Correlation effects in low dimensional quantum systems

This thesis primarily consists of two parts. In the first one, we benchmarked the performance of the functionals in density functional theory which have the asymptotically correct behavior. We compare the performance of the two different asymptotically corrected schemes, the long-range corrected (LC) hybrid scheme and the asymptotically corrected (AC) model potential scheme, in a wide range of applications. These schemes aim at producing the correct asymptotic behavior, which is known to be important to the Rydberg excitations and the frontier orbital energies of molecules. Based on our investigation, we found that the LC hybrid scheme outperforms the AC model potential scheme almost on all the properties studied, and we conclude that the LC hybrid scheme provides a promising direction for the future development of new exchange-correlation functionals. Besides, in the process of the benchmark, I established a molecular database ``IP131 full set', which consists of 131 experimental geometries of atoms and molecules, and their reference values of 131 experimental vertical ionization energies, 131 CCSD(T) calculated vertical electron affinities, 131 CCSD(T) calculated fundamental gaps and 113 CCSD(T) calculated atomization energies. The IP131 full set can be used in the benchmarks for other Ab Initio calculation methods, or in the parameter fitting in the development of density functionals in the future. In the second part, we adopted the Schrodinger equation numerical solutions to improve the applicability of the normal mode analysis in the case of TMA-H+-H2O, for the motivation that the theoretical simulated infrared spectrum of TMA-H+-H2O given by the normal mode analysis is far from the experiment results. Normal mode analysis is a convenient and widely adopted method that people usually use it to analyse the vibrational motions of multi-body mechanical systems, both classical and quantum. However, normal mode analysis is a approximate method and its assumptions, which make the analysis procedures simple and systematically solvable, also make the normal mode analysis not applicable to some vibrational motions in some mechanical systems. Our main modification method in this work is to regard the vibrational normal modes as some low dimensional quantum systems and adopt the Schrodinger equation numerical solutions in these quantum systems to replace the results from the normal mode analysis. By doing so, we can not only get a closer result to the experimental infrared spectrum, but also have a deeper understanding of the capability and applicability of normal mode analysis. Once in the future we encounter same problems when using the normal mode analysis in different systems, we may again adopt the procedures developed here to make it better.

Optimal control theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory that is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin maximum principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin maximum principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.

We study the pairwise entanglement close to separable ground states of a class of one-dimensional quantum spin models. At $T=0$ we find that such ground states separate regions, in the space of the Hamiltonian parameters, which are characterized by qualitatively different types of entanglement, namely parallel and antiparallel entanglement; we further demonstrate that the range of the concurrence diverges while approaching separable ground states, therefore evidencing that such states, with uncorrelated fluctuations, are reached by a long range reshuffling of the entanglement. We generalize our results to the analysis of quantum phase transitions occurring in bosonic and fermionic systems. Finally, the effects of finite temperature are considered: At $T>0$ we evidence the existence of a region where no pairwise entanglement survives, so that entanglement, if present, is genuinely multipartite.

Brief Introduction of Physics Researches in Chinese Academy of Sciences

The synchronization of oscillator ensembles is pervasive throughout nonlinear science, from classical or quantum mechanics to biology, to human assemblies. Traditionally, the main focus has been the identification of threshold parameter values for the transition to synchronization as well as the nature of such transition. Here, we show that considering an oscillator lattice as a discrete growing interface provides unique insights into the dynamical process whereby the lattice reaches synchronization for long times. Working on a generalization of the celebrated Kuramoto model that allows for odd or non-odd couplings, we elucidate synchronization of oscillator lattices as an instance of generic scale invariance, whereby the system displays space-time criticality, largely irrespective of parameter values. The critical properties of the system (like scaling exponent values and the dynamic scaling Ansatz which is satisfied) happen to fall into universality classes of kinetically rough interfaces with columnar disorder, namely, those of the Edwards-Wilkinson (equivalently, the Larkin model of an elastic interface in a random medium) or the Kardar-Parisi-Zhang (KPZ) equations, for Kuramoto (odd) coupling and generic (non-odd) couplings, respectively. From the point of view of kinetic roughening, the critical properties we find turn out to be quite innovative, especially concerning the statistics of the fluctuations as characterized by their probability distribution function (PDF) and covariance. While the latter happens to be that of the Larkin model irrespective of the symmetry of the coupling, in the generic non-odd coupling case the PDF turns out to be the Tracy-Widom distribution associated with the KPZ nonlinearity. This brings the synchronization of oscillator lattices into a remarkably large class of strongly-correlated, low-dimensional (classical and quantum) systems with strong universal fluctuations.

Recent advances of experimental nanofabrication techniques draw increasing attention to the non-equilibrium behavior of low dimensional systems. Of particular interest are the strongly interacting one dimensional systems whose description in out of equilibrium situations remains a theoretical challenge. This thesis contributes to the understanding of important elementary processes in the non-equilibrium physics of one dimensional electron systems. Tunneling of an electron into a Luttinger liquid leads to partitioning of its charge and energy into counter-propagating modes. This thesis studies the partitioning of the energy which had previously remained unexplored. It turns out that energy partitioning is essentially independent of the charge partitioning and one can even reach conditions such that energy and charge propagate in opposite directions. Another important difference is their experimental accessibility. In contrast to the charge, energy partitioning provides a measurable characteristic of the tunneling process even in dc setups and we propose experimental geometries that allow for tuning and detecting energy partitioning. At higher excitation energies it becomes necessary to include curvature effects of the electron dispersion. Another part of this thesis discusses the consequences of curvature induced three-body collisions on the relaxation in quantum wires. This is particularly interesting due to the integrability of the Luttinger model which does not allow for thermalization within this paradigm of one dimensional systems. In this thesis we derive energy relaxation rates due to three-body processes beyond the Luttinger model within a well-defined perturbative approach. It turns out that the electron spin and the long range Coulomb interaction are important ingredients for a quantitative description of recent experiments which we provide in this thesis. Furthermore, we study the influence of three-body collisions on the energy relaxation in integer quantum Hall edge states. We specifically address different interaction induced edge reconstruction scenarios and find that edge reconstruction strongly enhances the energy relaxation. This is particularly pronounced when the reconstruction creates additional counter-propagating modes. Finally, we discuss another system which is crucially controlled by non-equilibrium effects. The so called nanoelectromechanical systems show a coupling between the electronic and mechanical degrees of freedom. The electron current can thus influence the mechanical motion which leads to a number of interesting applications. Previous theoretical studies on the basis of non-equilibrium Green's functions showed that these current induced forces can be expressed in terms of intuitive scattering matrix expressions. This thesis sheds considerable light on this observation by providing a much more satisfactory and concise derivation of the scattering theory of current induced forces.

Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigenstates
\nof a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external magnetic
\nfield is discussed. Being smooth at the QD boundaries and of finite depth and range, this potential can
\nonly confine a finite number of excess electrons thus forming a realistic model of a QD with smooth interface
\nbetween the QD and its embedding environment. It is argued that the method of intermediate problems,
\nwhich provides convergent improvable lower bound estimates for eigenvalues of linear half-bound Hermitian
\noperators in Hilbert space, can be fused with the classical Rayleigh-Ritz variational method and stochastic
\nvariational method thus resulting in an efficient tool for analytical and numerical studies of the energy spectrum
\nand eigenstates of the Gaussian quantum dots, confining small-to-medium number of excess electrons,
\nwith controllable or prescribed precision.

We report the results of neutron scattering and nuclear magnetic resonance (NMR) measurements on the two-dimensional quantum spin system SrCu2(B03)2. The dispersion curve of the first excited state obtained by inelastic neutron scattering is almost flat. The Cu-NMR spectrum splits into four lines by the nuclear spin-spin coupling mediated by the electron spins of Cu ions which form dimers. These results are evidences of the characteristics suggested by theoretical studies. The Cu spectrum at 27.6, where the 1/8 plateau is observed in the magnetization curve, has a large number of peaks distributed over a wide frequency range. By comparing the NMR and theoretical results, the spin structure of the 1/8 plateau phase is determined. These results are good examples of the cooperative study by neutron scattering and NMR.

Harmonic inversion of Chebyshev correlation and cross-correlation functions by the filter diagonalization method (FDM) is one of the most efficient ways to accurately compute the complex spectra of low dimensional quantum molecular systems. This explains the growing popularity of the FDM in the past several years. Some of its most attractive features are the predictable convergence properties and the lack of adjusting parameters. These issues however are often misunderstood and mystified. We discuss the questions relevant to the optimal choices for the FDM parameters, such as the window size and the number of basis functions. We also demonstrate that the cross-correlation approach (using multiple initial states) is significantly more effective than the conventional autocorrelation approach (single initial state) for the common case of a non-uniform eigenvalue distribution.

In the fractional dimensional space, the ground state properties of the charged-boson fluid are studied in a theory which goes beyond the random phase approximation by including correlations through a static local-field-factor. The structure factor and static local field-factor are obtained in the self-consistent method. Qualitative agreement with the Monte Carlo results on static screening is achieved by imposing self-consistency on the compressibility of the fluid in addition to self-consistency on the structure factor. Using the structure factors obtained in these methods, several properties of the charged boson system such as the energy spectrum of elementary excitations, the pair-correlation function, the ground state energy, the pressure, the chemical potential and the compressibility are obtained in several dimensions including both integer and fractional values. Results obtained in the later method in two and three dimensional systems are close to the Monte Carlo and hypernetted-chain methods. The Wigner crystallisation in the lower dimensional system is found at higher density of bosons. However, it vanishes in any lower dimensional system whose dimension lies below 2.

With the evolution of quantum computing, researchers now-a-days tend to incline to find solutions to NP-complete problems by using quantum algorithms in order to gain asymptotic advantage. In this paper, we solve $k$-coloring problem (NP-complete problem) using Grover's algorithm in any dimensional quantum system or any $d$-ary quantum system for the first time to the best of our knowledge, where $d \ge 2$. A newly proposed comparator-based approach helps to generalize the implementation of the $k$-coloring problem in any dimensional quantum system. Till date, $k$-coloring problem has been implemented only in binary and ternary quantum system, hence, we abide to $d=2$ or $d=3$, that is for binary and ternary quantum system for comparing our proposed work with the state-of-the-art techniques. This proposed approach makes the reduction of the qubit cost possible, compared to the state-of-the-art binary quantum systems. Further, with the help of newly proposed ternary comparator, a substantial reduction in quantum gate count for the ternary oracle circuit of the $k$-coloring problem than the previous approaches has been obtained. An end-to-end automated framework has been put forward for implementing the $k$-coloring problem for any undirected and unweighted graph on any available Near-term quantum devices or Noisy Intermediate-Scale Quantum (NISQ) devices or multi-valued quantum simulator, which helps in generalizing our approach.

Magnetic properties of interfaces and multilayers based on thin antiferromagnetic oxide films

Critical behavior of low dimensional magnetic systems

Two-dimensional magnetic materials with tunable physical parameters are emerging as potential candidates for topological phenomena as well as applications in spintronics. The famous Mermin-Wagner theorem states that spontaneous spin symmetry cannot be broken at finite temperature in low dimensional magnetic systems which forbids the possibility of a transition to a long-range ordered state in a two-dimensional magnetic system at finite temperature. Though, there are some exceptions to Mermin-Wagner theorem in particular low dimensional magnetic systems with topologically ordered phase transitions. Here, we present an in-depth temperature dependent analysis for the bulk single crystals of two-dimensional (Mn1−xNix)2P2S6 with x = 1, 0.7, 0.3, 0 using the Raman spectroscopy supported by first-principles calculations of the phonon frequencies. We observed multiple phase transitions with tunability as a function of doping associated with the short and long-range spin-spin correlations. First transition at ~ 150 K to ~ 170 K for x = 0 to x = 0.7, and second one from ~ 60 K to ~ 153 K. Quite interestingly, a third transition is observed at low temperature (much below their respective TN) ~ 24 K to ~ 60 K and is attributed to the potential topological phase transition. These transitions are marked by the distinct changes observed in the temperature evolution of the phonon self-energy parameters, modes intensity and dynamic Raman susceptibility.

We have discussed the time evolution of squeezed states coupled to an environment. By way of example we showed that the influence of the initial preparation may decay on a time scale much longer than the time scale for the classical relaxation to equilibrium. We have proposed two preparations (2 and 4) with the same reduced density matrix at t=0 where the initial correlations between system and bath influence the average relaxation for extremely long times at low temperatures. In another example (1 and 3) the correlation effects do not show up in the average relaxation of 〈p〉 and 〈q〉 but they are apparent in the time evolution of the second moments. These effects cannot be obtained within a weak coupling theory and become more pronounced as temperature is decreased. The commonly used approximation of factorizing initial states is then not necessarily well justified. It is expected that this behavior is of general relevance for dissipative quantum systems and not restricted to the special case considered here.