The decay of quantum systems with a small number of open channels

Abstract

Recent developments in the quantum mechanical description of open systems are presented. In particular, the specific properties of the deep quantum region are pointed out, where the number of open channels is small. Based on the statistical assumptions of Random Matrix Theory, the solution of the Schrödinger equation describing generic open systems is reviewed. The connection between the scattering matrix and the decay behavior is established. On its basis, the non–exponential decay law for systems with a small number of open channels is derived. As a simple physical system possessing both a theoretical description and an experimental realization, scattering systems with leads are studied. Recent experimental results obtained from the scattering of microwaves on such cavities are shown to provide a detailed confirmation of the theoretical predictions for the decay law.

A second main topic is the behavior of quantum systems with overlapping resonances. For systems with a small number of open channels it is shown how this regime can be reached by enlarging the coupling strength between system and environment. Using Random Matrix considerations, the essential features of such systems are presented, and the differences between the decay of chaotic and of regular systems are pointed out. A class of simple scattering systems with overlapping resonances is introduced and its S-matrix properties are discussed. Finally, an extremum principle determining the gross features of the scattering matrix is suggested and investigated for the case of one open channel.

Introduction

Decay phenomena take place whenever a quantum system is coupled to its environment. Typical examples range from highly excited nuclei over mesoscopic or macroscopic cavities with attached wave guides to the overall interaction of (quantum) systems with the cosmic background radiation responsible for de-coherence phenomena. These systems are open in the sense that the (originally) bound states can be populated by a pulse coming, via some channels, from the “outside”. Vice versa, an initial population of bound states can decay into these channels.

The theoretical description of decay phenomena dates back to the early 1930s. In the pioneering work of Weisskopf and Wigner [1] it has been shown that an individual state generically dissolves at an exponential rate into the continuum of unbound states coupled to it. Later, a detailed investigation of decay properties of quantum systems has been performed in the context of nuclear reaction theory, see the classical textbook by Mahaux and Weidenmüller [2] and its precursors [3], [4], [5], [6], [7], as well as in the field of quantum chemistry [8].

In recent years, decaying quantum systems have been studied from the point of view of scattering on systems with (infinitely) high potential walls. Such systems can be realized in the form of “billiards” or microwave cavities, which allow for a detailed investigation both experimentally and theoretically. One can show that the scattering properties of such billiards are closely related to the spectral properties of the corresponding closed system [9], [10], [11]. Nowadays, one of the most active fields in quantum physics is the analysis (and design) of mesoscopic structures. Here, the openness is realized by attaching leads to a restricted area of low-dimensional electron gas, which gives rise to a variety of quantum transport phenomena, see the recent report by Beenakker [12].

Open systems allow for a dual description: on the one hand, they can be considered from the “inside” point of view, treating the coupling to the environment as a — not necessarily small — perturbation. From this point of view, one can study the (discrete) eigenvalues of the system, the widths of resonances and the resulting decay properties. On the other hand, open systems allow to take the “outside” point of view, considering the system as a perturbation of the environment. The typical quantity to be investigated from this point of view is the scattering matrix (S-matrix), i.e. the amplitude for passing from a given incoming channel to a certain outgoing channel as a function of energy.

The duality between the inside and the outside points of view has been used in order to investigate properties belonging to the system by doing scattering experiments on it. For billiard systems, a detailed study of the relation between the eigenphases of the S-matrix and the spectrum of the bound system has been initiated by the work of Smilansky et al. [13], [14]. These studies show that (under certain smoothness conditions for the boundary of the billiard) there is a one-to-one correspondence between the spectrum of the bound system and that of eigenphases of the S-matrix: Whenever the system is (at a given energy E) “transparent”, i.e. the S-matrix has an eigenvalue equal to 1, the closed system has a bound state, and vice versa [15], [16], [17].

There is a second reflection of this inside–outside duality, leading to a direct connection between the S-matrix and the decay function, i.e. the probability to find the system in a bound state after it has been excited by a pulse from the outside [18]. This connection can be used, e.g., in order to interpret experimental observations on super-conducting microwave cavities [19] in terms of quantum decay.

The results presented in this review will be derived mostly for a model which has turned out to be most appropriate for describing open quantum systems in terms of Random Matrix Theory (RMT). Its formulation starts from the assumption that the Hamiltonian describing the quantum system itself plus its environment can be decomposed into a part relevant for the system, another part describing the structure of the decay channels themselves and a third one responsible for the interaction. A comprehensive discussion of this approach can be found in the book by Mahaux and Weidenmüller [2]; for a systematic overview see the recent review by Guhr et al. [20].

The application of RMT has allowed to obtain a large number of powerful and general results both for the description of closed and of open quantum systems. Prominent examples are the derivation of analytical expressions for many statistical properties of spectra like nearest-neighbor spacing distributions and number variances [21], [22], [23], two-point correlation functions [24] or the amplitudes of wave functions [25]. For open systems, the distributions of decay and transition amplitudes have been obtained in certain limiting cases [26], [27], [28].

Random Matrix Models turn out to successfully describe the properties of quantum systems the classical counterpart of which is chaotic [29], [30], [31]; for an excellent overview, see [32]. Moreover, during the past decade the development of RMT has largely profited from the broad application of super-symmetric methods. Starting from the pioneering work of Verbaarschot, Weidenmüller and Zirnbauer [24] a huge amount of new results has been obtained; for a contemporary review on this subject see Ref. [20].

Recently, strong support for the development of RMT and its relevance for the description of chaotic quantum systems resulted from experiments on microwave cavities [10], [11], [33], [34], [35]; for a review see Ref. [36]. Taking into account the equivalence between the time-independent Schrödinger equation and the Helmholtz equation, describing the behavior of electromagnetic fields in conducting cavities, one can use such (macroscopic) experiments in order to study the properties of purely quantum systems.

A central point in the presented review will be the discussion of systems with overlapping resonances, i.e. with resonances having (on average) a separation in energy which is small compared to their typical width. Historically, this situation has been discussed first for the case of many decay channels where the widths of resonances are large because they can decay via many different decay paths. This picture describes the typical behavior of strongly interacting many-body systems [2]. As an immediate consequence of such a large resonance overlap characteristic signatures (Ericson fluctuations [37], [38]) in the cross section have been recognized.

Only recently, it has been proven rigorously [27], [28] that overlapping resonances can occur, as well, if the number of open channels is small. This is due to the fact that, quantum mechanically, the “openness” is a two-dimensional quantity: it can be enlarged either by increasing the number of open channels or by enhancing the (average) coupling strength between bound states and a given decay channel.

Note, that the case of a large number of channels corresponds to the quasi-classical regime. A good illustration for this fact provide systems with leads. In this case, each decay channel corresponds to a transverse mode in the wave guide. Obviously, in the semiclassical limit, ℏ→0, the number of open transverse modes goes to infinity, for any given diameter of the lead, and for any given energy of the wave propagating through the lead. On the other hand, the few-channel case corresponds to the deep quantum region. In the present review, we shall concentrate just on this situation omitting almost completely the presentation of semiclassical methods for the description of open quantum systems; from the broad spectrum of excellent books and review articles on this subject the works of Berry [39], Gutzwiller [40] and Smilansky [41] can be recommended.

This review is organized as follows. In Section 2 the Hilbert space description of open quantum systems is introduced. A generic Hamiltonian for a system with N bound states coupled to M decay channels is provided. A formal solution of the corresponding Schrödinger equation is obtained and an intuitive scheme for deriving the corresponding scattering matrix is given. For pedagogical reasons, two simple cases corresponding to systems with one and two resonances are considered. On the basis of these studies, a first expectation concerning the effect of overlap of different resonances is formulated.

Using the results of the previous section, in Section 3 the decay properties of quantum systems are derived. After establishing a general relation between the decay function and the S-matrix, the decay of chaotic systems is studied in detail. Using the results of Ref. [24] for the S-matrix correlation function, the decay law for an arbitrary number of open channels and for arbitrary coupling strength between bound states and continua is presented. In particular, the generically non-exponential character of quantum decay is pointed out.

In Section 4 the properties of decaying quantum systems are discussed under the assumption that the degree of overlap can be made arbitrarily large without enlarging the number of open decay channels. This corresponds to the regime of strong coupling. It is shown that this assumption leads to a separation of scales of resonance widths and, correspondingly, to a separation of time scales in the decay of the system. Analytical expressions are obtained for the pole structure of the S-matrix and for its averaged value. The differences between regular and chaotic systems in the regime of strong coupling are indicated, and results for the distribution of decay widths and other statistical quantities are presented.

Section 5 collects results on a class of dynamical systems illustrating the conclusions obtained so far. For this purpose, scattering systems with leads are analyzed in detail. As a starting point, the scattering on quasi-one-dimensional systems (“graphs”) is investigated. An explicit expression for the S-matrix of such systems is given. On its basis, the coupling matrix elements between the bound system and the decay channels are derived analytically. In order to illustrate the effect of strong coupling in such systems, we investigate periodic quasi-one-dimensional structures permitting locally an arbitrarily large degree of resonance overlap. The discussion of microwave experiments verifying the decay law derived in Section 3 concludes this section.

In Section 6, the results of the preceeding sections are discussed from a different point of view. Namely, it is pointed out, that the distribution of resonances, as obtained by diagonalizing the effective Hamiltonian, is favorable for the system in the sense of realizing its slowest possible decay. A corresponding extremum principle for quantum decay is formulated, and a numerical illustration for the case of one open channel is presented.

Finally, Section 7 contains a summary of the presented results and indicates some possible further experimental verifications.