arXiv subjects

Biao Wu

Publications and source records attributed to Biao Wu.

At least 109 records · Page 6Linked to original sources

arXiv2018

Quantum ergodicity and mixing and their classical limits with quantum kicked rotor

We study the ergodicity and mixing of quantum kicked rotor (QKR) with two distinct approaches. In one approach, we use the definitions of quantum ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)], which involve only eigen-energies (Floquet quasi-energies for QKR). In the other approach, we study ergodicity and mixing with quantum Poincarè section, which is plotted with a method that maps a wave function unitarily onto quantum phase space composed of Planck cells. Classical Poincarè section can be recovered with the effective Planck constant gradually diminishing. We demonstrate that the two approaches can capture the quantum and classical characteristics of ergodicity and mixing of QKR, and give consistent results with classical model at semiclassical limit. Therefore, we establish a correspondence between quantum ergodicity (mixing) and classical ergodicity (mixing).

quant-ph↗

arXiv2018

Lorentz quantum mechanics

We present a theoretical framework called Lorentz quantum mechanics, where the dynamics of a system is a complex Lorentz transformation in complex Minkowski space. In contrast, in usual quantum mechanics, the dynamics is the unitary transformation in Hilbert space. In our Lorentz quantum mechanics, there exist three types of states, space-like, light-like, and time-like. Fundamental aspects are explored in parallel to the usual quantum mechanics, such as matrix form of a Lorentz transformation, construction of Pauli-like matrices for spinors. We also investigate the adiabatic evolution in this mechanics, as well as the associated Berry curvature and Chern number. Three typical physical systems, where this Lorentz quantum dynamics can arise, are presented. They are one dimensional fermion gas, Bose-Einstein condensate (or superfluid), and one dimensional antiferromagnet.

quant-ph↗

arXiv2017

Quantum phase space with a basis of Wannier functions

A quantum phase space with Wannier basis is constructed: (i) classical phase space is divided into Planck cells; (ii) a complete set of Wannier functions are constructed with the combination of Kohn's method and Löwdin method such that each Wannier function is localized at a Planck cell. With these Wannier functions one can map a wave function unitarily onto phase space. Various examples are used to illustrate our method and compare it to Wigner function. The advantage of our method is that it can smooth out the oscillations in wave functions without losing any information and is potentially a better tool in studying quantum-classical correspondence. In addition, we point out that our method can be used for time-frequency analysis of signals.

quant-ph↗

arXiv2017

Majorana modes in solid state systems and its dynamics

We review the properties of Majorana fermions in particle physics and point out that Majorana modes in solid state systems are significantly different. The key reason is the concept of anti-particle in solid state systems is different from its counterpart in particle physics. We define Majorana modes as the eigenstates of Majorana operators and find that they can exist both at edges and in the bulk. According to our definition, only one single Majorana mode can exist in a system no matter at edges or in the bulk. Kitaev's spinless $p$-wave superconductor is used to illustrate our results and the dynamical behavior of the Majorana modes.

cond-mat.mes-hall↗

arXiv2017

Roles of energy eigenstates and eigenvalues in equilibration of isolated quantum systems

We show that eigen-energies and energy eigenstates play different roles in the equilibration process of an isolated quantum system. Their roles are revealed numerically by exchanging the eigen-energies between an integrable model and a non-integrable model. We ?find that the structure of eigenenergies of a non-integrable model characterized by non-degeneracy ensures that quantum revival occurs rarely whereas the energy eigenstates of a non-integrable model suppress the fluctuations for the equilibrated quantum state. Our study is aided with a quantum entropy that describes how randomly a wave function is distributed in quantum phase space. We also demonstrate with this quantum entropy the validity of Berry's conjecture for energy eigenstates. This implies that the energy eigenstates of a non-integrable model appear indeed "random".

quant-ph↗

arXiv2017

Matter wave interference of dilute Bose gases in the critical regime

Ultra-cold atomic gases provide new chance to study the universal critical behavior of phase transition. We study theoretically the matter wave interference for ultra-cold Bose gases in the critical regime. We demonstrate that the interference in the momentum distribution can be used to extract the correlation in the Bose gas. A simple relation between the interference visibility and the correlation length is found and used to interpret the pioneering experiment about the critical behavior of dilute Bose gases [Science {\bf 315}, 1556 (2007)]. Our theory paves the way to experimentally study various types of ultra-cold atomic gases with the means of matter wave interference.

cond-mat.quant-gas↗

arXiv2017

On a conjecture of Hefetz and Keevash on Lagrangians of intersecting hypergraphs and Turán numbers

Let $S^r(n)$ be the $r$-graph on $n$ vertices with parts $A$ and $B$, where the edges consist of all $r$-tuples with $1$ vertex in $A$ and $r-1$ vertices in $B$, and the sizes of $A$ and $B$ are chosen to maximise the number of edges. Let $M_t^r$ be the $r$-graph with $t$ pairwise disjoint edges. Given an $r$-graph $F$ and a positive integer $p\geq |V(F)|$, we define the {\em extension} of $F$, denoted by $H_{p}^{F}$ as follows: Label the vertices of $F$ as $v_1,\dots,v_{|V(F)|}$. Add new vertices $v_{|V(F)|+1},\dots,v_{p}$. For each pair of vertices $v_i,v_j, 1\le i<j \le p$ not contained in an edge of $F$, we add a set $B_{ij}$ of $r-2$ new vertices and the edge $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$ 's are pairwise disjoint over all such pairs $\{i,j\}$. Hefetz and Keevash conjectured that the Turán number of the extension of $M_2^r$ is ${1 \over r}n\cdot{{r-1\over r}n \choose r-1}$ for $r \ge 4$ and sufficiently large $n$. Moreover, if $n$ is sufficiently large and $G$ is an $H_{2r}^{M_2^r}$-free $r$-graph with $n$ vertices and ${1 \over r}n\cdot{{r-1\over r}n \choose r-1}$ edges, then $G$ is isomorphic to $S^r(n)$. In this paper, we confirm the above conjecture for $r=4$.

math.CO↗

arXiv2017

Dense $3$-uniform hypergraphs containing a large clique

An $r$-uniform graph $G$ is dense if and only if every proper subgraph $G'$ of $G$ satisfies $λ(G') < λ(G)$, where $λ(G)$ is the Lagrangian of a hypergraph $G$. In 1980's, Sidorenko showed that $π(F)$, the Turán density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all dense $F$-hom-free $r$-uniform hypergraphs. This connection has been applied in estimating Turán density of hypergraphs. When $r=2$, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when $r\ge 3$, it becomes much harder to estimate the Lagrangians of $r$-uniform hypergraphs and to characterize the structure of all dense $r$-uniform graphs. The main goal of this note is to give some sufficient conditions for $3$-uniform graphs with given substructures to be dense. For example, if $G$ is a $3$-graph with vertex set $[t]$ and $m$ edges containing $[t-1]^{(3)}$, then $G$ is dense if and only if $m \ge {t-1 \choose 3}+{t-2 \choose 2}+1$. We also give sufficient condition condition on the number of edges for a $3$-uniform hypergraph containing a large clique minus $1$ or $2$ edges to be dense.

math.CO↗

arXiv2016

Exact results for polaron and molecule in one-dimensional spin-1/2 Fermi gas

Using exact Bethe ansatz (BA) solutions, we show that a spin-down fermion immersed into a fully polarized spin-up Fermi sea with a weak attraction is dressed by the surrounding spin-up fermions to form the one-dimensional analog of a polaron. As the attraction becomes strong, the spin-down fermion binds with one spin-up fermion to form a tightly bound molecule. Throughout the whole interaction regime, a crossover from the polaron to a molecule state is fully demonstrated through exact results of the excitation spectrum, the effective mass, binding energy and kinetic energy. Furthermore, a clear distinction between the polaron and molecule is conceived by the probability distribution, single particle reduced density matrix and density-density correlations, which are calculated directly from the Bethe ansatz wave function. Such a polaron-molecule crossover presents a universal nature of an impurity immersed into a fermionic medium with an attraction in one dimension.

cond-mat.quant-gas↗

arXiv2016

Construction of Maximally Localized Wannier Functions

We present a general method of constructing maximally localized Wannier functions. It consists of three steps: (1) picking a localized trial wave function, (2) performing a full band projection, and (3) orthonormalizing with the Lowdin method. Our method is capable of producing maximally localized Wannier functions without further minimization, and it can be applied straightforwardly to random potentials without using supercells. The effectiveness of our method is demonstrated for both simple bands and composite bands.

cond-mat.other↗

arXiv2016

Turan numbers of extensions of some sparse hypergraphs via Lagrangians

Given a positive integer $n$ and an $r$-uniform hypergraph (or $r$-graph for short) $F$, the Turan number $ex(n,F)$ of $F$ is the maximum number of edges in an $r$-graph on $n$ vertices that does not contain $F$ as a subgraph. The extension $H^F $ of $F$ is obtained as follows: For each pair of vertices $v_i,v_j$ in $F$ not contained in an edge of $F$, we add a set $B_{ij}$ of $r-2$ new vertices and the edge $\{v_i,v_j\} \cup B_{ij}$, where the $B_{ij}$ 's are pairwise disjoint over all such pairs $\{i,j\}$. Let $K^r_p$ denote the complete $r$-graph on $p$ vertices. For all sufficiently large $n$, we determine the Turan numbers of the extensions of a $3$-uniform $t$-matching, a $3$-uniform linear star of size $t$, and a $4$-uniform linear star of size $t$, respectively. We also show that the unique extremal hypergraphs are balanced blowups of $K^3_{3t-1}, K^3_{2t}$, and $K^4_{3t}$, respectively. Our results generalize the recent result of Hefetz and Keevash [7].

math.CO↗

arXiv2016

Ergodicity and Mixing in Quantum Dynamics

After a brief historical review of ergodicity and mixing in dynamics, particularly in quantum dynamics, we introduce definitions of quantum ergodicity and mixing using the structure of the system's energy levels and spacings. Our definitions are consistent with usual understanding of ergodicity and mixing. Two parameters concerning the degeneracy in energy levels and spacings are introduced. They are computed for right triangular billiards and the results indicate a very close relation between quantum ergodicity (mixing) and quantum chaos. At the end, we argue that, besides ergodicity and mixing, there may exist a third class of quantum dynamics which is characterized by a maximized entropy.

cond-mat.stat-mech↗

arXiv2016

Observation of Quantum Equilibration in Dilute Bose Gases

We investigate experimentally the dynamical relaxation of a non-integrable quantum many-body system to its equilibrium state. A Bose-Einstein condensate is loaded into the first excited band of an optical lattice and let to evolve up to a few hundreds of milliseconds. Signs of quantum equilibration are observed. There is a period of time, roughly 40 ms long, during which both the aspect ratio of the cloud and its momentum distribution remain constant. In particular, the momentum distribution has a flat top and is not a Gaussian thermal distribution. After this period, the cloud becomes classical as its momentum distribution becomes Gaussian.

cond-mat.quant-gas↗

arXiv2016

Observation of quantum dynamical oscillations of ultracold atoms in the F and D bands of an optical lattice

We report the observation of quantum dynamical oscillations of ultracold atomic gases in the F and D bands of a single-well optical lattice. We are able to control the Bragg reflections at the Brillouin zone edge up to the third order. As a result, we can switch the quantum dynamics from oscillations across both the F and D bands to oscillations only within the F-band. Our capability to observe these remarkable oscillations comes from the innovative non-adiabatic technique which allows us to load ultracold atoms efficiently to the G-band of an optical lattice.

cond-mat.quant-gas↗

arXiv2016

Quantum-Classical Correspondence Principle for Work Distributions in a Chaotic System

We numerically study the work distributions in a chaotic system and examine the relationship between quantum work and classical work. Our numerical results suggest that there exists a correspondence principle between quantum and classical work distributions in a chaotic system. This correspondence was proved for one-dimensional (1D) integrable systems in a recent work [Jarzynski, Quan, and Rahav, Phys. Rev. X 5, 031038 (2015)]. Our investigation further justifies the definition of quantum work via two point energy measurements.

cond-mat.stat-mech↗

arXiv2016

Ehrenfest breakdown of the mean-field dynamics of Bose gases

The mean-field dynamics of a Bose gas is shown to break down at time $τ_h = (c_1/γ) \ln N$ where $γ$ is the Lyapunov exponent of the mean-field theory, $N$ is the number of bosons, and $c_1$ is a system-dependent constant. The breakdown time $τ_h$ is essentially the Ehrenfest time that characterizes the breakdown of the correspondence between classical and quantum dynamics. This breakdown can be well described by the quantum fidelity defined for reduced density matrices. Our results are obtained with the formalism in particle-number phase space and are illustrated with a triple-well model. The logarithmic quantum-classical correspondence time may be verified experimentally with Bose-Einstein condensates.

cond-mat.quant-gas↗

arXiv2015

Construction of Wannier Functions in Disordered Systems

We propose a general method of constructing Wannier functions in disordered systems directly out of energy eigenstates. This method consists of two successive operations: (i) a phase transformation setting the proper localization center; (ii) the mixing of adjacent states in energy to sufficiently minimize the spread of the Wannier functions. The latter operation can be well approximated by a band matrix, further facilitating the calculation. Detailed implementation of our method is illustrated with one dimensional systems; the generalization to higher dimensions is straightforward.

cond-mat.dis-nn↗

arXiv2015

Optimizing Quantum Adiabatic Algorithm

In quantum adiabatic algorithm, as the adiabatic parameter $s(t)$ changes slowly from zero to one with finite rate, a transition to excited states inevitably occurs and this induces an intrinsic computational error. We show that this computational error depends not only on the total computation time $T$ but also on the time derivatives of the adiabatic parameter $s(t)$ at the beginning and the end of evolution. Previous work (Phys. Rev. A \textbf{82}, 052305) also suggested this result. With six typical paths, we systematically demonstrate how to optimally design an adiabatic path to reduce the computational errors. Our method has a clear physical picture and also explains the pattern of computational error. In this paper we focus on quantum adiabatic search algorithm although our results are general.

quant-ph↗