Alexej I. Streltsov

In this Colloquium, the wave-function-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. The historical developments that have led to the formulation of the MCTDH-X method and motivate the necessity for wave-function-based approaches are briefly described. The derivation of the unified MCTDH-F and MCTDH-B equations of motion for complete and also specific restricted configuration spaces are sketched. The strengths and limitations of the MCTDH-X approach are assessed via benchmarks against an exactly solvable model and via convergence checks. Applications to some instructive and experimentally realized quantum many-body systems are highlighted: the dynamics of atoms in Bose-Einstein condensates in magneto-optical and optical traps and of electrons in atoms and molecules. The current development and frontiers in the field of MCTDH-X are discussed: theories and numerical methods for indistinguishable particles, for mixtures of multiple species of indistinguishable particles, the inclusion of nuclear motion for the nonadiabatic dynamics of atomic and molecular systems, as well as the multilayer and second-quantized-representation approaches, and the orbital-adaptive time-dependent coupled-cluster theory.

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 treatment in [Cohen and Lee, J. Math. Phys. {\bf 26}, 3105 (1985)], closed form expressions for the ground-state energy, wave-function, and 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 hereby prove that the exact ground-state energy and lowest-order intra-species and inter-species densities 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, the separability of the mixture's center-of-mass coordinate is used 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. Our analytical results show that many-body correlations exist in a mixture of Bose-Einstein condensates made of any number of particles. Implications are briefly discussed.

The variance of a many-particle operator is a very sensitive probe of many-body correlations. In contrast to the expectation value of such operators which depends on the condensation fraction of the Bose-Einstein condensate (BEC), their variance depends on the absolute number of depleted particles. Consequentially, the many-body correlations incorporated within the variance can be observed even in the limit of an infinite number of particles (keeping the interaction parameter, i.e., the product of the number of particles times the scattering length, constant) when the system becomes 100% condensed. In the present work we show both analytically and numerically the effect of many-body correlations on the variance and uncertainty product for an out-of-equilibrium BEC. Our work demonstrates the need to go beyond a mean-field description of the condensate in order to correctly describe these quantities even when only a single particle is depleted and the system is by definition 100% condensed.

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.

A unified view on linear response of interacting systems utilizing multiconfigurational time-dependent Hartree methods is presented. The cases of one-particle and two-particle response operators for identical particles and up to all-system response operators for distinguishable degrees-of-freedom are considered. The working equations for systems of identical bosons and identical fermions, as well for systems of distinguishable particles are explicitly derived. These linear-response theories – applicable for discrete excitation spectra – provide numerically exact excitation energies and system's properties, when numerical convergence is achieved in the calculations.

A unified view on linear response of interacting systems utilizing multicongurational time-dependent Hartree methods is presented. The cases of one-particle and two-particle response operators for identical particles and up to all-system response operators for distinguishable degrees-of-freedom are considered. The working equations for systems of identical bosons (LR-MCTDHB) and identical fermions (LR-MCTDHF), as well for systems of distinguishable particles (LR-MCTDH) are explicitly derived. These linear-response theories provide numerically-exact excitation energies and system's properties, when numerical convergence is achieved in the calculations.