The Density Matrix

Forster s theory of resonance energy transfer depends implicitly on thermal equilibration to trap the excitation on the acceptor. But the Forster theory is silent concerning the oscillations predicted by Eqs. (10.4a-10.5), and it does not address how rapidly the system equilibrates with the surroundings it simply assumes that equilibration occurs rapidly compared to the rate at which the excitation can return to the donor. 

Clearly, we need a more complete theory that bridges the gap between the Forster theory and the coherent oscillations described by Eqs. (10.4a-10.5). If the oscillations of the individual systems in an ensemble start out in phase, we would like to understand how this coherence decays, and how this decay affects the observed properties of the ensemble. To address such questions, we must deal with large ensembles of molecules that interact stochastically with their surroundings. We will develop some basic tools for this in the next three sections. 

Consider, again, an individual system with wavefunction P(i) = C (i)V iCjgp l-i(ffnt/ + Ci)]+C2(t) p2 xphi( 2ztM + C2)h where V l and V a are spatial wavefunctions for two basis states with energies 7/n and H22, and is a phase shift that depends on when state n is created. Let s combine the time-dependent coefficients Cl and C2 with the corresponding time-dependent exponential factors by defining a new set of coefficients Ci and C2  

In principle, we can find the expectation value of any dynamic property A of the system by evaluating the expression 

Equation (10.11) can be written more succinctly if we define a matrix p whose terms are 

Suppose that an assembly of N identical spin systems is considered. This allows a quantum statistical description of a spin system, for example, the kth spin system in the ensemble. If the spin system is in a state of wavefunction or ket ipk) the expectation value of a physical variable given by its operator Q is 

In general, the ket is time dependent and may be expanded using a complete orthonormal basis set of m stationary kets 0j) = z)  

This leads to a definition for the density operator tr, whose matrix elements in the orthonormal basis i) are 

It is deduced from Eq. (2.5) that = aji, i.e., the density operator is Hermitian and has real eigenvalues. In particular, its diagonal elements 

SO that if neither an nor Gjj vanish, aij do not vanish. This is a consequence of the fact that the ket ) is a superposition of the basis kets. If the ensemble is a statistical mixture described by kets the off-diagonal elements aij may vanish even if Ci and Cj are non-zero. This arises if the phases and are distributed at random in the mixture, which is the so-called hypothesis of random phases [2.1]. If the aij do not vanish, the phases axe therefore not random, and some coherence exists among various kets of the mixture. The spin system described by this density matrix is said to contain a coherent superposition of the quantum states z) and j). The concept of quantum coherence between two states z) and j) plays an important role in modern NMR spectroscopy. 

Fortunately there is a simple mathematical formalism that gives us the best of the quantum and classical approaches. By recasting the time-dependent Schrodinger equation into a form using a so-called density operator, physicists have long been able to follow the development of a quantum system with time. This formalism 

do we not ignore the preceding approaches and go directly to the density matrix The answer is that the density matrix method, while conceptually simple, becomes very tedious and gives very litde physical insight into processes that occur. We will come back to the density matrix in Chapter 11, but only after we have developed a better physical feeling for what happens in NMR experiments. 

As soon as large ensembles of particles with statistical populations of the eigenstates and incoherent exchange and relaxation processes between these states are investigated, quantum statistical tools are necessary to describe the system. In this situation the quantum mechanical density operator p has to be employed. For the coherent evolution of the density operator under the influence of a Hamiltonian H, the following differential equation is found [80] 

In this equation we have followed the NMR convention and set the constant fi = 1. This is equivalent to measuring energies in angular frequency units. Employing a suitable set of base functions of the Hilbert space, this equation can be converted into a set of linear differential equations for the matrix elements of p. In the case of a single parr of tunnel levels the Hamiltonian of the two levels with their tunnel splitting can be treated as a two-level system, employing fictitious spin 1/2 operators, describable by the Hamiltonian H 

is the unity matrix of Hilbert space. In the case of the tunnel Hamiltonian it is simply 

In QIP one frequently has to deal with situations where the state vector of the system is not known, but only a set of possible states V, ), each of which with a probability p, to occur. The set [pi, IV, ) is said to be a statistical ensemble. The appropriate mathematical tool to deal with such cases is the density matrix, p, defined as  

The density matrix is a positive operator, that is, its eigenvalues are real and nonnegative. Indeed, for any state (p). 

A quantum state is said to be pure if and only if Tr(p ) = 1. This can be proved using directly the definition of p  

The equality in (3.6.5) is only satisfied if pi = 0 to every i, except for a particular state j such as Pj = 1. Only an operator with the properties listed above can be considered a density matrix. 

When dealing with composite systems, the density operator of the subsystems can be obtained through the partial trace operation over the density operator of the whole system. The partial trace operation is a sum over all the possible states of one subsystem. For instance, if is the density operator of a composite system ab), the density operator of each subsystem is given by  

---

A more intuitive, and more general, approach to the study of two-level systems is provided by the Feynman-Vemon-Flellwarth geometrical picture. To understand this approach we need to first introduce the density matrix. 

Moreover, we will write the density matrix for the system as... 

In the above discussion of relaxation to equilibrium, the density matrix was implicitly cast in the energy representation. However, the density operator can be cast in a variety of representations other than the energy representation. Two of the most connnonly used are the coordinate representation and the Wigner phase space representation. In addition, there is the diagonal representation of the density operator in this representation, the most general fomi of p takes the fomi... 

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

Figure Al.6.32. (a) Initial and (b) final population distributions corresponding to cooling, (c) Geometrical interpretation of cooling. The density matrix is represented as a point on generalized Bloch sphere of radius R...

Tr(p ). For an initially thennal state the radius < 1, while for a pure state = 1. The object of cooling is to manipulate the density matrix onto spheres of increasingly larger radius. 

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 ... 

A similar expression applies to the density matrix, from its correspondence with the propagator. For example,... 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

The eigenfiinctions of a system of two particles are detemiined by their positions x and j, and the density matrix is generalized to... 

In the presence of a potential function U(x,y), the density matrix in the high-temperature approximation has the fomi... 

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

The main cost of this enlianced time resolution compared to fluorescence upconversion, however, is the aforementioned problem of time ordering of the photons that arrive from the pump and probe pulses. Wlien the probe pulse either precedes or trails the arrival of the pump pulse by a time interval that is significantly longer than the pulse duration, the action of the probe and pump pulses on the populations resident in the various resonant states is nnambiguous. When the pump and probe pulses temporally overlap in tlie sample, however, all possible time orderings of field-molecule interactions contribute to the response and complicate the interpretation. Double-sided Feymuan diagrams, which provide a pictorial view of the density matrix s time evolution under the action of the laser pulses, can be used to detenuine the various contributions to the sample response [125]. 

The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

In Liouville space, both the density matrix and the operator are vectors. The dot product of these Liouville space... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

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

N() -e that the summations are over the N/2 occupied orbitals. Other properties can be cali ulated from the density matrix for example, the electronic energy is ... 

Living calculated the integrals, we are now ready to start the SCF calculation. To formulate the Fock mahix it is necessary to have an initial guess of the density matrix, P. The simplest approach is to use the null matrix in which all elements are zero. In this initial step the Fock nulrix F is therefore equal to... 

The dipole moment operator is a sum of one-electron operators r , and as such the electronic conlribution to the dipole moment can be written as a sum of one-electron contributions. The eleclronic contribution can also be written in terms of the density matrix, P, as follows ... 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

Mulliken population analysis is a trivial calculation to perform once a self-consistent field has been established and the elements of the density matrix have been determined. 

---