Reduced density operator defined

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

The first-order reduced density operator y can be defined in terms of its kernel function37... 

Previously we have considered the reduced density matrices and the equations of motion. These quantities are the representation of the reduced density operators (Bogolyubov,6 Gurov7) defined by... 

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the... 

In general the exciton dynamics exhibits both coherent and incoherent behaviour, where the incoherence arises from the couphng of the system to a dissipative environment. This is conveniently modelled by an equation of motion for the reduced density operator, p, defined by... 

In the localized exciton basis m), defined by eqn (9.17), the matrix elements of the reduced density operator are... 

The density operator pit) has been formulated for the entire quantum-mechanical system. For magnetic resonance applications, it is usually sufficient to calculate expectation values of a restricted set of operators which act exclusively on nuclear variables. The remaining degrees of freedom are referred to as lattice . The reduced spin density operator is defined by ait) = Tri p(f), where Tri denotes a partial trace over the lattice variables. The reduced density operator can be represented as a vector in a Liouville space of dimension... 

As the foundation of quantum statistical mechanics, the theory of open quantum systems has remained an active topic of research since about the middle of the last century [1-40]. Its development has involved scientists working in fields as diversified as nuclear magnetic resonance, quantum optics and nonlinear spectroscopy, solid-state physics, material science, chemical physics, biophysics, and quantum information. The key quantity in quantum dissipation theory (QDT) is the reduced system density operator, defined formally as the partial trace of the total composite density operator over the stochastic surroundings (bath) degrees of freedom. 

The sequence begins with a 90° pulse of width T90 and a rotating-frame phase direction assumed along the x axis. The rotating-frame Hamiltonian during this RF pulse is defined by H o=- hyBiIxy where Bi is the amplitude of the rotating flux density component of the RF field. The reduced density operator immediately after this pulse reads... 

The structure and roles of membrane microdomains (rafts) in cell membranes are under intensive study but many aspects are still unresolved. Unlike in synthetic bilayers (Fig. 2-2), no way has been found to directly visualize rafts in biomembranes [22]. Many investigators operationally define raft components as those membrane lipids and proteins (a) that remain insoluble after extraction with cold 1% Triton X-100 detergent, (b) that are recovered as a low density band that can be isolated by flotation centrifugation and (c) whose presence in this fraction should be reduced by cholesterol depletion. 

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

It is worth mentioning that constant term disappears in Eq. (6) because of the suitable choice of the basis in the form (4). In the following, the evolution of the density matrix of the relevant system will be examined by means of the standard projection technique. The matrix elements pa/it) of the reduced density matrix operator are defined as follows... 

What did we achieve so far We have an equation, (10.133) or (10.134), for the time evolution of the system s density operator. All terms in this equation are strictly defined in the system sub-space the effect of the bath enters through correlation functions of bath operators that appear in the system-bath interaction. These correlation functions are properties of the unperturbed equilibrium bath. Another manifestation of the reduced nature of this equation is the appearance of... 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

The dynamics quantities in the HEOM formalism are a set of well-defined auxiliary density operators (ADOs), (p (r) = 0,1,...,, in which Po(t) = p t) is just the reduced system density operator. The hierarchy construction resolves not just system-bath coupling strengths but, more importantly, also memory time (l/yo) scales. ... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Because the many-particle Lagrangian density L° reduces to a sum of singleparticle operators, one may define an effective single-particle Lagrangian density iP°(r, t) by the usual recipe of summing over all spins and integrating over the coordinates of all the electrons but one, a step which expresses the result in terms of the charge density p(r, t) as... 

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