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Density Matrix
Published in David K. Ferry, An Introduction to Quantum Transport in Semiconductors, 2017
The last line introduces the Liouville equation, which is the equation of motion for the density matrix. The last form, however, is somewhat different in that a superoperator has been introduced. In this case, the superoperator is a “commutator generating superoperator,” but in general these functions are higher-order operators that reside in a Hilbert space of operators. This space is often called the Liouvillian space. Now it is important to note from the second line of Eq. (7.6) that the Hamiltonian operates on different wavefunctions in the two terms of the commutator that follows. This means that the derivatives operate on specific wavefunctions, and the explicit form of Eq. (7.6) is given by Eq. (2.69), which is () iℏ∂ρ∂t=[−ℏ22m*(∂2∂r2−∂2∂r′2)+V(r)−V(r′)]ρ(r,r′,t)
Non-classicality of two superconducting-qubits interacting independently with a resonator cavity: trace-norm correlation and Bures-distance entanglement
Published in Journal of Modern Optics, 2021
A.-B. A. Mohamed, H. A. Hessian
By using Equations (3) and (4), the solution of Equation (2) is given by If the eigenvectors and the eigenvalues of the Hamiltonian are Consequently, after writing the initial qubit-resonator density matrix in terms of the above eigenvectors and using the superoperator , the matrices are evolved as With and .
Sub-recoil-limit laser cooling via interacting dark-state resonances
Published in Journal of Modern Optics, 2019
Vase Moeini, Seyedeh Hamideh Kazemi, Mohammad Mahmoudi
In the presence of dissipation, the evolution of the system is described by the Born–Markov quantum master equation for the density matrix Where the Linblad superoperator describes incoherent processes, and , , and are the wave-vectors associated with , , and the corresponding normalized dipole radiation pattern projected along the x direction, respectively. The operator, having dimension of in order to preserve the interpretation of as a density operator, is often referred to as a jump operator.
EPR spectroscopy and molecular dynamics modelling: a combined approach to study liquid crystals
Published in Liquid Crystals, 2018
Here ρ is a spin density matrix whose evolution in time is described by Equation (2) and is a superoperator of the interaction Hamiltonian (1), expressed in the units of ℏ, defined by its action on the density matrix ρ as: . In the case of the nitroxide spin label of the electron spin coupled to a nuclear spin of 14N, the SLE in the Langevin form is reduced to a system of nine coupled differential equations. The form (2) requires an explicit computational model for the stochastic process to define the time propagation of the spin density matrix. This is achieved by the use of stochastic dynamical trajectories (DT) required for the explicit time variation of A DT can be generated by different means and level of modelling. For instance, using an appropriate Langevin dynamical equation [19,20] applied to a rigid rod undergoing rotational tumbling in an ordering potential, a Brownian dynamics (BD) trajectories can be generated. From direct integration of the Langevin equation, the stochastic trajectory of the orientations of the magnetic axes of the probe defined by three Euler angles is obtained and employed in Equation (2). Alternatively, DTs can be generated from MD simulations of actual molecular structures at either fully atomistic (FA) or coarse-grained (CG) levels. Once propagation of the spin density matrix is completed, EPR line shapes are obtained by applying a Fourier-Laplace transform to the time-dependent averaged transverse magnetisation into the frequency of field domain according to the following equation: