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Korelacije u jako-međudjelujućim višečestičnim jednodimenzionalnim sustavima

Karlo Lelas

Published Jul 2, 2012• 0 citations
OpenAlex

Abstract

In quantum many-body systems interactions between particles create correlations and particles can not be described independently. As a consequence the wave function of the system is a complicated object holding a large amount of information and in most cases it is not possible to obtain it exactly or analytically. Studying correlations in the context of strongly interacting onedimensional (1D) systems is interesting because in this regime we can exactly determine many particle wave function through Fermi-Bose (FB) mapping. FB mapping is in the essence of a Tonks Girardeau (TG) gas proposed in 1960, a theoretical model of 1D Bose gas with strongly repulsive contact interactions between particles. TG gas is experimentally realized in 2004 with rubidium atoms conned in tight atomic wave-guides and it is still a topic of experimental studies. Our goal is the study of correlations in TG gas through the formalism of the single particle density matrix. Single particle density matrix, among other important observable's, contains information about the momentum distribution which can be measured in experiments. We explored correlations in dark soliton-like excitations of the TG gas and found that these correlations give rise to a specific form of the momentum distribution which can be used to experimentally detect these states. In addition to the TG gas, FB mapping was used to construct exact wave functions of 1D Bose-Fermi mixtures (fermions are spin polarized). We derived the formula for the single particle density matrix of the mixture and used it to study correlations and momentum distribution of the mixture in the ground state of double well potential. Due to integrability, these systems are convenient to study stability to perturbations of many-body quantum states in different interaction regimes. We explored stability of non-equilibrium dynamics of 1D interacting Bose gases with respect to small spatial noise potential using the concept of Loschmidt echo (quantum fidelity). We used this concept to explore the pinning quantum phase transition. We found that the ground state fidelity and the Loschmidt echo can be used for diagnostics of the pinning quantum phase transition in experimentally relevant conditions.

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Pinning quantum phase transition in a Tonks-Girardeau gas: Diagnostics by ground-state fidelity and the Loschmidt echo

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We study the pinning quantum phase transition in a Tonks-Girardeau gas, both in equilibrium and out of equilibrium, using the ground-state fidelity and the Loschmidt echo as diagnostic tools. The ground-state fidelity will have a dramatic decrease when the atomic density approaches the commensurate density of one particle per lattice well. This decrease is a signature of the pinning transition from the Tonks to the Mott insulating phase. We study the applicability of the fidelity for diagnosing the pinning transition in experimentally realistic scenarios. We find that fidelity can predict the particle number(s) at which the pinning occurs. In addition, we explore the out-of-equilibrium dynamics of the gas following a sudden quench with a lattice potential. We find that all properties of the ground-state fidelity are reflected in the Loschmidt echo dynamics, i.e., in the nonequilibrium dynamics of the Tonks-Girardeau gas initiated by a sudden quench of the lattice potential.

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In quantum many-body systems interactions between particles create correlations and particles can not be described independently. As a consequence the wave function of the system is a complicated object holding a large amount of information and in most cases it is not possible to obtain it exactly or analytically. Studying correlations in the context of strongly interacting onedimensional (1D) systems is interesting because in this regime we can exactly determine many particle wave function through Fermi-Bose (FB) mapping. FB mapping is in the essence of a Tonks Girardeau (TG) gas proposed in 1960, a theoretical model of 1D Bose gas with strongly repulsive contact interactions between particles. TG gas is experimentally realized in 2004 with rubidium atoms conned in tight atomic wave-guides and it is still a topic of experimental studies. Our goal is the study of correlations in TG gas through the formalism of the single particle density matrix. Single particle density matrix, among other important observable's, contains information about the momentum distribution which can be measured in experiments. We explored correlations in dark soliton-like excitations of the TG gas and found that these correlations give rise to a specific form of the momentum distribution which can be used to experimentally detect these states. In addition to the TG gas, FB mapping was used to construct exact wave functions of 1D Bose-Fermi mixtures (fermions are spin polarized). We derived the formula for the single particle density matrix of the mixture and used it to study correlations and momentum distribution of the mixture in the ground state of double well potential. Due to integrability, these systems are convenient to study stability to perturbations of many-body quantum states in different interaction regimes. We explored stability of non-equilibrium dynamics of 1D interacting Bose gases with respect to small spatial noise potential using the concept of Loschmidt echo (quantum fidelity). We used this concept to explore the pinning quantum phase transition. We found that the ground state fidelity and the Loschmidt echo can be used for diagnostics of the pinning quantum phase transition in experimentally relevant conditions.
In one-dimensional (1D) quantum gases, the momentum distribution (MD) of the atoms is a standard experimental observable, routinely measured in various experimental setups. The MD is sensitive to correlations, and it is notoriously hard to compute theoretically for large numbers of atoms NNN, which often prevents direct comparison with experimental data. Here we report significant progress on this problem for the 1D Tonks-Girardeau (TG) gas in the asymptotic limit of large NNN, at zero temperature and driven out of equilibrium by a quench of the confining potential. We find an exact analytical formula for the one-particle density matrix $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathrm{\ensuremath{\Psi}}}({x}^{\ensuremath{'}})\ensuremath{\rangle}$ of the out-of-equilibrium TG gas in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, valid on distances $|x\ensuremath{-}{x}^{\ensuremath{'}}|$ much larger than the interparticle distance. By comparing with time-dependent Bose-Fermi mapping numerics, we demonstrate that our analytical formula can be used to compute the out-of-equilibrium MD with great accuracy for a wide range of momenta (except in the tails of the distribution at very large momenta). For a quench from a double-well potential to a single harmonic well, which mimics a ``quantum Newton cradle'' setup, our method predicts the periodic formation of peculiar, multiply peaked, momentum distributions.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
A Fermi-Bose mapping method is used to determine the exact ground states of several models of mixtures of strongly interacting ultracold gases in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). We detail the case of a Bose-Fermi mixture with TG boson-boson (BB) and boson-fermion (BF) interactions. Exact results are given for density profiles in a harmonic trap, single-particle density matrices, momentum distributions, and density-density correlations. Since the ground state is highly degenerate, we analyze the splitting of the ground manifold for large but finite BB and BF repulsions.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems. We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the 1/k41/k^41/k4 momentum tail. An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling. In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.
We study the pinning quantum phase transition in a Tonks-Girardeau gas, both in equilibrium and out of equilibrium, using the ground-state fidelity and the Loschmidt echo as diagnostic tools. The ground-state fidelity will have a dramatic decrease when the atomic density approaches the commensurate density of one particle per lattice well. This decrease is a signature of the pinning transition from the Tonks to the Mott insulating phase. We study the applicability of the fidelity for diagnosing the pinning transition in experimentally realistic scenarios. We find that fidelity can predict the particle number(s) at which the pinning occurs. In addition, we explore the out-of-equilibrium dynamics of the gas following a sudden quench with a lattice potential. We find that all properties of the ground-state fidelity are reflected in the Loschmidt echo dynamics, i.e., in the nonequilibrium dynamics of the Tonks-Girardeau gas initiated by a sudden quench of the lattice potential.