Dynamics of interacting bosons using the Herman–Kluk semiclassical initial value representation

Abstract

Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if timedependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman-Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n. The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit (  ¥ n). An error is, however, incurred in next-to-next-to-leading order (small parameter n 1 ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also

The semiclassical propagator in Fock space: dynamical echo and many-body interference

“…Following the steps learned from the study of quantum interference in mesoscopic systems, we then calculated the Fock-space version of the mesoscopic dynamical echo, and found a universal enhancement of the return probability due to the interplay between interference and interactions. This prediction, successfully confirmed in numerical calculations presented in [5], shows the power of the semiclassical thinking in many-body systems, a road that has been shown to be useful for other set-ups like the 1-site Bose-Hubbard system [44], the WKB approach for the 2-site case of [45][46][47][48][49], bosonic transport in optical lattices [50], connecting soliton-like solutions of discrete nonlinear equations with properties of the quantum spectra [51], investigating spectral statistics of fully chaotic systems [52][53][54][55] and going beyond the classical truncated Wigner method to describe dynamical processes in many-body systems [56]. While the extension of our methods to non-relativistic Fermionic fields on the lattice (partially initiated in [57]) is the subject of work in progress, the ultimate goal of addressing fully fledged quantum fields in the continuum, or equivalently an infinite number of possible single-particle orbitals, will face fundamental aspects of such systems in the presence of interactions 2 like renormalization issues, well beyond the present tools of semiclassical analysis.…”

Simulation of the quantum dynamics of indistinguishable bosons with the method of coupled coherent states

“…We also wish to consider condensates in multi-well traps, such as a multi-site ver-sion of the Bose-Hubbard model studied in Ref. [61], and other interesting systems that have previously been studied by ML-MCTDHB [45][46][47][48][49][50][51][52].…”

“…Due to the use of coherent states in CCSB and their relation to the creation and annihilation operators of second quantisation, together with the fact that systems with a large number of particles tend towards classical behaviour and arXiv:1811.06293v3 [quant-ph] 28 Jun 2019 the basis in CCSB is guided by classical-like trajectories, suggest that the method will be particularly suited to such systems. Indeed, recent semiclassical coherent state work with the Herman-Kluk method on indistinguishable bosons demonstrates this hypothesis [60,61]. CCSB is fully quantum however, as with standard CCS, and it has previously been shown that a local quadratic approximation of the Hamiltonian into the CCS equations yields the coherent state matrix of the Herman-Kluk propagator [62].…”

“…The first model problem consists of a bosonic bath with an impurity, demonstrating that the method is capable of studying multi-component bosonic systems and opening up the possibility of studying multi-atomic Bose-Einstein condensates [64], spinor Bose-Einstein condensates [65], dark-bright solitons [66], and Bose-polarons [67]. The second model problem consists of a collection of indistinguishable bosons in a harmonic trap, demonstrating the propensity of the method to study systems of bosons in optical lattices [68], for example with the Bose-Hubbard model [60,61,69], and the possibility to study bosons in a single well that is deformed into a double well, such as that in Ref. [22], and observed in experimental bosonic Josephson junctions [70,71].…”

Gouy’s Phase Anomaly in Electron Waves Produced by Strong-Field Ionization

“…This idea can be transferred to other semiclassically-treated problems, e.g. in the context of molecular spectroscopy [47,59,60] or interacting bosons [61], with the advantage of a clearer view on physical processes compared to previously used initial-value representations.…”