Molecular photoexcitation in a medium: Density operator approach

The excitation of a molecule in a medium, by absorption of a light pulse, is described in terms of density operators and the Liouville–von Neumann equation for the whole system. This is divided into a primary region containing the molecule and its close neighbors to which it may be bonded or strongly interacting, plus a secondary region including the remaining medium. A time-dependent self-consistent field procedure factors the density operator into primary and secondary components, each of which interacts with the applied light field. Direct absorption and indirect absorption are described with superoperators in a Liouville space formalism. Equations for the response of the primary region are developed assuming a stochastic medium. The treatment allows for dissipation due to collective electronic or vibrational excitations in the medium. Applications to photodesorption of molecules from metal surfaces are briefly discussed. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 367–375, 2000

The first part of the chapter is devoted to the discussion of density operators. We first introduce the definition of a density operator and the unitary evolution of a quantum state. We then examine the measurement of a quantum system using the density operator approach. We present the density operator associated with a qubit and a harmonic oscillator at thermal equilibrium. The second part of the chapter focuses on bipartite systems. We show that the quantum state of a subsystem is given by the partial trace of the density operator of the global system. Finally, we explain the concept of purification of a density operator

We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach. The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems. We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position. Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space. We present analytical solutions in L2(ℝ) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.

In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem.

In the scheme of the density and phase operator approach, the momentum distribution nk and the ground-state energy Eo are obtained by employing the structure factor and the radial distribution function calculated by Chang and Campbell for the Morse dipole-dipole potential. The condensate fraction, the ratio of the occupation number of the single-particle zero-momentum state No/N amounts to 0.096. The momentum distribution diverges as k- 1 in the low-wave number limit. The ground-state energy becomes Eo= -6.9 NK at the mean density po =0.02185.A-•.

Various theoretical methods address transport effects in quantum dots beyond single-electron tunneling while accounting for the strong interactions in such systems. In this paper we report a detailed comparison between three prominent approaches to quantum transport: the fourth-order Bloch-Redfield quantum master equation (BR), the real-time diagrammatic technique (RT), and the scattering rate approach based on the T-matrix (TM). Central to the BR and RT is the generalized master equation for the reduced density matrix. We demonstrate the exact equivalence of these two techniques. By accounting for coherences (nondiagonal elements of the density matrix) between nonsecular states, we show how contributions to the transport kernels can be grouped in a physically meaningful way. This not only significantly reduces the numerical cost of evaluating the kernels but also yields expressions similar to those obtained in the TM approach, allowing for a detailed comparison. However, in the TM approach an ad hoc regularization procedure is required to cure spurious divergences in the expressions for the transition rates in the stationary (zero-frequency) limit. We show that these problems derive from incomplete cancellation of reducible contributions and do not occur in the BR and RT techniques, resulting in well-behaved expressions in the latter two cases. Additionally, we show that a standard regularization procedure of the TM rates employed in the literature does not correctly reproduce the BR and RT expressions. All the results apply to general quantum dot models and we present explicit rules for the simplified calculation of the zero-frequency kernels. Although we focus on fourth-order perturbation theory only, the results and implications generalize to higher orders. We illustrate our findings for the single impurity Anderson model with finite Coulomb interaction in a magnetic field.

Using the formal analogy between a certain class of Maxwell equations and the Schrdinger equation, we derive the effective Hamiltonian operator that governs the propagation of electromagnetic (EM) wave modes inside nonconducting linear media, which include a large range of nanophotonic and plasmonic waveguides. It turns out that this Hamiltonian is essentially non-Hermitian, and thus requires a special treatment. We formulate the density operator approach for dynamical systems with non-Hermitian Hamiltonians, and derive a master equation that describes the statistical ensembles of EM wave modes. The method provides a theoretical instrument which can be used when designing the next generation of quantum EM devices for sensitive and non-invasive measurements.

To fully account for electron-vibrational coupling and vibrational relaxation in the course of electron motion through a molecular wire a density operator approach is utilized. If combined with a particular projection operator technique a generalized master equation can be derived which governs the populations of the electronic wire states. The respective memory kernels are determined beyond any perturbation theory with respect to the electron-vibrational coupling and can be classified via so-called Liouville space pathways. An ordering of the different contributions to the current-voltage characteristics becomes possible by introducing an electron transmission coefficient which describes ballistic as well as inelastic electron transport through the wire. The general derivations are illustrated by numerical calculations which demonstrate the drastic influence of the electron-vibrational coupling on the wire transmission coefficient as well as on the current-voltage characteristics.

We study the temperature dependence of energy diffusion in two chaotic gapped\nquantum spin chains, a tilted-field Ising model and an XZ model, using an open\nsystem approach. We introduce an energy imbalance by coupling the chain to\nthermal baths at its boundary and study the non-equilibrium steady states of\nthe resulting Lindblad dynamics using a matrix product operator ansatz for the\ndensity matrix. We define an effective local temperature profile by comparing\nlocal reduced density matrices in the steady state to those of a uniform\nthermal state. We then measure the energy current for a variety of driving\ntemperatures and extract the temperature dependence of the energy diffusion\nconstant. For the Ising model, we are able to study temperatures well below the\nenergy gap and find a regime of dilute excitations where rare three-body\ncollisions control energy diffusion. A kinetic model correctly predicts the\nobserved exponential increase of the energy diffusion constant at low\ntemperatures. For the XZ model, we are only able to access intermediate to high\ntemperatures relative to the energy gap and we show that the data is well\ndescribed by an expansion around the infinite temperature limit. We also\ndiscuss the limitations of the particular driving scheme and suggest that lower\ntemperatures can be accessed using larger baths.\n

In the opening chapters of the book, we saw how a variety of conservative phenomena could be described through the Poisson bracket of classical physics. As already mentioned, the majority of systems with which engineers and physicists must deal reveal dissipative phenomena inherent to their nature. In the last chapter we offered a concise overview of nonequilibrium thermodynamics as traditionally applied to describe, close to equilibrium, dissipative dynamic phenomena. In this chapter, we lay the groundwork for the incorporation of non-conservative effects into an equation of Hamiltonian form, and show several well-known examples in way of proof that such a thesis is, in fact, tenable. The unified formulation of conservative and dissipative processes based on the fundamental equations of motion is not a new idea. Among the most complete treatments is the one offered by the Brussels school of thermodynamics [Prigogine et al., 1973; Prigogine, 1973; Henin, 1974] based on a modified Liouville/von Neumann equation. This is a seminal work where, using the mathematical approach of projection operators and relying only on first principles, it is demonstrated how large isolated dynamic systems may present dissipative properties in some asymptotic limit. A key characteristic of their theory is that dissipative phenomena arise spontaneously without the need of any macroscopic assumptions, including that of local equilibrium [Prigogine et al., 1973, p. 6]. The main value, however, is mostly theoretical, demonstrating the compatibility of dissipative, irreversible, processes with the reversible dynamics of elemental processes through a “symmetry-breaking process.” The value of the theory in applications is limited since it relies on a quantum mechanical equation for the density matrix ρ which is, in general, very difficult to solve except for highly simplified problems [Henin, 1974]. In a nutshell, we hope to offer with the present work a macroscopic equivalent of the Brussels school theory which, at the expense of the introduction of the local equilibrium assumption, attempts to unify the description of dynamic and dissipative phenomena from a continuum, macroscopic viewpoint. The main tool to achieve that goal is the extension of the Poisson bracket formalism, analyzed in chapter 5, to dissipative continua.

We present a theoretical framework for the evolution of the internal state of a fast highly charged one-electron ion traversing an amorphous solid. We employ an open quantum system approach which incorporates the complex array of collisions with electrons and ionic cores in the solid within the framework of system-reservoir interactions. Interactions with the solid environment and the radiation field are treated on the same footing and the quantum master equation for the reduced density matrix of the electronic state of the ion is approximated by a Lindblad equation. The latter allows the solution of this multistate problem in terms of Monte Carlo sampling of quantum trajectories. Similarities and extensions to methods used in quantum optics and previously employed in ion-solid interactions are discussed. Our focus is on the transient buildup and destruction of coherences by stochastic processes. We apply our method to the study of coherence properties of the internal state of a fast ${\mathrm{Kr}}^{35+}$ ion traversing carbon foils. Simulations exhibit clear signatures of partially coherent transitions and are found to be in good agreement with experimental data.

A static observer with a finite lifetime has causal access to only a limited region of spacetime known as the causal diamond. The presence of an apparent horizon in the causal diamond, due to the observer's finite lifetime, is the origin of an Unruh-like thermal effect. Thus, even though the observer is static and the background is flat, the finite-lifetime observer experiences a thermal bath in the Minkowski vacuum. In this article, we provide an open quantum systems approach that yields a complete thermal characterization via the observer's steady-state density matrix, which is shown to be thermal with a temperature inversely proportional to its lifetime. This associated diamond temperature agrees with the established result derived from other methods. Moreover, our approach is particularly useful for designing entanglement harvesting protocols in the causal diamond. In addition, we introduce an insightful procedure that defines diamond coordinates using conformal transformations, and which leads to a more direct derivation of the thermal properties.

By employing the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory within an open quantum system framework, we derive a Lindblad equation governing the evolution of the heavy-quarkonium reduced density matrix, accurate to next-to-leading order (NLO) in the ratio of the state’s binding energy to the medium’s temperature [1]. The derived NLO Lindblad equation provides a more reliable description of heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological applications, we numerically solve this equation using the quantum trajectories algorithm. By averaging over Monte Carlo-sampled quantum jumps, we obtain solutions without truncation in the angular momentum quantum number of the considered states. Our analysis highlights the importance of quantum jumps in the nonequilibrium evolution of bottomonium states within the quark-gluon plasma [2]. Additionally, we demonstrate that the quantum regeneration of singlet states from octet configurations is essential to explain experimental observations of bottomonium suppression. The heavy-quarkonium transport coefficients used in our study align with recent lattice QCD determinations.

Test case generation, selection and coverage from natural language

The definition of memory in operational approaches to quantum\nnon-Markovianity depends on the statistical properties of different sets of\noutcomes related to successive measurement processes performed over the system\nof interest. Using projectors techniques we develop a perturbation theory that\nenables to expressing both joint probabilities and outcome correlations in\nterms of the unperturbed system density matrix propagator. This object defines\nthe open system dynamics in absence of measurement processes. Successive series\nterms, which are scaled by the system-environment interaction strength, consist\nin a convolution structure involving system propagators weighted by higher\norder bath correlations. The formalism is corroborated by studying different\ndynamics that admit an exact description. Using the perturbative approach,\nunusual memory effects induced by the interplay between the system-environment\ninteraction and measurement processes are found in finite temperature\nreservoirs.\n