DISSIPATIVE SCATTERING AND TRANSPORT IN MESOSCOPIC SYSTEMS

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.

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.

Because dissipative processes imply irreversibility and, therefore, a preferred direction in time, it is generally thought that quantum dynamical semigroups are the basic tools to introduce rigorously the dissipation in quantum mechanics. The most general form of the generators of such semigroups was given by Lindblad1. This formalism has been studied for the case of damped harmonic oscillators2,3 and applied to various physical phenomena, for instance, to the damping of collective modes in deep inelastic collisions in nuclear physics4. According to the axiomatic theory of Lindblad1, the usual von Neumann-Liouville equation ruling the time evolution of closed quantum systems is replaced in the case of open systems by the following time-homogeneous quantum mechanical Markovian master equation for the density operator ρ:KeywordsHarmonic OscillatorDensity OperatorWigner FunctionCollective ModeOpen Quantum SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.

In this work, we present a quantization scheme for the damped harmonic oscillator (QDHO) using a framework known as momentous quantum mechanics. Our method relies on a semiclassical dynamical system derived from an extended classical Hamiltonian, where the phase-space variables are given by expectation values of observables and quantum dispersions. The significance of our study lies in its potential to serve as a foundational basis for the effective description of open quantum systems (OQS), and the description of dissipation in quantum mechanics. By employing the Bateman’s dual model as the initial classical framework, and undergoing quantization, we demonstrate that our description aligns exceptionally well with the well-established Lindblad master equation for the QDHO. Furthermore, the robust formulation of our approach suggests a broad applicability in the context of OQS, extending implementations into more general systems, making it a versatile and powerful tool to study diverse phenomena.

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.

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

Equilibration implies asymptotic stationarity for open quantum systems

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.

To emphasize the semantic impact of local semantic and grammatical information among adjacent words in the input text, we establish a constraint functions-based quantum-like tensor compression sentence representation model by integrating the concept of extending the pure state-based density matrix to the mixed-state projection operator in quantum mechanics. The provided model highlights the semantic significance of mixed word associations in the input text, simultaneously reducing the reliance on information derived solely from dictionary statistics. We combine the correlation coefficient with the attention mechanism to establish the correlation coefficient between words. The quantum-like sentence representation based on pure state density matrix is extended to the projection operator of mixed states. Combining the acquisition of maximum in convex optimization, a constraint functions-based quantum-like text representation pruning model is established to reduce redundant information caused by dimensional expansion of tensor operations. The experimental results on SICK-2014, STS-benchmark, and STS-companion show that the provided model is more effective than the mainstream models in mining semantic information, especially more sensitive to the negative semantics of sentences.

Decoherence in nanostructures and quantum systems

We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here $H$ is the Hamiltonian of the system and $\mathcal{D}$ is the dissipator. We consider the case where the system consists of two parts, the ``boundary'' $A$ and the ``bulk'' $B$, and $\mathcal{D}$ acts only on $A$, so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where ${\mathcal{I}}_{B}$ is the identity superoperator on part $B$. Let ${\mathcal{D}}_{A}$ be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with $H$ must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any ${\mathcal{D}}_{A}$ that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between $A$ and $B$. Related results for nonergodic cases are also discussed.

In this work, we study the impact of quantum entanglement on the two-point correlation function and the associated primordial power spectrum of mean square vacuum fluctuation in a bipartite quantum field theoretic system. The field theory that we consider is the effective theorry of axion field arising from Type IIB string theory compactified to four dimensions. We compute the expression for the power spectrum of vacuum fluctuation in three different approaches, namely (1) field operator expansion (FOE) technique with the quantum entangled state, (2) reduced density matrix (RDM) formalism with mixed quantum state and (3) the method of non-entangled state (NES). For massless axion field, in all these three formalism, we reproduce, at the leading order, the exact scale invariant power spectrum which is well known in the literature. We observe that due to quantum entanglement, the sub-leading terms for these thee formalisms are different. Thus, such correction terms break the degeneracy among the analysis of the FOE, RDM and NES formalisms in the super-horizon limit. On the other hand, for massive axion field we get a slight deviation from scale invariance and exactly quantify the spectral tilt of the power spectrum in small scales. Apart from that, for massless and massive axion field, we find distinguishable features of the power spectrum for the FOE, RDM, and NES on the large scales, which is the result of quantum entanglement. We also find that such large-scale effects are comparable to or greater than the curvature radius of the de Sitter space. Most importantly, in near future if experiments probe for early universe phenomena, one can detect such small quantum effects. In such a scenario, it is possible to test the implications of quantum entanglement in primordial cosmology.