arXiv subjects

Frank Grossmann

Publications and source records attributed to Frank Grossmann.

At least 19 recordsLinked to original sources

arXiv2024

Quench dynamics of interacting bosons: generalized coherent states versus multi-mode Glauber states

Multi-mode Glauber coherent states (MMGS) as well as Bloch states with zero quasi-momentum, which are a special case of generalized coherent states (GCS), are frequently used to describe condensed phases of bosonic many-body systems. The difference of two-point correlators of MMGS and GCS vanishes in the thermodynamic limit. Using the established expansion of GCS in terms of MMGS, we derive a Fourier-type relation between the (auto-)correlation functions of the two different time-evolved states. This relation reveals that the (auto-)correlation and thus the dynamical free energy density for the two cases are still different, even in the thermodynamic limit, due to the lack of the U(1) symmetry of the MMGS. Analytic results for the deep lattice model of interacting bosons for increasing filling factors show multiple sharp structures in the dynamical free energy-density of increasing complexity. These are explained using the evolution of Husimi functions in phase space.

quant-ph↗

arXiv2024

Coupled coherent states method for tunneling dynamics: an interpretative study

Numerical solutions of the time-dependent Schrödinger equation based on the variational principle may offer physical insight that cannot be gained by a solution using fixed grids in position and momentum space. Here we focus on the tunneling dynamics in a quartic double-well and the use of classical, trajectory-guided coherent states to gain insight into the workings of the coupled coherent states method developed by Shalashilin and Child [J. Chem. Phys. {\bf 113}, 10028 (2000)]. It is shown that over-the-barrier classical trajectories, alone, can accurately describe the tunneling effect.

quant-ph↗

arXiv2023

Revealing quantum effects in bosonic Josephson junctions: a multi-configuration atomic coherent states approach

The mean-field approach to two-site Bose-Hubbard systems is well established and leads to nonlinear classical equations of motion for the population imbalance and the phase difference. It can, e.g., be based on the representation of the solution of the time-dependent Schrodinger equation either by a single Glauber state or by a single atomic (SU(2)) coherent state [S. Wimberger et al., Phys. Rev. A 103, 023326 (2021)]. We demonstrate that quantum effects beyond the mean-field approximation are easily uncovered if, instead, a multi-configuration ansatz with a few time-dependent SU(2) basis functions is used in the variational principle. For the case of plasma oscillations, the use of just two basis states, whose time-dependent parameters are determined variationally, already gives good qualitative agreement of the phase space dynamics with numerically exact quantum solutions. In order to correctly account for more non-trivial effects, like macroscopic quantum self trapping, moderately more basis states are needed. If one is interested in the onset of spontaneous symmetry breaking, however, a multiplicity of two gives a big improvement towards the exact result already. In any case, the number of variational trajectories needed for good agreement with full quantum results is orders of magnitude smaller than in the semiclassical case, which is based on multiple mean-field trajectories.

quant-ph↗

arXiv2023

Entanglement in the full state vector of boson sampling

The full state vector of boson sampling is generated by passing S single photons through beam splitters of M modes. The initial Fock state is expressed withgeneralized coherent states, and an exact application of the unitary evolution becomes possible. Due to the favorable polynomial scaling in M , we can investigate Renyi entanglement entropies for moderate particle and huge mode numbers. We find (almost) Renyi index independent symmetric Page curves with maximum entropy at equal partition. Furthermore, the maximum entropy as a function of mode index saturates as a function of M in the collision-free subspace case. The asymptotic value of the entropy increases linearly with S. Furthermore, we show that the build-up of the entanglement leads to a cusp at subsystem size equal to S in the asymmetric entanglement curve. The maximum entanglement is reached surprisingly early before the mode population is distributed over the whole system.

quant-ph↗

arXiv2022

Non-Markovian Vibrational Relaxation Dynamics at Surfaces

Vibrational dynamics of adsorbates near surfaces plays both an important role for applied surface science and as model lab for studying fundamental problems of open quantum systems. We employ a previously developed model for the relaxation of a D-Si-Si bending mode at a D:Si(100)-(2$\times$1) surface, induced by a "bath" of more than $2000$ phonon modes [U. Lorenz, P. Saalfrank, Chem. Phys. {\bf 482}, 69 (2017)], to extend previous work along various directions. First, we use a Hierarchical Effective Mode (HEM) model [E.W. Fischer, F. Bouakline, M. Werther, P. Saalfrank, J. Chem. Phys. {\bf 153}, 064704 (2020)] to study relaxation of higher excited vibrational states than hitherto done, by solving a high-dimensional system-bath time-dependent Schrödinger equation (TDSE). In the HEM approach, (many) real bath modes are replaced by (much less) effective bath modes. Accordingly, we are able to examine scaling laws for vibrational relaxation lifetimes for a realistic surface science problem. Second, we compare the performance of the multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) approach with the recently developed coherent-state based multi-Davydov D2 {\it ansatz} [N. Zhou, Z. Huang, J. Zhu, V. Chernyak, Y. Zhao, {J. Chem. Phys.} {\bf 143}, 014113 (2015)]. Both approaches work well, with some computational advantages for the latter in the presented context. Third, we apply open-system density matrix theory in comparison with basically "exact" solutions of the multi-mode TDSEs. Specifically, we use an open-system Liouville-von Neumann (LvN) equation treating vibration-phonon coupling as Markovian dissipation in Lindblad form to quantify effects beyond the Born-Markov approximation.

physics.chem-ph↗

arXiv2021

Quantum approach to the thermalization of the toppling pencil interacting with a finite bath

We investigate the longstanding problem of thermalization of quantum systems coupled to an environment by focusing on a bistable quartic oscillator interacting with a finite number of harmonic oscillators. In order to overcome the exponential wall that one usually encounters in grid based approaches to solve the time-dependent Schrödinger equation of the extended system, methods based on the time-dependent variational principle are best suited. Here we will apply the method of coupled coherent states [D. V. Shalashilin and M. S. Child, J. Chem. Phys. {\bf 113}, 10028 (2000)]. By investigating the dynamics of an initial wavefunction on top of the barrier of the double well, it will be shown that only a handful of oscillators with suitably chosen frequencies, starting in their ground states, is enough to drive the bistable system close to its uncoupled ground state. The long-time average of the double-well energy is found to be a monotonously decaying function of the number of environmental oscillators in the parameter range that was numerically accessible.

quant-ph↗

arXiv2021

Exact variational dynamics of the multimode Bose-Hubbard model based on SU(M) coherent states

We propose a variational approach to the dynamics of the Bose-Hubbard model beyond the mean field approximation. To develop a numerical scheme, we use a discrete overcomplete set of Glauber coherent states and its connection to the generalized coherent states studied in depth by Perelomov [A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag (Berlin, 1986)]. The variational equations of motion of the generalized coherent state parameters as well as of the coefficients in an expansion of the wavefunction in terms of those states are derived and solved for many-particle problems with large particle numbers S and increasing mode number M. For M = 6 it is revealed that the number of parameters that have to be propagated is more than one order of magnitude less than in an expansion in terms of Fock states.

quant-ph↗

arXiv2020

Apoptosis of moving, non-orthogonal basis functions in many-particle quantum dynamics

Due to the exponential increase of the numerical effort with the number of degrees of freedom, moving basis functions have a long history in quantum dynamics. In addition, spawning of new basis functions is routinely applied. Here we advocate the opposite process: the programmed removal of motional freedom of selected basis functions. This is a necessity for converged numerical results with respect to the size of a non-orthogonal basis, because generically two or more states approach each other too closely early on, rendering unstable the matrix inversion, required to make the equations of motion explicit. Applications to the sub-Ohmic spin-boson model as well as to polaron dynamics in a Holstein molecular crystal model demonstrate the power of the proposed methodology.

quant-ph↗

arXiv2018

Herman-Kluk propagator is free from zero-point energy leakage

Semiclassical techniques constitute a promising route to approximate quantum dynamics based on classical trajectories starting from a quantum-mechanically correct distribution. One of their main drawbacks is the so-called zero-point energy (ZPE) leakage, that is artificial redistribution of energy from the modes with high frequency and thus high ZPE to that with low frequency and ZPE due to classical equipartition. Here, we show that an elaborate semiclassical formalism based on the Herman-Kluk propagator is free from the ZPE leakage despite utilizing purely classical propagation. This finding opens the road to correct dynamical simulations of systems with a multitude of degrees of freedom that cannot be treated fully quantum-mechanically due to the exponential increase of the numerical effort.

physics.chem-ph↗

arXiv2018

Simplified Approach to the Mixed Time-averaging Semiclassical Initial Value Representation for the Calculation of Dense Vibrational Spectra

We present and test an approximate method for the semiclassical calculation of vibrational spectra. The approach is based on the mixed time-averaging semiclassical initial value representation method, which is simplified to a form that contains a filter to remove contributions from approximately harmonic environmental degrees of freedom. This filter comes at no additional numerical cost, and it has no negative effect on the accuracy of peaks from the anharmonic system of interest. The method is successfully tested for a model Hamiltonian, and then applied to the study of the frequency shift of iodine in a krypton matrix. Using a hierarchic model with up to 108 normal modes included in the calculation, we show how the dynamical interaction between iodine and krypton yields results for the lowest excited iodine peaks that reproduce experimental findings to a high degree of accuracy.

physics.chem-ph↗

arXiv2017

Application of the Mixed Time-averaging Semiclassical Initial Value Representation method to Complex Molecular Spectra

The recently introduced mixed time-averaging semiclassical initial value representation molecular dynamics method for spectroscopic calculations [M. Buchholz, F. Grossmann, and M. Ceotto, J. Chem. Phys. 144, 094102 (2016)] is applied to systems with up to 61 dimensions, ruled by a condensed phase Caldeira-Leggett model potential. By calculating the ground state as well as the first few excited states of the system Morse oscillator, changes of both the harmonic frequency and the anharmonicity are determined. The method faithfully reproduces blueshift and redshift effects and the importance of the counter term, as previously suggested by other methods. Differently from previous methods, the present semiclassical method does not take advantage of the specific form of the potential and it can represent a practical tool that opens the route to direct ab initio semiclassical simulation of condensed phase systems.

quant-ph↗

arXiv2017

Including temperature in a wavefunction description of the dynamics of the quantum Rabi model

We present a wavefunction methodology to account for finite temperature initial conditions in the quantum Rabi model. The approach is based on the Davydov-Ansatz together with a statistical sampling of the canonical harmonic oscillator initial density matrix. Equations of motion are gained from a variational principle and numerical results are compared to those of the thermal Hamiltonian approach. For a system consisting of a single spin and a single oscillator and for moderate coupling strength, we compare our new results with full quantum ones as well as with other Davydov-type results based on alternative sampling/summation strategies. All of these perform better than the ones based on the thermal Hamiltonian approach. The best agreement is shown by a Boltzmann weighting of individual eigenstate propagations. Extending this to a bath of many oscillators will, however, be very demanding numerically. The use of any one of the investigated stochastic sampling approaches will then be favorable.

quant-ph↗

arXiv2017

Polaron dynamics with off-diagonal coupling: beyond the Ehrenfest approximation

Treated traditionally by the Ehrenfest approximation, dynamics of a one-dimensional molecular crystal model with off-diagonal exciton-phonon coupling is investigated in this work using the Dirac-Frenkel time-dependent variational principle with the multi-D$_2$ {\it Ansatz}. It is shown that the Ehrenfest method is equivalent to our variational method with the single D$_2$ {\it Ansatz}, and with the multi-D$_2$ {\it Ansatz}, the accuracy of our simulated dynamics is significantly enhanced in comparison with the semi-classical Ehrenfest dynamics. The multi-D$_2$ {\it Ansatz} is able to capture numerically accurate exciton momentum probability and help clarify the relation between the exciton momentum redistribution and the exciton energy relaxation. The results demonstrate that the exciton momentum distributions in the steady state are determined by a combination of the transfer integral and the off-diagonal coupling strength, independent of the excitonic initial conditions. We also probe the effect of the transfer integral and the off-diagonal coupling on exciton transport in both real and reciprocal space representations. Finally, the variational method with importance sampling is employed to investigate temperature effects on exciton transport using the multi-$\rm D_2$ {\it Ansatz}, and it is demonstrated that the variational approach is valid in both low and high temperature regimes.

physics.chem-ph↗

arXiv2015

Mixed Semiclassical Initial Value Representation Time-Averaging Propagator for Spectroscopic Calculations

A mixed semiclassical initial value representation expression for spectroscopic calculations is derived. The formulation takes advantage of the time-averaging filtering and the hierarchical properties of different trajectory based propagation methods. A separable approximation is then introduced that greatly reduces (about an order of magnitude) the computational cost compared with a full Herman-Kluk time-averaging semiclassical calculation for the same systems. The expression is exact for the harmonic case and it is tested numerically for a Morse potential coupled to one or two additional harmonic degrees of freedom. Results are compared to full Herman-Kluk time-averaging calculations and exact quantum wavepacket propagations. We found the peak positions of the mixed semiclassical approximations to be always in very good agreement with full quantum calculations, while overtone peak intensities are lower with respect to the exact ones. Given the reduced computational effort required by this new mixed semiclassical approximation, we believe the present method to make spectroscopic calculations available for higher dimensional systems than accessible before.

quant-ph↗

arXiv2015

A semiclassical hybrid approach to linear response functions for infrared spectroscopy

Based on the integral representation of the semiclassical propagator of Herman and Kluk (HK), and in the limit of high temperatures, we formulate a hybrid expression for the correlation function of infrared spectroscopy. This is achieved by performing a partial linearization inside the integral over the difference of phase space variables that occurs after a twofold application of the HK propagator. A numerical case study for a coupled anharmonic oscillator shows that already for a total number of only two degrees of freedom, one of which is treated in the simplified manner, a substantial reduction of the numerical effort is achieved.

quant-ph↗

arXiv2012

Obtaining Maxwell's equations heuristically

Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light together with Galileo invariance of the Lorentz force allows us to introduce arbitrary electrodynamic units naturally.

physics.class-ph↗

arXiv2012

High harmonic spectra via dominant interaction Hamiltonians

We formulate the concept of dominant interaction Hamiltonians to obtain an integrable approximation to the dynamics of an electron exposed to a strong laser field and an atomic potential leading to high harmonic generation. The concept relies on local information in phase space to switch between the interactions. This information is provided by classical integrable trajectories from which we construct a semiclassical wave function. The high harmonic spectrum obtained is in excellent agreement with the accurate quantum spectrum. The separation in the atomic potential and laser coupling interactions should facilitate the calculation of high harmonic spectra in complex systems.

physics.atom-ph↗

arXiv2011

Spectra of Harmonium in a magnetic field using an initial value representation of the semiclassical propagator

For two Coulombically interacting electrons in a quantum dot with harmonic confinement and a constant magnetic field, we show that time-dependent semiclassical calculations using the Herman-Kluk initial value representation of the propagator lead to eigenvalues of the same accuracy as WKB calculations with Langer correction. The latter are restricted to integrable systems, however, whereas the time-dependent initial value approach allows for applications to high-dimensional, possibly chaotic dynamics and is extendable to arbitrary shapes of the potential.

quant-ph↗