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.
We report on an implementation of the multiconfigurational time-dependent Hartree method (MCTDH) for spin-polarized fermions (MCTDHF). Our approach is based on a mapping for operators in Fock space that allows a compact and efficient application of the Hamiltonian and solution of the MCTDHF equations of motion. Our implementation extends, builds on and exploits the recursive implementation of MCTDH for bosons (R-MCTDHB) package. Together with R-MCTDHB, the present implementation of MCTDHF forms the MCTDH-X package. We benchmark the accuracy of the algorithm with the harmonic interaction model and a time-dependent generalization thereof. These models consider parabolically trapped particles that interact through a harmonic interaction potential. We demonstrate, that MCTDHF is capable of solving the time-dependent many-fermion Schrödinger equation to an in principle arbitrary degree of precision and can hence yield numerically exact results even in the case of Hamiltonians with time-dependent one-body and two-body potentials. As an application we study the problem of two initially parabolically confined and charged fermions tunneling through a barrier to open space. We demonstrate the validity of a model proposed previously for the many-body tunneling to open space of bosonic particles with contact interactions [Proc. Natl. Acad. Sci. USA 109, 13521-13525 (2012)]. The many-fermion tunneling can be built up from sequentially happening single-fermion tunneling processes. The characteristic momenta of these processes are determined by the chemical potentials of trapped subsystems of smaller particle numbers: the escaped fermions convert the different chemical potentials into kinetic energy. Using the two-body correlation function, we present a detailed picture of the sequentiality of the process and are able to tell tunneling from over-the-barrier escape.
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 nonadiabatic 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.
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