One of the benefits of the time-irreversible open-system boundary conditions is that they provide an alternative to the use of periodic boundary conditions in the analysis of quantum-transport phenomena. The great disadvantage of periodic boundary conditions is that they do not address the case in which the potential varies significantly across a system. That is, their use restricts one to the study of low-field phenomena. It has been pointed out (Yennie, 1987, footnote 11 acknowledging private discussion with M. Weinstein) that quasi-periodic boundary conditions ( i.e., periodic within a phase factor which can be removed by a gauge transformation) are necessary if the momentum operator is to be Hermitian on a finite domain. The present work demonstrates that far-from-equilibrium phenomena can be modeled by employing a non-Hermitian momentum superoperator.
The connection between symmetries and conservation laws is undoubtedly one of the most fundamental results of the quantum theory. However, if one is faced with the task of describing the behavior of a nonconservative system, the inability to modify or violate the conservation laws becomes an obstacle to defining a realistic model, rather than a benefit. The problem is that one wants a model whose solutions stably approach a steady state, which requires complex-valued eigenvalues, but the expectation values of physical observables should be real. The present analysis of open-system models demonstrates that these conflicting requirements can be accommodated at the kinetic level, because the roles of generating the dynamical evolution and evaluating observables are filled by different superoperators. If we reexamine the models described above, we find that the dynamic effects such as generating time evolution or moving density by current flow are described by commutator superoperators, and these are the superoperators which become non-Hermitian when one incorporates interactions with the outside world. The measurement of the expectation values of observables is done by anticommutator superoperators, and these, with proper attention to the definition of the domain and boundary conditions, remain Hermitian. This separation of function has been noted by Prigogine (1980) in the superoperators generated by the Hamiltonian. In the open-system model the momentum superoperators appear in similar roles, and this demonstrates the existence of a more general underlying structure in the kinetic theory.
Let us consider the superoperators and derived from the momentum operator. We have already observed that the kinetic energy term of the Liouville equation (2.3) can be written as (3.21). will be Hermitian if we restrict our attention to density matrices whose off-diagonal elements approach zero for large (so that integration by parts may performed without a surface contribution in an integral over ). Such density matrices describe normal systems (as opposed to superconducting, or with some other long-range coherent effect) at nonzero temperature. In such normal cases produces the real-valued factor p in the drift term (4.38). generates the gradient in and is thus the superoperator which is rendered non-Hermitian by the boundary conditions (4.36).
We can also see the dichotomy of function between and by examining the elementary quantum continuity equation, which is conventionally written
By now we should readily recognize the presence of in the current density operator J. In fact, the current density is much more naturally regarded as a superoperator,
and we see in the role of measuring an observable. At the kinetic level the continuity equation is linear in terms of the density matrix and is simply the Liouville equation evaluated along the diagonal .
The continuity equation is of course just the zeroth-order moment equation of the Liouville equation. The higher-order moments of the Liouville equation may be obtained by operating on the equation with (or ) and evaluating the resulting expression along the diagonal. Let us denote the evaluation of an operator kernel for by angle brackets, . This is equivalent to the phase-space procedure of multiplying by some power of p and then integrating over all p, so that the corresponding expression for the Wigner function is
The moment equations we will derive are a special case of those which have been discussed by a number of authors (Frölich, 1967; Putterman, 1974; Iafrate, Grubin, and Ferry, 1981; Kreuzer, 1981), because we will not consider two-body or dissipative interactions. The objective is to demonstrate the role of the anticommutator superoperators in this procedure, a point which has not been previously articulated.
As a starting point from which to derive the moment equations, let us rewrite the Liouville equation in superoperator notation, making use of the factorization (3.21):
The manipulations required to generate the moment equations may be considerably simplified by using some superoperator relations to evaluate the effect of on the potential and its derivatives. To derive the necessary expressions, let us consider an operator , which is diagonal in position space. The commutators of the derived superoperators and with are then
where indicates the superoperator derived from the spatial derivative . Note that for any such operator, and for any operator . Now we may readily derive the moment equations. The zeroth moment is thus
which is a familiar form of the continuity equation. If a collision term is included in the kinetic equation, it must have a form such that if the theory is to satisfy the continuity equation. This means that in the density matrix representation [a condition satisfied by the Fokker-Planck operator (3.23)] or with , for the Wigner function (see Appendix 14).
The first moment equation is readily found to be
where is the momentum flux density. [For two- or three-dimensional models, the direct product of the two vectors is taken and will be a tensor (Landau and Lifshitz, 1959).] Equation (6.96) is identical to its classical counterpart. If we integrate it with respect to q (assuming that the domain is rectangular in the coordinates and extends over 0 < q < l), we obtain a generalization of Ehrenfest's theorem to the case of an open system:
The last two terms represent the effect of opening the system: a flux of momentum density through the boundaries of the system will affect the current flow within the system. To make contact with hydrodynamics, we would follow the standard kinetic-theory manipulations (Kreuzer, 1981, ch. 8) and define a kinetic pressure tensor and separate into .
Continuing with the above procedure, we may derive a second-moment equation by operating on (6.93) with to obtain
Quantum corrections in the form of terms containing will begin to appear in the third and higher moment equations, as one would expect from the Wigner-Moyal expansion (4.41). However, the second moment equation presents something of an ambiguity. We might also derive it by operating on (6.93) with the derived from the kinetic energy operator T. These are not at all the same superoperators:
Putterman (1974) displays both of these forms and notes that both lead to the same bulk properties, thus any physical difference must appear in a surface contribution. It is not the purpose of the present discussion to investigate these issues in detail, but only to demonstrate that anticommutator superoperators appear naturally in any attempt to evaluate expectation values in kinetic theory.
The same dichotomy between commutator and anticommutator superoperators can be seen in the case of the superoperators generated by the Hamiltonian H. Of course is just the Liouville superoperator , and we have examined at length the need for a departure from Hermiticity in the case of . We have not yet encountered a need for the anticommutator . One place it does occur is in a generalization of the Bloch equation (3.16) to the case of an open system. If one attempts to compute an equilibrium density matrix as a finite segment of a much larger system by modifying the boundary conditions on in the Bloch equation, one quickly discovers that product must be symmetrized to obtain sensible answers. Thus, the Bloch equation becomes
If the time-reversible open-system boundary conditions (3.19) are applied to the Bloch equation, one obtains a quite useful method for evaluating the equilibrium density matrix (in contrast to the disastrous effect these boundary conditions have upon the time evolution). Taking into account our particle-density normalization of , the correct Bloch equation is
with the initial condition
When this equation is integrated, the resulting densities in regions of constant potential are found to be equal to the semiclassically expected value . An example of an equilibrium density matrix obtained from such a calculation is illustrated in Fig. 22.
Figure 22. Equilibrium density matrix obtained by numerically integrating the generalized Bloch equation (6.102) subject to the reversible open-system boundary conditions (3.19). The potential, displayed above, represents the sort of features which are now realizable using semiconductor heterostructure technology. The chemical potential is indicated by the dashed line. The calculation employed parameters appropriate for the AlGaAs system at 77 K. The three energy barriers create two identical ``quantum wells,'' bounded by contacting layers. The lowest energy states in these wells are pushed toward higher energy by size quantization, which reduces the electron density in the wells via the Boltzmann factor. The shallow peaks off the diagonal measure the correlation between the phase of the electron at different positions, and indicate in the present case that the symmetric combination of the well states has a greater occupation factor than the antisymmetric combination.
Here we see that again the anticommutator superoperator appears in the role of evaluating an observable, in this case for the purpose of evaluating the energy and thus the occupation probability of the possible states. We would expect that, for this purpose, ought to be Hermitian. Its Hermiticity in fact depends upon the shape of the domain when the boundary conditions (3.19) are applied. Because is an elliptic operator, it is easy to show that it will be Hermitian when the domain is rectangular in the coordinates, so that the gradient in (3.19) is normal to the system boundary. It is not Hermitian when applied to a domain which is square in the coordinates, as in the calculation illustrated in Fig. 22. However, the departure from Hermiticity is small, and the results are physically quite reasonable. I have not yet implemented a program to perform such a calculation on a rectangular domain in , but this would be the proper way to proceed to evaluate the equilibrium density matrix using open-system boundary conditions.
Having noted that appears in the evaluation of the equilibrium density matrix, we can address a point raised by Dahl (1981). It is that , by itself, does not define a unique eigenvalue problem in the wavefunction space of a quantum system; but together with , it does define such a problem. This consideration enters the present problem only for bound states localized within the open system (Carruthers and Zachariasen, 1983). As noted earlier, such states would lead to a nontrivial null space of . The occupation of such states would have to be determined as an initial condition, such as an equilibrium distribution evaluated using .