In this paper we attempt to unify the theoretical treatment of a variety of cooperative interaction phenomena occurring in resonantly interacting radiation-matter systems. The system consists of a multimode optical field interacting with
N resonant atoms in a region of large spatial extent. The field and atoms may or may not be coupled to reservoirs. We derive a set of coupled master equations for the reduced matter and radiation density operators of the composite system in the first Born approximation, using a method due to Bogoliubov. These equations describe within a single framework phenomena involving superradiant emission and coherent propagation. In the first place, our theory generalizes existing formulations of the superradiance problem by treating the radiation field as part of the dynamical system rather than as a reservoir. Hence, our equations describe stimulated interaction effects, which are entirely absent in the usual superradiance master equations. Moreover, our equations contain terms describing the self-consistent interaction of an average field with a set of polarizable atoms, as in the coupled Bloch and Maxwell equations of the theory of coherent pulse propagation, and enable us to describe fluctuations, arising from incoherent emission, about the average behavior described by the Bloch-Maxwell theory. We solve formally the dynamical equation for the radiation density operator and obtain a displaced multivariate Gaussian distribution centered on the solution of the Maxwell wave equation, which has as parameters the first and second moments of the matter density operator. Then we derive and discuss the field and atomic-moment equations, both in the wave-number and position representations. The moment equations enable us to show the relationship between descriptions of superradiant damping based on the Bloch equations and those based on the existence of correlated atomic states, including both as special cases. We introduce reduced matter moments, which allow us to separate incoherent- and coherent-emission effects and enable us to discuss the question of the closure of the moment equations. Then we consider cases where the behavior of the atomic system is dominated by a strong coherent pump wave or where the atoms are in a symmetric correlated state and show how the slowly-varying-envelope approximation simplifies the radiation-moment equations in these two cases. In the former case, we obtain an approximate theory which consists, in lowest order, of the Bloch-Maxwell equations with radiative-damping terms, plus an equation describing the time development of the correlation function for field fluctuations. Returning to our general formalism, we relate the usual parameters of the theory of radiative decay and of resonant interactions to the more general quantities appearing in our spatial analysis. Finally, we discuss the importance of the stimulated-emission terms appearing in our theory and describe several extensions, including the introduction of inhomogeneous broadening.
Abstract In this work, we consider a pair of qubits controlled by coherent and incoherent controls. The dynamics of the two-qubit system is driven by a Gorini–Kossakowsky–Sudarchhan–Lindblad master equation where coherent control enters into the Hamiltonian and incoherent control inters into both the Hamiltonian (via Lamb shift) and the dissipative superoperator. Two classes of interaction between the system and the coherent field are considered. For this system, we analyze the control problem of generating a given target density matrix which is formulated as minimizing the Hilbert–Schmidt distance between the final density matrix and the target density matrix. Incoherent control is modeled as a sum of constant in time Gaussians with centers related with the transitions frequencies between the energy levels of the qubits. Coherent control in general formulation is considered as measurable function and in numerical experiments as piecewise constant function with constraints on magnitudes and variations. Finite-dimensional numerical optimization is performed using the dual annealing method; the corresponding results are described for some initial and target density matrices and for some set of the parameters of the control problem.
It has been asked by different authors whether the two classes of Schatten-
p-norm-based functionals ${C}_{p}(\ensuremath{\rho})={min}_{\ensuremath{\sigma}\ensuremath{\in}\mathcal{I}}{\ensuremath{\parallel}\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}\ensuremath{\parallel}}_{p}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p}(\ensuremath{\rho})={\ensuremath{\parallel}\ensuremath{\rho}\ensuremath{-}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}\ensuremath{\parallel}}_{p}$ with $p\ensuremath{\ge}1$ are valid coherence measures under incoherent operations, strictly incoherent operations, and genuinely incoherent operations, respectively, where
I is the set of incoherent states and $\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}$ is the diagonal part of density operator $\ensuremath{\rho}$. Of these questions, all we know is that ${C}_{p}(\ensuremath{\rho})$ is not a valid coherence measure under incoherent operations and strictly incoherent operations, but all other aspects remain open. In this paper, we prove that (1) ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{1}(\ensuremath{\rho})$ is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but not a valid coherence measure under incoherent operations, (2) ${C}_{1}(\ensuremath{\rho})$ is not a valid coherence measure even under genuinely incoherent operations, and (3) neither ${C}_{p>1}(\ensuremath{\rho})$ nor ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p>1}(\ensuremath{\rho})$ is a valid coherence measure under any of the three sets of operations. This paper not only provides a thorough examination on the validity of taking ${C}_{p}(\ensuremath{\rho})$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p}(\ensuremath{\rho})$ as coherence measures, but also finds an example that fulfills the monotonicity under strictly incoherent operations but violates it under incoherent operations.