Quantum coherence dynamics of spin pairs in many-spin cluster

We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of Görling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant λ naturally arises. We then find the reduced density matrix ρ_{λ}(r,r^{'},t), which, by construction, has the λ independent diagonal elements ρ_{λ}(r,r,t)=n(r,t), n(r,t) being the particle density. The off-diagonal elements of ρ_{λ}(r,r^{'},t) contribute importantly to the processes unaccessible via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in λ, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi's golden rule formula.

DISSIPATIVE SCATTERING AND TRANSPORT IN MESOSCOPIC SYSTEMS

The energy of any atom or molecule with an even number N of electrons is shown to be an exact functional of a single geminal where the functionals for both the kinetic energy and the external potential are explicitly known. We derive the foundations for geminal functional theory (GFT) through a generalized constrained search and the use of two theorems which demonstrate that all one-particle properties of atoms and molecules with even N may be parametrized by a single geminal [A. J. Coleman, Int. J. Quantum Chem. 63, 23 (1997); D. W. Smith, Phys. Rev. 147, 896 (1966)]. By generalizing constrained search to optimize the universal functionals with respect to the 2-RDM (two particle reduced density matrix) rather than the wave function, we closely connect the one-density, the 1-RDM (one-particle reduced density matrix), and the geminal functional theories with 2-RDM minimization of the energy. Constrained search with the 2-RDM emphasizes that all energy functional methods must implicitly account for the N-representability of the 2-RDM within their universal functionals. An approximate universal functional for GFT, equivalent to a variational ansatz using the antisymmetrized geminal power wave function, yields energies that are significantly better than those from Hartree–Fock and yet rigorously above the exact energy.

With the purpose to reveal consistency between multiple quantum (MQ) coherences and entanglement, we investigate numerically the dynamics of these phenomena in one-dimensional linear chains and ring of nuclear spins 1/2 coupled by dipole---dipole interactions. As opposed to the calculation of the MQ coherence intensity based on the density matrix describing the spin system as a whole, we consider the "differentiated" intensity related only to the chosen spin pair based on the reduced density matrix. It is shown that the entanglement and the MQ coherence have similar dynamics only for nearest neighbors while we did not obtained any consistency for remote spins.

We discuss the general features of the time-evolving reduced density matrix (RDM) of multistate systems coupled to dissipative environments and show that many important aspects of the dynamics are visualized effectively and transparently through coherence maps, defined as snapshots of the real and imaginary components of the RDM on the square grid of system sites. In particular, the spread, signs, and shapes of the coherence maps collectively characterize the state of the system and the nature of the dynamics, as well as the equilibrium state. The topology of the system is readily reflected in its coherence map. Rows and columns show the composition of quantum superpositions, and their filling indicates the extent of the surviving coherence. Linear combinations of imaginary RDM elements specify instantaneous population derivatives. The main diagonal comprises the incoherent component of the dynamics, while the upper/lower triangular areas give rise to coherent contributions that increase the purity of the RDM. In open systems, the coherence map evolves to a band surrounding the principal diagonal whose width decreases with increasing temperature and dissipation strength. We illustrate these behaviors with examples of 10-site model molecular aggregates with Frenkel exciton couplings, where the electronic states of each monomer are coupled to harmonic vibrational baths.

FeynDyn: A MATLAB program for fast numerical Feynman integral calculations for open quantum system dynamics on GPUs

Quantum correlations and coherence generated between two free spinless particles in the lattice, interacting with a common quantum phonon bath, are studied. The reduced density matrix is solved using the Markov approach. We show that the bath induces correlations between the particles. The coherence induced by the bath is studied, calculating off-diagonal elements of the density matrix, spatiotemporal dispersion, purity and quantum mutual information. We find a characteristic time-scale pointing out when this coherence is maximum. In addition, a Wigner-like distribution in the phase-space (lattice) is introduced as an indirect indicator of the quantumness of total correlations and coherence induced by the thermal bath. The negative volume of the Wigner function also shows a behavior which is in agreement with the time-scale that we have found. A Gaussian distribution for the profile of particles is not obtained, and interference patterns are observed as the result of bath-induced coherence. As the temperature of the bath vanishes, the ballistic behavior of the tight-binding model is recovered. The geometric quantum discord is calculated, to characterize the nature of the correlations.

This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.

Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.

We study the long-time average of the reduced density matrix (RDM) of a two-level system as the central system, which is locally coupled to a many-body quantum chaotic system as the environment, under an overall Schrödinger evolution. A phenomenological relation among elements of the RDM is proposed for a dissipative interaction in the strong coupling regime and is tested numerically with the environment as a defect Ising chain, as well as a mixed-field Ising chain.

In this work, the use of the Boltzmann collision operator for dissipative quantum transport is analyzed. Its mathematical role on the description of the time-evolution of the density matrix during a collision can be understood as processes of adding and subtracting states. We show that unphysical results can be present in quantum simulations when the old states (that built the density matrix associated to an open system before the collision) are different from the additional states generated by the Boltzmann collision operator. As a consequence of the Fermi Golden rule, the new generated sates are usually eigenstates of the momentum or kinetic energy. Then, the different time-evolutions of old and new states involved in a collision process can originate negative values of the charge density, even longer after the collision. This unphysical feature disappears when the Boltzmann collision operator generates states that were already present in the density matrix of the quantum system before the collision. Following these ideas, in this paper, we introduce an algorithm that models phonon-electron interactions through the Boltzmann collision operator without negative values of the charge density. The model only requires the exact knowledge, at all times, of the states that build the density matrix of the open system.

Equilibration implies asymptotic stationarity for open quantum systems

We have studied the effect of plasma energy on the absorption coefficient of metallic photonic crystals doped with an ensemble of three-level quantum dots, which are interacting with each other via dipole-dipole interaction. The quantum dots are also interacting with coupled plasma-photon modes present in the system. A probe laser field is applied in order to study the absorption coefficient. We also consider the effect of quantum interference in our simulations, whereby two absorbed photons interfere with one another. Here the density matrix method has been used to calculate the steady-state and transient behavior of the absorption coefficient for the system. Two different field configurations are considered in our numerical simulations. In the first configuration, a probe field couples the ground state and two closely excited states. Absorption occurs due to transitions from the ground state to the two excited states. It is found that the position of the transparent peak moves when the plasma energy is changed. In other words, changing the plasma energy causes the system to switch between a transparent and an absorbing state. The strong coupling between plasmons and the quantum dots is responsible for this phenomenon. In the second configuration, the probe field couples with only one excited state, while a pump field couples to the other excited state. The transition between excited states is dipole forbidden. We observed that the peak in the absorption profile splits into two and also that the system exhibits gain with inversion due to the change in the plasma frequency, which is caused by quantum interference and coherence. These are interesting results and can be used make nanoscale plasma devices.

In this work, we apply a new variational principle that allows us to find the distribution function of electron in an arbitrary atom. This principle is based on the method of density matrix. Using the density matrix, we found the energy of the electrons in an atom, entropy and thermodynamic potential. The main idea of this article is that all two-electron matrices must be anti-symmetric. This refers to the two-electron Hamiltonian. This applies to Slater two-particle wave function. We found the equation for the distribution function of electrons at the quantum orbits. This equation can be obtained in two cases. In the first case, based on the density matrix of the first order, this equation is obtained only for the distribution function. In the second case two equations are obtained, which include distribution function and correlation function. The distribution function of electrons in the atom is similar to the function of Fermi–Dirac for electrons in a solid. In this work we apply a new approach to calculate the energy levels of electron in an arbitrary atom. This approach is also based on the method of density matrix.

The quantum entanglement, discord, and coherence dynamics of two spins in the model of a spin coupled to a spin bath through an intermediate spin are studied. The effects of the important physical parameters including the coupling strength of two spins, the interaction strength between the intermediate spin and the spin bath, the number of bath spins and the temperature of the system on quantum coherence and correlation dynamics are discussed in different cases. The frozen quantum discord can be observed whereas coherence does not when the initial state is the Bell-diagonal state. At finite temperature, we find that coherence is more robust than quantum discord, which is better than entanglement, in terms of resisting the influence of environment. Therefore, quantum coherence is more tenacious than quantum correlation as an important resource.