Ensemble density matrix

Bloch s Equation for the Density Matrix.—Bloch s equation will be derived here both for the equilibrium ensemble density matrix of Eq. (8-204), and for the equilibrium grand ensemble density matrix of Eq. (8-219). 

This density can be obtained from the ensemble density matrix... 

This follows directly from the fact that if n B then there is a potential v which generates a ground state ensemble density matrix D[n which yields n. So... 

The first relation follows from the fact that if the density n is a pure state v-representable density then the minimizing density matrix for FL is a pure state density matrix. The second relation also easily follows. We take n to be an E-V-density which is not a PS-V-density. There is a ground state ensemble density matrix D[n for which we have... 

We may then define an ensemble density matrix p the elements of which are... 

The density matrices play a special role in statistical mechanics but also in any situation in which we possess incomplete information about some general system. In that case, the system will not be in a pure state and will thus not be represented by a wavefunction instead, the mixed state must be represented by a density matrix. The density matrix will then refer to an ensemble of identical systems of which a fractional number wk are in the definite state and the ensemble density matrix will be... 

Atomic coherences appear here as non-zero off-diagonal elements and X +. To solve for x we use the ensemble density matrix and compute the elements from the equations of motion (see e.g. Cohen-TannoudjiL77])... 

Next we proceed to develop the theory o resonance fluorescence experiments using the ensemble density matrix to describe the system of atoms. The important concepts of optical and radio-frequency coherence and of the interference of atomic states are discussed in detail. As an illustration of this theory general expressions describing the Hanle effect experiments are obtained. These are evaluated in detail for the frequently employed example of atoms whose angular momentum quantum numbers in the ground and excited levels are J =0 and Jg=l respectively. Finally resonance fluorescence experiments using pulsed or modulated excitation are described. 

From equations (15.9) and (15.10) we see that the mean value of the observable can be expressed in terms of the ensemble density matrix by... 

Modulation of light from incoherent sources. The light beats observed in resonance fluorescence experiments should be clearly distinguished from those observed by Forrester et aZ.(1955). In that experiment an exceedingly weak modulation was detected in the light emitted from a conventional mercury lamp. In this source the atoms are excited by random collisions and the ensemble density matrix possesses zero hertzian coherence. The beat frequencies observed were due to the mixing at the detector of the radiation frequencies emitted by different atoms and... 

These experiments are in fact entirely analagous to the Hanle effect or zero-field level-crossing experiments involving excited atoms discussed in Chapter 15. The coherent polarization of the pumping light referred to the quantization axis Oz in Fig.17.12 prepares the atoms in a coherent superposition of ground-state Zeeman sub-levels. The ensemble density matrix now has finite off-diagonal elements... 

We see that the fluorescence and absorption monitoring operators have very similar forms. However, because equation (17.45) involves the ground-state part of the ensemble density matrix, the absorption monitoring technique offers a more direct way of measuring the parameters of interest in many optical pumping experiments. 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The coordinate representation of the density matrix, in the canonical ensemble, may be written... 

The expectation value of the density operator, and, indeed, all the components of the density matrix, are stationary in time for an ensemble set up in terms of energy eigenstates. IT we use occupation number representation to set up the density matrix, it is at once seen from Eq. (8-187) that it also is independent of time ... 

Next, consider an ensemble defined in configuration space, so that the density matrix has the form of Eq. (8-190). We assume that the eigenvectors X> are not eigenvectors of the hamiltonian. We have... 

Density matrix and ensembles, 465 Bloch s equation, 475 Derogatory form, 73... 

A final study that must be mentioned is a study by Hartmann et al. [249] on the ultrafast spectroscopy of the Na3F2 cluster. They derived an expression for the calculation of a pump-probe signal using a Wigner-type density matrix approach, which requires a time-dependent ensemble to be calculated after the initial excitation. This ensemble was obtained using fewest switches surface hopping, with trajectories initially sampled from the thermalized vibronic Wigner function vertically excited onto the upper surface. 

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. 

The diagonal element of the density matrix, W(n) — a nan is the probability that a system chosen at random from the ensemble occurs in the state characterized by n, and implies the normalization... 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

These definitions are easily generalized from a pure state, described by to ensemble states, described by a system density matrix V, for which an expectation value is... 

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