Mixed density operator

Quantum statistical mechanics with the concepts of mixed states, density operators and the Liouville equation. 

The mixed state TDDFT of Rajagopal et al. (38) differs from our formulation in the aspects mentioned alx)ve and in the nature of the operator space where the supervectors reside. A particularly notable distinction is in the use of the factorization D = QQ of the state density operator that leads to unconstrained variation over the space of Hilbert-Schmidt operatOTS, rather than to a constrained variaticxi of the space of Trace-Class operators. 

The quantum-mechanical equivalent of phase density is known as the density matrix or density operator. It is best understood in the case of a mixed ensemble whose systems are not all in the same quantum state, as for a pure ensemble. 

In order to have an expression for the mixed-system exchange energy (192) in terms of the single-particle KS orbitals we need such an expression for the ensemble DM - the diagonal of pP. When the ensemble density operator is characterized by the following decompc ition into its pure-state contributions... 

This has the advantage of treating all relevant states I at once, and allows them to mix over time as determined by the L-vN equation. The density operator can be represented by a density matrix (DM) p(t) by expanding the states rji(q, t) in an electronic basis set (<7), in which case the matrix L-vN equation for the electronic DM must be solved coupled to the hamiltonian equations for the nuclear motions. Details of this procedure have been given in a previous publication.[32]... 

A systematic route to achieve a mixed quantum classical description of EET may start with the partial Wigner representation p(R,P t) of the total density operator referring to the CC solvent system. R and P represent the set of all involved nuclear coordinates and momenta, respectively. However, p(R,P t) remains an operator in the space of electronic CC states (here 4>o and the different first order of the -expansion one can change to electronic matrix elements. Focusing on singly excited state dynamics we have to consider pmn( It, / /,) = 4>m p R, P t) 4>n) which obeys the following equation... 

A version of the reduced density operator to be used in the mixed quantum classical description may be obtained if we replace p(t) by the pure state... 

So far we have treated only the case of wave packets constructed from pure states. Consider now the control of a molecular system in a mixed state in which the initial states are distributed at a finite temperature. The time evolution of the system density operator pit) is determined by the Liouville equation,... 

A fundamental limitation to coherent population control is that it is impossible to transfer 100% of the population in a mixed state. That is, the maximum value of the population transferred cannot exceed the maximum of the initial population distribution of a system without any dissipative process such as spontaneous emission. This result can be simply verified using the unitary property of the density operator, pit) = U(t, to)pito)U t, to), where p(to) is the diagonalized density operator at t = to, Uit, to) is the time-evolution operator given by... 

The norm of the density operator o- = (Tr o-V ) is always conserved under the unitary transformation of Eq. (71). However, in general, additional constants of motion exist. If the effective mixing Hamiltonian is composed exclusively of zero-quantum operators, it commutes with the z component of the total angular momentum operator ... 

This implies that the multiple-quantum order p of the density operator is preserved during the mixing process (vide infra Bazzo and Boyd, 1987). Equation (207) also implies that the expectation value of is constant during the mixing period. For example, in the case of a two-spin system the expectation value is a constant of motion (see Section II). As a... 

The most important criteria for experimental Hartmann-Hahn mixing sequences are their coherence-transfer properties, which can be assessed based on the created effective Hamiltonians, propagators, and the evolution of the density operator. Additional criteria reflect the robustness with respect to experimental imperfections and experimental constraints, such as available rf amplitudes and the tolerable average rf power. For some spectrometers, simplicity of the sequence can be an additional criterion. Finally, for applications with short mixing periods, such as one-bond heteronuclear Hartmann-Hahn experiments, the duration Tj, of the basis sequence can be important. 

The maximum entropy principle can then be formulated as follows Assume that a given quantum system is in a pure state but that this pure state is unknown. Assume furthermore that the only knowledge about the system is some nonpure state with density operator D. Then the probability of finding the pure state in some subset of all pure states is described by the probability distribution p of pure states having maximum entropy with respect to equipartition. The probability distribution p is chosen from all the probability distributions p of pure states to yield the given density operator via mixing in the sense of Eq. (37). 

A theoretical treatment of the DREAM adiabatic homonuclear recoupling experiment has been given using Floquet theory. An effective Hamiltonian has been derived analytically and the time evolution of the density operator in the adiabatic limit has been described. Shape cycles have been proposed and characterized experimentally. Application to spin-pair filtering as a mixing period in a 2D correlation experiment has been explored and the experimental results have been compared to theoretical predictions and exact numerical simulations. 

An important observation is that several, but not all, properties of the pure state density operator remain also in mixed states ... 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

It is frequently stated that a mixed density operator refers to an ensemble made up of systems each of which is in a pure state. Such a statement, as pointed out by Park (9), is meaningless. In quantum theory, the only experimentally observed reality is that which is revealed by the statistics of measurements performed on an ensemble of identical systems prepared in a specified manner. If a given preparation results in a mixed density operator, then this operator represents the only meaningful reality of the state. Park points out that a general quantum ensemble characterized by a density operator can be numerically (as opposed to operationally) subdivided in an infinite variety of ways into pure or mixed subensembles, namely,... 

It has been established in Sect.6 that whenever it becomes necessary to employ a mixed (rather than pure) ensemble, with density operator (44), the interatomic spin-coupling density Q ri,r2) is everywhere zero and thus yields zero for the expectation value of the spin scalar product Sx Sb there can be no spin coupling between A and B. Two regimes may thus be distinguished ... 

This is a very useful expression for considering it associated with the mono-density operators when the many-fermionic systems are treated, although similar procedure applies for mixed (sample) states as well. There is immediate to see that for Af formally independent partitions the Hilbert space corresponding to the iV-mono-particle densities on pure states, we individually have, see Eqs. (4.176), (4.181), (4.182) and (4.184),... 

According to (29), this sum over a continuum involves the level density function Pp Ep), which is derived by the standard quantum statistical arguments and is generally smooth on the energy scale of interest. As regards the formal representation of the state of the reservoir, we may note that it cannot be strictly considered as a pure state like (31), but rather as a mixed state, described by the reduced density operator... 

The second important limiting case concerns chaotic light as emitted from eonventional sources like gas discharge lamps, or thermal sources like thermal cavities or filament lamps, where special consideration must be given to classical, statistical aspects of the pulses and beams (Glauber, 1965,1970 Loudon, 1973). Formally, instead of the pure-state description based on Eqs. (54)-(57), we must then consider the incident light pulses or beam as an ensemble of photons in a mixed state represented by a statistical density operator p(fo) (Dicke and Wittke, 1961 Loudon, 1973). 

EPR entanglement in off-resonantly driven mixing systems [38], First we consider the Lambda system, as shown in Fig. 1. An ensemble of N atoms is placed in a two-mode optieal cavity. Two metastable states of the a-atom are denoted by 1 ) and 2 ) and the excited state is indicated by 3 ). Two external driving fields of eircular frequencies codi and are applied to the dipole-allowed transitions 1, 2 )- 3 ). Two cavity fields of cireular frequencies i and 2 are generated from these two transitions, respectively. The master equation for the density operator p of the atom-field system is written in Ae dipole approximation and in an appropriate rotating frame as [82-84]... 

The density operator p(f) is a generalized wave function which describes a so-called mixed state. This state corresponds to a statistical ensemble of quantum-mechanical objects, which in our case is a collection of nuclei with magnetic moments. Therefore, the statistics specific of these objects and the statistics of an ensemble are simultaneously present. 

Let us start with a short discussion about the kind of transformation we are seeking for. As described in the Chapter 2, the density matrix corresponding to a pure state is a projector, which satisfies the following properties (Chapter 3) p = p" and Tr(/o ) = 1. On the other hand, for a statistically mixed state, p p and Tr(p ) < 1. Now, let us look at a density operator that is obtained from a mixed state operator p by a unitary transformation, p = UpU. The question is whether this operator can or cannot be a pure state operator. The trace and idempotency properties for the transformed operator become ... 

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