QUANTUM FLUCTUATIONS OF ELEMENTARY EXCITATIONS IN DISCRETE MEDIA

Elementary excitations (electrons, holes, polaritons, excitons, plasmons, spin waves, etc.) on discrete substrates (e.g., polymer chains, surfaces, and lattices) may move coherently as quantum waves (e.g., Bloch waves), but also incoherently (“hopping”) and may lose their phases due to their interaction with their substrate, for example, lattice vibrations. In the frame of Heisenberg equations for projection operators, these latter e ﬀ ects are often phenomenologically taken into account, which violates quantum mechanical consistency, however. To restore it, quantum mechanical ﬂuctuating forces (noise sources) must be introduced, whose properties can be determined by a general theorem. With increasing miniaturization, in the nanotechnology of logical devices (including quantum computers) that use interacting elementary excitations, such ﬂuctuations become important. This requires the determination of quantum noise sources in composite quantum systems. This is the main objective of my paper, dedicated to the memory of Ilya Pri-gogine.


Introduction
Thermodynamics impresses us again and again by its great generality. Its laws apply to all kinds of matter and all kinds of aggregations (provided the systems are in or close to thermal equilibrium). Thus it is no surprise that Ilya Prigogine, who was a master in thermodynamics and its applications to physical and chemical processes, used thermodynamics as a starting point for his explorations into the fascinating field of open systems with their ability to form dissipative structures. This is witnessed for instance by his books jointly with Glansdorff [2] as well as with Nicolis [5]. An excellent account of various approaches to the physics of open nonequilibrium systems can be found in the book by Babloyantz [1]. In fact, as it happens quite often in science, a new field may be explored from several starting positions. Thus in the field of nonequilibrium processes, my own approach was based on quantum field theory and quantum statistics. The present paper, that I dedicate to the memory of Ilya Prigogine, is a late outflow of my early steps in quantum statistics of systems away from thermal equilibrium. My paper is also motivated by the progress made in information technology, and here again especially in nanotechnology. So far, information processing is based on logical elements that use large numbers of elementary excitations such as electrons so that quantum fluctuations can be ignored, at least in many cases. With increasing miniaturization, however, the quantum domain becomes important and thus quantum fluctuations can no more be ignored. Therefore, my contribution tries to show how quantum fluctuations can be calculated in a unique way for all kinds of elementary excitations and their interactions that may be used in logical elements (devices).

Elementary excitations
The formalism I am going to develop in this paper is applicable to electrons, holes, excitons (Wannier, Frenkel), plasmons, spin waves, and so forth. They may be delocalized or localized, for example, at quantum dots. In the following, I will base the analysis on localized states from which running elementary excitations can be built up by means of suitable superpositions. In order to realize logical elements characterized by their truth table, one may start from rate equations for electron densities that are for instance of the typical form where p and l represent gain and losses, respectively. A truth table for the logical operation "and" can be realized by where n 1 and n 2 are incoming currents of the channels 1 and 2, and n r is the resulting occupation number in the outflow. Quite clearly, n r can be produced only if both n 1 and n 2 are unequal to zero. When we proceed to small particle numbers, quantum fluctuations will play an important role and the obvious question arises how we can replace the phenomenological rate equations (2.2) by fully quantum mechanical equations.

General approach
Here I will proceed in two steps. First, I will consider a single quantum system with levels i = 1,...,N, and later composite quantum systems. We describe its dynamics by means of projection operators P i j that project the system from state j to state i. If we wish to use creation and annihilation operators for particles, we may represent P i j in the form a + i a j , where a + ,a are the creation and annihilation operators, respectively. The use of P is more general because it implies that we are dealing with elementary excitations, for instance, with polarons that are electrons surrounded by their ionic cloud. The same remark holds for other elementary excitations. If not otherwise stated, we have localized states in mind. In the following, we use the fundamental quantum mechanical property of projection operators P i j P lk = δ jl P ik . (3.1)

Hermann Haken 171
Our approach will be based on the Heisenberg equations of motion that are in the form where H represents the Hamiltonian. Most importantly in all practical applications, especially in nonequilibrium systems, the quantum system under consideration is coupled to reservoirs that give rise to damping and pumping and perhaps other incoherent effects. Thus it is absolutely necessary for a complete description to include these reservoirs. As can be shown, these reservoirs with their numerous variables can be eliminated, which gives rise to projection operator equations in the Heisenberg form but with additional terms: The detailed derivation of the operator L r,i j may be tedious, but another approach has turned out to be successful. Namely, the damping and pumping terms, and so forth, can be introduced in a phenomenological manner as I will show by means of an example. The quantum statistical average over P ii can be interpreted as particle number at point i: Then in a phenomenological way, one may describe the hopping process of that particle along a chain with sites i by means of the rate equations This is, however, a quantum statistically averaged equation where the quantum fluctuations have been lost. Our main purpose will be to restore the quantum mechanical consistency as expressed by (3.1). Another example is provided by the coherent motion along a chain where the Hamiltonian can be described by The Heisenberg equation of motion reads where we have used the property P lm ,P i j = δ mi P l j − δ jl P im (3.8) that derives from (3.1). The additional operator L r,i j stems from incoherent processes due to the interaction with reservoirs such as lattice vibrations. Taking only the nearest neighbour interaction in a linear chain into account, (3.7) can be evaluated as Again the incoherent processes can be incorporated in addition to the coherent processes determined by the Hamiltonian (3.6) by choosinḡ Equation (3.11) describes phase-destroying processes. In order to restore quantum mechanical consistency, we have to add fluctuating forces Γ i j so that the total Heisenberg equations acquire the form where, however, in the second term on the right-hand side the averagedP is to be replaced by P. We now turn to the explicit determination of the fluctuating forces Γ. [3,4] We denote quantum statistical averages by square brackets. We assume that, for instance, the following averaged equations are given phenomenologically or partly phenomenologically and partly from first principles:

Haken-Weidlich theorem
where the elements M do not depend on P, but may depend on variables of other quantum systems. As one may show, the solutions to (4.1) do not obey the quantum mechanical consistency relations (3.1). To restore quantum mechanical consistency, we introduce the equation We assume that the averages vanish: and that the fluctuating forces are δ-correlated in time: This is the only assumption to be made in the present context. In many cases, it is fulfilled if for instance the reservoirs are broadband or the relaxation time of the fluctuating where M (1) stems from i/ [H,P i j ]. As can be shown, the terms M (1) cancel each other so that it is sufficient to determine the strengths of the fluctuating forces by using M (2) instead of M in (4.5). We illustrate our result by means of an example that is self-explanatory: (4.7)

Composite quantum systems
Logical elements are realized by means of the interaction or transformations of different quantum systems as can be seen, for example, from (2.2). We must observe, however, that such relations can be translated into quantum mechanics in several ways depending on the experimental setup. For instance, the particle numbers can be translated into particle number operators according to n j l −→ P j 1,1 , (5.1) but they can also be translated into probability amplitudes according to where b j is an annihilation operator of a particle of the kind j. In the latter case, the interaction stems from the Hamiltonian of the form The details of these translations will be published elsewhere. Here, however, we want to concentrate on the central issue, namely, how to generalize the Haken-Weidlich theorem to composite quantum systems. This requires the introduction of the appropriate multiplication rules of projection operators. First, we adopt the already known rule where the upper index l refers to the specific subsystem. Similarly we have However, what is new is the relation for the composite system given by From (5.5), (5.6), and (5.7), we may deduce the following multiplication rules: For the following, we need a concise notation so that we introduce the following abbreviations: P K ĩ, j for K = 1, ĩ = 1, and j = j; for K = 2, ĩ = 1, and j = j; and for K = 0, ĩ = i 1 i 2 , and j = j 1 j 2 . With its help, we can cast the relations (5.5), (5.6), (5.7), (5.8), (5.9), and (5.10) in the concise form P K ĩ, j P L ĩ , j = h KLV ĩ, j;ĩ , j ;ĩ , j · P V ĩ j , (5.11) where, for instance, if (K,L) = 1,1, h 111 i j,i j ,i j = δ ji δ i i δ j j holds. The basic idea is now similar to that of Section 4. We assume that the averaged equations The formal solution of (5.15) reads where G is the Green's function with the property and A h a solution to the homogeneous equation (5.15). We consider

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009