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Physical Review A

Shape resonances in ground-state diatomic molecules: General trends and the example of RbCs

B. E. Londoño, Jorge Mahecha, Eliane Luc-Koenig, A. Crubellier

Published Jul 29, 2010• 30 citations
Publisher

Abstract

The presence of shape resonances due to tunneling through the centrifugal barrier modifies strongly the dynamics of cold atom scattering. As shown on the example of the ground and lowest triplet electronic states of the $^{85}\mathrm{Rb}$$^{133}\mathrm{Cs}$ molecule, the crucial parameter is, as usual for cold collisions, the scattering length. A general description of shape resonances of diatomic molecules is given from three simple single channel asymptotic models, whose respective performances are discussed. The first model, which consists of a ${R}^{\ensuremath{-}6}$ potential limited at short range by a repulsive wall, positioned to reproduce the $s$-wave scattering length, accounts satisfactorily for the main system-independent properties of shape resonances. Introduction in the model of energy- and angular-momentum-dependent nodal lines specific to the inner part of the potential greatly improves its efficiency. When the energy and angular momentum dependence of the nodal lines cannot be deduced from full potential calculations or from experiment, a rough, but universal, estimate of these properties is obtained by extending the ${R}^{\ensuremath{-}6}$ behavior of the potential up to the origin.

DOI
10.1103/physreva.82.012510

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We observe the time dependence of a cold-atom collision in a pulsed photoassociation experiment. For a $g$-wave shape resonance in the ${}^{85}\mathrm{Rb}{+}^{85}\mathrm{Rb}$ system we measure the time needed to build up the resonant state by tunneling through the centrifugal barrier. Combining this with time-independent ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ photoassociation we determine the resonance energy and find evidence for the decay of the shape resonance into inelastic channels. We also determine the ${}^{85}\mathrm{Rb}+{}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}{+}^{87}\mathrm{Rb}$ ${C}_{6}$ coefficient and triplet scattering lengths without relying on ab initio calculations.

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Historical Notes on Feshbach and Shape Resonances

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In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.

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Analytical calculation of cold-atom scattering

V. V. Flambaum, G. F. Gribakin, C. Harabati

The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (${a}_{0}$ is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length $a,$ which showed that it has 75% preference for positive values for $n=6,$ characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of $a,$ ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where ${F}_{n},$ ${G}_{n},$ and ${H}_{n}$ depend only on \ensuremath{\gamma}. Thus, we know the $s$-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the $s$-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.

Physical Review A•Mar 1, 1999•180 citations
The presence of shape resonances due to tunneling through the centrifugal barrier modifies strongly the dynamics of cold atom scattering. As shown on the example of the ground and lowest triplet electronic states of the ^{85}\mathrm{Rb}$$^{133}\mathrm{Cs} molecule, the crucial parameter is, as usual for cold collisions, the scattering length. A general description of shape resonances of diatomic molecules is given from three simple single channel asymptotic models, whose respective performances are discussed. The first model, which consists of a R\ensuremath−6{R}^{\ensuremath{-}6}R\ensuremath−6 potential limited at short range by a repulsive wall, positioned to reproduce the sss-wave scattering length, accounts satisfactorily for the main system-independent properties of shape resonances. Introduction in the model of energy- and angular-momentum-dependent nodal lines specific to the inner part of the potential greatly improves its efficiency. When the energy and angular momentum dependence of the nodal lines cannot be deduced from full potential calculations or from experiment, a rough, but universal, estimate of these properties is obtained by extending the R\ensuremath−6{R}^{\ensuremath{-}6}R\ensuremath−6 behavior of the potential up to the origin.
Non-resonant light interacting with diatomics via the polarizability anisotropy couples different rotational states and may lead to strong hybridization of the motion. The modification of shape resonances and low-energy scattering states due to this interaction can be fully captured by an asymptotic model, based on the long-range properties of the scattering (Crubellier et al 2015 New J. Phys. 17 045020 ). Remarkably, the properties of the field-dressed shape resonances in this asymptotic multi-channel description are found to be approximately linear in the field intensity up to fairly large intensity. This suggests a perturbative single-channel approach to be sufficient to study the control of such resonances by the non-resonant field. The multi-channel results furthermore indicate the dependence on field intensity to present, at least approximately, universal characteristics. Here we combine the nodal line technique to solve the asymptotic Schrödinger equation with perturbation theory. Comparing our single channel results to those obtained with the full interaction potential, we find nodal lines depending only on the field-free scattering length of the diatom to yield an approximate but universal description of the field-dressed molecule, confirming universal behavior.
We observe the time dependence of a cold-atom collision in a pulsed photoassociation experiment. For a ggg-wave shape resonance in the 85Rb+85Rb{}^{85}\mathrm{Rb}{+}^{85}\mathrm{Rb}85Rb+85Rb system we measure the time needed to build up the resonant state by tunneling through the centrifugal barrier. Combining this with time-independent 85Rb{}^{85}\mathrm{Rb}85Rb and 87Rb{}^{87}\mathrm{Rb}87Rb photoassociation we determine the resonance energy and find evidence for the decay of the shape resonance into inelastic channels. We also determine the 85Rb+85Rb{}^{85}\mathrm{Rb}+{}^{85}\mathrm{Rb}85Rb+85Rb and 87Rb+87Rb{}^{87}\mathrm{Rb}{+}^{87}\mathrm{Rb}87Rb+87Rb C6{C}_{6}C6​ coefficient and triplet scattering lengths without relying on ab initio calculations.
In his interesting Reference Frame piece (Physics Today, August 2004, page 12), Daniel Kleppner relates Herman Feshbach’s reaction to the term “Feshbach resonance” and gives a capsule account of the topical and vital role this phenomenon plays in today’s explorations of cold collisions of fermions and bosons, and of condensates of atoms and molecules. Readers may be interested in the earlier history and why atomic physicists began to use a term that Feshbach himself dismissed as jargon.In the Los Alamos experiment of Howard Bryant’s group on photo-absorption by the negative ion of hydrogen, two prominent resonances were seen at photon energies of about 10.95 eV. This energy marks the position of the first excited state of the hydrogen atom above the single bound state of the negative ion.One narrow resonance occurs just below the threshold of that excitation, whereas the second, broad one occurs just above. Their physical mechanisms are different and, following the paradigm set by that historic experiment, it has become customary to refer to the two types of resonance, both common in atomic and molecular physics, as Feshbach and shape resonance, respectively. That is, a resonant coupling just below the relevant threshold, leading to narrow profiles, is distinguished from those lying just above the threshold to which they are most strongly coupled, and typically broad.The two types of resonances are most naturally viewed in state and coordinate space, respectively. Feshbach resonances are thought of in terms of the superposition of two quantum states, one with discrete and one with continuum character, leading to a temporary quasi-bound state that manifests itself as a resonance in that continuum channel. The independent and almost simultaneous work in nuclear physics by Feshbach using the language of projection operators and in atomic physics by Ugo Fano using the language of superposition of wave functions serves for their technical description.Shape resonances, on the other hand, may be associated with the shape of a trapping potential in real, coordinate space. Most often, the trapping potential is an angular momentum potential that combines with the internal one to create intervening barriers to low-energy particles. The resulting pictures often display “two-valley” potentials. In the case of the negative ion of hydrogen, such a barrier has been described in terms of the hyperspherical coordinates of that three-body system. Examples of such shape resonances abound in low-energy scattering of electrons from molecules.For more discussion and references, including to recent applications in mesoscopic condensed matter systems, see my pedagogical mini-review in Physica Scripta, volume 69, page C10, 2004.© 2005 American Institute of Physics.
The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.
A (diatomic) shape resonance is a metastable state of a pair of colliding atoms quasi-bound by the centrifugal barrier imposed by the angular momentum involved in the collision. The temporary trapping of the atoms' scattering wavefunction corresponds to an enhanced atom pair density at low interatomic separations. This leads to larger overlap of the wavefunctions involved in a molecule formation process such as photoassociation, rendering the process more efficient. However, for an ensemble of atoms, the atom pair density will only be enhanced if the energy of the resonance comes close to the temperature of the atomic ensemble. Herein we explore the possibility of controlling the energy of a shape resonance by shifting it toward the temperature of atoms confined in a trap. The shifts are imparted by the interaction of non-resonant light with the anisotropic polarizability of the atom pair, which affects both the centrifugal barrier and the pair's rotational and vibrational levels. We find that at laser intensities of up to 5×1095\times 10^{9}5×109 W/cm2^22 the pair density is increased by one order of magnitude for 87^{87}87Rb atoms at 100μ100 μ100μK and by two orders of magnitude for 88^{88}88Sr atoms at 20μ20 μ20μK.
The interaction between atoms behaves as $\ensuremath{-}\ensuremath{\alpha}{/r}^{n}$ at large distances and, owing to the large reduced mass \ensuremath{\mu} of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\ensuremath{\gamma}=\sqrt{2\ensuremath{\mu}\ensuremath{\alpha}}/\ensuremath{\Elzxh}\ensuremath{\gg}{a}_{0}^{(n\ensuremath{-}2)/2}$ (a0{a}_{0}a0​ is the Bohr radius) and the semiclassical zero-energy phase $\ensuremath{\Phi}\ensuremath{\gg}1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length a,a,a, which showed that it has 75% preference for positive values for n=6,n=6,n=6, characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range and show that it is a function of a,a,a, ${r}_{e}{=F}_{n}\ensuremath{-}{G}_{n}{/a+H}_{n}{/a}^{2},$ where Fn,{F}_{n},Fn​, Gn,{G}_{n},Gn​, and Hn{H}_{n}Hn​ depend only on \ensuremath{\gamma}. Thus, we know the sss-phase shift at low momenta $k\ensuremath{\ll}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ from the expansion $k\mathrm{cot}{\ensuremath{\delta}}_{0}\ensuremath{\simeq}\ensuremath{-}1/a+\frac{1}{2}{r}_{e}{k}^{2}.$ At $k\ensuremath{\gg}{\ensuremath{\gamma}}^{\ensuremath{-}2/(n\ensuremath{-}2)}$ the phase shift is obtained semiclassically as ${\ensuremath{\delta}}_{0}=\ensuremath{\Phi}+\ensuremath{\pi}/4\ensuremath{-}{I}_{n}{\ensuremath{\gamma}}^{2/n}{k}^{(n\ensuremath{-}2)/n},$ where ${I}_{n}=[n/(n\ensuremath{-}2)]\ensuremath{\Gamma}((n\ensuremath{-}1)/n)\ensuremath{\Gamma}((n+2)/2n)/\sqrt{\ensuremath{\pi}}.$ Therefore, \ensuremath{\gamma} and \ensuremath{\Phi} determine the sss-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.