Matrices density

The density matrix method is useful in treating relaxation processes, linear and non-linear laser spectroscopies and non-equilibrium statistical mechanics. In this chapter, the definition of density matrix and the equation of motion (EOM) it follows are introduced. The projection operator technique, which makes the density matrix method a very powerful tool in non-equilibrium statistical mechanics, is presented. 

For simplicity, consider a two-state system described by wavefunctions ua(q) and ub(q). To describe the dynamics of the system the time-dependent Schrodinger equation has to be solved  

For the pure system, the density matrix elements are defined by [1-6] 

25) is often referred to as generalized master equation (GME). It should be noted that Eq. (2.20) or (2.25) describes the time evolution of an isolated system. 

The density matrix (the thermal average density matrix operator) is introduced through 

The partition function Z is the trace (sum of diagonal elements) of a matrix for which the diagonal elements are 

Thermodynamic functions of magnetism in terms of the canonical partition function 

Such a matrix would be obtained as the representation in the orthogonal eigenstates of a Hamiltonian H. Thus Z is no longer an operator it is a number serving as the normalisation constant. The above density matrix is normalised 

Thus the thermal average energy is the internal energy 

When an eigenfunction is intrinsic, the projection matrix is itself a density matrix. Equation (7) then leads to 

It is apparent that the projeetion matriees are orthogonal, and using the spectral resolution theorem we may write the representation matrix of H as 

When power m increases from 0 to 2 in eq. (11), the following simultaneous equations are defined  

The terms on the left-hand side of these equations eorrespond to the power of the Hamiltonian operators. The right-hand side of (11) eorresponds to the progression of X I D, (i = p.q.r). For example, if the system eonsists of six eigenvalues, density matrices can be obtained using six equations. The simultaneous equations that appear in eq. (12) are written coneisely as the produet of the vector and matrix  

The vector of the density matrices,, can be calculated through the inverse of Vandermonde matrix Aj, that is, 

The realization of such coherent systems requires special experimental preparations that, however, can be achieved with several techniques of coherent laser spectroscopy (Chap. 12). An elegant theoretical way of describing observable quantities of a coherently or incoherently excited system of atoms and molecules is based on the density-matrix formalism. 

Let us assume, for simplicity, that each atom of the ensemble can be represented by a two-level system (Sect. 2.7), described by the wave function 

The diagonal elements paa and pbb represent the probabilities of finding the atoms of the ensemble in the level a) and b), respectively. 

If the phases 0 of the atomic wave function (2.124) are randomly distributed for the different atoms of the ensemble, the nondiagonal elements of the density matrix (2.125) average to zero and the incoherently excited system is therefore described by the diagonal matrix 

If definite phase relations exist between the wave functions of the atoms, the system is in a coherent state. The nondiagonal elements of (2.125) describe 

The density matrix p is defined by the product of the two state vectors 

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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. 

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

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. 

Fundamentally, the conditions for lasing are detemiined unambiguously once the populations and coherences of the system density matrix are known. Yet, we have been unable to find in the literature any simple criterion for lasing in multilevel systems in temis of the system density matrix alone. Our conjecture is that entropy, as... 

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... 

The usual context for linear response theory is that the system is prepared in the infinite past, —> -x, to be in equilibrium witii Hamiltonian H and then is turned on. This means that pit ) is given by the canonical density matrix... 

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 ... 

A3.13.1). From [38]. The two-level structure (left) has two models I I = const and random signs (upper part), random V.j but V < V.j < (lower part). The right-hand side shows an evolution with initial diagonal density matrix (upper part) and a single trajectory (lower part). 

In any case, the polarizing action upon the material by a given generator must be followed in more detail. In density matrix evolution, each specified field action transfonns either the ket or the bra side of the density... 

Fane U 1957 Description of states in quantum mechanics by density matrix and operator techniques Rev. Mod. Phys. 29 74-93... 

Redfleld A G 1996 Relaxation theory density matrix formulation Encyclopedia of Nuclear Magnetic Resonance ed D M Grant and R K Harris (Chichester Wiley) pp 4085-92... 

Stepisnik J 1981 Analysis of NMR self-diffusion measurements by a density-matrix oaloulation Physica B/C 104 350-64... 

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. 

In the usual preparatioii-evohition-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except hi special cases. Starthig from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators,... 

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