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arXiv:1803.08387

Numerically exact approach to few-body problems far from a perturbative regime

Marcin Płodzień, Dariusz Wiater, Andrzej Chrostowski, Tomasz Sowiński

Published Mar 22, 2018• 24 citations
arXivHTMLPDF

Abstract

Recent developments of experimental techniques in the field of ultra-cold gases open a path to study the crossover from 'few' to 'many' on the quantum level. In this case, accurate description of inter-particle correlations is very important since it is believed that they can be utilized by quantum engineers in quantum metrology, quantum thermometry, quantum heat engines, {\it etc}. Unfortunately, a theoretical description of these correlations is very challenging since they are far beyond any variational approaches. By contrast, the exact many-body description rapidly hits numerical limitations due to an exponential increase of the many-body Hilbert space. In this work, we brush up a very effective method of constructing a many-body basis which originates in the physical argumentation. We show that, in contrast to the commonly used approach of a straightforward cut-off, it enables one to perform exact calculations with very limited numerical resources. As examples, we study quantum correlations in systems of spinless bosons and two-component mixtures of fermions confined in a one-dimensional harmonic trap being far from the perturbative regime.

DOI
10.48550/arxiv.1803.08387

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Recent developments of experimental techniques in the field of ultra-cold gases open a path to study the crossover from 'few' to 'many' on the quantum level. In this case, accurate description of inter-particle correlations is very important since it is believed that they can be utilized by quantum engineers in quantum metrology, quantum thermometry, quantum heat engines, {\it etc}. Unfortunately, a theoretical description of these correlations is very challenging since they are far beyond any variational approaches. By contrast, the exact many-body description rapidly hits numerical limitations due to an exponential increase of the many-body Hilbert space. In this work, we brush up a very effective method of constructing a many-body basis which originates in the physical argumentation. We show that, in contrast to the commonly used approach of a straightforward cut-off, it enables one to perform exact calculations with very limited numerical resources. As examples, we study quantum correlations in systems of spinless bosons and two-component mixtures of fermions confined in a one-dimensional harmonic trap being far from the perturbative regime.
Cold atomic gases have provided us with a great number of opportunities for studying various physical systems under controlled conditions that are seldom offered in other fields. We are thus at the point where one can truly do quantum simulation of models that are relevant for instance in condensed-matter or high-energy physics, i.e. we are on the verge of a ’cool’ quantum simulator as envisioned by Feynman. One of the avenues under exploration is the physics of one-dimensional systems. Until recently this was mostly in the many-body limit but now experiments can be performed with controllable particle numbers all the way down to the few-body regime. After a brief introduction to some of the relevant experiments, I will review recent theoretical work on one-dimensional quantum systems containing bosons, fermions, or mixtures of the two, with a particular emphasis on the case where the particles are held by an external trap.
Cold atomic gases have provided us with a great number of opportunities for studying various physical systems under controlled conditions that are seldom offered in other fields. We are thus at the point where one can truly do quantum simulation of models that are relevant for instance in condensed-matter or high-energy physics, i.e. we are on the verge of a ’cool’ quantum simulator as envisioned by Feynman. One of the avenues under exploration is the physics of one-dimensional systems. Until recently this was mostly in the many-body limit but now experiments can be performed with controllable particle numbers all the way down to the few-body regime. After a brief introduction to some of the relevant experiments, I will review recent theoretical work on one-dimensional quantum systems containing bosons, fermions, or mixtures of the two, with a particular emphasis on the case where the particles are held by an external trap.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. This Review collects the predictions coming from a family of exact solutions which allows us to obtain the many-body wavefunction of strongly correlated quantum fluids confined by a tight waveguide and subjected to any form of longitudinal confinement. It directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved. The exact solution applies to bosons, fermions, and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. The Review illustrates the idea of the solution, presents the key theoretical achievements, and the main experiments on strongly correlated one-dimensional quantum gases.
Cold atomic gases have become a paradigmatic system for exploring fundamental physics, which at the same time allows for applications in quantum technologies. The accelerating developments in the field have led to a highly advanced set of engineering techniques that, for example, can tune interactions, shape the external geometry, select among a large set of atomic species with different properties, or control the number of atoms. In particular, it is possible to operate in lower dimensions and drive atomic systems into the strongly correlated regime. In this review, we discuss recent advances in few-body cold atom systems confined in low dimensions from a theoretical viewpoint. We mainly focus on bosonic systems in one dimension and provide an introduction to the static properties before we review the state-of-the-art research into quantum dynamical processes stimulated by the presence of correlations. Besides discussing the fundamental physical phenomena arising in these systems, we also provide an overview of the calculational and numerical tools and methods that are commonly used, thus delivering a balanced and comprehensive overview of the field. We conclude by giving an outlook on possible future directions that are interesting to explore in these correlated systems.
We systematically investigate and illustrate the complete ground-state phase diagram for a one-dimensional, three-species mixture of a few repulsively interacting bosons trapped harmonically. To numerically obtain the solutions to the many-body Schrödinger equation, we employ the improved Exact Diagonalization method (Anh-Tai et al 2023 SciPost Phys. 15 048), which is capable of treating strongly-correlated few-body systems from first principles in an efficiently truncated Hilbert space. We present our comprehensive results for all possible combinations of intra- and interspecies interactions in the extreme limits that are either the ideal limit (g = 0) or close to the hard-core limit (). These results show the emergence of unique ground-state properties related to correlations, coherence and spatial localization stemming from strongly repulsive interactions.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. In particular I will present large-size calculations of the frequency- and momentum-resolved spectral function of strongly repulsive lattice bosons and a symmetry-breaking exact solution for a Bose-Bose mixture. Our approach directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved.