic2013pra] can be identified in such cases by giving them equal amplitude near the origin and choosing a 90 degree phase difference. For instance, in the WKB approximation which is often adequate to specify boundary conditions in cold collision problems, but not essential, these analytic solutions have the following behavior in channel at small :
(8a)  
(8b) 
where the point is some fixed small distance, and the phase is in general independent of energy and and is chosen according to some convenient criterion. A choice particularly advantageous for at is the one suggested by [ruzic2013pra], which picks the unique phase such that at zero channel energy, the exact zeroenergy independent solution approaches at at channel energy . Because the potential energy is deep in the region where the boundary conditions are chosen, far deeper than the energy range of interest in cold or ultracold collisions for practically any diatomic system, will be extremely smooth functions of energy over a broad range from well above a dissociation threshold to well below. The usual set of real regular and irregular functions from scattering theory that oscillate 90 degrees out of phase at infinity with equal amplitudes are more convenient to use for defining physical reaction matrices. For any long range field one must find the energy and dependent constant coefficients relating to , a relationship written as:
(9a)  
(9b) 
The physical meaning of these parameters has been expounded in various references, although the notations and conventions are not always uniform among the various QDT references. For instance, the long range QDT parameter can be viewed as containing the energyrescaling factor needed such that the solution has the usual energynormalized amplitude at . As such, it contains the threshold law physics, i.e. the Wigner threshold laws and their generalizations appropriate to different long range potentials. The parameter in this notation connects with the parameter in the notation of [mies1984jcpa], where also the long range parameter is written as . The significance of the and functions for cold atomic collisions is discussed by [julienne1989josab, julienne2007acp] along with illustrative figures of their behavior. The parameter is interpreted qualitatively as reflecting the fact that two linearly independent solutions that oscillate 90 degrees out of phase at small generally cannot generally retain this phase difference all the way to . The asymptotic solutions of scattering theory (reaction matrices, etc.) require solutions 90 degrees out of phase at infinity, and relevant phase correction parameter or equivalently must be determined for each long range field of interest.
A major difference between MQDT and ordinary scattering theory is that in multichannel quantum defect theory, one initially postpones enforcement of larger boundary conditions in the closed channels, and works with an enlarged channel space that includes closed () as well as open () channels. The linearlyindependent shortrange (sr) reaction matrix solutions in this enlarged channel space have the following form outside the radius beyond which the potential assumes its purely longrange form:
(10) 
These solutions are sometimes called “unphysical” because they diverge exponentially in the closed () channels at any given energy, and to obtain the physical wavefunctions one must find superpositions of these linearly independent solutions that eliminate the exponentially growing terms. This is straightforward once one writes the coefficient of the exponentially growing terms, namely
(11) 
And the linear algebra that produces the physical reaction is simple, as has been derived in many other references, giving
(12) 
In this form, the appearance of FanoFeshbach resonances is manifestly clear through the matrix inversion that becomes singular near closedchannel resonances. The smoothness of implies that it is often energyindependent and even fieldindependent to an excellent approximation over a broad range, i.e. over 0.11 K in energy and over hundreds of gauss. Moreover the eigenvectors are expected on physical grounds to be approximately given by the recoupling transformation coefficients connecting shortrange channels to the different coupling scheme appropriate at long range. Specifically, in this short range coupling scheme, the nuclear spins and electronic spins of the individual atoms are first coupled to total nuclear and electronic spins and in the presence of strong electronic interaction, then couple to the total hyperfine angular momentum and its projection for the much weaker hyperfine interactions. At zero Bfield, the asymptotic coupling that is relevant has good atomic hyperfine quantum numbers . But when one must diagonalize the BreitRabi Hamiltonian [breit1931pr, nafe1948pr]:
(13) 
to find the atomic dissociation thresholds and eigenvectors and as well, in order to construct a first order approximation to the asymptotically correct channel states. In Eq. (13) , are electron and nuclear factors, is the Bohr magneton. The projection of these onto the shortrange eigenchannels gives a unitary matrix that plays the role of a frame transformation which can give an effective approximation scheme that does not require solution of any coupled equations. Example applications of this multichannel quantum defect theory with frame transformation (MQDTFT) are discussed by [burke1998prlb, gao2005pra, Pires2014pra]. Note that similar content is expressed in the “threeparameter” description of twobody FanoFeshbach resonances as developed by [Hanna2009pra].
2.3 Numerical predictions of Feshbach resonances
Although the MQDT has proved to be a very powerful tool to study lowenergy scattering problems and to make quantitative predictions in many systems, in heavier atomic systems such as Cs, the importance of dipolar interaction [stoof1988prb, moerdijk1995pra], second order spinorbit coupling [chin2004pra, mies1996J.Res.Natl.Inst.Stand.Technol, leo1998prl, kotochigova2000pra], and higher order dispersive potential (, , …) [leo2000prl] make the Feshbach spectrum more complicated and difficult to be treated by the MQDT via frame transformation. It is therefore essential to numerically integrate the Schrödinger equation to study the Feshbach physics in such systems.
Numerical studies of atomic collisions relevant to ultracold experiments were firstly done for hydrogen atoms [Lagendijk1986prb, stoof1988prb]. Later studies then focused on alkali atoms [tiesinga1992pra, tiesinga1993pra, moerdijk1994prl, moerdijk1995pra, boesten1996pra, houbiers1998pra, burke1999pra, chin2004pra] with their cooling down to quantum degeneracy, and more recently on ultracold lanthanizes — Er and Dy [kotochigova2011pccp, petrov2012prl, frisch2014nature]. When including the spin degrees of freedom, ultracold atomic collisions can be solved as a standard coupled channel problem with typically a few to tens of channels. In the cases where the magnetic dipoledipole interaction and/or the anisotropy of electronic interactions are not negligible, orbital angular momentum ceases to be a good quantum number and therefore needs to be coupled. This can lead to thousands of coupled channels in case of highly magnetic atoms like Dy or Er [kotochigova2011pccp, petrov2012prl, frisch2014nature, baumann2014pra] and therefore makes scattering studies quite numerically demanding, if scattering over a wide range of magnetic field is to be investigated.
For collisions between alkali atoms that have been widely studied, only the lowest two BornOppenheimer potentials and are involved. Although these potentials are labeled by the total electronic spin , it is more convenient to expand the scattering wave function in individual atom’s electronic and nuclear spin basis where the Hamiltonian matrix can be efficiently evaluated (see, for example, Ref. [kohler2006rmp, chin2010rmp, berninger2013pra, hutson2008pra] for implementations). In the asymptotic region where interactions between atoms vanish, the scattering solution is projected to the eigenbasis of individual atoms — the atomic hyperfine states (see Sec. 2.2) — where the scattering matrix is extracted.
The main challenge in the numerical study of ultracold collisions is the large integration range in the atomic separation, since the range should be at least a few times of the de Broglie wavelength. This typically means a range over Bohr and could be even larger when close to a Feshbach resonance. At small interatomic distance the BornOppenheimer potentials are generally deep enough for the scattering wave function to have hundreds of nodes, which require a good number of radial points or radial elements to represent. Although it is not a real issue for modern computers to solve such a problem in the simplest scenario, the calculations start becoming cumbersome when different partial waves are coupled by, for instance, the magnetic dipoledipole interaction.
To improve numerical efficiency, propagation methods are widely used in such scattering studies and also used in finding weaklybound state energies of Feshbach molecules. For scattering calculations, a practical choice is the eigenchannel matrix propagation method [aymar1996rmp], which is numerically stable and has the flexibility of being adaptive to different radial representations.
3 van der Waals physics for three atoms
3.1 Universal threebody parameter
The threebody parameter is central in studies of lowenergy threebody physics. The origin of the parameter can be traced back to the early discovery of a peculiar quantum behavior for three particles — the Thomas collapse [thomas1935pr]. In 1935, Thomas discovered that the ground state energy of three identical bosons diverges to as the range of pairwise interactions shrinks to zero — even if one keeps the twobody binding energy a constant in this limiting process.
The threebody parameter is essentially introduced to “regularize” the unphysical divergence of the threebody spectrum. In the effective field theory (EFT) [braaten2006prep] or in the momentumspace Faddeev equation [faddeev1961jetp], the Thomas collapse is manifested as a logrithmic divergence of the threebody spectrum when the interparticle momentum is taken to the “ultraviolet” limit [amado1971plb, amado1972prd, bedaque1999prl]. This divergence can be formally avoided by adding a threebody force that effectively introduces a cutoff in the interparticle momentum, which assumes one form of the threebody parameter.
The threebody parameter can be more intuitively understood in the hyperspherical coordinates [wang2010thesis]. In fact, this was the way used by Efimov to discover the Efimov effect [efimov1973npa]. The hyperspherical corrdinates are defined for many particles in analogy to the spherical coordinates for one particle, where only one of the cooridinates — the hyperradius
(14) 
— represents distance, whereas all other coordinates are defined as “angles”. The threebody mass can be defined as
(15) 
The advantage of the hyperspherical coordinates is primarily from being a universal breakup coordinate for all defragmentation processes. The hyperangles, represented by a collective of coordinates , can have different definitions according to the choice of orthogonal coordinates [wang2010thesis].
Fixing in the limit is essentially equivalent to keeping the twobody scattering length unchanged. To study the behavior of three particles in the universal limit where , it is easier to consider the special case with and . In this case it is easy to show [efimov1973npa] that the hyperadial motion separates from the hyperangular motions. The hyperangular motions can be solved analytically and lead to the following hyperradial equation:
(16) 
where are universal constants [efimov1973npa]. For identical bosons and some combinations of massimbalanced threebody systems, the lowest is negative — we denote it as with . Eq. (16) then leads to the wellknown “falltothecenter” problem [perelomov1970theomathphys] — the threebody energy has no lower bound. The problem comes, obviously, from the effective hyperradial potential
(17) 
being too singular near the origin. To avoid this threebody collapse, some regularization is needed such that the potential does not extend all the way to . A simple cure is to give a cutoff to the potential, which is equivalent to giving an earlier mentioned momentumspace cutoff in the EFT. In either case, a cutoff leads to a finite threebody ground state energy , or a binding wavenumber which is another form of the threebody parameter.
Although the situation is not realized in nature, the unitarity condition where is now routinely realized in ultracold atoms by using Feshbach resonances [chin2010rmp]. In this case, the longrange potential leads to an infinite number of threebody bound states — Efimov states — with energies following a geometric scaling , which can happen even when none of the twobody subsystems is really bound ( implies ). This is known as the Efimov effect [efimov1970plb, efimov1973npa] in quantum physics.
In case of finite , the potential still exists but only extends to a distance , so there will be only a finite number of Efimov states. As demonstrated in Fig. 2, however, when increases more Efimov states become bound either below the threebody breakup threshold or the atomdimer threshold, for or , respectively. The values of where the first Efimov state becomes bound, () or (), as well as the position of the first threebody recombination minimum [braaten2006prep, wang2013amop] at , can also represent the position of the ground Efimov state and are therefore often referred to as the threebody parameter as well.
Generally speaking, the above mentioned representations of the threebody parameter are related by the universal relations. For identical bosons these are known as [braaten2006prep, gogolin2008prl]
(18) 
In realistics systems such as ultracold atoms, the ground Efimov state is often “contaminated” by the details of shortrange interactions that deviate from the effective potential, so large deviations from the universal relations are often reported in the experimental observations at low scattering lengths [kraemer2006nature, knoop2009natphys, pollack2009science, ferlaino2011fewbodysys, dyke2013pra].
With the concept that the Efimov potential extends to small and keeps attractive even when it is modified by shortrange interactions, it is natural to expect that the ground Efimov state energy depends strongly on how this potential behaves from all the way up to , or in the very least, on the semiclassical WKB phase in this range of the potential. It can be shown [wang2010thesis] that when changes by the whole Efimov spectrum is overall shifted by , whereas the lowenergy threebody observables are kept unchanged. In atomic systems as shortrange interactions are strong enough to hold hundreds to thousands of rovibrational states already in the twobody level, a less than 1 percent change in the interactions would be sufficient to change the shortrange phase by many multiples of , which makes the prediction of the threebody parameter practically almost impossible. In ultracold experiments, when the magnetic field is scanned through different Feshbach resonances, the number of twobody bound states changes so that the WKB phase in the twobody interactions (at zero energy) changes through multiple of ’s. An obvious guess on the change of threebody phase is in a range greater than the change in the twobody phase. Similarly, if experiments are done in different atomic hyperfine states it had been expected that the change in the interactions is also significant enough to give completely different threebody phases.
Another layer of complication is that in realistic threebody systems there are usually many atomdimer breakup thresholds. In the hyperspherical representation they are manifested by atomdimer channels, whose potential energies go asymptotically to the dimer energies. These potentials should in principle couple to the Efimov potential when , which seems to make the prediction of a threebody parameter even less likely.
Finally as shown in Fig. 3, the nonadditive threebody forces for alkali atoms, which differ drastically for different atomic species or different atomic spin states, are often stronger than the sum of twobody interactions near the distance of chemical bond ( Bohr). If threebody forces do contribute to the threebody shortrange phase, there would be even less hope for the prediction of a threebody parameter due to the finite precision at which the potential surfaces can be calculated.
Based on the above considerations, it has been generally believed [braaten2006prep, dincao2009jpb] that the threebody parameter should not carry any universal properties. However, the experiments with Li atoms [gross2009prl, gross2010prl] showed the first contradiction to this expectation: the threebody parameter was measured to be independent of the atoms’ hyperfine state. More surprisingly, the Innsbruck group later measured for Cs near Feshbach resonances with quite different characters but found a persistent value of [berninger2011prl]
(19) 
Moreover, the magnitude of the threebody recombination induced loss rate has been observed to be similar near the Efimov resonances in different regions of magnetic fields [ferlaino2011fewbodysys]. After the Innsbruck experiment, more experimental results including the earlier ones (shown in Table 3.1), are found to be consistent with this “universal” value of when it is cast in the unit of .
The clear disagreement between the experiments and the theoretical expectation triggered a good deal of interest in understanding the physics behind this universality. A key question that needs to be addressed here is the insensitivity of the threebody parameter to the complexity of the threebody dynamics — vibrational and rotational excitations — at small distances, which occurs in the case of alkali atoms where the interactions are strong enough to have hundreds or thousands of rovibrational states already on the twobody level.
This rather mysterious universality was soon understood by an effective threebody barrier in the hyperspherical coordinates whose repulsive wall is located near . This was demonstrated by [wang2012prl] where all the twobody rovibrational states from rather deep van der Waals potentials are included in their calculations. Naidon, et al. [naidon2014prl] also show such a “universal” barrier by including only the vibrational states. Intuitively, this barrier gives a shortrange cutoff to the Efimov potential and “protects” the Efimov states from the influence of threebody forces at small distances. This barrer leads to the suppression of the threebody amplitude inside the barrier, which is in fact not a necessary condition for a universal threebody parameter — as we will discuss below for the heteronuclear systems with extreme mass ratios. The crux for the universality lies in the universal position of the hyperradial node or quasinode (a nodelike structure where the amplitude is small but finite) that can be either understood by the rise of the effective barrier or by the sharp increase in the interaction strength [wang2012prl, chin2011arXiv]. This nodal structure has also been interpreted as a consequence of universal threebody correlation for van der Waals interactions, and can be generalized to other types of shortrange interactions except for square well potentials [naidon2014prl].
Similar to the homonuclear threebody systems, a heteronuclear threebody system can also have the Efimov effect, particularly when two of the particles are heavy (identical) and the scattering length between heavy () and light () particles is large [efimov1973npa]. In such a system the Efimov scaling factor is small [efimov1973npa, dincao2006pra], excited Efimov states are therefore easy to form and this is often called an “Efimovfavored” scenario.
Since the scaling behavior of the abovedmentioned hyperradial potentials depends on the mass ratio [efimov1973npa, dincao2006pra], the universality in the threebody parameter found in homonuclear systems cannot be readily carried over to heteronuclear systems. Nevertheless, Wang, et al. have shown [wang2012prlb] that the threebody parameter in a heteronuclear atomic system is also universally determined by without being affected by the details of shortrange interactions. In the “Efimovfavored” systems, the universality of the threebody parameter is particularly understood in a simple picture when the motion of the light atom is treated by the BornOppenheimer approximation [wang2012prlb].
In this case the Efimov physics is dictated by the BornOppenheimer potential , where is the distance between the atoms. Here the origin of the universal threebody parameter is a combination of the universal property of the Efimov behavior in at large and that of van der Waals behavior at small . This can be easily seen by writing as
(20) 
where is the “bare”  interaction and is the interaction induced by . The index labels the quantum states of when atoms are fixed in space. The channel relevant to the Efimov effect is the highest state with symmetry (zero angular momentum projection along the  axis and the wave function is symmetric upon inversion of its coordinates). It is wellknown that in this channel, has the behavior
(21) 
in the range , where is the mass of , is the van der Waals length between and , is a universal constant. At , behaves in a complicated way due to avoided crossings with other channels, but has a magnitude on the order of the van der Waals energy , where is the reduced mass of and .
On the other hand, in the range , where is the distance where the electronic exchange interaction starts becoming significant, with often satisfied in atomic systems. In such a scenario, the potential is dominated by the behavior when but is otherwise dominated by the van der Waals interaction when . Importantly, thanks to the universal properties of van der Waals interaction (see Sec. 2.1) the nodal structure of the BornOppenheimer radial wave function in the van der Waals dominant region is completely determined by , or , without refering to the interactions at distances near or smaller than . Moreover, this nodal structure “anchors” the position of the next node in the Efimov region, which determines , or the threebody parameter. Such “universal” nodal determination is demonstrated in Fig. 4 for a threeatom system with extreme mass ratio — YbYbLi. In Fig. 4, the formation of a regular nodal structure in the region is clearly shown when the van der Waals interaction extends to smaller distances but with fixed. In fact, this nodal structure closely follows that in the zeroenergy wave function in a pure potential with the same .
Briefly speaking, the universality of the threebody parameter in an “Efimovfavored” system can be understood as the following: all the nonuniversal ingredients in near are “dissolved” by the strong van der Waals interaction in , whereas the nonuniversal ingredients in are completely “absorbed” in . Using the above picture, a universal threebody parameter can be obtained analytically, which depends only on , , and the mass ratio [wang2012prlb].
In the “Efimovunfavored” heteronuclear systems where , Ref. [wang2012prlb] shows that the threebody parameter is still universal, but with a mechanism similar to the identical boson case, i.e., with the existence of a universal repulsive barrier.
Finally, it should be noted that the above theoretical analyses are based on singlechannel atomic interactions and are therefore not expected to be applicable to narrow Feshbach resonances. Experimental observations on identical bosons, however, have shown consistency with the universal threebody parameter for relatively narrow ones in Li and K (see Table 3.1). These interesting observations are yet to be understood by future studies. In addition, in recent ultracold experiments on the CsCsLi system [tung2014arXiv, pires2014prl] where the CsLi Feshbach resonance is relatively narrow, the observed threebody parameter is also consistent with the theory for broad Feshbach resonance — a prediction based on Ref. [wang2012prlb]. The physics behind these unexpected agreements is still under investigation, and may be understood by the threebody spinor models introduced in Sec. 3.2, where the multichannel character of the Feshbach resonances is properly represented.
3.2 Threebody physics with spinors — multichannel models
Although the situation of large scattering length is almost always produced by multichannel phenomena in nature, the majority of studies on Efimov physics so far are performed by singlechannel interactions. Such treatment is adequate near broad, or openchannel dominant Feshbach resonances where studies [kohler2006rmp] have shown that the twobody physics in those cases can be well represented by models with singlechannel interactions.
Generally, an isolated Feshbach resonance can be characterized by two parameters: the background scattering length and the parameter. A broad or narrow Feshbach resonance can be distinguished by or . Today the question on Efimov physics and threebody scaling properties is more focused on the cases of nonbroad Feshbach resonances where is not satisfied so that a singlechannel treatment is expected to fail. As such Feshbach resonances are commonly accessed in ultracold experiments, it is particularly desirable to understand the threebody physics in such cases.
The success of the zerorange interaction model in predicting the Efimov effect [efimov1970plb] make it attractive to extend the model to a multichannel version. Efimov made an early effort in this direction to understand how the Efimov effect can manifest in the presence of the nuclear spin [bulgac1976snjp]. Some other recent efforts focused more on reproducing the energy dependence of twobody scattering near narrow Feshbach resonances by generalizing the zerorange boundary conditions or by introducing a twobody “molecular state” into the threebody Hamilton [petrov2004prl, lee2007pra, gogolin2008prl, massignan2008pra, jonalasinio2010prl, pricoupenko2010pra]. These theories all introduce a parametrization by the effective range or equivalently the parameter [petrov2004prl, gogolin2008prl]. For atoms with van der Waals interactions is defined as ; near narrow Feshbach resonances there is further the relation . Such parametrization, though simple and elegant and is expected to work in the extreme limit and , is not sufficiently capable of representing van der Waals physics for three atoms.
More sophisticated multichannel zerorange models treat the closedchannel Hilbert space in equal footing with the openchannel space, rather than simply a bound “molecular state”. A relatively straightforward way to do this is to write the threebody wave function as a superposition of open and closedchannel components and , and match separately in the open and closedchannel zerorange boundary conditions [sorensen2012pra, sorensen2013jpb, sorensen2013fewbodysys]. A more rigorous way is to build the internal degrees of freedom — such as spins — directly to individual atoms, as the multichannel zerorange model by [mehta2008pra].
Thanks to their simplicity, the multichannel zerorange models have recently been applied to fit threebody recombination losses in ultracold experiments [sorensen2013fewbodysys] and to study spinor condensates [colussi2014fewbodysys]. It should be noted, however, that the zerorange boundary condition for the closedchannel wave function implies a weaklybound twobody Feshbach state in the closed channel, which is rarely the case for realistic systems. In addition, the multichannel zerorange models still have the Thomas collapse [sorensen2013fewbodysys], which also limits their predictive power for ultracold experiments.
To better understand the role of van der Waals physics in ultracold collisions of three atoms, it is desirable to have a multichannel model that has finiterange van der Waals interactions built in. Moreover, the connection found by [wangj2012pra] between a wave twobody van der Waals state and a threebody state suggests that a correct characterization of the highlying twobody van der Waals spectrum is also important for predicting ultracold threebody physics in atoms.
Recently, significant progress has been made by [wang2014natphys, wangUnpub] down this road, where the authors have developed a threebody multichannel model with van der Waals interactions. By allowing atoms in their model to carry multiple spin states and interact via spincoupled van der Waals potentials, the authors have successfully reproduced/predicted many experimentally observed threebody loss features without fitting parameters, particularly in the case of Cs where threebody physics has been extensively studied in Innsbruck experiments [ferlaino2011fewbodysys]. After the groundbreaking discovery of a universal threebody parameter, this theoretical work shows that without addressing the unknown shortrange details, ultracold threebody physics in atoms can be quantitatively predicted. This pushes the front of fewbody research to a much further position.
In Wang, et al.’s work, three atoms are represented by the Hamiltonian [wang2014natphys]
(22)  
Here is the kinetic energy for the relative motion of three atoms, is the spin state of atom , , or , which can take , , , … in a multispin model. Also, is the magnetic moment for an individual atom, and is the zerofield, singleatom energies that represent the hyperfine splittings. The pairwise twobody interactions are in a symmetrized twobody spin basis ( is the symmetrization operator), with 612 LennardJones potentials in the diagonals. Thanks to the universal properties of twobody van der Waals physics [gao2000pra], the nearthreshold twobody spectrum can be reasonably reproduced by the LennardJones potentials that are deep enough for a few bound states. By including more spin states this Hamiltonian is capable of representing complicated Feshbach physics in cold atoms. For instance, isolated Feshbach resonances can be represented by including two spin states for each atom, whereas overlapping Feshbach resonances can be represented by three spin states.
With this model, Wang, et al. have studied an interesting case in Cs where an Efimov doublet was observed near a narrow wave Feshbach resonance [berninger2011prl, ferlaino2011fewbodysys]. Their calculations successfully reproduced the positions of the Efimov resonances in the doublet and showed how the magnitude of the loss rates can be predicted correctly when a 2spin model is replaced by a 3spin model, where the overlapping character of the wave Feshbach resonance is better represented [wang2014natphys]. Moreover, even though the theory agrees with the experiment in that the positions of the Efimov resonances in this case are all near the universal value, Wang, et al.’s theory further points out that the universal resonance positions are actually a consequence of being near the universal value () and should not be considered as general [wangUnpub].
The above mentioned multichannel threebody models can also be applied to heteronuclear atomic systems. For instance, to study the CsCsLi Efimov physics in the ultracold CsLi admixtures realized in the Chicago group [tung2014arXiv] and the Heidelburg group [pires2014prl], a 2spin model can be used to treat the CsLi interactions [tung2014arXiv]. With the great variety of heteronuclear combinations that are accessible to future experiments, theoretical studies with multichannel models are just in the beginning. In this context, many questions on the van der Waals universality still remain to be answered.
Finally, it should mentioned that the magnetic Feshbach resonances are by far not the only place where multichannel physics is important. In the emerging fields of ultracold studies where light is used to manipulate the atomic interactions, such as the optical Feshbach resonance [chin2010rmp] and synthesized spinorbit interactions [zhai2012intjmpb], new, interesting fewbody phenomena due to multichannel physics is already start being predicted or observed (see Sec. 6).
4 van der Waals physics in the chemical reaction of compact molecules
We have been discussing the weakly bound states of three atoms, or the collisions of an atom with a weakly bound dimer. Now we turn to collisions or chemical reactions of ordinary molecules, for example, in their ground vibrational state. Such molecules are much more complex than atoms, having additional degrees of freedom, and their chemical reaction dynamics provides an excellent example of fewbody dynamics. While we may expect that molecular dynamics is too complex to show any universality, we will show here that there are some cases, perhaps even widespread, where universality is found, that is, insensitivity of the reaction rate coefficient to the details of shortrange chemical interactions.
It is wellknown that some chemical reactions are described by a universal classical model when there is a unit probability of a reaction occurring if the two reactant species come in close contact with one another. The collision cross section is then determined by the long range potential: it is only necessary to count those classical trajectories of the colliding species that are captured by the long range potential such that the collision partners spiral in to close contact with each other. The Langevin model [langevin1905acp] is a good example of this for ionmolecule reactions, and the related Gorin model [gorin1938actphys, gorin1939jcp] describes barrierless reactions between neutral chemical species that collide via a van der Waals potential. [fernandezramos2006cr] describes the Langevin and Gorin models in the context of chemical kinetics.
In the low temperature regime such classical capture models need to be adapted to the quantum mechanical nearthreshold dynamics governed by quantum threshold laws and resonant scattering. This can be readily done based on QDT treatments of neutral [gao1996pra] or ionic reactions [jachymski2013prl]. We concentrate on the neutral case here, which has also been reviewed by [quemener2012cr]. It is important to note that QDT theories are not unique, but can be implemented in different ways, in either their analytic or numerical versions, depending on how the reference solutions to the problem are set up. We concentrate here on the particular QDT implementation of Mies [mies1984jcpa, mies1984jcpb, julienne1989josab]. Gao [gao2008pra, gao2010prla, gao2010prlb, gao2013pra] obtains similar results with his implementation of the theory.
As discussed in Sec. 2.2, the essence of QDT for molecular reactions is the separation of the interactions into short range and long range regions, with the former characterized by a few quantum defect parameters that allow for chemical reaction at short range. Generalizing the classical Gorin or Langevin models to the quantum threshold is especially simple in a QDT framework [idziaszek2010pra]. The approach of the two colliding species is governed by long range quantum dynamics, whereas the short range region is characterized by perfectly absorbing, or “black hole” boundary conditions that allow no backscattering of the incoming partial wave from the short range region. On the other hand, the long range region can quantum reflect the incoming flux in a way strongly dependent on partial wave , allowing for quantum tunneling to short range in the case of centrifugal barriers when . Following Gao [gao2010prla] we will call QDT models with such “black hole” boundary conditions quantum Langevin (QL) models.
In general, the short range region may not exhibit unit reactivity, but may reflect some of the incoming flux back into the long range entrance channel. The interference of incoming and outgoing flux establishes a phase shift of the long range standing wave that is related to the scattering length. In this “grey hole” partially reactive case, two dimensionless quantum defect parameters are needed: one, , related to the phase shift and one, , related to the short range reactivity. The probability of short range reaction is [jachymski2013prl]. The rate constants for elastic scattering or for inelastic or reactive scattering loss from the entrance channel are determined from the diagonal matrix element , which can be specified for any partial wave and by the complex phase shift , or alternatively, by a complex energydependent scattering length defined by [idziaszek2010prl]:
(23) 
[idziaszek2010prl] give the universal van der Waals  and wave in QDT form, which reduce in the limit to (in van der Waals units):
(24) 
In the specific “black hole” case of the QL model, where , the complex scattering lengths take on an especially interesting universal threshold form, independent of and with equal magnitudes for their real and imaginary parts: and . The corresponding elastic and reactive/inelastic loss rate constants are determined respectively from and . The loss rate constants for the lowest contributing partial waves near the threshold are
(25) 
where the symmetry factor in the case of identical particles in identical internal states and otherwise; identical bosons only have even collisions and identical fermions only odd ones, whereas nonidentical particles have both. The numerical factor in the wave expression is wave are summed, assuming them to have identical complex scattering lengths, as would be the case for a rotationless molecule (total angular momentum ). Thus, the threshold thermally averaged universal van der Waals rate constants for wave collisions is the constant expression in Eq. 25(a), and for wave collisions of identical fermions it is , where all three components of the
(26) 
where is the Boltzmann constant and is the temperature. Thus, threshold wave collisions for “black hole” collisions are universally specified by the vdW length and wave ones by the van der Waals volume .
The predictions of these universal rate constants for the QL model agree well with measurements on threshold chemical reaction rates of KRb fermions in their , state. The wave expression in Eq. 26 agrees within experimental uncertainty when the fermions are in a single spin state, verifying the linear variation of rate with . The wave expression in Eq. 25(a) agrees within a factor of 2 or better when the fermions are in different spin states or when they react with K atoms [idziaszek2010prl].
[quemener2010praa, ni2010nature] developed an approximate threshold model to extend the Gorin model when an electric field is introduced to cause the molecules to have a laboratoryframe dipole moment. [idziaszek2010pra] extended this method using a hybrid numerical quantum defect theory based on a sum of van der Waals and dipolar long range potentials to predict the variation of reaction rates of identical fermions with electric field, in excellent agreement with experiment. These quantum defect treatments can readily be adapted to quasi1D (“tube” geometry with tight confinement in two directions) [olshanii1998prl] or quasi2D (“pancake” geometry with tight confinement in one direction) [petrov2001pra] collisions of reduced dimensionality D due to quantum confinement by optical lattice structures. The threedimensional (3D) boundary conditions due to short range interactions is incorporated in the QDT parameters and , and the asymptotic boundary conditions can be taken to be appropriate to the situation of reduced dimension due to quantized tight confinement [naidon2007njp]. [micheli2010prl] extend the universal vdW rate constants in Eqs. 25(a) and 26 to quasi1D and quasi2D geometry, and additionally develop numerical methods for “black hole” collisions in quasi2D geometry when the molecules have a dipole moment; [quemener2011pra] also give similar calculations for KRb, and [quemener2011prab] and [julienne2011pccp] discuss the universal collisions of polar molecules containing other alkalimetal atoms. These calculations explain experiments with KRb fermions in “pancake” quasi2D geometry, verifying earlier predictions [buchler2007prl] that aligning the dipoles perpendicular to the plane of the “pancake” suppresses the reaction rate, instead of increasing rapidly with dipole strength as in ordinary threedimensional collisions.
QDT models can be readily extended to include more partial waves as temperature increases. Both [gao2010prla, gao2010prlb] and [jachymski2013prl] have done this to get universal QL rate constants for both the van der Waals and the ionatom cases ( 6 and 4), including the quantum effect of centrifugal barrier tunneling. The universal QL model goes to the classical Gorin or Langevin models when the temperature goes above a characteristic temperature associated with the long range energy scale. The QDT model can also be implemented for the more general “grey hole” case, where both and are needed. Jachymski et al [jachymski2013prl] apply the general QDT theory to explain the merged beam experiments on the Penning ionization of S metastable He atoms with Ar atoms [henson2012science] up to collision energies on the order of 10 K. In this case, where both atoms are in states and there is a single potential, the contributions from all partial waves, including a prominent shape resonance for , can be explained by a single wave complex scattering length parameterized by , . In the more general case of molecules with anisotropic potentials, we can expect to need dependent QDT parameters. Gao’s discussion of the general case [gao2008pra, gao2010prla] also describes the role or resonance states, which allow incoming and outgoing scattering flux to make multiple passes between the inner and outer regions, quite unlike the “black hole” case, where resonances are suppressed by the total absorption at short range.
The review by [quemener2012cr] examines the applicability of the universal QL model to a variety of molecular collisions governed by the van der Waals potential. They review work on the vibrational quenching collisions of atom with () or () molecules, where and represent alkalimetal atoms and is an excited vibrational level. Both measured () quenching in magnetooptical traps around 100 K (see their Fig. 21) and calculated threshold quenching of () (see their Fig. 41) tend to be within around a factor of 2 to 4 from the predictions of universal QL model. This suggests that vibrational quenching in general tends to have a high probability of a short range quenching event, that is, has a relatively large not far from unity. The question that remains to be answered is why the quenching rate coefficients for different species and levels fluctuate by several factors from the predictions of the QL model.
Finally, it is important to note that real molecular collisions are likely to involve a very dense set of resonance states associated with the various vibrational and rotational degrees of freedom in the molecule. The Bohn group [mayle2012pra, mayle2013pra, croft2014pra] has described how the density of resonances are so high that statistical random matrix theories are needed to describe their effects even at ultralow temperatures. Recent cold atoms experiments with the strongly magnetic dipolar atomic species Er have uncovered a high density of Feshbach resonance states characterized by a WigerDyson distribution of resonance spacings that is a characteristic of random matrix theory [frisch2014nature]; similar experiments are being carried out with Dy atoms [baumann2014pra]. If there are many resonances within the thermal spread , the molecules are predicted to stick together in a longlived collision complex with a universal collision rate given by the QL model [mayle2013pra]. If nonreactive molecules in such a collision complex should undergo short range loss processes with unit probability and never returned outgoing flux to the entrance channel, then they would undergo loss processes at the same rate as a highly reactive collision. The question of the magnitude and control of collision rates for the quenching of internal vibration, rotation, or spin for ultracold molecules having dense sets of resonance levels, as well as the nature of their threebody collisions, remains an open research area experimentally and theoretically.
5 Dipolar physics for two and three atoms
As discussed in the previous sections, the recent experimental development in producing ultracold groundstate polar molecules [quemener2012cr] or ultracold highly magnetic atoms [hancox2004nature, werner2005prl, griesmaier2005prl, stuhler2005prl, lahaye2007nature, kanjilal2008pra, lahaye2008prl, koch2008naturephys, connolly2010pra, lu2011prl, lu2012prl, aikawa2012prl, aikawa2014prl, petrov2012prl, frisch2014nature] has stimulated a lot of interest in the study of fewbody systems with the longrange dipolar interactions. To the simplest level, dipolar atoms and molecules can both be modeled as point dipoles with a “permanent” dipole moment that can be aligned by external fields. As has been discussed in Sec. 4, however, the elastic properties of such models do not hold for reactive molecules due to the highly inelastic chemical processes that occur when molecules are close together, and the random matrix theory arguments [mayle2012pra, mayle2013pra, croft2014pra] predict that even for nonreactive molecules the elasticity is practically nonobservable. We therefore restrict the application of our discussion below to atoms only where the elasticity of the shortrange interactions is still preserved at current experimental conditions.
5.1 Universal dipolar physics for two fieldaligned dipoles
Generally speaking, the dynamics of dipoles include rotation of individual dipoles where the angular momentum of a dipole is exchanged with its environment. In practice, however, it is highly challenging to include the rotational degrees of freedom in studies of fewdipole physics even for point dipoles. Nevertheless, from the point of view of universal physics it is more preferential to “freeze” the rotation of individual dipoles by orienting them with external fields and study the scaling behavior of dipolar systems with the anisotropic, longrange interactions.
For oriented dipoles, their interaction potential can be simply written as
(27) 
where the dipole length is the characteristic length scale for the dipolar interaction [bohn2009njp], which connects to the induced dipole moment by
(28) 
and is tunable by an external fields [roudnev2009jpb]. The unit vector is along the direction of the external field. For magnetic dipolar atoms can saturate to as large as a few hundred Bohr, whereas for alkali dipolar molecules the saturation value is between Bohr to Bohr. Without a shortrange cutoff, the dipole potential is too singular at the origin — the system collapses again due to the “falltothecenter” problem [perelomov1970theomathphys]. In reality the “shortrange” interaction always deviates from Eq. (27) and is regular at the origin. A nonuniversal shortrange cutoff funtion needs to be introduced in theoretical studies such that
(29) 
In spite of the nonuniversal behavior at small distances, universal scaling properties with the strength of the dipolar interaction are in the central of the study. Ref. [wang2012pra] shows that unlike shortrange interactions, the dipole potential leads to a lowenergy expansion in the scattering phase shift, whose real part of characterizes the crosssection where is not changed during a collision and has the form
(30) 
for any partial wave , with the power of wavenumber increases by 1 for consecutive terms. Moreover, it is found that the expansion coefficients may or may not have shortrange dependence according to [wang2012pra]. For all the coefficients are shortrange dependent. For , however, shortrange dependence only starts from the term and beyond. Therefore, for bosons nonuniversal behavior is generally expected in lowenergy scattering, whereas for fermions universal expressions for scattering observeables can be derived [bohn2009njp, wang2012pra]. The imaginary part of characterizes changing collisions, which has in the leading order and has the leading coefficient that has been analytically derived [wang2012pra].
In analogy to the magnetic Feshbach resonance, formation of dipolar resonances by tuning is also of great interest. It is known that the dipolar couplings between different lead to both shape and Feshbach resonances [deb2001pra, ticknor2005pra, ticknor2007pra, kanjilal2008pra, roudnev2009pra, roudnev2009jpb] and the relative positions between these resonances are universal [kanjilal2008pra, roudnev2009pra, roudnev2009jpb] — i.e., independent of shortrange details. Due to the anisotropy of the dipolar interaction. However, the different partial wave characters that dipolar resonances bear make them line up with quite irregularly. Nevertheless, such knowledge is still important for the development of manybody dipolar theory, where the renormalization theory can be readily applied and system independent properties can be derived [beane2001pra].
Being the connector between microscopic and macroscopic phenomena, fewbody studies, in particular twobody studies, play the role of developing simple models that can be used in studies of manybody physics. To this end, zerorange models that can correctly reproduce lowenergy scattering properies have been developed in both 2D [kanjilal2006pra, shih2009pra, dincao2011pra] and 3D [yi2000pra, yi2001pra, derevianko2003pra, derevianko2005pra]. In view of stable dipolar gases against inelastic collisions, elastic twodipole physics in 2D or quasi2D geometries has been studied in great details in Refs. [ticknor2009pra, ticknor2010pra, armstrong2010epl, ticknor2011pra, dincao2011pra, volosniev2011jpb, ticknor2012pra, zinner2012fbs, koval2014pra].
For studies of Efimov physics, however, zerorange models do not have enough information in the scaling behavior of nearthreshold bound states, which is crucial for studies of the scaling laws. Numerical study based on the potential in Eq. (27), however, indicates that the characteristic size of the nearthreshold dipolar states scales like and their energies scales with [wang2012pra].
In studies on the transition from the weak dipole regime () to the strong dipole regime (), the concept of angular momentum mixture is often used to indicate the level of anisotropy of a dipolar state. Although this characterization is useful in connecting the ordinary molecular physics to the dipolar physics when the dipolar interaction is not very strong (or the dipoles are not strongly aligned), it becomes less useful in the strong dipole regime where lots of angular momenta make contributions and dipolar states take characters of “pendulum” states [wang2012pra]. Nevertheless, the expectation value of the square of the angular momentum in this regime follows a universal scaling
(31) 
where is the vibrational quantum number of the pendulum states and is the magnetic quantum number.
5.2 Universal threedipole physics
The universal properties found in twodipole physics have clearly stimulated interest in the universal physics for three dipoles. To begin with, we would first like to point out the nontrivial aspects in threedipole physics, particularly those relevant to Efimov physics:

The prediction of the Efimov effect was based on isotropic interactions where the total orbital angular momentum is conserved. In the case of dipoles, however, it is not clear how the couplings between partial waves may impact the Efimov effect.

The Efimov effect also assumes a scattering length only in the wave collisions, whereas the longrange dipolar interaction brings in equally defined scattering lengths for each partial wave.
With the above question marks, a clear, definitive answer to the threedipole Efimov effect can only be given by a quantitative threebody theory where dipolar interactions are properly treated. To this end, the hyperspherical method would be a good choice for the study, where the existence of the Efimov effect can be directly read off from the longrange scaling behavior of the hyperradial potentials [wang2010thesis].
By numerically solving the Schrödinger equation in the hyperspherical coordinates, Wang, et al. showed [wang2011prl] that the Efimov effect does exist for three bosonic dipoles near an wave dipolar resonance where the wave scattering length goes through a pole. It is also found that the dipolar Efimov states follow the same scaling properties as the nondipolar states. More interestingly, contrary to a common expectation that the threebody parameter is the most likely nonuniversal because of the irregular pattern in the twodipole spectrum (see Sec. 5.1), the ground Efimov state energy is found to be universally determined by without referring to the shortrange details. This finding is similar to what has been discussed in Sec. 3.1, but was discovered before the concept of universal threebody parameter being built. In this case, the universal threebody parameter is also manifested by a repulsive barrier in the hyperradial potential, but appears near . The dipolar Efimov state is still wave dominant, the three dipoles in this state therefore prefer to stay far away so that its size is much bigger than . At the dipolar resonance, the position of the ground Efimov state, given by the real part of its energy, scales like
(32) 
Interestingly, applying the universal relation between and the Efimov resonance position leads to
(33) 
which has a numerical factor very close to that in the van der Waals case [see Eq. (19)]. In addition to the position of the ground Efimov state, its width also shows an overall scaling behavior, which leads to a quasiuniversal scaling for the decay rate of an Efimov state. This decay is characterized by the parameter [braaten2006prep] which can be expressed as at . Based on the numerical results [wang2011prl] the dipolar has a value around 0.084. However, the width of the dipolar Efimov state is less universal than its position — with a finite sample of values in and some variations in the form of shortrange cutoff, [wang2011prl] shows that can vary up to 60% depending on which dipole resonance the threebody calculation was carried out for.
The stability of ultracold bosonic dipolar gases strongly depends on the dipolar Efimov physics discussed above. In an early study, Ticknor and Rittenhouse [ticknor2010prl] performed scaling analysis on threedipole loss rates and derived scaling law when and scaling law when . Based on the knowledge in Efimov physics and the scaling behavior of two dipoles discussed in Sec. 5.1, Wang, et al. [wang2011prl] have obtained the same scaling laws. These scaling laws indicate significant losses for a 3D dipolar gas in the strong dipole regime. Neverthless, a stable dipolar gas can be prepared in optical lattices or reduceddimension traps [danzl2009njp, danzl2010naturephys, chotia2012prl, hummon2013prl, yan2013nature, takekoshi2014arXiv].
For fermionic dipoles where wave scattering is absent, Ref. [wang2011prlb] showed the existence of a wave scattering length does not modify the longrange scaling behavior. The threshold law for three fermionic dipoles is therefore the same as the nondipole case [esry2001pra]. Near a wave dipolar resonance, however, different from nondipole fermions where no threebody state exists, there is one, and just one universal threedipole state with binding energy that scales with as
(34) 
A threebody resonance is therefore expected between two twobody wave resonances. In contrast to the bosonic dipoles, the binding of fermionic dipoles is mainly from the anisotropy of the dipolar interaction and therefore has a size smaller than . This dipolar state has a preferential spatial distribution as shown in Fig. 5. In the most probable configuration the bond lengths were numerically found to be universally determined by as , , and [wang2011prlb].
Near a wave resonance where the scattering volume goes through a pole, the scaling law of the threebody loss rate has been extracted numerically [wang2011prlb] as . Even though there is the suppression at ultracold temperatures, grows very quickly with and implies shortlived fermionic dipolar gases on resonance. Away from resonance, the loss is expected to have scaling and still implies reduced stability in the strong dipole regime.
6 Fewbody physics with other types of longrange interactions
Longrange interactions bring in longrange correlations — the craddle of universal physics. In fewbody systems, longrange correlations often lead to binding of exotic quantum states. Such states have strong impact on the dynamics of atoms and molecules at ultracold temperatures, and provide convenient means in quantum controls in ultracold chemistry. An example is the the engineering of molecular states, as demonstrated by Innsbruck group to steer diatomics through highlying rovibrational states using magnetic Feshbach resonances [mark2007pra, lang2008naturephys, zenesini2014pra]. Although the control over triatomics and polyatomic molecular states is still in the beginning stage, microwave association of Efimov trimers has been experimentally achieved by [lompe2010science] and Machtey, et al. [machtey2012prl]. For future experimental development, it is important to understand the properties of exotic fewbody states with other types of longrange interactions, as well as possible controls that are realizable. In this section we give a brief review over such systems.
Coulomb interactions. Ultracold fewbody systems with Coulomb force is very rich in exotic quantum physics. In a system of identical charges, although interesting phenomena such as ion Coulomb crystal [mavadia2013naturecomm] is expected in the manybody physics level, fewbody physics is less intriguing because of the lack of binding. Systems with opposite charges, on the other hand, are much more complicated and have been one of the main subjects in studies of atoms and molecules for many decades. In such systems, the existence of Rydberg series in the twobody subsystems and the emergence of quantum chaos near a threebody breakup threshold [xu2008pra] often bring difficulties in the quantum analysis of scaling laws. Such scaling laws are very important for atoms forming in cold plasma and has been more widely studied by semiclassical analyses [wannier1953pr, macek1995prl, macek1996pra] and classical MonteCarlo simulations [mansbach1969pr, pohl2008prl]
Attractive potential. Depending on its strength, the attractive potential can either be supercritical with infinite number of twobody bound states or subcritical where no twobody state is bound. If we write the potential as
(35) 
for two identical bosons of mass , the super and subcritical behaviors are determined by and . In the supercritical case, as mentioned earlier the attractive is too singular at the origin to have a lower bound for the ground state energy — a shortrange cutoff is therefore necessary for a system to be physical. In the subcritical case, although the attractive singularity does not cause any ill behavior for two bosons, Ref. [guevara2012prl] has numerically showed that the threeboson system suffers a collapse in the ground state energy when is the range . With a shortrange cutoff, this collapse is avoided but the threeboson system is no less interesting — it has an infinite number of bound states as a result of the longrange form of a numerically observed, effective hyperradial potential
(36) 
with in the range of where the nonregularized system collapses. The parameter is the characteristic length scale for the shortrange cutoff. Both and depend on , whereas depends also on the detail of the shortrange cutoff and is therefore nonuniversal [guevara2012prl]. The collapse of a nonregularized system can be understood by Eq. (36) when the limit is taken — diverges for every . So unlike the Thomas collapse, this collapse cannot be removed by simply introducing a shortrange threebody force.
For fermions, the supercritical behavior occurs when due to their wave barrier. In the subcritical case, it is numerically found [guevara2012prl] that the effective hyperradial potential for three fermions has the asymptotic behavior
(37) 
with when . This again leads to a collapse of the threebody system without proper regularization in . Unlike the case of bosons, though, taking the limit here does not lead to a collapse in , therefore a threebody force at small is sufficient to avoid the collapse of the system. Here the bizarreness is that three fermions have an infinite number of bound states when , where the effective twobody interactions (including the wave barrier) are all repulsive!
Although the attractive interaction doesn’t exist between fundamental particles, similar type of interactions can be found between composite particles. One example is the interaction between a charge and dipole, although it is not obvious what changes the anisotropy could make in the threebody physics. Also, in such systems there can be at most two pairs of interactions, so that the results discussed above for identical particles do not apply directly. Another possible scenario is a system of three heavy and one light atoms with resonant heavylight interactions (infinite heavylight scattering length). In such systems the effective interaction between the heavy ones (induced by the light particle) is also attractive [see Eq. (21)]. Here, though, the question is that if the way to get the effective interaction — the BornOppenheimer approximation — is good enough to study the new, exotic threebody states in the subcritical regime, where relatively weak effective interactions, or moderate mass ratios, are needed. Another caution is that the effective interactions in this system do not exist for all geometries of the heavy particles [amado1973prd], where the possible effects are yet to be investigated.
Ionatom interactions. The attractive interaction between an ion and a neutral atom is of much longer range than the van der Waals interaction between neutral atoms. The direct consequence is that the density of state in the nearthreshold twobody spectum for the ionatom interaction is much higher that the van der Waals systems, and the characteristic length scale
(38) 
for the interaction is typically on the order of Bohr, which is much greater than in atomic systems. Nevertheless, the nearthreshold twobody physics for the attractive interaction is essentially the same as that for the van der Waals interaction [watanabe1980pra, gao2010prla, idziaszek2011njp, gao2013pra, li2014pra], and so is expected for Efmov physics (assuming attractive interactions for all pairs) [naidon2014prl]. In realistic systems, however, inelastic collisions due to electron exchange are often nonnegligible, which is beyond the physics that singlechannel attractive interactions describe. Recent experimental studies [harter2012prl], though, have suggested that in a system of one Rb ion and two neutral Rb atoms the ion behaves more like a “catalyst” in the process of threebody recombination, and rarely appear in the recombination products. The threebody physics in the ionneutral hybrid system, including electron exchange and the physics of longrange attractive interaction, is therefore an important topic for future studies to understand the dynamics in ionneutral admixtures.
Spinorbit interactions. The term “Spinorbit interaction”, or “Spinorbit coupling”, is used both in the context of atomic physics and condensed matter physics. Here we refer to its latter meaning, namely, the interaction between a particle’s spin or pseudospin and its linear momentum in the following form: