## Final Report Summary - STRONG NEUTRAL-ION (Strong interactions of periodically-driven trapped ions, cold atoms and molecules)

Within this project we developed theoretical tools for analyzing periodically driven phases of cold matter in a nonperturbative way, with a focus on a general understating of the structure of solution sets in the parameter space. Such an understanding allows us to present original ideas for reaching new regimes of many-body physics, with specific application to cooling molecules and atoms to cold and ultracold temperatures.

The motivation for this study stems in recent advancements in technological capabilities to trap, cool, and manipulate ions and atoms cooled close to the quantum ground state of the motion. Trapped ions are strongly interacting via Coulomb repulsion, while neutral atoms interact more weakly when they collide at close range. A new frontier is currently opening up with molecules – both charged and neutral. Methods of cooling and trapping of molecules are being studied theoretically and experimentally, with many potential applications.

The project objectives and the achieved results are summarized in the following.

Objective 1 – Formulating a theory describing cold particles in AC fields.

For obtaining this objective, we have studied three fundamental aspects of the dynamics of particles in Alternating Current (AC) fields.

First we set out to obtain solutions of the exact quantum mechanical wavefuctions of a particle interacting with a driven quantum well. The novelty of our study lies in treating on equal footing the interaction potential and the time-dependent drive, in a 3-dimensional setup where the axial driving force breaks the spherical symmetry of the potential. Such a description applies generally to any two-body problem driven by a periodic force, with an approximate scale separation. This allowed us to capture nonperturbative effects such as high-frequency stabilization and nonmonotonous decay rate of the bound particles, and the angular distribution of escaping bound particles. This general framework can be applied to understand the interaction of atoms and ions in hybrid trap, as we employ in Objective 2, as well as periodic driving of Feshbach resonances, as detailed below, molecules in AC traps, and more.

In parallel with this study, we have studied the classical dynamics of an ion trapped in surface Paul traps, that are becoming increasingly common due to their simple architecture and potential for miniaturization. One of the most fundamental challenges with those traps, however, is the so-called “anomalous” heating of the ion due to noise from the electrodes. The motion in this type of trap is ruled by nonlinear, periodically driven equations. We have conducted a thorough study of single ion dynamics in such traps, using dynamical-system theory methods to classify the integrable and chaotic motion of the ion, depending on the trap parameters. With a detailed mapping of the Hamiltonian phase-space structure, we proceeded to treat both anomalous heating and laser cooling dynamics of the ion, within a stochastic framework. We developed a new theoretical tool for analyzing the interplay of nonlinear resonances of the ion’s motion in the time-dependent trap, with the realistic noise with a “colored” spectrum. This study, performed in collaboration with Prof. Denis Ullmo and researchers within the lab and from the ion storage group at NIST, Boulder, gives in a systematic understanding of Hamiltonian, heating and cooling dynamics of an ion in surface traps. First experiments, lead by Dietrich Leibfried at NIST, were performed to probe experimentally the theory that we developed, and the collaboration is ongoing.

The third study undertaken considers a periodic driving of the Feshbach resonance of two ultracold atoms within the zero-range approximation. Within this approximation, that is relevant to cold and also degenerate gases, we consider a nonperturbative modification of the scattering properties of the atoms by the periodic drive. This collaboration within the lab with Prof. Dmitry Petrov and Dr. Andrew Sykes continues. The problem of atom-atom scattering with periodic driving has broad importance due to the growing interest in this tool for modifying macroscopic properties of physical many-body systems.

Objective 2 – Understanding low energy solutions, excitations and dissipative dynamics in hybrid ion-atom systems.

Using theoretical and numerical models that we developed in Objective 1, we have studied the interaction of cold atoms with an ion trapped in a Paul trap. Starting with the exact scattering states of cold atoms from the driven ion, we can derive a self-consistent solution for the atoms co-trapped with an oscillating ion. The limit of small interatomic interaction is studied together with its interplay with the time-dependent drive. This study is continuing, and is expected to be completed in the near future, with the aim of clarifying the effect of strong periodic drive of the two-body problem, as it enters into the many-body dynamics in the form of approximations such as the Gross-Pitaevskii equation.

A second study within Objective 2 concerns the possibility of sympathetic cooling of heavy molecular ions in linear Paul traps. Using numerical simulation and a simplified analytic model, we have found, in a project performed in collaboration with researchers in the lab and within the outreach program “Coolaboration”, how heavy molecular ions can be efficiently cooled down to crystallization temperatures. This collaboration is ongoing and is expected to be completed in the near future. Our results could find wide applications in spectroscopy and analysis of molecules of chemical and biological importance.

Objective 3 – Studying solitons and vortices in hybrid neutral-ion systems.

Solitons, being nonperturbative solutions of nonlinear equations, and vortices, that carry a topological charge and can be modified significantly with strong nonlinearity, are not amenable to analytic treatment except in rare cases. In hybrid neutral-ion systems, the involved physics is extremely complex due to the mutual presence of a few effects: the breaking of rotational invariance due to the axially symmetric drive, the role of damping and fluctuations due to the periodic driving and cooling, and the scattering of atoms off the soliton. We have aimed to study the dynamics introduced by all of these three effects separately, each in a simpler physical setup, that allows one to capture some of the involved complexity.

In the first study, in collaboration with Prof. Denis Ullmo from our lab, and with Prof. Victor Fleurov and Prof. Shimshon Barad from the Tel Aviv University, we considered a vortex that is trapped behind a finite potential barrier in 2 dimensions, with an elliptical tunneling region, surrounded by a second circular trap well. Using numerical simulations we analyzed the tunneling dynamics of a vortex of high topological charge, following a preparation of the vortex and a quench of its tunneling barrier. The macroscopic vortex is first stabilized by the trap. It then breaks into separate singly charged vortices, that escape the well simultaneously with an equal number of oppositely charged vortices tunneling inwards, leading to a spontaneous flip of the vorticity (e.g. from clockwise to counterclockwise rotation). Using semiclassical methods and phase-space analysis in terms of quantized billiards, we showed that this topological charge inversion mechanism is coherent and can be completely controlled (suppressed or enhanced) without stirring the vortex directly, only by changing the size of the external well. The results of this research are genuinely interdisciplinary, and find application also in the optics community, since we present a mechanism coupling two fundamental types of waveguide modes – “whispering gallery” and “bouncing ball” modes. We are continuing this study and developing the new tools required in order to complete the analysis in the regime of strong nonlinearity of an atomic condensate, or a strong optical Kerr nonlinearity.

In the second study, in collaboration with the group of Prof. Giovanna Morigi in Saarbrucken, Dr. Thomás Fogarty, and Dr. Cecilia Cormick, we show the possibility of ground state cooling of an ion crystal, using discrete solitons (kinks) with trapped ions inside an optical cavity. We found that a soliton, formed due to a forced incommensurability of the average inter-ion separation and the optical potential wavelength, can modify the spectrum of linearized excitations in the system in a way that allows the simultaneous cooling of the entire crystal to its motional ground state. These results could find applications in optomechanical cooling of solids, and contribute to ideas concerning the cooling of solitons.

The third study is a collaboration with the experiment group of Prof. Tobias Schaetz in Freiburg. We have studied a discrete soliton formed spontaneously in a self-assembled ion crystal, and its dynamics subject to an external periodic drive and laser-induced cooling and fluctuations. In the presence of a high frequency signal, nondirectional energy from the drive is converted by the soliton via a nonlinear mechanism to heat in the form of linearized phonon excitations in the crystal. The low frequency scattered phonons drive the soliton above its trap barrier by thermal activation. In the presence of damping and fluctuations, the discrete soliton can be directed towards one end of its channel at an externally controlled rate. Therefore the presented mechanism could serve as a model for soliton-based transport of mass, electric charge, or possibly other topological charges. These results are particularly interesting in the context of the growing field of Brownian ratchets, also referred to as Brownian motors.

Objective 4 – Studying Floquet topological insulators.

Within this objective we have studied how the results of the previous sections, in particular the accurate quantum wavefunctions of particles subject to a periodic drive, can be applied to improve the understanding of driven many body systems. Our study into this fundamental problem of the interplay of an external drive with many-body physics continues, and the fellow will continue to pursue this direction in his future research.

Outreach activities.

As the main outreach activity, the fellow has initiated the program “Coolaboration”, that is entering its third year of activity. Within this program, PhD students guide high school or undergraduate students, in an original research project performed in the course of two academic years. The program runs in France and Israel in parallel, with the students in the different countries working on similar research subjects and forming a “mini-collaboration” by exchanging ideas and materials. In addition the students communicate their studies and results to their peers in outreach activities organized at their schools. Thus the scientific activity acts as a bridge between students from different cultures, between universities and schools, and between students and their local community. At the time of writing, 4 PhD students and 14 high school and undergraduate students have taken part in the program in total.

The motivation for this study stems in recent advancements in technological capabilities to trap, cool, and manipulate ions and atoms cooled close to the quantum ground state of the motion. Trapped ions are strongly interacting via Coulomb repulsion, while neutral atoms interact more weakly when they collide at close range. A new frontier is currently opening up with molecules – both charged and neutral. Methods of cooling and trapping of molecules are being studied theoretically and experimentally, with many potential applications.

The project objectives and the achieved results are summarized in the following.

Objective 1 – Formulating a theory describing cold particles in AC fields.

For obtaining this objective, we have studied three fundamental aspects of the dynamics of particles in Alternating Current (AC) fields.

First we set out to obtain solutions of the exact quantum mechanical wavefuctions of a particle interacting with a driven quantum well. The novelty of our study lies in treating on equal footing the interaction potential and the time-dependent drive, in a 3-dimensional setup where the axial driving force breaks the spherical symmetry of the potential. Such a description applies generally to any two-body problem driven by a periodic force, with an approximate scale separation. This allowed us to capture nonperturbative effects such as high-frequency stabilization and nonmonotonous decay rate of the bound particles, and the angular distribution of escaping bound particles. This general framework can be applied to understand the interaction of atoms and ions in hybrid trap, as we employ in Objective 2, as well as periodic driving of Feshbach resonances, as detailed below, molecules in AC traps, and more.

In parallel with this study, we have studied the classical dynamics of an ion trapped in surface Paul traps, that are becoming increasingly common due to their simple architecture and potential for miniaturization. One of the most fundamental challenges with those traps, however, is the so-called “anomalous” heating of the ion due to noise from the electrodes. The motion in this type of trap is ruled by nonlinear, periodically driven equations. We have conducted a thorough study of single ion dynamics in such traps, using dynamical-system theory methods to classify the integrable and chaotic motion of the ion, depending on the trap parameters. With a detailed mapping of the Hamiltonian phase-space structure, we proceeded to treat both anomalous heating and laser cooling dynamics of the ion, within a stochastic framework. We developed a new theoretical tool for analyzing the interplay of nonlinear resonances of the ion’s motion in the time-dependent trap, with the realistic noise with a “colored” spectrum. This study, performed in collaboration with Prof. Denis Ullmo and researchers within the lab and from the ion storage group at NIST, Boulder, gives in a systematic understanding of Hamiltonian, heating and cooling dynamics of an ion in surface traps. First experiments, lead by Dietrich Leibfried at NIST, were performed to probe experimentally the theory that we developed, and the collaboration is ongoing.

The third study undertaken considers a periodic driving of the Feshbach resonance of two ultracold atoms within the zero-range approximation. Within this approximation, that is relevant to cold and also degenerate gases, we consider a nonperturbative modification of the scattering properties of the atoms by the periodic drive. This collaboration within the lab with Prof. Dmitry Petrov and Dr. Andrew Sykes continues. The problem of atom-atom scattering with periodic driving has broad importance due to the growing interest in this tool for modifying macroscopic properties of physical many-body systems.

Objective 2 – Understanding low energy solutions, excitations and dissipative dynamics in hybrid ion-atom systems.

Using theoretical and numerical models that we developed in Objective 1, we have studied the interaction of cold atoms with an ion trapped in a Paul trap. Starting with the exact scattering states of cold atoms from the driven ion, we can derive a self-consistent solution for the atoms co-trapped with an oscillating ion. The limit of small interatomic interaction is studied together with its interplay with the time-dependent drive. This study is continuing, and is expected to be completed in the near future, with the aim of clarifying the effect of strong periodic drive of the two-body problem, as it enters into the many-body dynamics in the form of approximations such as the Gross-Pitaevskii equation.

A second study within Objective 2 concerns the possibility of sympathetic cooling of heavy molecular ions in linear Paul traps. Using numerical simulation and a simplified analytic model, we have found, in a project performed in collaboration with researchers in the lab and within the outreach program “Coolaboration”, how heavy molecular ions can be efficiently cooled down to crystallization temperatures. This collaboration is ongoing and is expected to be completed in the near future. Our results could find wide applications in spectroscopy and analysis of molecules of chemical and biological importance.

Objective 3 – Studying solitons and vortices in hybrid neutral-ion systems.

Solitons, being nonperturbative solutions of nonlinear equations, and vortices, that carry a topological charge and can be modified significantly with strong nonlinearity, are not amenable to analytic treatment except in rare cases. In hybrid neutral-ion systems, the involved physics is extremely complex due to the mutual presence of a few effects: the breaking of rotational invariance due to the axially symmetric drive, the role of damping and fluctuations due to the periodic driving and cooling, and the scattering of atoms off the soliton. We have aimed to study the dynamics introduced by all of these three effects separately, each in a simpler physical setup, that allows one to capture some of the involved complexity.

In the first study, in collaboration with Prof. Denis Ullmo from our lab, and with Prof. Victor Fleurov and Prof. Shimshon Barad from the Tel Aviv University, we considered a vortex that is trapped behind a finite potential barrier in 2 dimensions, with an elliptical tunneling region, surrounded by a second circular trap well. Using numerical simulations we analyzed the tunneling dynamics of a vortex of high topological charge, following a preparation of the vortex and a quench of its tunneling barrier. The macroscopic vortex is first stabilized by the trap. It then breaks into separate singly charged vortices, that escape the well simultaneously with an equal number of oppositely charged vortices tunneling inwards, leading to a spontaneous flip of the vorticity (e.g. from clockwise to counterclockwise rotation). Using semiclassical methods and phase-space analysis in terms of quantized billiards, we showed that this topological charge inversion mechanism is coherent and can be completely controlled (suppressed or enhanced) without stirring the vortex directly, only by changing the size of the external well. The results of this research are genuinely interdisciplinary, and find application also in the optics community, since we present a mechanism coupling two fundamental types of waveguide modes – “whispering gallery” and “bouncing ball” modes. We are continuing this study and developing the new tools required in order to complete the analysis in the regime of strong nonlinearity of an atomic condensate, or a strong optical Kerr nonlinearity.

In the second study, in collaboration with the group of Prof. Giovanna Morigi in Saarbrucken, Dr. Thomás Fogarty, and Dr. Cecilia Cormick, we show the possibility of ground state cooling of an ion crystal, using discrete solitons (kinks) with trapped ions inside an optical cavity. We found that a soliton, formed due to a forced incommensurability of the average inter-ion separation and the optical potential wavelength, can modify the spectrum of linearized excitations in the system in a way that allows the simultaneous cooling of the entire crystal to its motional ground state. These results could find applications in optomechanical cooling of solids, and contribute to ideas concerning the cooling of solitons.

The third study is a collaboration with the experiment group of Prof. Tobias Schaetz in Freiburg. We have studied a discrete soliton formed spontaneously in a self-assembled ion crystal, and its dynamics subject to an external periodic drive and laser-induced cooling and fluctuations. In the presence of a high frequency signal, nondirectional energy from the drive is converted by the soliton via a nonlinear mechanism to heat in the form of linearized phonon excitations in the crystal. The low frequency scattered phonons drive the soliton above its trap barrier by thermal activation. In the presence of damping and fluctuations, the discrete soliton can be directed towards one end of its channel at an externally controlled rate. Therefore the presented mechanism could serve as a model for soliton-based transport of mass, electric charge, or possibly other topological charges. These results are particularly interesting in the context of the growing field of Brownian ratchets, also referred to as Brownian motors.

Objective 4 – Studying Floquet topological insulators.

Within this objective we have studied how the results of the previous sections, in particular the accurate quantum wavefunctions of particles subject to a periodic drive, can be applied to improve the understanding of driven many body systems. Our study into this fundamental problem of the interplay of an external drive with many-body physics continues, and the fellow will continue to pursue this direction in his future research.

Outreach activities.

As the main outreach activity, the fellow has initiated the program “Coolaboration”, that is entering its third year of activity. Within this program, PhD students guide high school or undergraduate students, in an original research project performed in the course of two academic years. The program runs in France and Israel in parallel, with the students in the different countries working on similar research subjects and forming a “mini-collaboration” by exchanging ideas and materials. In addition the students communicate their studies and results to their peers in outreach activities organized at their schools. Thus the scientific activity acts as a bridge between students from different cultures, between universities and schools, and between students and their local community. At the time of writing, 4 PhD students and 14 high school and undergraduate students have taken part in the program in total.