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Physical Review A

Quantum state preparation of homonuclear molecular ions enabled via a cold buffer gas: An ab initio study for the H2+ and the D2+ case

Stephan Schiller, I. V. Kortunov, Mario Hernández Vera, Franco Antonio Gianturco, Humberto da Silva

Published Apr 13, 2017• 26 citations
Publisher

Abstract

Precision vibrational spectroscopy of the molecular hydrogen ions is of significant interest for determining fundamental constants, for searching for new forces, and for testing quantum electrodynamics calculations. Future experiments can profit from the ability of preparing molecular hydrogen ions at ultralow kinetic energy and in preselected internal states, with respect to vibration, rotation, and spin degrees of freedom. For the homonuclear ions (${\mathrm{H}}_{2}{}^{+}, {\mathrm{D}}_{2}{}^{+}$), direct laser cooling of the rotational degree of freedom is not feasible. We show by quantum calculations that rotational cooling by cold He buffer gas is an effective approach. For this purpose we have computed the energy-dependent cross sections for rotationally elastic and inelastic collisions, ${\mathrm{h}}_{2}{}^{+}(v=0,\phantom{\rule{0.16em}{0ex}}N)+\mathrm{He}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow} {\mathrm{h}}_{2}{}^{+}(v=0,\phantom{\rule{0.16em}{0ex}}{N}^{\ensuremath{'}})+\mathrm{He}$ (where $\mathrm{h}=\mathrm{H},\phantom{\rule{0.16em}{0ex}}\mathrm{D})$, using ab initio coupled-channel calculations. We find that rotational cooling to the lowest rotational state is possible within tens of seconds under experimentally realistic conditions. We furthermore describe possible protocols for the preparation of a single quantum state, where also the spin state is well defined.

DOI
10.1103/physreva.95.043411

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Cooling all degrees of freedom of a molecule, a cluster, or even a nanoparticle which is suspended in a vacuum, is an experimental challenge. Without suitable schemes, cold or ultracold chemical reactions are not feasible. Methods such as laser based preparation of very slow atoms, decelerating molecules to low velocities with electric fields or freezing molecular ions into Coulomb crystals, are generally not suitable to cool the vibrational or rotational motion of molecules. This contribution describes a new method in which a beam of slow atoms or molecules (H, He, H2, or D2) is used for cooling charged particles confined in a multi-electrode rf trap. For reaching sub-K temperatures, the fast part of a cold effusive beam is removed with a shutter before the slow remaining neutrals interact with the ion cloud. The development of a pulsed cold beam source is discussed as well as suitable methods for determining the ion temperature. A challenging application is to prepare internally cold CH5+ for spectroscopy or chemistry. New experimental results for hydrogen abstraction in collisions with slow H atoms are reported at energies of a few meV. For evaluating these measurements and for predicting effective rate coefficients at lower energies, the kinematic conditions of the slow neutral beam-ion trap arrangement have been analyzed in detail. The potential of cooling ions such as protonated methane or H3+ with slow energy selected H atoms is briefly mentioned. An interesting process is the formation of weakly bound ions such as H4+ or CH6+ via radiative or ternary association. Such ions are ideal candidates for preparing the corresponding collision complexes very close (microeV) to the dissociation continuum using infrared transitions.

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•Jun 1, 2011•0 citations

Quantum dynamics and cooling kinetics of BN− anions via buffer gases in ion traps

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Following the previous study with an extensive range of quantum calculations involving different electronic states of the BN- anion [Dulitz et al., Phys. Scripta 100, 055411 (2025)], we now extend that work by modeling the quantum dynamics of the collision cooling of its rotational states in order to investigate possible paths for bringing this molecular anion down to temperatures of a few Kelvins. This specific ionic system is of direct interest when modeling experiments in cold ion traps where He or Ar atoms can function as the chief buffer gases that drive the anions down to the low trap temperatures. We employ accurate, ab initio calculations of the potential energy surfaces for the title system in its ground electronic state, interacting with either He or Ar atoms. We then obtain a wide range of inelastic cross sections and the ensuing rate coefficients in order to model the quantum kinetics of the time evolution of the cooling steps under different temperature and trap conditions. The results are analyzed and employed to estimate the cooling efficiency paths provided by various trap arrangements for the title anion. The results show that-using either of the two investigated species-the buffer gas cooling process very efficiently brings the anions to their lowest rotational states. These findings are very promising for future applications in the field of anion laser cooling.

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Quantum Control of a Single H2+ Molecular Ion

David Holzapfel

In this thesis I describe how we achieve control over the quantum state of a single H2+ hydrogen molecular ion. Scientific progress is based on the comparison of theories with experimental measurements. For the simplest molecular quantum systems, such as the hydrogen molecular ions H2+ or HD+, the theoretical predictions of quantum electrodynamics are very accurate. In order to test the theory or to extract fundamental constants from the combination of theory and experiments, the experiments must also achieve high precision. In the recent past, hydrogen molecular ions were mainly measured in ensembles. For example, measurements exist both on molecular beams and with dissociative detection in ion traps. However, the spectroscopy of single ions is one of the most precise measurement methods in physics, which was demonstrated by the spectroscopy of a single ytterbium ion in an ion trap. To perform spectroscopy of a single ion in this way, it must have a suitable energy structure for laser cooling, state preparation and fluorescence detection. This is not the case for many ions, including H2+. Quantum logic spectroscopy (QLS) is a technique that makes it possible to perform measurements on an ion without such an energy structure. The ion to be measured (spectroscopy ion) is trapped in an ion trap with a second ion (helper ion), which has the properties described in the previous paragraph. The shared modes of motion between the two ions allow cooling, preparation and detection of the spectroscopy ion via the helper ion. This was demonstrated with an aluminum spectroscopy ion co-trapped with a magnesium helper ion. QLS can also be used for molecules. This has been shown with MgH+, CaH+ and N2+. We implement QLS to enable spectroscopy of a single H2+ ion. For this we trap H2+ together with a 9Be+ ion. We cool the joint motion of the ions to their ground state. We combine buffer gas cooling with a laser and microwave pulse based state preparation sequence to prepare the H2+ molecule in a pure quantum state. This allows us to perform high contrast QLS on a single H2+ ion. Furthermore, we can change and control the quantum state in the hyperfine structure using microwave pulses. We demonstrate the power of our technique by performing microwave spectroscopy on a magnetic field insensitive hyperfine transition. We use QLS to read out the result of the measurement. Within half an hour, we can achieve a statistical uncertainty of 2 Hz. This paves the way for future high-precision measurements on H2+. The techniques described here could potentially be applied to other light molecular ions.

•Jan 1, 2025•0 citations
Precision vibrational spectroscopy of the molecular hydrogen ions is of significant interest for determining fundamental constants, for searching for new forces, and for testing quantum electrodynamics calculations. Future experiments can profit from the ability of preparing molecular hydrogen ions at ultralow kinetic energy and in preselected internal states, with respect to vibration, rotation, and spin degrees of freedom. For the homonuclear ions (H2+,D2+{\mathrm{H}}_{2}{}^{+}, {\mathrm{D}}_{2}{}^{+}H2​+,D2​+), direct laser cooling of the rotational degree of freedom is not feasible. We show by quantum calculations that rotational cooling by cold He buffer gas is an effective approach. For this purpose we have computed the energy-dependent cross sections for rotationally elastic and inelastic collisions, h2+(v=0,N)+He\ensuremath→h2+(v=0,N\ensuremath′)+He{\mathrm{h}}_{2}{}^{+}(v=0,\phantom{\rule{0.16em}{0ex}}N)+\mathrm{He}\phantom{\rule{0.16em}{0ex}}\ensuremath{\rightarrow} {\mathrm{h}}_{2}{}^{+}(v=0,\phantom{\rule{0.16em}{0ex}}{N}^{\ensuremath{'}})+\mathrm{He}h2​+(v=0,N)+He\ensuremath→h2​+(v=0,N\ensuremath′)+He (where h=H,D)\mathrm{h}=\mathrm{H},\phantom{\rule{0.16em}{0ex}}\mathrm{D})h=H,D), using ab initio coupled-channel calculations. We find that rotational cooling to the lowest rotational state is possible within tens of seconds under experimentally realistic conditions. We furthermore describe possible protocols for the preparation of a single quantum state, where also the spin state is well defined.
Using the ab initio computed potential energy surface for the electronic interaction of the ${\mathrm{MgH}}^{+}(^{1}\mathrm{\ensuremath{\Sigma}}$) ion with the He(1S^{1}S1S) atom, we calculate the relevant state-changing rotationally inelastic collision cross sections from a quantum treatment of the multichannel scattering problem. We focus on the quantum dynamics at the translationally low energies for the present partners discussed in earlier, cold ion trap experiments (see below), which we wish to model in detail. The corresponding state-changing rates computed between the lower rotational states of the molecular ion are employed to describe the time-evolution kinetics followed by recent experiments on Coulomb-crystallized ${\mathrm{MgH}}^{+}(^{1}\mathrm{\ensuremath{\Sigma}}$), where the ions are rotationally cooled by micromotion tuning after being uploaded into the trap of He as a buffer gas. The present computational modeling of the final ion rotational temperatures in the experiments turns out to agree very well with the observations and points at a fast equilibration between rotational and thermal temperatures of the ions.
Cooling all degrees of freedom of a molecule, a cluster, or even a nanoparticle which is suspended in a vacuum, is an experimental challenge. Without suitable schemes, cold or ultracold chemical reactions are not feasible. Methods such as laser based preparation of very slow atoms, decelerating molecules to low velocities with electric fields or freezing molecular ions into Coulomb crystals, are generally not suitable to cool the vibrational or rotational motion of molecules. This contribution describes a new method in which a beam of slow atoms or molecules (H, He, H2, or D2) is used for cooling charged particles confined in a multi-electrode rf trap. For reaching sub-K temperatures, the fast part of a cold effusive beam is removed with a shutter before the slow remaining neutrals interact with the ion cloud. The development of a pulsed cold beam source is discussed as well as suitable methods for determining the ion temperature. A challenging application is to prepare internally cold CH5+ for spectroscopy or chemistry. New experimental results for hydrogen abstraction in collisions with slow H atoms are reported at energies of a few meV. For evaluating these measurements and for predicting effective rate coefficients at lower energies, the kinematic conditions of the slow neutral beam-ion trap arrangement have been analyzed in detail. The potential of cooling ions such as protonated methane or H3+ with slow energy selected H atoms is briefly mentioned. An interesting process is the formation of weakly bound ions such as H4+ or CH6+ via radiative or ternary association. Such ions are ideal candidates for preparing the corresponding collision complexes very close (microeV) to the dissociation continuum using infrared transitions.
Stationary molecules in well-defined internal states are of broad interest in both physics and chemistry. Through high-resolution spectroscopy, fundamental physics can be tested and lead to, e.g., improved values of the electron-to-proton mass ratio [1], new boundaries on the electron electric dipole moment [2] and observation of the potential evolution of the fine structure constant over time [3]. Translationally and internally cold molecules have as well been considered promising candidates for qubits in new quantum computing senarios [4], and molecular ions could in the future likely become an excellent alternative to atomic qubits in the realization of a practical ion trap based quantum computer due to favourable internal state decoherence rates. In chemistry, state prepared molecular targets are an ideal starting point for uni-molecular reactions, including coherent control of photofragmentation through the application of various laser sources [5,6]. In cold bi-molecular reactions, where the effect of even tiny potential barriers becomes significant, experiments with state prepared molecules can yield important information on the details of the potential curves of the molecular complexes [7,8,9]. Furthermore, in order to learn more about the chemistry in interstellar clouds, astrochemists can benefit greatly from direct measurements on cold reactions in laboratories [9]. Working with MgH+ molecular ions in a linear Paul trap, we routinely cool their translational degree of freedom by sympathetic cooling with Doppler laser cooled Mg+ ions. Giving the time for the molecules to equilibrate internally to the room temperature blackbody radiation, the vibrational degree of freedom will freeze out, leaving only the rotational degree of freedom to be cooled. We report here on the implementation of a new technique for laser-induced rotational ground-state cooling of vibrationally and translationally cold MgH+ ions [10]. The scheme is based on excitation of a single rovibrational transition [11], and it should be generalizable to any diatomic polar molecular ion, given appropriate mid-infrared laser sources such as a quantum cascade laser are available. In recent experiments, a nearly 15-fold increase in the rotational ground-state population was obtained, with the resulting ground-state population of 36,7±1,2 %, equivalent to that of a thermal distribution at about 20 K. The obtained cooling results imply that, through this technique, cold molecular-ion experiments can now be carried out at cryogenic temperatures in room-temperature set-ups. References [1] Koelemeij, J. C. J., Roth, B., Wicht, A., Ernsting, I. and Schiller, S., Phys. Rev. Lett. 98, 173002 (2007). [2] Hudson, J. J., Sauer, B. E., Tarbutt, M. R. and Hinds, E. A., Phys. Rev. Lett. 89, 023003 (2002). [3] Hudson, E. R., Lewandowski, H. J., Sawyer, B. C. and Ye, J, Phys. Rev. Lett. 96, 143004 (2006). [4] DeMille, D, Phys. Rev. Lett. 88, 067901 (2002). [5] Rice, S. A. and Zaho, M. Optical control of molecular dynamics (Wiley, New York, 2000). [6] Shapiro, M. and Brumer, P. W. Principles of the Quantum Control of Molecular Processes (Wiley, New York, 2003). [7] Willitsch, S., Bell, M. T., Gingell, A. D. and Softley, T. P., Phys. Chem. Chem. Phys. 10, 7200 ( 2008). [8] Krems, R. V., Phys. Chem. Chem. Phys. 10, 4079 (2008). [9] Smith, I. W. M., Low temperatures and cold molecules (Imperial College Press, London, 2008). [10] Staanum, P. F., Højbjerre, K., Skyt, P.S., Hansen, A. K. and Drewsen, M., Nat. Phys. 6, 271 (2010). [11] Vogelius, I. S., Madsen, L. B. and Drewsen, M., Phys. Rev. Lett. 89, 173003 (2002).
Following the previous study with an extensive range of quantum calculations involving different electronic states of the BN- anion [Dulitz et al., Phys. Scripta 100, 055411 (2025)], we now extend that work by modeling the quantum dynamics of the collision cooling of its rotational states in order to investigate possible paths for bringing this molecular anion down to temperatures of a few Kelvins. This specific ionic system is of direct interest when modeling experiments in cold ion traps where He or Ar atoms can function as the chief buffer gases that drive the anions down to the low trap temperatures. We employ accurate, ab initio calculations of the potential energy surfaces for the title system in its ground electronic state, interacting with either He or Ar atoms. We then obtain a wide range of inelastic cross sections and the ensuing rate coefficients in order to model the quantum kinetics of the time evolution of the cooling steps under different temperature and trap conditions. The results are analyzed and employed to estimate the cooling efficiency paths provided by various trap arrangements for the title anion. The results show that-using either of the two investigated species-the buffer gas cooling process very efficiently brings the anions to their lowest rotational states. These findings are very promising for future applications in the field of anion laser cooling.
The anisotropic interactions of three polar molecules CO, HF, and LiH with He atoms (in their ground electronic states), obtained from accurate ab initio calculations that explicitly include their vibration-to-translation coupling terms are analyzed in detail, to compare their relative features. The quantum scattering calculations of their rotovibrational inelastic cross sections are conducted using a recently proposed multichannel treatment, the modified variable phase method, that has been implemented by the authors and applied here to ultralow collision energies. A comparison of the different relaxation efficiencies exhibited by the three title molecules in releasing their internal vibrational energy during ultracold collisions with 4 He and 3 He buffer gas is performed in detail and specific suggestions for experimental choices are extracted from these findings.
In this thesis I describe how we achieve control over the quantum state of a single H2+ hydrogen molecular ion. Scientific progress is based on the comparison of theories with experimental measurements. For the simplest molecular quantum systems, such as the hydrogen molecular ions H2+ or HD+, the theoretical predictions of quantum electrodynamics are very accurate. In order to test the theory or to extract fundamental constants from the combination of theory and experiments, the experiments must also achieve high precision. In the recent past, hydrogen molecular ions were mainly measured in ensembles. For example, measurements exist both on molecular beams and with dissociative detection in ion traps. However, the spectroscopy of single ions is one of the most precise measurement methods in physics, which was demonstrated by the spectroscopy of a single ytterbium ion in an ion trap. To perform spectroscopy of a single ion in this way, it must have a suitable energy structure for laser cooling, state preparation and fluorescence detection. This is not the case for many ions, including H2+. Quantum logic spectroscopy (QLS) is a technique that makes it possible to perform measurements on an ion without such an energy structure. The ion to be measured (spectroscopy ion) is trapped in an ion trap with a second ion (helper ion), which has the properties described in the previous paragraph. The shared modes of motion between the two ions allow cooling, preparation and detection of the spectroscopy ion via the helper ion. This was demonstrated with an aluminum spectroscopy ion co-trapped with a magnesium helper ion. QLS can also be used for molecules. This has been shown with MgH+, CaH+ and N2+. We implement QLS to enable spectroscopy of a single H2+ ion. For this we trap H2+ together with a 9Be+ ion. We cool the joint motion of the ions to their ground state. We combine buffer gas cooling with a laser and microwave pulse based state preparation sequence to prepare the H2+ molecule in a pure quantum state. This allows us to perform high contrast QLS on a single H2+ ion. Furthermore, we can change and control the quantum state in the hyperfine structure using microwave pulses. We demonstrate the power of our technique by performing microwave spectroscopy on a magnetic field insensitive hyperfine transition. We use QLS to read out the result of the measurement. Within half an hour, we can achieve a statistical uncertainty of 2 Hz. This paves the way for future high-precision measurements on H2+. The techniques described here could potentially be applied to other light molecular ions.