Time-dependent Hartree Approximation Research Articles

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the time-dependent Dirac–Frenkel variational principle. Our main result addresses a general setting with smooth potentials where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations. We also discuss the case of Coulomb potentials, though treated differently (using Strichartz estimates and a classical fixed point argument).

Various variational approximations of quantum dynamics

We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan [Mol. Phys. 8, 39–44 (1964)], minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer and Saraceno [Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics Vol. 140 (Springer, Berlin, 1981)], imposes the stationarity of an action functional. We characterize both principles in terms of metric and symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.

Dynamical approximations for composite quantum systems: assessment of error estimates for a separable ansatz

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work Burghardt et al (2021 J. Phys. A: Math. Theor. 54 414002). Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The time-dependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.

On Weak-Field (One-Photon) Coherent Control of Photoisomerization.

Photochemistry induced by phase-coherent laser light is an intriguing topic. The possibility of weak-field (one-photon) phase-only control of photoisomerization is controversial. Experimental studies on the weak-field coherent control of cis-trans isomerization have led to conflicting results. Here we address this issue by quantum dynamical calculations, focusing on a mechanism where different "phase-shaped" wave packets are quickly stabilized ("dumped") in the trans configuration because of prompt energy dissipation. We systematically investigate different relaxation rates with the system-bath dynamics described within the time-dependent Hartree approximation leading to a friction-type force. We find evidence for phase control of trans-isomer yields (about 5%) in this model with pure energy dissipation given sufficiently strong dampening.

Quantum breathers and modulational instability in quantum zigzag spin chains with alternative combination ferromagnetic and antiferromagnetic interactions in an external magnetic field

Abstract Quantum breathers properties and modulational instability in quantum zigzag spin chains with alternative combination ferro-antiferromagnetic interactions are carried out. For a few number of bosons, by using the number states method and numerical diagonalization, the energy spectrum, multi-quantum breathers states and theirs new properties are found. For a large number of bosons, based on the time dependent Hartree approximation and semi-discrete multiple-scale method, we provide a detailed analytical analysis, discuss conditions to obtain solitons bright localized solutions and predict the formation of instability regions in the system. In both configurations, we have constructed quantum breathers states and deduced their quantum energies. Furthermore the modulational instability analysis are done and the formation conditions of appearance instabilities regions predicted by our analytical investigations are found and are in full agreement. Convincing quantitative agreement has been found between the analytical approaches and the numerical simulations for the modulational instability. Experimental data on Rb2Cu2Mo3O12 are parameters used in this work.

Quantum optimal control of multiple weakly interacting molecular rotors in the time-dependent Hartree approximation.

We perform quantum optimal control simulations, based on the Time-Dependent Hartree (TDH) approximation, for systems of three to five dipole-dipole coupled OCS rotors. A control electric field is used to steer all of the individual rotors, arranged in chains and regular polygons in a plane, toward either identical or unique objectives. The goal is to explore the utility of the TDH approximation to model the field-induced dynamics of multiple interacting rotors in the weak dipole-dipole coupling regime. A stochastic hill climbing approach is employed to seek an optimal control field that achieves the desired objectives at a specified target time. We first show that multiple rotors in chain and polygon geometries can be identically oriented in the same direction; these cases do not significantly depend on the presence of the dipole-dipole interaction. Additionally, in particular geometrical arrangements, we demonstrate that individual rotors can be uniquely manipulated toward different objectives with the same field. Specifically, it is shown that for a three rotor chain, the two end rotors can be identically oriented in a specific direction while keeping the middle rotor in its ground state, and for an equilateral triangle, two rotors can be identically oriented in a specific direction while the third rotor is oriented in the opposite direction. These multirotor unique objective cases exploit the shape of the field in coordination with dipole-dipole coupling between the rotors. Comparisons to numerically exact calculations, utilizing the TDH-determined fields, are given for all optimal control studies involving systems of three rotors.

Modulational Instability and Quantum Discrete Breather States of Cold Bosonic Atoms in a Zig-Zag Optical Lattice

A theoretical study on modulational instability and quantum discrete breather states in a system of cold bosonic atoms in zig-zag optical lattices is presented in this work. The time-dependent Hartree approximation is employed to deal with the multiple body problem. By means of a linear stability analysis, we analytically study the modulational instability, and estimate existence conditions of the bright stationary localized solutions for different values of the second-neighbor hopping constant. On the other hand, we get analytical bright stationary localized solutions, and analyze the influence of the second-neighbor hopping on their existence conditions. The predictions of the modulational instability analysis are shown to be reliable. Using these stationary localized single-boson wave functions, the quantum breather states corresponding to the system with different types of nonlinearities are constructed.

Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving next-nearest neighbor interactions

The dynamics and modulation instability of an ultracold gas of bosonic atoms in an optical lattice can be portrayed by a Bose–Hubbard model and the system parameters are mastered by laser light. Based on the time dependent Hartree approximation combined with the semi-discrete multiple-scale method, the equation of motion for single-boson wave function is found analytically, the existence conditions of appearance of bright stationary localized solitons solutions of this quantum Bose–Hubbard model are discussed. We find that the introduction of the next nearest neighbor interactions (NNNI) may change the stability property of the plane waves and may predict the formation of modulational instability in the wave number k = kmax and k = keBZ in the system. With the help of stationary localized single-boson wave functions obtained, the quantized energy level and the quantum breather state are determined. The performance of the analytical results are checked by numerical calculations. Furthermore, we have shown that the presence of the NNNI affect significatively the shape of the region of modulational instability and it is responsible of the appearance of new region of modulational instability that occurs for the k = kmax carrier wave. The formation conditions of the modulational instability region predicted by the analytical analysis of stationary localized solutions in the wave number k = kmax and k = keBZ are in good agreement with the forecast respectively.

Bright solitons in ultracold atoms

We review old and recent experimental and theoretical results on bright solitons in Bose-Einstein condensates made of alkali-metal atoms and under external optical confinement. First we deduce the three-dimensional Gross-Pitaevskii equation (3D GPE) from the Dirac-Frenkel action of interacting identical bosons within a time-dependent Hartree approximation. Then we discuss the dimensional reduction of the GPE from 3D to 1D, deriving the 1D GPE and also the 1D nonpolynomial Schr\"odinger equation (1D NPSE). Finally, we analyze the bright solition solutions of both 1D GPE and 1D NPSE and compare these theoretical predictions with the available experimental data.

Quantum soliton in 1D Heisenberg spin chains with Dzyaloshinsky-Moriya and next-nearest-neighbor interactions.

We report in this work, an analytical study of quantum soliton in 1D Heisenberg spin chains with Dzyaloshinsky-Moriya Interaction (DMI) and Next-Nearest-Neighbor Interactions (NNNI). By means of the time-dependent Hartree approximation and the semi-discrete multiple-scale method, the equation of motion for the single-boson wave function is reduced to the nonlinear Schrödinger equation. It comes from this present study that the spectrum of the frequencies increases, its periodicity changes, in the presence of NNNI. The antisymmetric feature of the DMI was probed from the dispersion curve while changing the sign of the parameter controlling it. Five regions were identified in the dispersion spectrum, when the NNNI are taken into account instead of three as in the opposite case. In each of these regions, the quantum model can exhibit quantum stationary localized and stable bright or dark soliton solutions. In each region, we could set up quantum localized n-boson Hartree states as well as the analytical expression of their energy level, respectively. The accuracy of the analytical studies is confirmed by the excellent agreement with the numerical calculations, and it certifies the stability of the stationary quantum localized solitons solutions exhibited in each region. In addition, we found that the intensity of the localization of quantum localized n-boson Hartree states increases when the NNNI are considered. We also realized that the intensity of Hartree n-boson states corresponding to quantum discrete soliton states depend on the wave vector.

Quantum Breathers in Anisotropy Ferromagnetic Chains with Second-Order Coupling

Under considering the next-nearest-neighbor interaction, quantum breathers in one-dimensional anisotropy ferromagnetic chains are theortically studied. By introducing the Dyson-Maleev transformation for spin operators, a map to a Heisenberg ferromagnetic spin lattice into an extended Bose–Hubbard model can be established. In the case of a small number of bosons, by means of the numerical diagonalization technique, the energy spectrum of the corresponding extended Bose–Hubbard model containing two bosons is calculated. When the strength of the single-ion anisotropy is enough large, a isolated single band appears. This isolated single band corresponds to two-boson bound state, which is the simplest quantum breather state. It is shown that the introduction of the next-nearest-neighbor interaction will lead to interesting band structures. In the case of a large number of bosons, by applying the time-dependent Hartree approximation, quantum breather states for the system is constructed. In this case, the effect of the next-nearest-neighbor interaction on quantum breathers is also analyzed.

Quantum breathers in Heisenberg ferromagnetic chains with Dzyaloshinsky-Moriya interaction.

We present an analytical study on quantum breathers in one-dimensional ferromagnetic XXZ chains with Dzyaloshinsky-Moriya interaction by means of the time-dependent Hartree approximation and the semidiscrete multiple-scale method. The stationary localized single-boson wave functions are obtained and these analytical solutions are checked by numerical simulations. With such stationary localized single-boson wave functions, we construct quantum breather states. Furthermore, the role of the Dzyaloshinsky-Moriya interaction is discussed.

Quantum Solitons and Breathers in an Anisotropic Ferromagnet with Octupole-Dipole Interaction

By using a full quantum approach based on the time-dependent Hartree approximation and the semidiscrete multiple-scale method, we study quantum nonlinear excitations in a one-dimensional ferromagnetic chain with octupole-dipole interaction and on-site uniaxial anisotropy. We find that quantum solitons and breathers can exist in the ferromagnetic chain, and analyze existence conditions of these excitations. Since the system states corresponding to quantum breathers are stationary states, we can get the energy level formula of such quantum breathers.

Response of solid Ne upon photoexcitation of a NO impurity: A quantum dynamics study

The ultrafast geometrical rearrangement dynamics of NO doped cryogenic Ne matrices after femtosecond laser pulse excitation is studied using a quantum dynamical approach based on a multi-dimensional shell model, with the shell radii being the dynamical variables. The Ne-NO interaction being only weakly anisotropic allows the model to account for the main dynamical features of the rare gas solid. Employing quantum wave packet propagation within the time dependent Hartree approximation, both, the static deformation of the solid due to the impurity and the dynamical response after femtosecond excitation, are analysed. The photoinduced dynamics of the surrounding rare gas atoms is found to be a complex high-dimensional process. The approach allows to consider realistic time-dependent femtosecond pulses and the effect of the pulse duration is clearly shown. Finally, using the pulse parameters of previous experiments, pump-probe signals are calculated and found to be in good agreement with experimental results, allowing for a clear analysis of the ultrafast mechanism of the energy transfer into the solid.

Quantum Theory of Solid State Plasma Dielectric Response

AbstractWe review the quantum mechanical derivation of the random phase approximation (RPA) for solid state plasmas, starting from the Hamilton equations for canonically paired “second quantized” creation and annhilation field operators of interacting quantum many‐body systems. Discussing variational differentiation, the coupled equations of motion for the quantum field operators are derived. The concept of Green's functions is reviewed and interpreted, first for retarded Green's functions, and their equations of motion are developed from the equations of motion for the field operators. Thermodynamic Green's functions are discussed, and their periodicity/antiperiodicity properties in imaginary time are carefully examined with discussion of Matsubara Fourier series and representation in terms of a spectral weight function. The analytic continuation from imaginary time to real time is treated. Finally, we define nonequilibrium Green's functions and discuss the linearized timedependent Hartree approximation leading to the random phase approximation. An interesting application to the case of Graphene in a perpendicular magnetic field is discussed in detail, along with applications to normal systems, in terms of attendant phenomenology involving electron‐hole pair excitations and plasmons (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Quantum dynamics through conical intersections in macrosystems: Combining effective modes and time-dependent Hartree

We address the nonadiabatic quantum dynamics of (macro)systems involving a vast number of nuclear degrees of freedom (modes) in the presence of conical intersections. The macrosystem is first decomposed into a system part carrying a few, strongly coupled modes, and an environment, comprising the remaining modes. By successively transforming the modes of the environment, a hierarchy of effective Hamiltonians for the environment can be constructed. Each effective Hamiltonian depends on a reduced number of effective modes, which carry cumulative effects. The environment is described by a few effective modes augmented by a residual environment. In practice, the effective modes can be added to the system’s modes and the quantum dynamics of the entire macrosystem can be accurately calculated on a limited time-interval. For longer times, however, the residual environment plays a role. We investigate the possibility to treat fully quantum mechanically the system plus a few effective environmental modes, augmented by the dynamics of the residual environment treated by the time-dependent Hartree (TDH) approximation. While the TDH approximation is known to fail to correctly reproduce the dynamics in the presence of conical intersections, it is shown that its use on top of the effective-mode formalism leads to much better results. Two numerical examples are presented and discussed; one of them is known to be a critical case for the TDH approximation.

Regularity of the multi-configuration time-dependent Hartree approximation in quantum molecular dynamics

We discuss the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrodinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclear wave function by a linear combination of products of functions depending only on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensional nonlinear partial differential equations and ordinary differential equations. We show that, with a smooth and bounded potential, the MCTDH equations are well-posed and retain high-order Sobolev regularity globally in time, that is, as long as the density matrices appearing in the method formulation remain invertible. In particular, the solutions are regular enough to ensure local quasi-optimality of the approximation and to admit an efficient numerical treatment.

A GAUSSIAN WAVEPACKET APPROACH FOR CURVE-CROSSING DYNAMICS

We present a Gaussian wavepacket approach for curve-crossing dynamics that only requires a single Gaussian wavepacket per surface. Unlike other Gaussian wavepacket approaches to curve-crossing dynamics, the present method does not rely upon probability density being built up on a non-adiabatically coupled surface by the break-up of an evolving wavepacket. Thus, our trial solution for the time-dependent Schrodinger equation comprised of a single Gaussian wavepacket per non-adiabatically coupled surface. We also present a generalization of the method to treat multi-dimensional systems that is based on the time-dependent Hartree approximation. We present numerical results for both single and multi-dimensional curve-crossing dynamics and compare them to the exact results.

Mean-field approach to excitations of a trapped Bose-Einstein condensate gas at finite temperatures

The monopole and quadrupole oscillations of a Bose-Einstein condensate (BEC) gas trapped in a spherically symmetric potential are calculated using the time-dependent Hartree approximation which enables us to use an orthonormal single-particle basis for the description of both the condensate and noncondensate components. It is found that the monopole mode has a much smaller damping rate than the quadrupole mode. The obtained temperature dependence of the frequency shifts and the damping rates of the collective oscillations are found to be consistent with experimental data obtained for BEC gases in anisotropic trapping potentials. A comparison between the time-dependent Hartree approximation and the time-dependent Hartree-Fock approximation is also made and it is found that there is a rather small difference.

Electron tunneling dynamics in anharmonic bath

Tunneling transition probability for a particle interacting with an anharmonic bath is found in a time-dependent Hartree approximation. The general expression is presented in terms of medium Keldysh functions that are assumed to be known. Furthermore, the transition probability is calculated in the noninteracting-blip approximation where the rate constant does not exhibit an activation dependence at high temperatures. The reorganization energy E(r) and the renormalized reaction heat epsilon are expressed in terms of the correlation matrix for a solvent and internal modes in both quantum and classical regimes. It is shown that E(r) and epsilon are temperature dependent.