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AIP conference proceedings

Time-Dependent Mean-Field Theory in Nuclear and Electronic Systems

Kazuhiro Yabana, Y. Kawashita, Tsunenori Inakura, Takashi Nakatsukasa, Sun‐Chan Jeong, Y. Utsuno, T. Motobayashi, A. Bracco

Published Jan 1, 2009• 0 citations
Publisher

Abstract

The time‐dependent mean‐field theory has a long history in describing many‐nucleon dynamics in nuclear physics, and is now widely applied to many‐electron dynamics in atomic, molecular, and condensed matter physics. We present our recent studies in these fields, applying similar computational methods to both nuclear and electronic systems. In the linear response regime, we show calculations of the optical response of a medium mass nucleus and a fullerene molecule C60. The applications in nonlinear regime are also briefly mentioned.

DOI
10.1063/1.3141682

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Time-dependent mean-field approximation for nuclear dynamical problems

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Physical Review C•Apr 1, 1980•64 citations
The time‐dependent mean‐field theory has a long history in describing many‐nucleon dynamics in nuclear physics, and is now widely applied to many‐electron dynamics in atomic, molecular, and condensed matter physics. We present our recent studies in these fields, applying similar computational methods to both nuclear and electronic systems. In the linear response regime, we show calculations of the optical response of a medium mass nucleus and a fullerene molecule C60. The applications in nonlinear regime are also briefly mentioned.
We compute nuclear response functions by solving the time-dependent A-body Schrödinger equation, recording the time-dependent transition moment and extracting spectral information via Fourier transforms. The solution of the time-dependent many-body problem accounts for correlations on top of the mean field by taking advantage of a time-dependent formulation of coupled-cluster theory. As a validation, we focus on electric dipole transitions in 4^44He and 16^{16}16O and compare moments of the response function distribution to the results of an equivalent static framework, finding negligible discrepancies. We investigate how proton and neutron densities evolve in time, and we see the traditional picture of soft and giant dipole resonances as collective oscillations of protons and neutrons emerging from our calculations in 16^{16}16O and 24^{24}24O. This method also allows us to investigate the behavior of the nucleus in the presence of a strong electric field. In that regime, the behavior of the system becomes chaotic. Qualitatively, the spectral information obtained in this limit is in line with previous time-dependent mean-field results.
Understanding quantum many-body systems with long-range or infinite-range interactions is of relevance across a broad set of physical disciplines, including quantum optics, nuclear magnetic resonance, and nuclear physics. From a theoretical viewpoint, these systems are appealing since they can be efficiently studied with numerics, and in the thermodynamic limit are expected to be governed by mean-field equations of motion. Over the past years the capabilities to experimentally create long-range interacting systems have dramatically improved permitting their control in space and time. This allows us to induce and explore a plethora of nonequilibrium dynamical phases, including time crystals and even chaotic regimes. However, establishing the emergence of these phases from numerical simulations turns out to be surprisingly challenging. This difficulty led to the assertion that mean-field theory may not be applicable to time-dependent infinite-range interacting systems. Here, we rigorously prove that mean-field theory in fact exactly captures their dynamics, in the thermodynamic limit. We further provide bounds for finite-size effects and their dependence on the evolution time.
The many approaches that have been pursued in seeking an understanding of nuclear rotational dynamics are reviewed and reassessed with a view to their development in the light of recent progress and the research tools that are now available. A motivation for this review is the widespread observation of nuclear shape coexistence and sequences of rotational states in all regions of the nuclear periodic table combined with the recognition that the study of the rotational dynamics of quantum fluids has led to significant advances in the quantum theory of many-boson systems. Recent experimental investigations of the rotational dynamics of a low-temperature 6^66Li gas indicate that its slow rotational flows are likewise the irrotational flows of a superfluid. In this context, the dynamics of rotating nuclei are of fundamental interest because the nucleus is a unique zero-temperature finite many-fermion quantum system. A promising approach is provided by algebraic mean-field theory which, as its name suggests, is a combination of algebraic and mean-field methods. Static mean-field theories play a central role in many-body theory by defining optimal independent-particle and independent quasi-particle basis states for the quantum mechanics of many-fermion systems. Their time-dependent extensions also lead, in the small-amplitude random-phase approximation, to the quantisation of the classical normal-mode vibrations of many-fermion systems about their static equilibrium states. This review shows that mean-field methods become significantly more powerful when combined with algebraic methods and an appropriate coupling scheme for the nuclear shell model.
Understanding quantum many-body systems with long-range or infinite-range interactions is of relevance across a broad set of physical disciplines, including quantum optics, nuclear magnetic resonance and nuclear physics. From a theoretical viewpoint, these systems are appealing since they can be efficiently studied with numerics, and in the thermodynamic limit are expected to be governed by mean-field equations of motion. Over the past years the capabilities to experimentally create long-range interacting systems have dramatically improved permitting their control in space and time. This allows to induce and explore a plethora of nonequilibrium dynamical phases, including time-crystals and even chaotic regimes. However, establishing the emergence of these phases from numerical simulations turns out to be surprisingly challenging. This difficulty led to the assertion that mean-field theory may not be applicable to time-dependent infinite-range interacting systems. Here, we rigorously prove that mean-field theory in fact exactly captures their dynamics, in the thermodynamic limit. We further provide bounds for finite-size effects and their dependence on the evolution time.
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.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.