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Physical review. E

Information theoretic measures for interacting bosons in optical lattice

Rhombik Roy, Barnali Chakrabarti, N. D. Chavda, M. L. Lekala

Published Feb 13, 2023• 15 citations
Publisher

Abstract

This work reports the different information theoretic measures, i.e., Shannon information entropy, order, disorder, complexity, and their dynamical measure for the interacting bosons in an optical lattice with both commensurate and incommensurate filling factor. We solve the many-body Schrödinger equation from first principles by multiconfigurational time-dependent Hartree method which calculates all the measures with high level of accuracy. We find for both relaxed state as well as quenched state the López-Ruiz-Mancini-Calbet (LMC) measure of complexity is the most efficient depictor of superfluid (SF) to Mott-insulator transition. In the quench dynamics, the distinct structure of LMC complexity can be used as a "figure of merit" to obtain the timescale of SF to Mott state entry, Mott holding time, and the Mott state to SF state entry in the successive cycles. We also find that fluctuations in the dynamics of LMC complexity measure for incommensurate filling clearly establish that superfluid to Mott-insulator transition is incomplete. We overall conclude that distinct structure in the complexity makes it more sensitive than the standard use of Shannon information entropy.

DOI
10.1103/physreve.107.024119

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Journal of Physics B Atomic Molecular and Optical Physics•Jun 13, 2019•17 citations
This work reports the different information theoretic measures, i.e., Shannon information entropy, order, disorder, complexity, and their dynamical measure for the interacting bosons in an optical lattice with both commensurate and incommensurate filling factor. We solve the many-body Schrödinger equation from first principles by multiconfigurational time-dependent Hartree method which calculates all the measures with high level of accuracy. We find for both relaxed state as well as quenched state the López-Ruiz-Mancini-Calbet (LMC) measure of complexity is the most efficient depictor of superfluid (SF) to Mott-insulator transition. In the quench dynamics, the distinct structure of LMC complexity can be used as a "figure of merit" to obtain the timescale of SF to Mott state entry, Mott holding time, and the Mott state to SF state entry in the successive cycles. We also find that fluctuations in the dynamics of LMC complexity measure for incommensurate filling clearly establish that superfluid to Mott-insulator transition is incomplete. We overall conclude that distinct structure in the complexity makes it more sensitive than the standard use of Shannon information entropy.
The present work reports on the dynamical measures of order, disorder and complexity for the interacting bosons in optical lattice. We report results both for the relaxed state as well as quench dynamics. Our key observations are: (1) Lattice depth can be taken as order-disorder parameter. (2) The superfluid to Mott insulator transition can be treated as `order-disorder' transition. Our main motivation is to find how the system organize by itself during quench and how it optimizes the complexity. We find dynamical measures of order and disorder are more sensitive tool than entropy measures. We specifically calculate the time scale of entry and exit of different phases during time evolution. Initially the system exhibits collapse revival trend, however gradually looses its ability to turn back to superfluid phase and finally Settle to Mott insulator phase.
We investigate the dynamics of a few bosons in an optical lattice induced by a quantum quench of a parameter of the many-body Hamiltonian. The evolution of the many-body wave function is obtained by solving the time-dependent many-body Schrödinger equation numerically, using the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). We report the time evolution of three key quantities, namely, the occupations of the natural orbitals, that is, the eigenvalues of the one-body reduced density matrix, the many-body Shannon information entropy, and the quantum fidelity for a wide range of interactions. Our key motivation is to characterize relaxation processes where various observables of an isolated and interacting quantum many-body system dynamically converge to equilibrium values via the quantum fidelity and via the production of many-body entropy. The interaction, as a parameter, can induce a phase transition in the ground state of the system from a superfluid (SF) state to a Mott-insulator (MI) state. We show that, for a quench to a weak interaction, the fidelity remains close to unity and the entropy exhibits oscillations. Whereas for a quench to strong interactions (SF to MI transition), the relaxation process is characterized by the first collapse of the quantum fidelity and entropy saturation to an equilibrium value. The dip and the non-analytic nature of quantum fidelity is a hallmark of dynamical quantum phase transitions. We quantify the characteristic time at which the quantum fidelity collapses and the entropy saturates.
Already a few bosons with contact interparticle interactions in small optical lattices feature a variety of quantum phases: superfluid, Mott-insulator, and fermionized Tonks gases can be probed in such systems. To detect these phases---pivotal for both experiment and theory---as well as their many-body properties we analyze several distinct measures for the one-body and many-body Shannon information entropies. We exemplify the connection of these entropies with spatial correlations in the many-body state by contrasting them to the Glauber normalized correlation functions. To obtain the ground state for lattices with commensurate filling (i.e., an integer number of particles per site) for the full range of repulsive interparticle interactions we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB) in order to solve the many-boson Schr\"odinger equation. We demonstrate that all emergent phases---the superfluid, the Mott insulator, and the fermionized gas can be characterized equivalently by many-body entropy measures and by Glauber's normalized correlation functions. In sharp contrast, single-particle entropy cannot capture these phases.
We investigate the dynamics of one-dimensional interacting bosons in an optical lattice after a sudden quench in the weakly interacting (Bose-Hubbard) and strongly interacting (sine-Gordon) regimes. While in a higher dimension, the Mott-superfluid phase transition is observed for weakly interacting bosons in deep lattices, in one dimension an instability is generated also for shallow lattices with a commensurate periodic potential pinning the atoms to the Mott state through a transition described by the sine-Gordon model. The present work aims at identifying the quench dynamics in both the Bose-Hubbard and sine-Gordon interaction regimes. We numerically exactly solve the time-dependent Schr\"odinger equation for a small number of atoms and obtain dynamical measures of several key quantities. We investigate the correlation dynamics of first and second order; both exhibit rich many-body features in the dynamics. We conclude that in both cases, dynamics exhibits collapse-revival phenomena, though with different timescales. We argue that the dynamical fragmentation is a convenient quantity to distinguish the dynamics especially near the pinning zone. To understand the relaxation process we measure the many-body information entropy. Bose-Hubbard dynamics clearly establishes the possible relaxation to the maximum entropy state. In contrast, the sine-Gordon dynamics is so fast that it does not exhibit any signature of relaxation in the present timescale of computation.
We explore the consequences of disorder on phase coherence in the Mott insulator phases in an optical lattice. Few bosons with contact interaction in small optical lattice can feature varieties of insulating phases---weakly interacting Mott in deep lattice, maximally fragmented and strongly interacting Mott in intermediate lattice, weak Mott with double filling and intrawell coherence, fermionized Mott with strong intrawell coherence. Utilization of the multiconfigurational time-dependent Hartree method for bosons to solve the many-boson Schr\"odinger equation, facilitates to understand the microscopic effect of disorder on the different kinds of Mott phases in the primary lattice. The many-body properties are analyzed by distinct measures of the reduced one-body density in real and momentum space, fragmentation, order parameter, variance of spatial single-shot measurements, compressibility, and the Glauber normalized correlation functions. We find very complex competition of localization due to disorder and Mott correlation. We observe distinct response of four different Mott phases in the disordered lattice. When the weakly interacting Mott exhibits the Bose glass phase, the strongly correlated and fully fragmented Mott exhibits leakage, melting and central localization. In contrast, weak Mott with double filling which is a spatially separated dimer in each site, exhibits only some dissociation of intrawell coherence and assist in the development of interwell coherence in the presence of strong disorder. For the fermionized Mott, when the density in each well is fragmented, strong disorder interferes with the intrawell correlation and the characteristic dip in each site starts to disappear leading to simple Mott localization with a pair of bosons.
Abstract The relaxation process of a few strongly interacting bosons in a triple well optical lattice is studied from the first principle using the multiconfigurational time-dependent Hartree method for bosons. We report the contrasting response of the system under two independent quench processes: an interaction quench and a lattice depth quench. We analyze the evolution of the reduced one-body density matrix, two-body density and the Shannon information entropy for a wide range of lattice depth and interaction strength parameters. For the strong interaction quench, we observe a very fast relaxation to the steady state. In contrast, for the lattice depth quench, we observe collapse–revival dynamics in all the key measures. We also provide the best fitting formulas for relaxation and revival time which follow power law decay.