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Physical review. B./Physical review. B

Controllable quantum walks, gravitational constant, and Casimir energy in one-dimensional three-boson systems with on-site interactions

Hongting Yang, Pinquan Qin

Published Jul 3, 2024• 1 citation
Publisher

Abstract

This study scrutinizes one-dimensional three-boson systems with on-site interactions, delving into perturbation theory, continuous-time quantum walks, and quantum Fisher information (QFI), among other areas. Notably, two critical on-site interaction strength points emerge. At these specific locations, the QFI reaches its zenith, exhibiting a proportionality to ${t}^{4}$. This substantively amplifies the precision in measuring both the Newtonian constant of gravitation and the E\"otv\"os parameter. The improvement surpasses models devoid of on-site interactions by 6 orders of magnitude. The methodology used to measure big $G$ can be adapted to quantify the Casimir energy of a many-particle system.

DOI
10.1103/physrevb.110.014302

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This study scrutinizes one-dimensional three-boson systems with on-site interactions, delving into perturbation theory, continuous-time quantum walks, and quantum Fisher information (QFI), among other areas. Notably, two critical on-site interaction strength points emerge. At these specific locations, the QFI reaches its zenith, exhibiting a proportionality to t4{t}^{4}t4. This substantively amplifies the precision in measuring both the Newtonian constant of gravitation and the E"otv"os parameter. The improvement surpasses models devoid of on-site interactions by 6 orders of magnitude. The methodology used to measure big GGG can be adapted to quantify the Casimir energy of a many-particle system.
Recent experiments on quantum walks (QWs) demonstrated a full control over the statistics-dependent walks of single particles and two particles in one-dimensional lattices. However, little is known about the general characterization of QWs at the many-body level. Here, we rigorously study QWs, Bloch oscillations, and the quantum Fisher information for three indistinguishable bosons and fermions in one-dimensional lattices using a time-evolving block decimation algorithm and many-body perturbation theory. We show that such strongly correlated QWs not only give rise to statistics-and-interaction-dependent ballistic transports of scattering states and of two- and three-body bound states but also allow a quantum enhanced precision measurement of the gravitational force. In contrast to the QWs of the fermions, the QWs of three bosons exhibit strongly correlated Bloch oscillations, which present a surprising time scaling t^{3} of the Fisher information below a characteristic time t_{0} and saturate to the fundamental limit of t^{2} for t>t_{0}.
By confining a Bose-Einstein condensate in a vertical lattice subjected to a gravitational potential, we analyze the quantum Fisher information to determine its scaling with respect to time, system size and particle number. Our results reveal that in the localized phase, on-site interactions UUU amplify the quantum Fisher information by a factor with respect to resonance condition U=mhU=mhU=mh where UUU is factor of gradient field amplitude hhh. This precision enhancement can be employed in gravitational acceleration measurements with a finite number of particles trapped in optical lattices.
Anyons, namely particles obeying fractional quantum statistics that interpolate between bosons and fermions, possess a lot of new and exotic physical properties related to the particle exchange statistics. In this work, we explore the few-body quantum dynamics and quantum correlations of indistinguishable anyons with on-site interactions in onedimensional lattices within the scheme of three-body continuous-time quantum walks. By employing a time-evolving block decimation (TEBD) algorithm, we numerically calculate the dynamical evolution process of the local density distribution of anyons among the whole lattice. Numerical simulations shown in the main text mainly focus on a threebody initial state as ψ|(t=0)＞=â†-1â†0â†1|0＞, in which three particles are located on neighbouring sites at lattice centre. This choice of initial state features that the three particles influence one another most strongly, while we have also implemented numerical simulations on other choices of three-body initial states as are discussed in appendix. It is shown that the local density distribution of anyons is dramatically altered by fractional particle statistics with new dynamical structure showing up during the time evolution. For free anyons, an inner cone emerges as the statistical parameter increases, while the outer cone remains robust all along. When the on-site interaction joins in, the structure of the inner cone is further modified with new features. Specifically, for interaction of finite strength, an exotic dynamical asymmetry in real space, is clearly demonstrated during the time evolution of the local density distribution for particles within the fractional statistics regime. However, for boson limit and pseudofermion limit, the time evolution of the local density distribution keeps symmetric as the three-body initial state. And remarkably, the dynamical asymmetry is interactiondependent manifested as the local density distribution of anyons favors opposite side of the lattice for repulsive and attractive interaction, respectively. Moreover, when the on-site interaction is further increased to hard-core limit, the dynamical asymmetry will then be largely suppressed. We also calculate the density-density correlations for anyons before they reach the lattice boundary to reveal the interesting effect of fractional statistics on quantum correlations. It is shown that the inner cone corresponds to co-walking of anyons, while the outer cone is related to individual walking and is immune to the variation of statistical parameter. Furthermore, the exotic real-space asymmetry originated from the interplay of fractional statistics and finite interaction is also shown up in the density-density correlations.
We address the precision of the estimation procedures for the minimum length arising from gravitational theories. In particular, we provide bounds on precision and assess the use of quantum probes to enhance the estimation performance. At first, we review the concept of minimum length and how it induces a perturbative term appearing in the Hamiltonian of any quantum system, which itself is proportional to a parameter depending on the minimum length. We then systematically study the effects of this perturbation on different state preparations and for several one-dimensional systems, and evaluate the quantum Fisher information in order to find the ultimate bounds to precision. Eventually, we investigate the role of dimensionality by analyzing the use of two-dimensional square well and harmonic oscillator systems to probe the minimal length. Our results show that quantum probes are convenient resources, providing potential enhancement in precision. Additionally, our results provide a set of guidelines to design future experiments to detect the minimum length.
In the adiabatic perturbation theory, Berry curvature is related to the generalized force, and the quantum metric tensor is linked with energy fluctuation. While the former is tested with numerous numerical results and experimental realizations, the latter is less considered. Quantum Fisher information, the key to quantum precision measurement, is a four times quantum metric tensor. It is difficult to relate the quantum Fisher information with some physical observable. One interesting candidate is the square of the symmetric logarithmic derivative, which is usually tough to obtain, both theoretically and experimentally. The adiabatic perturbation theory enlightens us to measure the energy fluctuation to directly extract the quantum Fisher information. In this article we first adopt an alternative way to derive the link of energy fluctuation to the quantum Fisher information. Then we numerically testify to the direct extraction of the quantum Fisher information based on adiabatic perturbation in two-level systems and simulate the experimental realization in a nitrogen-vacancy center with experimentally practical parameters. Statistical models such as the transverse-field Ising model and Heisenberg spin chains are also discussed to compare with the analytical result and show the level crossing, respectively. Our discussion will provide a practical scheme to measure the quantum Fisher information and will also be a benefit to quantum precision measurement and the understanding of the quantum Fisher information.
Quantum Fisher Information (QFI) is a ubiquitous quantity with applications ranging from quantum metrology and resource theories to condensed matter physics. In equilibrium local quantum many-body systems, the QFI of a subsystem with respect to an extensive observable is typically proportional to the subsystem's volume. Specifically, in large subsystems at equilibrium, the QFI per unit volume squared becomes negligible. We reveal a natural mechanism that amplifies the QFI in a quantum spin chain with a zero-temperature ordered phase. At zero or sufficiently low temperatures, a transient localized perturbation enhances the QFI, causing it to scale quadratically with the subsystem's length. Furthermore, this enhancement can be controlled through more general localized kicking protocols. We also revisit the behavior of the quantum Fisher information after a global quench in the thermodynamic limit, focusing on the generation of localized -- confined within compact subsystems -- multipartite entanglement. We show that the density of localized multipartite entanglement approaches zero at late times, but there is an optimal time frame proportional to the length in which subsystems fall into macroscopic quantum states. We test our predictions against numerical data obtained using a novel technique, based on a remarkable identity between quantum Fisher information and Wigner-Yanase-Dyson skew information, that allows one to compute the quantum Fisher information with respect to the order parameter in noninteracting spin chains.