Quasi-1D ultracold rigid-rotor collisions : reactive and non-reactive cases

TL;DR

This paper investigates how quasi-1D confinement and electric fields influence ultracold polar molecule collisions, showing that long-range repulsive interactions can suppress reactive and loss processes, aiding experimental control.

Contribution

It introduces a model for polar molecule collisions in quasi-1D traps considering rigid rotors and compares it to the fixed-dipole approximation, highlighting loss suppression mechanisms.

Findings

Long-range repulsive interactions suppress short-range reactive processes.

Electric field-induced dipoles can prevent two-body reactions.

Shielding effect reduces loss rates in ultracold molecule experiments.

Abstract

We study polar alkali dimer scattering in a quasi-1D geometry for both reactive and non-reactive species. Elastic and reactive rates are computed as a function of the amplitude of a static electric field within a purely long-range model with suitable boundary conditions at shorter range. We describe the diatomic molecules as rigid rotors and results are compared to the fixed-dipole approximation. We show in particular that for molecules with a sufficiently strong induced dipole moment oriented perpendicular to the trap axis, the long-range repulsive interaction leads to the suppression of short-range processes. Such shielding effect occurs for both reactive and non-reactive molecules, preventing two-body reactions as well as losses due to "sticky processes" [Phys. Rev. A 85, 062712 (2012)] from occurring. The present results demonstrate the possibility to suppress loss rates in current ultracold molecule experiments using 1D confinement.

Tables

Table 1. Table 1: Relevant numerical parameters used in our calculation, as defined in the text. The reference collision energy E coll subscript 𝐸 coll E_{\rm coll} is 50 nK. Two confinement strengths, intermediate and strong, are indicated for the LiRb dimer.

	$\overline{a}\left(a_0\right)$	$a_{\mathrm{ho}}\left(a_0\right)$	${\nu }_{⟂}$(kHz)	$E_{\mathrm{coll}}/\hslash {\omega }_{\mathrm{perp}}$	$d_{\mathrm{perm}}$(a.u.)	$B_v$(GHz)Lepers et al. (2013)
39K87Rb	117	955	10	1.6[-2]	0.242	1.13
7Li39K	224	2236	5	3.3[-2]	1.410	7.69
7Li87Rb	325	2473	2	8.3[-2]	1.645	6.46
strong	-	341	100	1.6[-3]	-
7Li133Cs	497	6408	0.2	8.3[-1]	2.201	5.62
23Na87Rb	355	2286	2	8.3[-2]	1.304	2.09

Equations

$$H=T_R+T_{\mathrm{rot}}+V_{\mathrm{trap}}+V_{\mathrm{int}}=-\frac{{\hslash }^2}{2\mu }\frac{d^2}{d{R}^2}+\frac{{\mathbf{\ell }}^2}{2\mu R^2}+\frac{1}{2}\mu {\omega }_{\perp }^2{\rho }^2+B_v{\mathbf{J}}_A^2+B_v{\mathbf{J}}_B^2+V_{\mathrm{int}}(\mathbf{R}).$$

$${\mathbf{V}}_{\mathrm{int}}(R)=\frac{{\mathbf{C}}_3}{R^3}+\frac{{\mathbf{C}}_6}{R^6}+{\mathbf{V}}_{\mathrm{Stark},A}+{\mathbf{V}}_{\mathrm{Stark},B}$$

$$⟨J_A^{\prime }{M}_A{J}_B^{\prime }{M}_B^{\prime }{l}^{\prime }{M}_l^{\prime }|\frac{{\mathbf{C}}_3}{R^3}|J_A{M}_A{J}_B{M}_{B}l{M}_l⟩=$$

$$⟨J_A^{\prime }{M}_A^{\prime }|{\mathbf{V}}_{\mathrm{Stark},A}|J_A{M}_A⟩=-\mathcal{E}⟨J_A^{\prime }{M}_A^{\prime }|d_{z,A}|J_A{M}_A⟩=\mathcal{E}d\sqrt{[l][l^{\prime }]}\left(\begin{array}{ccc}{J}_A^{\prime }& 1& J_A\\ 0& 0& 0\end{array}\right)\left(\begin{array}{ccc}{J}_A^{\prime }& 1& J_A\\ -M_A^{\prime }& 0& M_A\end{array}\right)$$

$$|{\Psi }_{J_A{M}_A{J}_B{M}_{B}l{M}_l}⟩=\frac{|J_A{M}_A{J}_B{M}_{B}l{M}_l⟩+(-1{)}^l|J_B{M}_B{J}_A{M}_{A}l{M}_l⟩}{\sqrt{2(1+{\delta }_{J_A{J}_B}{\delta }_{M_A{M}_B})}}.$$

$$|{\Psi }_{J_A{M}_A{J}_B{M}_{B}l{M}_l}^{ϵ}⟩=\frac{|{\Psi }_{J_A{M}_A{J}_B{M}_{B}l{M}_l}⟩+(-1{)}^{ϵ+M_A+M_B+M_l}|{\Psi }_{J_A-M_A{J}_B-M_{B}l-M_l}⟩}{\sqrt{2(1+{\delta }_{M_{A}0}{\delta }_{M_{B}0}{\delta }_{M_{l}0})}}.$$

$$⟨{\tilde{J}}_A^{\prime }{\tilde{J}}_B^{\prime }{l}^{\prime }|H|{\tilde{J}}_A{\tilde{J}}_{B}l⟩=\sum _{J_A{J}_B{J}_A^{\prime }{J}_B^{\prime }}{\alpha }_{J_A}{\alpha }_{J_B}{\alpha }_{J_A^{\prime }}{\alpha }_{J_B^{\prime }}⟨J_A^{\prime }{J}_B^{\prime }{l}^{\prime }|H|J_A{J}_{B}l⟩,$$

$$C_6^r=\sum _{({\tilde{J}}_A,{\tilde{J}}_B)\ne (\tilde{0},\tilde{0})}\frac{|⟨{\tilde{J}}_A,{\tilde{J}}_B|{\mathbf{d}}_A.{\mathbf{d}}_B|\tilde{0},\tilde{0}⟩{|}^2}{\Delta E_A+\Delta E_B}$$

$$C_6^r(\mathcal{E}=0)=\frac{d_{\mathrm{perm}}^4}{4B_v}$$

$$⟨{\tilde{J}}_A{\tilde{J}}_{B}l|V_{\mathrm{corr}}|{\tilde{J}}_A{\tilde{J}}_{B}l⟩=\sum _{{\tilde{J}}_A^{\prime }{\tilde{J}}_B^{\prime }>\tilde{J}}\frac{|⟨{\tilde{J}}_A^{\prime }{\tilde{J}}_B^{\prime }l|V|{\tilde{J}}_A{\tilde{J}}_{B}l⟩{|}^2}{\Delta E_A+\Delta E_B}$$

Full text

I Introduction

Ultracold molecules have added a new twist to the field of cold atoms Carr et al. (2009); Quéméner and Julienne (2012). First groundbreaking experiments were carried out several years ago at JILA, where a dense gas of ultracold fermionic KRb dimers was produced by two photon association Ni et al. (2008). This molecular species is reactive, a feature that has been explored in the context of quantum-state controlled chemical reactions Ospelkaus et al. (2010). However, from the standpoint of many-body studies the reactivity of KRb is in general a drawback leading to fast particle loss from the trap.

More recently, different groups have reported the production of non-reactive ultracold alkali dimers Takekoshi et al. (2014); Molony et al. (2014); Park et al. (2015); Guo et al. (2016); Rvachov et al. (2017); Ye et al. (2018). Such molecules, if prepared in the absolute single-particle energy state, are strictly stable under two-body collisions. However, there is experimental evidence that even under these favorable conditions inelastic losses still occur at fast pace Molony et al. (2014); Park et al. (2015); Guo et al. (2016); Ye et al. (2018). This is true in particular for bosonic molecules, whereas fermionic samples appear to be more stable Rvachov et al. (2017). A mechanism that might explain the observed losses is related to the existence of Fano-Feshbach resonances available with high density at thermal energies even in an ultracold gas. Presence of such resonances increases the collision lifetime of a pair of molecules, creating a tetrameric complex. Such metastable complex can be lost from the trap since it is not necessarily trapped in the optical lattice and can undergo recombination to deeper levels by colliding with a third molecule, a process often referred to as “sticky collision” Mayle et al. (2012, 2013). Excitation of the complex by the trapping laser has also been recently proposed as an alternative mechanism that might explain the observed loss rates Christianen et al. .

In order to increase the sample lifetime, it is therefore important to devise strategies to prevent molecules from approaching at short distances, where detrimental inelastic processes take place. Unfortunately, in three dimensions there always exists an attractive head-to-tail reaction path leading to the short range. This circumstance has been shown to result in a strong dependence $\sim d^{6}$ of the reactive rate on the induced dipole moment $d$ Quéméner and Bohn (2010). More subtle quantum effects have been proposed to control the reaction dynamics of alkali dimers prepared in rotationally excited states Wang and Quéméner (2015). Microwave shielding leading to a dramatic lifetime increase has also been recently theoretically studied Lassablière and Quéméner (2018); Karman and Hutson (2018).

The situation is different if molecules are confined in tight traps, for instance by optical means. In fact, confinement can then be easily designed in such a way that molecule pairs will tend to collide in a repulsive side-to-side configuration, the head-to-tail pathway being energetically unfavored by the confining potential. This approach has been demonstrated both experimentally and theoretically to lead to a drastic increase of the lifetime of reactive polar molecules trapped in one and two spatial dimensions  Micheli et al. (2010); de Miranda et al. (2011); Chotia et al. (2012); Simoni et al. (2015).

Our previous study of reactive collisions in one-dimensional optical tubes has addressed highly reactive species Simoni et al. (2015). Therein, polar molecules have been simply described as fixed dipoles of magnitude equal to the induced dipole moment. Resonances have been found to be strongly quenched by inelastic processes and have no influence on the scattering cross sections. Aim of the present work is twofold. First we relax the fixed-dipole approximation by introducing explicitly the rotational degrees of freedom of the colliding diatoms. Secondly, in addition to studying reactive collisions we introduce a model for non-reactive molecules, in which scattering cross sections can indeed be dominated by dense resonance spectra. We compare both reactive and non-reactive rigid-rotor models with the fixed-dipole approximation. As a main result, we show that resonances in elastic cross sections and thus possibly complex-mediated collisions can be suppressed using the dipole-dipole repulsion induced by an applied electric field.

The paper is organized as follows. Section II introduces the formalism and our numerical approach. Section III presents the long-range multipolar expansion of the molecule-molecule interaction. Results for different dialkali species and collision energies are presented as a function of an applied electric field in Sec. IV. A short conclusion summarizes this work.

II Formalism

We consider two molecules A and B confined in a quasi one-dimensional geometry by a potential approximated as an harmonic trap. The quadratic nature of the confining potential allows one to separate center-of-mass and relative motion. The rigid-rotor Hamiltonian describing the collision in relative coordinates includes the kinetic energy, the confinement potential, and an interaction term $V_{\rm int}$ defined below in Eq. (2) which includes inter-molecular forces and the Stark interaction energy with the external electric field. In this work we will describe the dimer molecules as rigid rotors, an approximation valid as long as the intermolecular distance remains large compared to the extent of the internal coordinates.

The kinetic energy of the dimer+dimer complex will be decomposed into a sum of three independent terms depending respectively on the relative distance $\mathbf{R}$ between the two centers of mass, the internal coordinate of the dimer A and the internal coordinate of dimer B van der Avoird et al. (1994). Furthermore, we will assume that the dimers remain in their vibrational ground level reducing the dimer contribution to the kinetic energy to a purely rotational term.

Putting all terms together, the total Hamiltonian reads :

[TABLE]

Here $\boldsymbol{\ell}$ is the orbital angular momentum of the relative motion, $\mathbf{J}_{A}$ and $\mathbf{J}_{B}$ are the angular momenta of A and B with rotational constant $B_{v}$, $\omega_{\perp}=2\pi\nu_{\perp}$ is the trap frequency and $\rho$ the distance to the trap axis $z$. We use for each molecule the spectroscopically determined rotational constant $B_{v}$ of the ground level $X^{1}\Sigma^{+};v=0$. Values and reference of the spectroscopic study can be found in Lepers et al. (2013). The transverse oscillator length characterizing the trap size will be defined as $a_{\rm ho}=\sqrt{\hbar/(\mu\omega_{\perp})}$. As demonstrated in three spatial dimensions, hyperfine interactions would greatly increase the complexity of the problem Mayle et al. (2012, 2013). However, we do not expect such added complexity to bring novel qualitative features to the main effects we wish to demonstrate and hyperfine interactions will be neglected in this work.

We will now extend to rigid rotors the computational approach developed for fixed dipoles in our previous work Simoni et al. (2015). Our method to construct the scattering wavefunction consists in a simultaneous expansion of the angular part of the solution in a suitable internal basis whereas the radial coordinate is discretized on a grid of points. The radial discretization is detailed in Sec. II.1. The angular basis comprises the orbital angular coordinate $\hat{R}$ as well as any internal degree of freedom of the diatomic, as described in Sec. II.2.

II.1 Radial discretization

The radial discretization is performed according to the spectral element approach Karniadakis and Sherwin (2005); Simoni et al. (2017). Briefly, we choose a minimum $r_{c}$ and a maximum $r_{\rm max}$ radial distance and define a solution interval $I=\left[r_{c},r_{\rm max}\right]$ which is in turn partitioned into a set of N non-overlapping sectors. A number of grid points and a basis set of Gauss-Lobatto cardinal functions associated with those points are assigned to each sector. The wavefunction is represented on the discrete basis and continuity of the wavefunction and of its radial derivative is enforced at the connection points between sectors. This strategy results in a highly sparse matrix in the grid indices. It should be remarked that our method allows different angular bases to be used in each sector. With the basis over all coordinates defined, we can rewrite the Schrödinger equation as a linear system that can be solved for a matrix solution $\Psi$ in the interval $I$ provided boundary conditions are assigned at the endpoints $r_{c}$ and $r_{\rm max}$.

At the right endpoint we impose $\Psi^{\prime}(r_{\rm max})=\mathbf{I}$, the matrix solution at $r_{\rm max}$ becomes then equal to the $R$-matrix defined as $\mathbf{R}\equiv\Psi\left(\Psi^{\prime}\right)^{-1}$. From $\mathbf{R}$ one can then extract the scattering matrix and hence all physical observables such as the elastic and reactive collision rates Micheli et al. (2010); Simoni et al. (2015).

We will impose two different boundary conditions at $r_{c}$. The first assumes that at short distance the molecules react with unit probability. We thus require that the spherical surface $R=r_{c}$ is totally absorbing , i.e. across the surface we only have incoming flux and no reflected outgoing flux. In practice we first define local adiabatic channels $|\alpha\rangle$ and corresponding energies $E_{\alpha}$ by diagonalizing the angular Hamiltonian $T_{\rm rot}+V_{\rm int}$ at distance $R=r_{c}$. Next, we assume that the wavefunction can be described by a pure incoming spherical wave in each channel $\alpha$ for $R\simeq r_{c}$. The logarithmic derivative $Z(r_{c})\equiv\Psi^{\prime}(r_{c})\Psi^{-1}(r_{c})$ is therefore diagonal with elements $(-ik_{\alpha}-\dfrac{1}{2}k_{\alpha}^{\prime}k_{\alpha}^{-1/2})$, where $k_{\alpha}=\sqrt{2\mu(E_{\rm coll}-E_{\alpha})/\hbar^{2}}$ is the channel wave vector and the derivative is taken with respect to $R$. This method has been shown to give accurate prediction for dialkali reactive species Idziaszek and Julienne (2010).

Our second approach amounts to the Dirichlet boundary condition $\Psi={\bf 0}$, i.e. the wavefunction is required to have a nodal surface for $R=r_{c}$. In this description we can observe resonances between the incoming open channel with collision energy $E_{\rm coll}$ and bound levels of closed channels with energy close to $E_{\rm coll}$. Since the radius $r_{c}$ is chosen arbitrarily and there is a-priori no reason for the wavefunction to have a nodal surface for $R=r_{c}$, we cannot predict the location of the resonances in terms of the amplitude of the electric field. As a matter of fact, the short-range collision parameters (more precisely, the quantum defects, see Mayle et al. (2012, 2013)) can be considered as stochastic variables arising from the extremely complex four-body dynamics taking place inside $r_{c}$. In this view, our specific Dirichlet condition can be considered as one possible realization of such complex process.

The specific value $r_{c}=40~{}a_{0}$ has been chosen so to satisfy the following criteria. Firstly, the rigid rotor model is expected to be accurate in the external region $R>r_{c}$. Second, motion is semiclassical near $r_{c}$ for the adsorbing model of Ref. Idziaszek and Julienne (2010) to apply. Finally, since we are interested in resonance spectra, the density of states near threshold must be the same for the Dirichlet “truncated potential” as for the full one. We have checked that for our choice of $r_{c}$ this holds true up to energies on the order of 1 K in the case of a single deep potential well with van der Waals tail.

The present approach can therefore expected to give useful insight into the collisions dynamics and it allow us to compare on an equal footing different approaches, namely the fixed-dipole and the rigid-rotor descriptions. Finally, note that for the sake of comparison we will use the non-reactive Dirichlet condition even for intrinsically reactive species such as LiCs or LiRb.

II.2 Angular basis

The angular part of the wavefunction in the rigid-rotor model is expanded in the so-called decoupled basis $|J_{A}M_{A},J_{B}M_{B},lM_{l}\rangle$, representation defined by the operators $\mathbf{J}^{2}_{A}$, $\mathbf{J}^{2}_{B}$, ${\boldsymbol{\ell}}^{2}$ with eigenvalues $\hbar^{2}J_{A}(J_{A}+1)$, $\hbar^{2}J_{B}(J_{B}+1)$, and $\hbar^{2}l(l+1)$ as well as by their projections on the laboratory axis with eigenvalues $M_{A}$, $M_{B}$ and $M_{l}$. The laboratory axis is taken as the axis of the confining optical tube. In parallel configuration where the electric field axis is aligned along the laboratory axis, the projection of the total angular momentum $M\equiv M_{A}+M_{B}+M_{l}$ is conserved, allowing us to work with a smaller basis set. For non-parallel configurations one has to take into account couplings between different $M$ due to the electric field.

The interaction potential matrix included in Eq. (1)

[TABLE]

comprises the dipole-dipole interaction ${\mathbf{C}}_{3}/R^{3}$, the van der Walls interaction ${\mathbf{C}}_{6}/R^{6}$, and the Stark term arising from the interaction between the molecular dipoles ${\mathbf{d}}_{A,B}$ and the electric field. The matrix elements of ${\mathbf{C}}_{3}$ and ${\mathbf{C}}_{6}$ depend on the angular momenta $J_{A}$, $J_{B}$ and $l$ as well as their projection on the laboratory axis. They are obtained using Wigner-Eckart theorem, their expression in the space fixed frame can be found in Avoird et al. (1980); Krems and Dalgarno (2004); Micheli et al. (2007); Bohn et al. (2009). For completeness, we report here the expression of the dipole-dipole interaction

[TABLE]

where the common abbreviation $[X]=2X+1$ has been used. The Stark term represents an internal interaction that acts on the degrees of freedom of the individual molecule. The term for molecule A (resp. B) is therefore diagonal in the quantum numbers of molecule B (resp. A) as well as in the orbital quantum numbers $lM_{l}$. The matrix elements are given explicitly for molecule of label A by

[TABLE]

in terms of the electric field intensity $\cal E$, a formally identical equation holding for B. The electronic van der Waals interaction is in general anisotropic Kotochigova (2010). However, we have taken explicitly into account the anisotropic contribution for the study of KRb+KRb and found it to give negligible corrections to the computed scattering observables. Therefore, only the isotropic contribution will be included in our model.

Few additional points must be taken into consideration. First, since we focus on collisions of identical diatomics we need to use symmetrized wavefunctions

[TABLE]

For the perpendicular electric field configuration, the Hamiltonian is also symmetric with respect to reflection across the plane orthogonal to the trap axis and containing the origin. We can construct a symmetrized wavefunction

[TABLE]

Lastly, while the decoupled basis is useful in the weak electric field regime, for higher amplitudes, different J levels are heavily mixed by the field, the system is better described in a dressed basis set Friedrich and Herschbach (1999); Avdeenkov et al. (2006). To obtain this basis we numerically diagonalize for each field amplitude the diatomic potential $B_{v}\mathbf{J}^{2}_{X}+V_{{\rm Stark},X}$ ($X=A,B$). We thus get the eigenfunctions $|\widetilde{J}{\widetilde{M}_{J}}\rangle=\sum\limits_{J}\alpha_{J}|JM_{J}\rangle$ which are in turn combined into the tetrameric wavefunction $|\widetilde{J}_{A}\widetilde{M}_{A}\widetilde{J}_{B}\widetilde{M}_{B}lM_{l}\rangle$. In order to compute the total potential matrix we first evaluate the matrix elements in the decoupled basis and then we numerically evaluate

[TABLE]

where the magnetic quantum numbers have been suppressed for notational convenience.

The rigid-rotor description is only used between $R=40~{}a_{0}$, chosen as the $r_{c}$ boundary value in this work, and $R=200~{}a_{0}$. For intermediate distances above $R=200~{}a_{0}$, dipole-dipole interaction between different rotational level is weak compared to the energy gap of those level. One can thus focus on the rotational ground level with $\widetilde{J}_{A}=0$ and $\widetilde{J}_{B}=0$, using perturbation theory to take into account excited levels; see Sec. III. Spherical harmonics $|lM_{l}\rangle$are still used to represent orbital motion.

Finally, in the long-range domain, typically $R>10^{4}~{}a_{0}$, the cylindrical confining potential of the 2D-trap overcomes the molecule-molecule interaction, limiting $\hat{R}$ to a small angular region around the pole. A development in spherical harmonic becomes increasingly inefficient and we use cylindrical grid basis functions projected on the spherical surface that stay localized near the pole as $R$ varies. As an order of magnitude, few hundreds spherical harmonics are needed to enforce continuity at the connection point between the intermediate and the long-range regions.

We will compare the rates obtained with the current approach to results from our previous model where fixed dipoles were used instead of rigid-rotors Simoni et al. (2015). In the latter, the interaction between the molecule in its rotational ground level and the Stark field is described as a point particle with a given induced permanent dipole moment. All rotational effects factor into a unique $C_{6}$ long-range coefficient. The total angular wavefunction is expanded on a basis of orbital spherical harmonics and the Hamiltonian only contains the orbital kinetic energy, the confinement potential and the interaction potential $V_{\rm int}$. Such point-particle model is equivalent to our intermediate-range rigid-rotor model where we include only the rotational ground state, though the correction to the $C_{6}$ coefficient needs to be taken the same in both models; See Sec. III.

III Induced dipole moment interaction

Some care must be taken in the definition of the $C_{6}$ matrix coefficient in order to have a correct description of the Van der Waals interaction. Two different approaches are used in this work.

In the first one, used for the intermediate range domain or for the point particle model, excited rotational states are not explicitly included in the basis. Resulting from a second order perturbation treatment, the $C_{6}$ coefficient is thus written as a sum over all excited rotational, vibrational and electronic levels of both molecules. Following Refs. Quéméner and Bohn (2011); Lepers et al. (2013) the $C_{6}$ can be decomposed into the sum of two contributions, a term $C_{6}^{e}$ including the electronic transitions contribution and one $C_{6}^{r}$ including the rotational transition contribution. Pure vibrational transitions as well as cross terms have been shown to be negligible Lepers et al. (2013). Values of the $C_{6}^{e}$ are taken from Lepers et al. (2013). The $C_{6}^{r}$ term is described in more detail in Sec. III.1.

In the second approach, used for the short range domain, the rotational states are explicitly included in the basis. The second order term is thus coming only from the electronic transitions $C_{6}^{e}$. An additional corrective term is however needed to prevent the appearance of unphysical levels; See Sec. III.2.

As mentioned above, in both cases the $C_{6}^{e}$ is taken as a purely isotropic term.

III.1 Rotational transition contribution

Due to the competition between the electric field and the rotational Hamiltonian, the rotational spectrum is heavily perturbed. A field free $C_{6}^{r}$ value would thus not be appropriate Julienne et al. (2011).

In more detail the expression of the rotational contribution is

[TABLE]

$\Delta E_{A}$ (resp. $\Delta E_{B}$) being the energy difference between the ground and the excited $J_{A}$ (resp. $J_{B}$) level. Dependence on the space-fixed projections $M_{A}$ and $M_{B}$ of the excited rotational levels are omitted in the formula for ease of reading. For the field free case $\widetilde{J}\equiv J$ and the only non-zero transition is from $J=0$ to $J=1$. We then obtain the expression for two identical molecules

[TABLE]

with $d_{\rm perm}$ the permanent dipole moment of the diatomic, expressed in its own molecular frame.

To obtain more accurate results for the collision rate we compute an improved $C_{6}^{r}$ value using Eq. (8) at each electric field amplitude. We note that for a non-zero field the heteronuclear molecules present an induced permanent dipole moment in the laboratory frame. In this case the contribution of matrix elements like $\langle\widetilde{0},\widetilde{1}|\mathbf{d}_{A}.\mathbf{d}_{B}|\widetilde{0},\widetilde{0}\rangle$ is non negligible. Those transition terms with one molecule in the ground level and the other in an excited one are related to the induction interaction $C_{6}^{\rm ind}$. While the remaining transitions involving two rotationally excited molecules added to the electronic $C_{6}^{e}$ form the dispersion coefficient $C_{6}^{\rm disp}$.

Rotational contributions have been shown to be small for heteronuclear molecules LiNa and KRb due to their weak permanent dipole moment Lepers et al. (2013). However for a molecule like LiCs they are by far the dominant term. In Fig. 1 we show the difference in $C_{6}$ coefficients for LiCs-LiCs interactions depending on the model used. Calculations were performed with a single $C_{6}^{e}$ term, a fixed field-free $C_{6}$ term, a field-dependent $C_{6}^{\rm disp}$ term and finally a field-dependent $C_{6}^{\rm disp}+C_{6}^{\rm ind}$ term. At weak fields the difference between the values obtained at the different levels of approximation is small and the collision rate is essentially the same for each model. At higher fields the dominant term is the $C_{3}$ dipole-dipole interaction and modifications of the $C_{6}$ coefficient have no major impact on our results. However the $C_{6}$ interaction is still relevant at short intermolecular distances.

Bottom panel in Fig. 1 shows the elastic collision rate computed using model potentials built with and without the induction interaction. Calculations are performed at low collision energy with the Dirichlet boundary condition at short range, such that scattering is purely elastic. The calculated collision rate presents several resonant features associated to the presence of quasi-bound states. The origin of these features will be discussed in more detail in Sec. IV. It is however important to remark here, as apparent from the figure, that their density at large electric fields is severely underestimated if one omits the induction interaction.

III.2 Unphysical states

When neglecting the rotation in the point-particle model we still need to compute a corrective $C_{6}$ term, which means that the modification of the rotational structure must be taken into account in the model in second order perturbation theory. In a similar fashion, the use of the rigid-rotor basis mentioned in Sec. II.2 leads to unphysical states if no correction is made Tscherbul and Dalgarno (2010); Suleimanov et al. (2012). Indeed due to computational limitations one has to truncate the basis, neglecting all the levels with $\widetilde{J}>\widetilde{J}_{max}$. In particular all the coupling matrix element of the type $\langle\widetilde{J}_{A}^{\prime}\widetilde{J}_{B}^{\prime}l^{\prime}|V|\widetilde{J}_{A}\widetilde{J}_{B}l\rangle$ with either $J_{A}^{\prime}$ or $J_{B}^{\prime}$ above the angular momenta $\widetilde{J}_{max}$ are cut off. Rotational levels with $J=\widetilde{J}_{max}$ are the most affected by this truncation Tscherbul and Dalgarno (2010); Suleimanov et al. (2012). To reduce the effect of this approximation on the ground state, and thus on the collision rate, one can simply increase the size of the basis.

Another approach is to use second order perturbation theory. In this approximation, the omitted couplings are assumed to be a perturbation to the levels included in our basis, giving rise to a corrective term $V_{\rm corr}$ to be added to the Hamiltonian. This term is diagonal with matrix elements

[TABLE]

where the sum is over every level above the cutoff angular momentum and $V$ is the dominant coupling term, the dipole-dipole interaction $C_{3}/R^{3}$ in our case. The square of the matrix element on the rhs leads to a $R^{-6}$ correction to the $C_{6}$ coefficient. In this work we take $\widetilde{J}_{max}$ equal to either 1 or 2, and the sum is carried out up to $\widetilde{J}=7$. The energy gap $\Delta E$ is taken as the diatomic energy gap, thus neglecting the molecule-molecule interaction.

IV Results

We perform calculations for different bosonic species under various trapping conditions, collision energy and electric field. Table 1 resumes the relevant characteristic physical parameters relevant for our calculations. The van der Waals length ${\bar{a}}=(2\mu C_{6}/\hbar^{2})^{1/4}/2$ given in the table represents the average scattering length for collisions in a pure $C_{6}/R^{6}$ potential and can be interpreted as the range of such potential Gribakin and Flambaum (1993). We focus here on an intermediate confinement regime, which we define following Ref. Simoni and Launay (2011) as $a_{\rm ho}\approx 10~{}\bar{a}$. The collision energy of the identical heteronuclear molecules will be fixed in most calculations to $50$ nK. With reference to the table one can remark that for a heavy molecule such as LiCs such collision energy can be considered as “hot” in terms of the trap level spacing. In fact, the gap with the first excited energy level of the transverse harmonic oscillator is only slightly larger than twice the collision energy.

We first consider the configuration where the electric field is parallel to the confinement axis. In this case, molecules tend to be in attractive head-to-tail configuration and to react at the cutoff radius $r_{c}$.

This intuitive picture is confirmed by the analysis of the adiabatic potentials, obtained by diagonalizing the total interaction potential at each value of the interparticle distance $R$. Note that at parallel configuration the projection of the orbital angular momentum on the trap axis is conserved and it has been fixed to $M_{l}=0$. One can observe in Fig. 2 that the lowest adiabatic curve, the one that to first approximation controls the dynamics, presents no potential barrier preventing the molecules from reaching the short-range reactive region. On the converse, the more excited adiabatic potentials present at short range $\sim 50a_{0}$ a barrier arising from the centrifugal potential. The rotational degrees of freedom correspond to the excited thresholds $\sim$GHz in the leftmost panel of Fig. 2. Each rotational manifold presents in turn a finer energy structure due to the transverse harmonic trap levels with equal spacing $h\nu_{\perp}$; See rightmost panel of Fig. 2.

Coming to the dynamics, we find that for the universal reactive model the effect of the rotationally closed channel is minor. In fact, as shown in the upper panel of Fig. 3, the elastic collision rates computed within the rigid rotor and the fixed dipole models are essentially identified. The difference is more pronounced when taking the Dirichlet boundary condition, as illustrated in the lower panel. When the scattering phase crosses a multiple of $\pi$, a broad zero crossing is observed in both the rigid-rotor and the fixed-dipole calculations. This happens near 2500 V/cm in both models, with a relative shift of few hundreds V/cm. A second zero occurs near 5000 V/cm. By analogy with the Ramsauer-Townsend effect, which occurs as a function of collision energy, in the following we will briefly refer to such zeros as Ramsauer minima.

Most importantly, for rigid rotor collisions we observe resonance effects, manifesting themselves as a series of narrow features superimposed to the slowly varying background. As in the case of atoms, such resonances can in principle be used to control the dimer-dimer scattering properties through an applied field. In this work we do not attempt a precise resonance assignment, which would require for instance quasi-bound-state or reduced adiabatic calculations. However, the fact that such resonances are absent in the fixed-dipole calculation strongly suggests the ones in the figure are of rotational origin, i.e. they can be assigned to some potential curve correlating with rotationally excited molecules. A related study of true (as opposed to quasi-) bound states of molecules in a quasi-1D geometry can be found in Dawid et al. (2018).

We now focus on a system where the electric field is perpendicular to the trap axis. It can be expected that in this configuration and with a strong enough induced permanent dipole moment the diatomic molecules will repel each other at long range Simoni and Launay (2011) and can be protected against short-range reactive collisions. Figure 4 shows the elastic and reactive collision rates for LiK molecules calculated with the fully absorbing boundary condition. Results obtained in the fixed-dipole approximation are virtually indistinguishable from the rigid-rotor model and are not shown. In particular, the reactive rate is suppressed by about three orders of magnitude in the considered range of $\cal E$, confirming that the shielding phenomenon is robust versus rotation. Also note from the figure that the elastic rate presents a much more pronounced minimum in the elastic cross section as compared to KRb. This happens since, due to the small magnitude of the reactive rate for large electric field, LiK collisions are essentially elastic and the minimum is not quenched by inelastic processes as it happens for KRb.

Not all molecules have indeed sufficiently strong dipole moment and thus dipole-dipole repulsion to suppress reactive processes. In general, molecules with a large intrinsic dipole moment and a small rotational constant are more polarizable, the induced electric dipole in the laboratory frame is larger and thus the shielding is more effective in these systems for a given electric field. To confirm this trend, we have carried out sample calculations with the trapping parameters from Tab. 1 corresponding to intermediate confinement, and found that the only molecules not having a strong enough permanent dipole moment to obtain significant shielding effects are LiNa and KRb.

We now consider non-reactive species, taking the NaRb dimer as an example. The long range adiabatic curves of the NaRb-NaRb tetrameric taken for different amplitudes of the static electric field are shown in Fig. 5. Each adiabatic potential correlates asymptotically with an energy level $h\nu_{\perp}(n+1)$ of the isotropic transverse harmonic oscillator with principal quantum number $n$ and degeneracy $(n+1)$. At shorter distance, the dipolar interaction becomes significant and breaks the isotropy of oscillator. The dipolar interaction contributes for instance an energy of $d^{2}/R^{3}$ for molecules oscillating in the plane perpendicular to the dipoles and $d^{2}/R^{3}\left[1-3\cos^{2}(\theta)\right]$ for molecules in the plane containing the dipoles and the trap axis, with $\theta$ the angle between $\mathbf{R}$ and $\mathbf{d}$. This anisotropy leads to the lifting of the asymptotic degeneracy clearly visible in the three panels of the figure as the intermolecular distance decreases.

Moreover, as expected at perpendicular configuration, a barrier to reaction is formed in the lowest adiabatic potential as the amplitude of the field increases. To experience a significant short-range dynamics the molecules would need to tunnel through this barrier, which for instance at $5$ kV/cm has height of 510 nK; See rightmost panel of Fig. 5. Under this field-induced shield it is interesting to compare the dynamics of a collision with a collision energy well below the maximum of the barrier ($50$ nK), and slightly over the top of the adiabatic potential barrier ($600$ nK).

Figure 6 shows the results of calculations performed with the Dirichlet boundary condition. With reference to the upper panel of the figure, one may once again remark a Ramsauer minimum near 1200 kV/cm at the lowest considered collision energy. The minimum shifts at larger electric fields, outside the range of the bottom panel of the figure, at larger collision energy. In fact, quite generally, a potential has less influence on faster particles and in order to have the accumulated phase shift go through $\pi$ larger values of $\cal E$ are needed. The main scattering feature is the dense spectrum of overlapping resonances observed at both considered collision energies. Such resonances arise from the coupling between the incoming channel and the closed channels, either correlating with the trap or with rotationally excited level.

Resonance observed in both panels have positions essentially independent of collision energy, as expected since a collision energy of the order of $nK$ is essentially negligible on the scale of the Stark shift $-d\cal E$. Note however that both the lineshape and the resonance widths vary significantly between the two panels of Fig. 6. Most importantly, at low collision energy (upper panel) resonance effects tend to be washed out in particular at strong fields, say above ${\cal E}\sim 2000$ kV/cm, since the adiabatic barrier becomes increasingly high and broad. In other terms, the presence of the barrier tends to keep the molecules far apart and prevents resonance effects, which are due to a short-range coupling between the open and the closed collision channels, from occurring. At larger collision energy barrier tunnelling is more effective and resonances only begin to disappear near the upper limit of the figure ${\cal E}\sim 5000$ kV/cm .

As conjectured in Mayle et al. (2012, 2013) resonant population of dense quasi-bound states increases the collision lifetime along with the probability of loss of the untrapped complex and of inelastic recombination via collision with a third body. These phenomena are suspected to be a significant limiting mechanism to the lifetime of ultracold quantum gases Molony et al. (2014); Park et al. (2015); Guo et al. (2016). According to the present results, resonances can be suppressed in 1D geometries. Confinement combined with a strong static electric field and low temperatures should thus allow one to shield non-reactive molecules from complex-mediated collisions in addition of shielding reactive molecules.

In order to confirm such conclusion on a different molecular species, we consider a non-reactive LiRb model; see Fig. 7. As for NaRb, at the smaller collision energy of the upper panel, the resonance width strongly decreases with $\cal E$, resonances first tend to become non-overlapping and their influence on the background cross section finally vanishes. The presence of a Ramsauer minimum, its shift, and the behavior of the resonance width with increasing collision energy (lower panel) follow the same trend observed in NaRb.

Figure 7 also shows an interesting comparison with the fixed-dipole approximation. Note that as expected the density of resonant features is larger for the rigid-rotor model due to the additional rotational degrees of freedom of the dimers. However, the density of resonances observed in the fixed-dipole model, thus purely due to trap confinement and tuned via the Stark energy shift, is significant in this system. Shielding becomes otherwise effective at comparable values of the electric field at each considered collision energy. It is worthwhile to compare the upper panel of Fig. 7 with Fig. 4 both referring to the same molecular system and physical parameters, but with different (non-reactive vs reactive) boundary conditions. Such comparison makes clear the fact that the scattering rate for the reactive LiRb model in Fig. 4 presents virtually no structure due to the absence of electrically tuned quasi-bound states but rather because resonance effects are strongly quenched by fast reactive decay.

Let us finally consider the effect of confinement. On physical grounds, for dipoles perpendicular to the axis, stronger transverse confinement should further help preventing particles from approaching. In fact, dipoles in side-by-side configuration repel, and can only get close by moving away from the trap axis towards an attractive head-to-tail configuration. In terms of adiabatic potentials, this means that the barrier in the lowest adiabatic curve will be stronger at a given electric field for a more confining than for a looser harmonic trap.

This picture is confirmed by Fig. 8, that shows the elastic collision rate for a non-reactive LiRb + LiRb collision model in a strong confinement regime. The resonance spectrum appears relatively sparse even at relatively low electric fields, meaning again that coupling between the open channel and quasi-bound states is weak since particles are prevented from reaching the short-range coupling region. This result extends to non-reactive systems the conclusion of Ref. Simoni and Launay (2011), that demonstrated within the fixed-dipole approximation a more marked suppression of reactive processes occurring in tight traps. As expected, inclusion of rotation does not seem to change this general conclusion, that basically depends on the presence of a potential barrier formed at long-range, in a region where the fixed-dipole approximation is an excellent one.

Overall, it seems possible to find conditions for stabilizing the gas against inelastic processes of both reactive or complex-mediated nature to the extent that a suitable balance between low temperatures, confinement and electric field intensity is found.

V Conclusions

We have presented a rigid-rotor model to study identical polar molecule collisions in a quasi-1D optical trap. Collisions of reactive molecules in the rotational ground state are well described in the fixed-dipole approximation. The present calculation confirms that for sufficiently strong induced dipole moments electrostatic repulsion between dipoles perpendicular to the trap axis leads to suppression of the reactive rates. For non-reactive molecules, the rotational degrees of freedom result in an increased density of Fano-Feshbach resonances. We demonstrate that the resonance widths decrease and resonance spectra dramatically decongest for increasing induced electric dipole moment of the molecules. This effect could be exploited to decrease the collision lifetime and possibly to suppress harmful processes due to the formation of an intermediate long-lived complex.

In perspective, it can also be interesting to model experiments where collision dynamics has been studied in the presence of an additional optical lattice along the axis of the tube Chotia et al. (2012). This computational task could be accomplished for instance by combining the present 3D solution strategy and the asymptotic reference Bloch functions constructed in Terrier et al. (2016). The effect of hyperfine interactions could also be included in the model to various levels of approximation.

Acknowledgements.

This work is supported by the Agence Nationale de la Recherche (Contract COLORI No. ANR-12-BS04-0020-01).