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Tunneling in a many-body system appears as one of the novel implications of quantum physics, in which particles move in space under an otherwise classically-forbidden potential barrier. Here, we theoretically describe the quantum dynamics of the tunneling phenomenon of a few intricate bosonic clouds in a closed system of a two-dimensional symmetric double-well potential. We examine how the inclusion of the transverse direction, orthogonal to the junction of the double-well, can intervene in the tunneling dynamics of bosonic clouds. We use a well-known many-body numerical method, called the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method. MCTDHB allows one to obtain accurately the time-dependent many-particle wavefunction of the bosons which in principle entails all the information of interest about the system under investigation. We analyze the tunneling dynamics by preparing the initial state of the bosonic clouds in the left well of the double-well either as the ground, longitudinally or transversely excited, or a vortex state. We unravel the detailed mechanism of the tunneling process by analyzing the evolution in time of the survival probability, depletion and fragmentation, and the many-particle position, momentum, and angular-momentum expectation values and their variances. As a general rule, all objects lose coherence while tunneling through the barrier and the states which include transverse excitations do so faster. Implications are briefly discussed., Comment: 20 Figures, 47 pages

Static and dynamic properties of Bose-Einstein condensates in annular traps are investigated by solving the many-boson Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree for bosons method. We concentrate on weakly-interacting bosons exhibiting low depletion. Analysis of the mean-field position variance, which accounts for the shape of the density only, and the many-body position variance, which incorporates a tiny amount of excitations through the reduced two-particle density matrix, shows that the former behaves essentially as a quasi-one-dimensional quantity whereas the latter as a two-dimensional quantity. This brings another dimension to the physics of bosons in ring-shaped traps., Comment: 24 pages, 8 figures

A mixture of two kinds of identical bosons held in a harmonic potential and interacting by harmonic particle–particle interactions is discussed. This is an exactly-solvable model of a mixture of two trapped Bose–Einstein condensates which allows us to examine analytically various properties. Generalizing the treatments in Cohen and Lee (1985) and Osadchii and Muraktanov (1991), closed form expressions for the mixture’s frequencies and ground-state energy and wave-function, and the lowest-order densities are obtained and analyzed for attractive and repulsive intra-species and inter-species particle–particle interactions. A particular mean-field solution of the corresponding Gross–Pitaevskii theory is also found analytically. This allows us to compare properties of the mixture at the exact, many-body and mean-field levels, both for finite systems and at the limit of an infinite number of particles. We discuss the renormalization of the mixture’s frequencies at the mean-field level. Mainly, we hereby prove that the exact ground-state energy per particle and lowest-order intra-species and inter-species densities per particle converge at the infinite-particle limit (when the products of the number of particles times the intra-species and inter-species interaction strengths are held fixed) to the results of the Gross–Pitaevskii theory for the mixture. Finally and on the other end, we use the mixture’s and each species’ center-of-mass operators to show that the Gross–Pitaevskii theory for mixtures is unable to describe the variance of many-particle operators in the mixture, even in the infinite-particle limit. The variances are computed both in position and momentum space and the respective uncertainty products compared and discussed. The role of the center-of-mass separability and, for generically trapped mixtures, inseparability is elucidated when contrasting the variance at the many-body and mean-field levels in a mixture. Our analytical results show that many-body correlations exist in a trapped mixture of Bose–Einstein condensates made of any number of particles. Implications are briefly discussed.

In this Colloquium, the wavefunction-based Multiconfigurational Time-Dependent Hartree approaches to the dynamics of indistinguishable particles (MCTDH-F for Fermions and MCTDH-B for Bosons) are reviewed. MCTDH-B and MCTDH-F or, together, MCTDH-X are methods for describing correlated quantum systems of identical particles by solving the time-dependent Schrödinger equation from first principles. MCTDH-X is used to accurately model the dynamics of real-world quantum many-body systems in atomic, molecular, and optical physics. The key feature of these approaches is the time-dependence and optimization of the single-particle states employed for the construction of a many-body basis set, which yields nonlinear working equations. We briefly describe the historical developments that have lead to the formulation of the MCTDH-X methods and motivate the necessity for wavefunction-based approaches. We sketch the derivation of the unified MCTDH-F and MCTDH-B equations of motion for complete and also specific restricted configuration spaces. The strengths and limitations of the MCTDH-X approach are assessed via benchmarks against an exactly solvable model and via convergence checks. We highlight some applications to instructive and experimentally-realized quantum many-body systems: the dynamics of atoms in Bose-Einstein condensates in magneto-optical and optical traps and of electrons in atoms and molecules. We discuss the current development and frontiers in the field of MCTDH-X: theories and numerical methods for indistinguishable particles, for mixtures of multiple species of indistinguishable particles, the inclusion of nuclear motion for the nonadia-batic dynamics of atomic and molecular systems, the so-called multilayer generalizations to the MCTDH-F and MCTDH-B methods, and the time-dependent orbital-adaptive coupled cluster theory are discussed.

Bose-Einstein condensates (BECs) offer a fruitful, often uncharted ground for exploring physics of many-particle systems. In the present year of the MCTDHB project at the HLRS, we maintained and extended our investigations of BECs and interacting bosonic systems using the MultiConfigurational Time-Dependent Hartree for Bosons (MCTDHB) method and running the MCTDHB and MCTDH-X software packages on the Cray XC40 system Hazel Hen. The results we disseminate within this report comprise: (i) Entropies and correlations of ultracold bosons in a lattice; (ii) Crystallization of bosons with dipole-dipole interactions and its detection in single-shot images; (iii) Management of correlations in ultracold gases; (iv) Pulverizing a BEC; (v) Dynamical pulsation of ultracold droplet crystals by laser light; (vi) Two-component bosons interacting with photons in a cavity; (vii) Quantum dynamics of a bosonic Josephson junction and the impact of the range of the interaction; (viii) Trapped bosons in the infinite-particle limit and their exact many-body wavefunction and properties; (ix) Angular-momentum conservation in a BEC and Gross-Pitaevskii versus many-body dynamics; (x) Variance of an anisotropic BEC; and (xi) excitation spectrum of a weakly-interacting rotating BEC showing enhanced many-body effects. These are all basic and appealing, sometimes unexpected many-body results put forward with the generous allocation of computer time by the HLRS to the MCTDHB project. Finally, expected future developments and research tasks are prescribed, too.

The ground state of a Bose-Einstein condensate in a two-dimensional trap potential is analyzed numerically at the infinite-particle limit. It is shown that the anisotropy of the many-particle position variance along the $x$ and $y$ axes can be opposite when computed at the many-body and mean-field levels of theory. This is despite the system being $100\%$ condensed, and the respective energies per particle and densities per particle to coincide., 9 pages, 6 figures

The out-of-equilibrium quantum dynamics of a Bose gas trapped in an asymmetric double well and interacting with a finite-range interaction has been studied in real space by solving the time-dependent many-body Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We have focused on the weakly interacting limit where the system is essentially condensed. We have examined the impact of the range of the interaction on the dynamics of the system, both at the mean-field and many-body levels. Explicitly, we have studied the maximal and the minimal values of the many-body position variance in each cycle of oscillation, and the overall pace of its growth. We find that the range of the interaction affects the dynamics of the system differently for the right well and the left well. We have also examined the infinite-particle limit and find that even there, the impact of the range of the interaction can only be described by a many-body theory such as MCTDHB.

The excitation spectrum of a highly-condensed two-dimensional trapped Bose-Einstein condensate (BEC) is investigated within the rotating frame of reference. The rotation is used to transfer high-lying excited states to the low-energy spectrum of the BEC. We employ many-body linear-response theory and show that, once the rotation leads to a quantized vortex in the ground state, already the low-energy part of the excitation spectrum shows substantial many-body effects beyond the realm of mean-field theory. We demonstrate numerically that the many-body effects grow with the vorticity of the ground state, meaning that the rotation enhances them even for very weak repulsion. Furthermore, we explore the impact of the number of bosons $N$ in the condensate on a low-lying single-particle excitation, which is describable within mean-field theory. Our analysis shows deviations between the many-body and mean-field results which clearly persist when $N$ is increased up to the experimentally relevant regime, typically ranging from several thousand up to a million bosons in size. Implications are briefly discussed.

The out-of-equilibrium quantum dynamics of an interacting Bose gas trapped in a 1D asymmetric double-well potential is studied by solving the many-body Schr\"odinger equation numerically accurately. We examine how the loss of symmetry of the confining trap affects the macroscopic quantum tunneling dynamics of the system between the two wells. In an asymmetric DW, the two wells are not equivalent anymore -the left well is deeper than the right one. Accordingly, we analyze the dynamics by initially preparing the condensate in both the left and the right well. We examined the frequencies and amplitudes of the oscillations of the survival probabilities, the time scale for the development of fragmentation and its degree, and the growth and oscillatory behavior of the many-body position and momentum variances. There is an overall suppression of the oscillations of the survival probabilities in an asymmetric double well. However, depending on whether the condensate is initially prepared in the left or right well, the repulsive inter-atomic interactions affect the survival probabilities differently. The degree of fragmentation depends both on the asymmetry of the trap and the initial well in which the condensate is prepared in a non-trivial manner. Overall, the many-body position and momentum variances bear the prominent signatures of the density oscillations of the system in the asymmetric double well as well as a breathing-mode oscillation. Finally, a universality of fragmentation for systems made of different numbers of particles but the same interaction parameter is also found. The phenomenon is robust despite the asymmetry of the junction and admits a macroscopically-large fragmented condensate characterized by a diverging many-body position variance., Comment: 42 pages, 12 figures

The dynamics of attractive bosons trapped in one dimensional anharmonic potentials is investigated. Particular emphasis is put on the variance of the position and momentum many-particle operators. Coupling of the center-of-mass and relative-motion degrees-of-freedom necessitates an accurate numerical treatment. The multiconfigurational time-dependent Hartree for bosons (MCTDHB) method is used, and high convergence of the energy, depletion and occupation numbers, and position and momentum variances is proven numerically. We demonstrate for the ground state and out-of-equilibrium dynamics, for condensed and fragmented condensates, for small systems and {\it en route} to the infinite-particle limit, that intriguing differences between the density and variance of an attractive Bose-Einstein condensate emerge. Implications are briefly discussed., Comment: 31 pages, 6 figures. arXiv admin note: text overlap with arXiv:1709.04223

The many-body physics of trapped Bose-Einstein condensates (BECs) is very rich and demanding. During the past year of the MCTDHB project at the HLRS we continued to shed further light on it with the help of the MultiConfigurational Time-Dependent Hartree for Bosons (MCTDHB) method and using the MCTDHB and MCTDH-X software packages. Indeed, our results on which we report below span a realm of many-body effects in fragmented, depleted, and even in fully condensed BECs. Our findings include: (1) fragmented superradiance of a BEC trapped in an optical cavity; (2) properties of phantom (fragmented) vortices in trapped BECs; (3) dynamics of a two-dimensional trapped BEC described by the Bose-Hubbard Hamiltonian with MCTDH-X; (4) overlap of exact and Gross-Pitaevskii wave-functions in trapped BECs; (5) properties of the uncertainty product of an out-of-equilibrium trapped BEC; (6) many-body excitations and de-excitations in trapped BECs and relation to variance; and (7) many-body effects in the excitation spectrum of weakly-interacting BECs in finite one-dimensional optical lattices. These are all appealing and fundamental many-body results made through the kind allocation of computer resources by the HLRS to the MCTDHB project. Finally, we put forward some future developments and research plans, as well as further many-body perspectives.

The out-of-equilibrium quantum dynamics of a bosonic Josephson junction (BJJ) with long-range interaction is studied in real space by solving the time-dependent many-body Schr\"odinger equation numerically accurately using the multiconfigurational time-dependent Hartree method for bosons. Having the many-boson wave-function at hand we can examine the impact of the range of the interaction on the properties of the BJJ dynamics, viz. density oscillations and their collapse, self trapping, depletion and fragmentation, as well as the position variance, both at the mean-field and many-body level. Explicitly, the frequency of the density oscillations and the time required for their collapse, the value of fragmentation at the plateau, the maximal and the minimal values of the position variance in each cycle of oscillation and the overall pace of its growth are key to our study. We find competitive effect between the interaction and the confining trap. The presence of the tail part of the interaction basically enhances the effective repulsion as the range of the interaction is increased starting from a short, finite range. But as the range becomes comparable with the trap size, the system approaches a situation where all the atoms feel a constant potential and the impact of the tail on the dynamics diminishes. There is an optimal range of the interaction in which physical quantities of the junction are attaining their extreme values., Comment: Contribution to the Special Issue of Chemical Physics dedicated to Professor Hans-Dieter Meyer on the occasion of his 70th birthday; few typos corrected

The anisotropy of a trap potential can impact the density and variance of a Bose-Einstein condensate (BEC) in an opposite manner. We exemplify this effect for both the ground state and out-of-equilibrium dynamics of structureless bosons interacting by a long-range inter-particle interaction and trapped in a two-dimensional single-well potential. We demonstrate that even when the density of the BEC is, say, wider along the $y$ direction and narrower along the $x$ direction, its position variance can actually be smaller and momentum variance larger in the $y$ direction than in the $x$ direction. This behavior of the variance in a many-particle system is counterintuitive. It suggests using the variance as a tool to characterize the strength of correlations along the $y$ and $x$ directions in a trapped BEC., Comment: 25 pages, 7 figures

In this work, we study many-body excitations of Bose-Einstein condensates (BECs) trapped in periodic one-dimensional optical lattices. In particular, we investigate the impact of quantum depletion onto the structure of the low-energy spectrum and contrast the findings to the mean-field predictions of the Bogoliubov-de Gennes (BdG) equations. Accurate results for the many-body excited states are obtained by applying a linear-response theory atop the MCTDHB (multiconfigurational time-dependent Hartree method for bosons) equations of motion, termed LR-MCTDHB. We demonstrate for condensates in a triple well that even weak ground-state depletion of around $1\%$ leads to visible many-body effects in the low-energy spectrum which deviate substantially from the corresponding BdG spectrum. We further show that these effects also appear in larger systems with more lattice sites and particles, indicating the general necessity of a full many-body treatment.

Here we report on further applications, developments, expansion, and proliferation of the Multi-Configurational Time-Dependent Hartree for Bosons (MCTDHB) method in the context of ultra-cold atomic systems. In this year we put our main efforts to understanding and generalizing vortices—two-dimensional (2D) and three-dimensional (3D) quantum objects carrying angular momentum—from the perspective of the many-body physics. We have studied static properties and quantum dynamics of vortices confined in simple parabolic traps and in circular traps. Particular emphasis has been put on the loss of coherence and build-up of the fragmentation. Complimentary, we continue to develop the MCTDHB method spanning several directions of the theoretical and computational physics as well as optimal-control theory: (a) the linear-response on-top of the MCTDHB method (LR-MCTDHB) has been reformulated in a compact block form, expanded for general inter-particle interactions, and benchmarked against the exactly-solvable harmonic-interaction model; (b) a new analysis tool capable of simulating the outcomes of typical shots generated in the experimental detection of ultra-cold atomic systems has been invented, tested, and applied; (c) a novel algorithm offering a direct quantitative measurement of the possible fragmentation in bosonic systems has been proposed and applied; (d) the optimal-control Chopped RAndom Basis (CRAB) algorithm has been merged with the MCTDHB package and applied to manipulate quantum systems. Implications and further perspectives and future research plans are briefly discussed and addressed.

The variance and uncertainty product of the position and momentum many-particle operators of structureless bosons interacting by a long-range inter-particle interaction and trapped in a single-well potential are investigated. In the first example, of an out-of-equilibrium interaction-quench scenario, it is found that, despite the system being fully condensed, already when a fraction of a particle is depleted differences with respect to the mean-field quantities emerge. In the second example, of the pathway from condensation to fragmentation of the ground state, we find out that, although the cloud's density broadens while the system's fragments, the position variance actually decreases, the momentum variance increases, and the uncertainty product is not a monotonous function but has a maximum. Implication are briefly discussed., Comment: 14 pages, 3 figures

During the past year of the MCTDHB project at the HLRS, we continued to strive and conquest further applications, developments, and expansion of the MultiConfigurational Time-Dependent Hartree for Bosons (MCTDHB) method in the context of ultracold atomic systems. We also announce the MCTDH-X package, the Multiconfigurational Time-Dependent Hartree for Indistinguishable Particles X package, which is able to treat identical bosons and fermions, with or without spin/internal degrees of freedom, alike. Here we report on a plethora of results and versatile applications which include: (i) single-shot imaging of fluctuating vortices in a fragmented Bose-Einstein condensate (BEC); (ii) the many-body tunneling and fragmetnation of vortices in 2D trapped BECs; (iii) the transition from vortices to solitonic vortices in 2D trapped BECs; (iv) the variance of a many-particle system being very sensitive to correlations even in the infinite-particle limit; (v) the consequences of the latter on the out-of-equilibrium uncertainty product of an evolving BEC; (vi) the mechanism of tunneling to open space of a few interacting polarized fermions; and (vii) composite fragmentation of multi-components BECs (i.e., with internal degrees of freedom). These are all exciting results made throughout the allocation of computer time by the HLRS to the MCTDHB project. Finally, further perspectives and future research plans are briefly discussed.

The tunneling process in a many-body system is a phenomenon which lies at the very heart of quantum mechanics. It appears in nature in the form of alpha-decay, fusion and fission in nuclear physics, photoassociation and photodissociation in biology and chemistry. A detailed theoretical description of the decay process in these systems is a very cumbersome problem, either because of very complicated or even unknown interparticle interactions or due to a large number of constitutent particles. In this work, we theoretically study the phenomenon of quantum many-body tunneling in a more transparent and controllable physical system, in an ultracold atomic gas. We analyze a full, numerically exact many-body solution of the Schr\"odinger equation of a one-dimensional system with repulsive interactions tunneling to open space. We show how the emitted particles dissociate or fragment from the trapped and coherent source of bosons: the overall many-particle decay process is a quantum interference of single-particle tunneling processes emerging from sources with different particle numbers taking place simultaneously. The close relation to atom lasers and ionization processes allows us to unveil the great relevance of many-body correlations between the emitted and trapped fractions of the wavefunction in the respective processes., Comment: 18 pages, 4 figures (7 pages, 2 figures supplementary information)

The MCTDHB package has been applied to study the physics of trapped interacting many-boson systems by solving the underlying time-dependent (as well as the time-independent) many-boson Schrodinger equation. Here we report on four studies where novel physical ideas and phenomena have been proposed and discovered: (a) Universality of the fragmentation dynamics in double wells – at long propagation times properties of the evolving system saturate to some asymptotic values; (b) Novel many-body spectral features in trapped systems – the newly-developed linear-response theory on-top of MCTDHB predicts the existence of low-lying excitations not described so far by the standard theory even in harmonic potentials; (c) Efficient protocol to control the many-particle tunneling dynamics to open space, by combining the effects of a threshold potential and inter-particle interaction; (d) Physics behind the formation of patterns in the ground states of trapped bosonic systems with strong finite- and long-range repulsive interactions and the origin of their dynamical stability. From the perspective of the required computational resources and numerical algorithms applied, each of these numerically-demanding studies has challenged different aspects of computational physics and mathematics: Long-time propagation – stability of the numerical methods used to integrate the MCTDHB equations-of-motion; Control of the tunneling dynamics – a very detailed study where an interplay of the parameters controlling the decay by tunneling dynamics is accompanied by a long-time propagation on huge spatial grids, which are needed to simulate open systems; Excited states of many-body systems – construction and diagonalization of complex non-hermitian linear-response matrices; Finite- and long-range interactions in 1D, 2D, and 3D setups – efficient methods and techniques for evaluation of involved high-dimensional integrals. Implications and further perspectives and future plans are briefly discussed and addressed.

A solvable model of a generic trapped bosonic mixture, N 1 bosons of mass m 1 and N 2 bosons of mass m 2 trapped in an harmonic potential of frequency ω and interacting by harmonic inter-particle interactions of strengths λ 1, λ 2, and λ 12, is discussed. It has recently been shown for the ground state [J. Phys. A 50, 295002 (2017)] that in the infinite-particle limit, when the interaction parameters λ 1(N 1 − 1), λ 2(N 2 − 1), λ 12 N 1, λ 12 N 2 are held fixed, each of the species is 100% condensed and its density per particle as well as the total energy per particle are given by the solution of the coupled Gross-Pitaevskii equations of the mixture. In the present work we investigate properties of the trapped generic mixture at the infinite-particle limit, and find differences between the many-body and mean-field descriptions of the mixture, despite each species being 100%. We compute analytically and analyze, both for the mixture and for each species, the center-of-mass position and momentum variances, their uncertainty product, the angular-momentum variance, as well as the overlap of the exact and Gross-Pitaevskii wavefunctions of the mixture. The results obtained in this work can be considered as a step forward in characterizing how important are many-body effects in a fully condensed trapped bosonic mixture at the infinite-particle limit.

In this work, we study the out-of-equilibrium many-body tunneling dynamics of a Bose-Einstein condensate in a two-dimensional radial double well. We investigate the impact of interparticle repulsion and compare the influence of angular momentum on the many-body tunneling dynamics. Accurate many-body dynamics are obtained by solving the full many-body Schr\"odinger equation. We demonstrate that macroscopic vortex states of definite total angular momentum indeed tunnel and that, even in the regime of weak repulsions, a many-body treatment is necessary to capture the correct tunneling dynamics. As a general rule, many-body effects set in at weaker interactions when the tunneling system carries angular momentum., Comment: 26 pages, 9 figures

Bose-Einstein condensates made of ultracold trapped bosonic atoms have become a central venue in which interacting many-body quantum systems are studied. The ground state of a trapped Bose-Einstein condensate has been proven to be 100% condensed in the limit of infinite particle number and constant interaction parameter [Lieb and Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. The meaning of this result is that properties of the condensate, noticeably its energy and density, converge to those obtained by minimizing the Gross-Pitaevskii energy functional. This naturally raises the question whether correlations are of any importance in this limit. Here, we demonstrate both analytically and numerically that even in the infinite particle limit many-body correlations can lead to a modification of the variance of any operator compared to that expected from the Gross-Pitaevskii result. The deviation of the variance stems from its explicit dependence on terms of the reduced two-body density matrix which otherwise do not contribute to the energy and density in this limit. This makes the variance a sensitive probe of many-body correlations even when the energy and density of the system have already converged to the Gross-Pitaevskii result. We use the many-particle position and momentum operators to exemplify this persistence of correlations. Implications of this many-body effect are discussed.

In the present work we show, analytically and numerically, that the variance of many-particle operators and their uncertainty product for an out-of-equilibrium Bose-Einstein condensate (BEC) can deviate from the outcome of the time-dependent Gross-Pitaevskii dynamics, even in the limit of infinite number of particles and at constant interaction parameter when the system becomes 100% condensed. We demonstrate our finding on the dynamics of the center-of-mass position--momentum uncertainty product of a freely expanding as well as of a trapped BEC. This time-dependent many-body phenomenon is explained by the existence of time-dependent correlations which manifest themselves in the system's reduced two-body density matrix used to evaluate the uncertainty product. Our work demonstrates that one has to use a many-body propagation theory to describe an out-of-equilibrium BEC, even in the infinite particle limit., Comment: 26 pages, 5 figures

Bosons in optical lattices and rings are attractive and active fields of research in cold-atom physics. Here, we apply our recently developed coupled-cluster approach for bosons in external traps to these systems, and extend it to the lowest-in-energy excited states with total quasi- or angular-momentum k. In the coupled-cluster approach the exact many-boson ground state is obtained by applying an exponential operator exp{T}, T = ∑ n = 1 N T n to the ground configuration, which is (usually) the state where the bosons occupy a single orbital. For excited states, a second exponential operator exp{T(k)}, T ( k ) = ∑ n = 1 N T n ( k ) is employed to accommodate the remaining excitations from the unperturbed excited configuration. Due to the conservation of momentum, T1 and T 1 ( k ) can vanish. Working equations for coupled-cluster (singles) doubles (CCD) are provided and their implications are briefly discussed.

The present Letter analyzes the ground-state of weakly-interacting cold bosonic atoms in three-dimensional (3D) multi-well traps. A physical repulsive pairwise interaction is considered. As the interparticle interaction is increased, fragmentation of condensates is established and discussed. Analogously, superfluid to Mott-insulator transitions are described by a continuum real-space approach. In a related context, we also discuss the fermionization of cold bosonic systems in and beyond 1D. Illustrative numerical examples for fragmentation in 3D and 2D and fermionization in 1D traps are provided.

The classification of a molecule's spatial symmetry is an essential initial step in predicting its vibrational spectra. When a (chemical) substitution is made, the relationship between the spatial symmetries of the molecule and its derivatives governs the differences between their corresponding vibrational spectra. These text-book statements of molecular spectroscopy have also found recent and successful applications with much larger molecules, namely single-walled nanotubes. The purpose of this work is to review the profound interplay between the spatial symmetry and the numbers of Raman-and infrared-active vibrations in single-walled carbon and boron nitride nanotubes.

The standard one-particle Green function (GF) is extended to account for scattering from open-shell many-body targets. This extended GF possesses a self-energy which is an exact optical potential (OP) for scattering from open-shell targets. Furthermore, to each scattering process with well-defined quantum numbers there is a specific OP, thus reducing the multichannel problem to single-channel ones. As an explicit example we work out, in detail, the scattering of spin-½ projectiles from open-shell targets.

A mixture of two kinds of identical bosons, species $1$ and species $2$, held in a harmonic potential and interacting by harmonic intra-species and inter-species particle-particle interactions is discussed. To prove Bose-Einstein condensation of the mixture three steps are needed. First, we integrate the all-particle density matrix, employing a four-parameter matrix recurrence relations, down to the lowest-order intra-species and inter-species reduced density matrices of the mixture. Second, the coupled Gross-Pitaevskii (mean-field) equations of the mixture are solved analytically. Third, we analyze the mixture's reduced density matrices in the limit of an infinite number of particles of both species $1$ and $2$ (when the interaction parameters, i.e., the products of the number of particles times the intra-species and inter-species interaction strengths, are held fixed) and prove that: (i) Both species $1$ and $2$ are 100\% condensed; (ii) The inter-species reduced density matrix per particle is separable and given by the product of the intra-species reduced density matrices per particle; and (iii) The mixture's energy per particle, and reduced density matrices and densities per particle all coincide with the Gross-Pitaevskii quantities. Finally, when the infinite-particle limit is taken with respect to, say, species $1$ only (with interaction parameters held fixed) we prove that: (iv) Only species $1$ is 100\% condensed and its reduced density matrix and density per particle, as well as the mixture's energy per particle, coincide with the Gross-Pitaevskii quantities of species $1$ alone; and (v) The inter-species reduced density matrix per particle is nonetheless separable and given by the product of the intra-species reduced density matrices per particle., Comment: 31 pages

A two-dimensional Bose-Einstein condensate (BEC) split by a radial potential barrier is investigated. We determine on an accurate many-body level the system's ground-state phase diagram as well as a time-dependent phase diagram of the splitting process. Whereas the ground state is condensed for a wide range of parameters, the time-dependent splitting process leads to substantial fragmentation. We demonstrate for the first time the dynamical fragmentation of a BEC despite its ground state being condensed. The results are analyzed by a mean-field model and suggest that a large manifold of low-lying fragmented excited states can significantly impact the dynamics of trapped two-dimensional BECs., 5+eps pages, 4 figures

Two generically different but universal dynamical quantum many-body behaviors are discovered by probing the stability of trapped fragmented bosonic systems with strong repulsive finite/long range inter-particle interactions. We use different time-dependent processes to destabilize the systems -- a sudden displacement of the trap is accompanied by a sudden quench of the strength of the inter-particle repulsion. A rather moderate non-violent evolution of the density in the first "topology-preserved" scenario is contrasted with a highly-non-equilibrium dynamics characterizing an explosive changes of the density profiles in the second scenario. The many-body physics behind is identified and interpreted in terms of self-induced time-dependent barriers governing the respective under- and over-a-barrier dynamical evolutions. The universality of the discovered scenarios is explicitly confirmed in 1D, 2D and 3D many-body computations in (a)symmetric traps and repulsive finite/long range inter-particle interaction potentials of different shapes. Implications are briefly discussed., (see Ancillary files with 2D and 3D movies in MPG format)

A scheme to control the many-boson tunneling process to open space is derived and demonstrated. The number of ejected particles and their velocities can be controlled by two parameters, the threshold of the potential and the interparticle interaction. Since these parameters are fully under experimental control, this is also the case for the number of ejected particles and their emission spectrum. The process of tunneling to open space can hence be used, for example, for the quantum simulation of complicated tunneling ionization processes and atom lasers. To understand the many-body tunneling process, a generalization of the model introduced in [Proc. Natl. Acad. Sci. USA, 109, 13521 (2012)] for tunneling in the absence of a threshold is put forward and proven to apply for systems with a non-zero threshold value. It is demonstrated that the model is applicable for general interparticle interaction strengths, particle numbers and threshold values. The model constructs the many-body process from single-particle emission processes. The rates and emission momenta of the single-particle processes are determined by the chemical potentials and energy differences to the threshold value of the potential for systems with different particle numbers. The chemical potentials and these energy differences depend on the interparticle interaction. Both the number of confined particles and their rate of emission thus allow for a control by the manipulation of the interparticle interaction and the threshold. Numerically exact results for two, three and one hundred bosons are shown and discussed. The devised control scheme for the many-body tunneling process performs very well for the dynamics of the momentum density, the correlations, the coherence and of the final state, i.e., the number of particles that remain confined in the potential., Comment: 36 pages, 14 figures

We investigate the elastic scattering of Bose-Einstein condensates at shallow periodic and disorder potentials. We show that the collective scattering of the macroscopic quantum object couples to internal degrees of freedom of the Bose-Einstein condensate such that the Bose-Einstein condensate gets depleted. As a precursor for the excitation of the Bose-Einstein condensate we observe wave chaos within a mean-field theory., Comment: 10 pages, 5 figures

A vortex in Bose-Einstein condensates is a localized object which looks much like a tiny tornado storm. It is well described by mean-field theory. In the present work we go beyond the current paradigm and introduce many-body vortices. These are made of {\it spatially-partitioned} clouds, carry definite total angular momentum, and are fragmented rather than condensed objects which can only be described beyond mean-field theory. A phase diagram based on a mean-field model assists in predicting the parameters where many-body vortices occur. Implications are briefly discussed., Comment: 5+eps+3 pages, 3+3 figures

The linear-response theory of the multiconfigurational time-dependent Hartree for bosons method for computing many-body excitations of trapped Bose-Einstein condensates [Phys. Rev. A {\bf 88}, 023606 (2013)] is implemented for systems with general interparticle interaction. Illustrative numerical examples for repulsive and attractive bosons are provided. The many-body linear-response theory identifies the excitations not unraveled within Bogoliubov--de Gennes equations. The theory is herewith benchmarked against the exactly-solvable one-dimensional harmonic-interaction model. As a complementary result, we represent the theory in a compact block-diagonal form, opening up thereby an avenue for treating larger systems., Comment: 11 pages, 2 tables

We extend the method for the formulation of selection rules for high harmonic generation spectra [Phys. Rev. Lett. 80, 3743 (1998)] beyond the dipole approximation and apply it to single-walled carbon nanotubes interacting with a circularly polarized laser field. Our results show that the carbon nanotubes can be excellent systems for a selective generation of high harmonics, up to the soft x-ray regime.

For three-dimensional many-electron time periodic Hamiltonians which are invariant under dynamical symmetry of order $N$ we prove that only the $(\mathrm{nN}\ifmmode\pm\else\textpm\fi{}1)\mathrm{th}$, $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},$ harmonics are generated. We discuss the application of the dynamical symmetry based selection rules to the generation of high harmonics by thin crystals. The derived selection rules are demonstrated numerically for a one-dimensional model, showing that the dynamically symmetric systems can be used not only as ``filters'' of the very high harmonics but also as their ``amplifiers.''

We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically-exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov-de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory., 44 pages, 7 figures, 1 table

We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-ranged Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. Complementary, we also explicitly verify by a variational treatment that in the zero-range limit the positive delta potential in two dimensions only reproduces the non-interacting results, if the Hilbert space in not truncated. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature., 19 pages, 5 figures, 1 table

The quantum many-body dynamics of indistinguishable, interacting particles are described by the time-dependent many-body Schrodinger equation (TDSE). The TDSE constitutes a difficult problem and is not solvable analytically in most cases. The present review article expedites and benchmarks the capabilities of a novel theoretical method, the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) that is designed for solving the TDSE of interacting bosonic particles. The MCTDHB package is a software implementation that solves the equations of motion of MCTDHB numerically. It is assessed with a benchmark versus an analytically treatable model of trapped interacting bosons (a closed system) that the MCTDHB is capable of solving the TDSE for bosons numerically exactly, i.e., to any desired numerical precision. Furthermore, the structure and parallelization of the MCTDHB package as well as an application to the tunneling to open space dynamics [Proc. Natl. Acad. Sci. USA, 109, 13521 (2012)] are discussed.

International audience; The exactly solvable quantum many-particle model with harmonic one-and two-particle interaction terms is extended to include time dependency. We show that when the external trap potential and interparticle interaction have a time dependency, the numerically exact solutions of the corresponding time-dependent many-boson Schrödinger equation are still available. We use these exact solutions to benchmark the recently developed multiconfigurational time-dependent Hartree method for bosons (MCTDHB) [Phys. Rev. Lett. 99, 030402 (2007); Phys. Rev. A 77, 033613 (2008)]. In particular, we benchmark the MCTDHB method for (i) the ground state; (ii) the breathing many-body dynamics activated by a quench scenario where the interparticle interaction strength is suddenly turned on to a finite value; (iii) the nonequilibrium dynamic for driven scenarios where both the trap-and interparticle-interaction potentials are time-dependent. Excellent convergence of the ground state and dynamics is demonstrated. The great relevance of the self-consistency and time adaptivity, which are the intrinsic features of the MCTDHB method, is demonstrated by contrasting the MCTDHB predictions and those obtained within the standard full configuration interaction method spanning a Fock space of the same size, but utilizing as one-particle basis set the fixed-shape eigenstates of the one-particle potential. Connections of the model's results to ultracold Bose-Einstein condensed systems are addressed.

Using the (t,t') method the time-dependent Schrodinger equation for two electronic potential surfaces which are coupled via a time-dependent (not necessarily time-periodic) field is transformed from the length to momentum, reduced momentum and acceleration gauge representations. Combining the (t,t') method and complex scaling enables the calculation of partial widths by the asymptotic analysis of the tail of the resonance wavefunction obtained in the length gauge after being transformed to the acceleration gauge such that a free particle is obtained as R (the reaction coordinate) to infinity regardless of the shape of the laser pulse. As an illustrative numerical example the partial widths of H2+ in high periodic fields are calculated.

Using the (t, t′) formalism [J. Chem. Phys. 99 (1993) 4590] combined with the complex coordinate method, exact (i.e. not perturbative) condition for control of a model chemical reaction is derived regardless of the field intensity and whether the electromagnetic field is time-periodic or not. We prove that upon breaking the dynamical symmetry H(p, x, t) = H(−p, −x, + T/2), the dissociation channel of the A + BA ← ABA → AB + A reaction can be controlled. It is shown that when the molecular/field interaction is given by pzf(t), where f(t) represents the electromagnetic field, the dissociation channel of the A + BC ← ABC → AB + C reaction can be controlled by breaking the dynamical symmetry property, f(t) = −f(t + T/2). For time-periodic fields T is the time period (i.e. one optical cycle) where for pulsed lasers it is the duration of the pulse. Numerical examples are given for symmetric and asymmetric model Hamiltonians subjected to two cw lasers. These numerical examples illustrate the role of the phase difference between the cw lasers, regardless of the laser intensity, in the coherent control procedure which was first proposed by Brumer and Shapiro [Chem. Phys. Letters 126 (1986) 54].