Final Report Summary - SUPERCOLD (Quantum scattering at ultracold temperatures: a few S-matrix columns is all we need)
Ultracold species will have a significant impact in high-precision measurements and metrology, the study of collective phenomena in condensed-matter systems and of quantum-controlled chemical reactions, and the development of quantum technologies. Studying low-temperature collisions is of central importance. Much of the information gathered over the years on molecular properties and interactions is the result of performing and analysing thousands of scattering experiments; ultracold species will not be an exception. Moreover, collisions are fundamental in reaching the ultracold regime since various cooling techniques rely on thermalisation with colder species. Furthermore, collisions often determine loss rates in traps and thus the lifetime of (ultra)cold samples.
The coupled-channel method has been a powerful theoretical tool for several decades. Its brute-force application to many low-temperature scattering problems, however, faces two key challenges: (1) the need for large basis sets that become computationally intractable, and (2) the need to explore a large, multidimensional parametric space to answer questions concerning real experiments. The "basis size" problem arises because many interactions that are negligible at higher temperatures become comparable to, or larger than, the kinetic energies involved. These include interactions associated with the electronic (fine) and nuclear (hyperfine) degrees of freedom. Taking all such terms into account significantly increases the size of the basis needed for convergence and has dramatic effects on the computing effort per scattering calculation. Accounting for hyperfine interactions is specially challenging, even if these may be key at ultracold temperatures: depending on the system, hyperfine terms can increase the computational cost per scattering calculation by one to four orders of magnitude. The "parameters" problem arises because these computationally expensive scattering calculations need to be performed at many different values of, e.g. the collision energies and external field(s) strength. Simultaneously, calculations need to explore the effect of inaccuracies in the interaction potentials and calculated parameters.
SUPERCOLD was designed to search for alternative formalisms to deal with these obstacles. Its main objectives are (the): (1) Implementation of alternative algorithms for S-matrix calculations---including code development, testing and optimisation; (2) Calculation of molecular interaction potentials and/or interaction parameters needed for their application; and (3/4) Original studies of molecular systems of experimental interest, excluding/including hyperfine couplings, using the developed methodology.
The project has achieved 3 significant results so far:
(1) Statistical formalism for (ultracold) chemical reactions in external fields.
The formalism was developed in collaboration with colleagues at Institut des Sciences Moléculaires in Bordeaux. It is based on the statistical approximation to state-to-state transition probabilities (modulus squared of S-matrix elements) and provides the first generally tractable, rigorous theoretical framework for computing statistical product-state distributions for ultracold reactions in fields. We showed that fields have two main effects on the products of a statistical reaction, by: (1) modifying the product energy levels thus potentially reshaping the product distributions; and/or (2) adding or removing product states by changing the reaction exothermicity. These effects are observed in Fig. 1, which depicts statistical translational energy distributions in the ultracold reaction 40K + 87Rb2 --> 40K87Rb + 87Rb in an external electric field. Our results are a key demonstration that external control of product distributions is possible, although limited, even for statistical processes. We additionally demonstrated how statistical predictions can be used to distinguish between different reaction mechanisms, e.g. we explored distributions when hyperfine degrees of freedom are "active" during the reaction and when they are simply "spectators". In general, statistical tests using predicted curves as prior distributions may be used to assess the involvement of different degrees of freedom in a reaction and its variation with varying conditions.
Such alternative, statistical formalism will provide fundamental tests of statistical concepts, which have been suggested may be broadly applicable to describe reaction rates and global properties for ultracold processes. Furthermore, it makes it possible to calculate benchmark distributions to critically evaluate possible departures from statistical behaviour and may thus become a powerful tool to rationalise all kinds of reactions. These are absolutely crucial developments, because they will help understanding product formation and control in ultracold reactions, and provide theoretical leadership for the world-wide effort towards exploiting the full potential of ultracold chemistry.
I developed the highly-optimised StatFields program package to derive statistical predictions in fields.
(2) Re-coupling methodology for approximate hyperfine effects for ultracold scattering in external fields.
I developed a new method to approximately account for the effects of hyperfine interactions in ultracold scattering calculations in external fields. The technique is based on rewriting S-matrix elements using a frame transformation (re-coupling scheme) and a Wigner correction to account for low-energy threshold behaviour. I have shown how hyperfine effects may be added a posteriori to hyperfine-free calculations in order to obtain approximate hyperfine state-to-state cross sections. The new method provides a natural explanation to the main hyperfine effects identified in the literature and can be used independently or combined with other approximations to significantly reduce the basis sizes in ultracold scattering calculations. Depending on the system considered, it may lead to one to four orders-of-magnitude savings in computing times, opening the door to widely studying hyperfine effects in ultracold molecular collisions.
As an example, Fig. 2 depicts state-to-state cross sections in ultracold 24Mg+14NH collisions using full close-coupling and approximate calculations with the new methodology. The agreement between these calculations is excellent, although full calculations require over 200 times more computing time than those using the new algorithm. Additional tests are currently underway in collaboration with Hutson's group at Durham University.
(3) Determination and statistical analysis of the Feshbach spectrum of alkali+lanthanide systems.
In collaboration with P. S. Zuchowski from Nicolas Copernicus University at Torun, Poland, I carried out high-level ab initio calculations on the interaction potentials for Li(2S)+Er(3H) systems. We then used these potentials in accurate coupled-channel calculations of the magnetic Feshbach spectra of various isotopologues and studied their most significant statistical properties. Our work is the first study of the magnetic Feshbach spectra of an ultracold mixture of alkali and a highly-magnetic lanthanide atom. It provides compelling and robust theoretical evidence that low-field resonances immune to background losses exist for Li+Er, with widths well within current experimental resolution. This is a critical new development in the search for suitable atomic pairs to create ultracold molecules with both electric and magnetic dipole moments. Furthermore, the Li+Er system is particularly appealing for studies of quantum many-body physics, high-precision spectroscopy and Efimov physics.
While results (1) and (2) above mainly account to alternative ways to derive S-matrix elements, and thus deal directly with the "basis size" challenge, these Li+Er study tackles more directly the "parameters" problem. As shown in Figs.3(a) and (b), the Li+Er spectra are predicted to have strikingly different statistical properties than those of other systems involving highly-magnetic atoms. In particular, the spectra are much less congested, while remaining conveniently dense, and exhibit non-chaotic properties. These features would facilitate identifying and addressing individual resonances. Based on these properties, we derived a simple model to predict resonance positions for different isotopologues from measurements on a reference system, which would greatly simplify designing experiments (and bound-state calculations) involving different Er bosonic isotopes and give key insight into non-adiabatic effects.
SUPERCOLD's public website: http://community.dur.ac.uk/m.l.gonzalez-martinez/supercold.html(se abrirá en una nueva ventana)
The coupled-channel method has been a powerful theoretical tool for several decades. Its brute-force application to many low-temperature scattering problems, however, faces two key challenges: (1) the need for large basis sets that become computationally intractable, and (2) the need to explore a large, multidimensional parametric space to answer questions concerning real experiments. The "basis size" problem arises because many interactions that are negligible at higher temperatures become comparable to, or larger than, the kinetic energies involved. These include interactions associated with the electronic (fine) and nuclear (hyperfine) degrees of freedom. Taking all such terms into account significantly increases the size of the basis needed for convergence and has dramatic effects on the computing effort per scattering calculation. Accounting for hyperfine interactions is specially challenging, even if these may be key at ultracold temperatures: depending on the system, hyperfine terms can increase the computational cost per scattering calculation by one to four orders of magnitude. The "parameters" problem arises because these computationally expensive scattering calculations need to be performed at many different values of, e.g. the collision energies and external field(s) strength. Simultaneously, calculations need to explore the effect of inaccuracies in the interaction potentials and calculated parameters.
SUPERCOLD was designed to search for alternative formalisms to deal with these obstacles. Its main objectives are (the): (1) Implementation of alternative algorithms for S-matrix calculations---including code development, testing and optimisation; (2) Calculation of molecular interaction potentials and/or interaction parameters needed for their application; and (3/4) Original studies of molecular systems of experimental interest, excluding/including hyperfine couplings, using the developed methodology.
The project has achieved 3 significant results so far:
(1) Statistical formalism for (ultracold) chemical reactions in external fields.
The formalism was developed in collaboration with colleagues at Institut des Sciences Moléculaires in Bordeaux. It is based on the statistical approximation to state-to-state transition probabilities (modulus squared of S-matrix elements) and provides the first generally tractable, rigorous theoretical framework for computing statistical product-state distributions for ultracold reactions in fields. We showed that fields have two main effects on the products of a statistical reaction, by: (1) modifying the product energy levels thus potentially reshaping the product distributions; and/or (2) adding or removing product states by changing the reaction exothermicity. These effects are observed in Fig. 1, which depicts statistical translational energy distributions in the ultracold reaction 40K + 87Rb2 --> 40K87Rb + 87Rb in an external electric field. Our results are a key demonstration that external control of product distributions is possible, although limited, even for statistical processes. We additionally demonstrated how statistical predictions can be used to distinguish between different reaction mechanisms, e.g. we explored distributions when hyperfine degrees of freedom are "active" during the reaction and when they are simply "spectators". In general, statistical tests using predicted curves as prior distributions may be used to assess the involvement of different degrees of freedom in a reaction and its variation with varying conditions.
Such alternative, statistical formalism will provide fundamental tests of statistical concepts, which have been suggested may be broadly applicable to describe reaction rates and global properties for ultracold processes. Furthermore, it makes it possible to calculate benchmark distributions to critically evaluate possible departures from statistical behaviour and may thus become a powerful tool to rationalise all kinds of reactions. These are absolutely crucial developments, because they will help understanding product formation and control in ultracold reactions, and provide theoretical leadership for the world-wide effort towards exploiting the full potential of ultracold chemistry.
I developed the highly-optimised StatFields program package to derive statistical predictions in fields.
(2) Re-coupling methodology for approximate hyperfine effects for ultracold scattering in external fields.
I developed a new method to approximately account for the effects of hyperfine interactions in ultracold scattering calculations in external fields. The technique is based on rewriting S-matrix elements using a frame transformation (re-coupling scheme) and a Wigner correction to account for low-energy threshold behaviour. I have shown how hyperfine effects may be added a posteriori to hyperfine-free calculations in order to obtain approximate hyperfine state-to-state cross sections. The new method provides a natural explanation to the main hyperfine effects identified in the literature and can be used independently or combined with other approximations to significantly reduce the basis sizes in ultracold scattering calculations. Depending on the system considered, it may lead to one to four orders-of-magnitude savings in computing times, opening the door to widely studying hyperfine effects in ultracold molecular collisions.
As an example, Fig. 2 depicts state-to-state cross sections in ultracold 24Mg+14NH collisions using full close-coupling and approximate calculations with the new methodology. The agreement between these calculations is excellent, although full calculations require over 200 times more computing time than those using the new algorithm. Additional tests are currently underway in collaboration with Hutson's group at Durham University.
(3) Determination and statistical analysis of the Feshbach spectrum of alkali+lanthanide systems.
In collaboration with P. S. Zuchowski from Nicolas Copernicus University at Torun, Poland, I carried out high-level ab initio calculations on the interaction potentials for Li(2S)+Er(3H) systems. We then used these potentials in accurate coupled-channel calculations of the magnetic Feshbach spectra of various isotopologues and studied their most significant statistical properties. Our work is the first study of the magnetic Feshbach spectra of an ultracold mixture of alkali and a highly-magnetic lanthanide atom. It provides compelling and robust theoretical evidence that low-field resonances immune to background losses exist for Li+Er, with widths well within current experimental resolution. This is a critical new development in the search for suitable atomic pairs to create ultracold molecules with both electric and magnetic dipole moments. Furthermore, the Li+Er system is particularly appealing for studies of quantum many-body physics, high-precision spectroscopy and Efimov physics.
While results (1) and (2) above mainly account to alternative ways to derive S-matrix elements, and thus deal directly with the "basis size" challenge, these Li+Er study tackles more directly the "parameters" problem. As shown in Figs.3(a) and (b), the Li+Er spectra are predicted to have strikingly different statistical properties than those of other systems involving highly-magnetic atoms. In particular, the spectra are much less congested, while remaining conveniently dense, and exhibit non-chaotic properties. These features would facilitate identifying and addressing individual resonances. Based on these properties, we derived a simple model to predict resonance positions for different isotopologues from measurements on a reference system, which would greatly simplify designing experiments (and bound-state calculations) involving different Er bosonic isotopes and give key insight into non-adiabatic effects.
SUPERCOLD's public website: http://community.dur.ac.uk/m.l.gonzalez-martinez/supercold.html(se abrirá en una nueva ventana)
