Delay-time distribution in the scattering of time-narrow wave packets (II) - Quantum Graphs
We apply the framework developed in the preceding paper in this series  to compute the time-delay distribution in the scattering of ultra short RF pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high quantum number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time delay distribution is established.
When an ultra-short pulse of radiation is scattered on a complex medium, the emerging radiation pulse is broadened in time and the pulse shape reflects the distribution of time-delays induced in the scattering process. This distribution can be intuitively explained as due to the existence of a large number of paths of varying lengths through which the radiation can traverse the scatterer. Recently, novel methods to produce ultra-short light pulses were introduced. They opened a new horizon for experiments where the distribution of delay-times induced by scattering from complex targets can be measured, with interesting and surprising results, see e.g., [2, 3, 4]. The ultra-short pulses are realized as broad-band coherent wave packets, which are presently available only for electromagnetic radiation, but not yet for sub-atomic particles such as e.g., electrons. However, work towards this end has already begun . These developments emphasize the need for theoretical tools to aid planning of new experiments and interpret the measured results.
The preceding paper in this series  presented a general theoretical framework for the computation of the delay-time distribution in scattering of short radiation pulses on complex targets. The ingredients which are needed in this theory are the scattering matrix where is the wave-number, the pulse (wave-packet) envelope and the dispersion relation . For scattering of electromagnetic radiation the latter is where denotes the velocity of light. In this case it is convenient to express the time by the optical path-length . The general expression for the delay-time distribution is then given by
if the delay is measured for pulses impinging in channel and detected in channel .
In the present paper, we apply this general formalism to scattering on quanum graphs [6, 7, 8, 9, 10]. We do so for two reasons: First, quantum graphs are known as a successful paradigm for scattering from complex targets while at same time they are analytically and numerically much more tractable. For example we will present in this paper a full analytic solution of a model which contains some essential ingredients for complex targets such as an exponetially increasing number of scattering trajectories and relevant quantum interferences between them. Thus, studying quantum graphs in the present context might reveal typical features which are difficult to decipher in more realistic systems. Second, quantum graphs are very good models for the scattering of radio frequency (RF) signals in networks of wave-guides. As a matter of fact, experiments on the delay-time distribution in such systems are presently performed in Maryland and Warsaw [11, 12].
In the following Section 1.3 the necessary definitions and tools from the theory of quantum graphs will be provided. Then in Section 1.4 this theory will be extended to scattering on graphs and an explicit formula for the scattering matrix will be discussed. In Section 2 we apply this formula to Eq. (1) and derive on this basis approximate expressions for the delay time distribution in the case of broad envelope functions corresponding to wave packets narrow in time.
Section 3 is devoted to the clasical analogue of the delay time distribution. In particular we show that for finite and connected graphs the classical delay distribution decays exponentially for long times and calculate the decay exponent. The classical delay distribution provides both, a simple short-time approximation to the fully coherent expression (1) and a reference result which allows to highlight quantum interference contributions to (1) for longer times. Moreover, as in the above mentioned experiments with RF radiation some decoherence cannot be avoided, a satisfactory theory might involve a crossover between our results for coherent and incoherent time delay.
1.3 Quantum graphs in a nut-shell
A graph consists of a finite set of vertices and edges . It will be assumed that is connected and simple (no parallel edges and no self connecting loops). The connectivity of is encoded in the adjacency matrix : if the vertices are connected and otherwise. The set of edges connected to the vertex is denoted by . The degree of the vertex is . When , the connecting edge will be endowed with two directions , the positive direction is chosen to point from the lower indexed vertex to the higher. A pair is a directed edge. The set of all directed edges will be denoted by and is its size. The reverse of is denoted by . When is a directed edge pointing from vertex (the origin of ) to (the terminus of ) we write and , respectively.
An alternative way to describe the connectivity of is in terms of the edge adjacency matrix of dimension :
The metric endowed to the graph is the natural one-dimensional Euclidian metric on every edge. The length of an edge is denoted by and is the set of these edge lengths. The edge lengths are assumed to be rationally independent. A graph is compact when all the edge lengths are finite. The lengths of the directed edges and its reverse are equal. Denote by the coordinate of a point on the edge , measured from the vertex with the smaller index and . A function is given in terms the functions so that if . The action of the Laplacian on for is and the domain of the Laplacian is . Assume is compact. Then, the Laplacian is self-adjoint if it acts on a restricted space of functions which satisfy appropriate boundary conditions. Frequently used boundary conditions are the Neumann conditions which require the function to be continuous at all the vertices, and for every vertex, , where the derivatives at are taken in the direction which points away from . The most general prescriptions for boundary conditions were first introduced and discussed in . The time dependent wave equation (with time ) is
with the boundary conditions specified above which must be satisfied for all . The stationary equation
can be solved only for a discrete, yet infinite set of wave numbers which is the spectrum of the stationary wave equation.
A useful method of computing the spectrum of the graph Laplacian is based on the following decomposition of the wave function. Consider the functions , where and are arbitrary complex numbers. These functions are the general solutions of on all the edges. The constants should be computed so that satisfies the boundary conditions at all the vertices. Consider all the edges which are connected to a vertex . For Neumann boundary conditions the continuity of the graph wave function at the vertex imposes independent requirements on the coefficients . Namely, , where denotes the value of at the vertex , where or depending on the orientation of . Again for Neumann boundary conditions, another relation among the is imposed by the requirement that the sum of the outgoing derivatives of the at the vertex vanishes. Therefore there are linear equations which the coefficients must satisfy. Hence, if one denotes the set of directed edges which point towards by and the complementary set of outgoing directed edges by , then the boundary conditions at provide a linear relation between the two subsets of coefficients:
The symmetric and unitary matrix of dimension is the vertex scattering matrix corresponding to Neumann boundary condition at the vertex. Other boundary conditions yield different vertex scattering matrices, and their unitarity is due to the fact that the underlying graph Laplacian is self adjoint. Using the vertex scattering matrices for all the vertices on the graph, one can construct a unitary matrix
where is the unit matrix in dimension . The unitarity of for real implies that all the eigenvalues of are on the unit circle. As varies, eigenvalues cross the real axis, where the secular equation is satisfied. Therefore the spectrum is real.
is referred to as the graph evolution operator in the quantum chaos literature. Its matrix elements provide the amplitudes for scattering from an edge directed to a vertex , to an edge directed away from . Their absolute squares can be interpreted as the probabilities that a classical particle confined to the graph and moving on the edge toward the vertex is scattered to the edge and moves away from it. Due to the unitarity of the matrix
does not depend on and is double Markovian: . The transition probability matrix allows to define a random walk on the graph. For the graphs considered here, satisfies the conditions of the Frobenius-Perron theorem and therefore the largest eigenvalue of is and it is single. Suppose that at time the probability distribution to find the walker on the directed edge is given by the vector . Then, at integer time the distribution will be and converges to equidistribution for large independently of the initial probability distribution. In other words, the classical evolution on the graph is ergodic. (Note that we will use the symbol for a discretized topological time while the continuous physical time is measured in terms of the path length as in Eq. (1).)
1.4 Scattering on quantum graphs
So far we discussed the wave equation and its classical limit on a compact graph. To turn this graph into a scattering system, we choose a subset of vertices , and at every vertex we add a semi-infinite edge (lead). is the number of leads. The directed edges on the lead attached to vertex are denoted by which points away from the vertex and which points towards it. The Laplacian is extended to the leads in a natural way, and the boundary conditions at the vertices are modified by replacing by . Measuring distances from the vertex outwards, the functions which are allowed on the lead take the form . The spectrum of the Laplacian for a scattering graph is continuous and covers the entire real line, possibly with a discrete set of embedded eigenvalues (See e.g., ).
Consider the matrix
where for are the vertex scattering matrices which are modified as explained above, and for in the complement of , they take the values of the vertex scattering matrices for the compact graph. Note that is a matrix, and its entries are indexed by the labels of the directed edges in the compact part of the graph, in the same way as the original matrix of Eq. (6). However, unlike , is not unitary, because some of its building blocks, namely the vertex scattering matrices are not unitary when they are restricted to the directed edges in the compact part of the graph.
The analogue of defined in (8) for the non-compact graph is
It is independent of and sub-Markovian since for the sums and are strictly less than . The Perron-Frobenius theorem guarantees that the spectrum of is confined to the interior of the unit circle. For a random walker whose evolution is dictated by , the probability to stay inside the compact part of the graph approaches zero after sufficiently long time. This is due to the walks which escape to the leads and never return.
Consider now a solution of the stationary wave equation for a given subject to the condition that the wave function on the leads has the form . The scattering matrix for a non compact graph is a unitary matrix of dimension which provides the vector of ”outgoing amplitudes” in terms of the vector of ”incoming amplitudes” . It follows from the linearity of the wave equation that
Here, is the back reflection amplitude, is the transmission amplitude from the lead to the edge in the compact part of the graph, and is the transmission amplitude from an edge in the compact graph to a lead . The first line in (1.4) expresses the fact that scattering proceeds by either reflecting from the incoming lead back to itself (the term outside the sum), or by penetrating to the compact part and scattering inside it several times before emerging outside. The contribution of the scattering process in the compact graph is provided by the expression in curly brackets. It can be rewritten as
Here, counts the number of vertices on a path connecting the entrance and exit vertices and . is the set of all the paths crossing vertices which start on and end at after traversing directed edges with . Each path is of length . The term occurs only when and are neighbors on . Then is the directed edge connecting to , consists of the single bond , and . For the amplitudes can be written as
The series in the first line of (1.4) converges to the expression in the second line for any real because is sub-unitary. The explicit form of the matrix provided in Eq. (1.4) will be used in the next sections. An alternative expression for which will not be used here can be found in .
2 Scattering of wave-packets and the delay-time distribution
Given a graph with leads to infinity as defined above, we consider a particular solution of the stationary wave equation with wave number , where the wave function consists of an incoming wave with unit amplitude in a single lead but outgoing waves in all the leads. Limiting our attention to a specific lead the wave function has the form
The last equality follows from the definition of the scattering matrix. A time-dependent solution describing the propagation of a wave packet is obtained by a superposition of functions with an envelope function . As in  is positive and normalized by . Assuming a linear dispersion relation (such as e.g. for electromagnetic waves in transmission lines), the intensity of the outgoing wave function in the position at time is
which is the analogue of equation (11) in . The unitarity of guarantees the conservation of probability.
For a Gaussian envelope,
and under the condition , one can approximate the delay-time distribution by (see (17) in )
In the above result, we did not include the reflections from the vertex (which correspond to zero delay). To render the discussion more transparent, we shall proceed in the limit where is very large, which allows to write the first square bracket above as a Kronecker and the last square bracket as a Dirac functions, resulting in
This expression can be simplified by recalling that the length of any path can be written as where are non negative integers whose sum is . Note that does not depend on the direction in which the edges are traversed. Because of the rational independence of the edge lengths, paths which share the same length must share also the same sequences , and they are distinct if they cross the same edges the same number of times but in different order. Figure 1 shows an example for such isometric but topologically distinct paths. Denote with . The set of isometric paths which share the code will be denoted by . Then
where the probabilities contain all interference effects between the isometric paths belonging to . The result of these interferences is determined by the phases of the individual amplitudes . These in turn depend on the phases of the elements of the vertex-scattering matrices encountered along the path but they are independent of the precise values of the edge lengths. Thus, the only information about the actual lengths of the graph edges in the delay time distribution comes from the Dirac delta functions concentrating at the path lengths .
It is convenient to define the cumulative probability
where is the Heavyside function. Clearly, is a non-decreasing function of s. On the other hand it depends parametrically on the edge lengths and is a non-increasing function of any , because these lengths appear only in the arguments of the Heavyside step functions. We can use this fact to bound from below and above by similar expressions with modified edge lengths. To this end define the function
where all edge lengths have been replaced by one and the same value . Note that this is a formal definition and not related to the delay distribution of a graph with equal edge lengths, because Eq. (23) was derived under the assumption of rationally independent lengths. Take now being the maximum of the edge lengths of the graph under consideration. For the same value of the arguments of the Heavyside functions in Eq. (24) are smaller (or equal) in comparison to Eq. (23) and thus in Eq. (24) less terms contribute. Repeating this argument for we see that the cumulative delay distribution can be bound from below and above by
Note that . Thus it suffices to calculate for integer values of . We refer to this quantity as the cumulative probability for the topological delay time , i.e. the number of edges along the walk. Since there is no metric information to consider, is typically easier to calculate than the full expression in Eq. (23).
To proceed, write , where
The partition of the delay-time distribution into the Diagonal part and the Non-Diagonal part separates the purely ”classical” contribution from the contribution from the interference of waves which propagate on isometric paths. The former will be studied in the next section. Sometimes, (when e.g., and the graph is not invariant under geometrical symmetries) the contribution of can be ignored upon further averaging. However this is not always the case, especially since the number of isometric trajectories may increase indefinitely with [15, 16, 17], and the sums do not necessarily vanish in spite of the fact that the individual contributions have complicated, seemingly random phases.
While some general properties of the classical time-delay distribution (26) can be derived as presented in the next section, there are no analogous results pertaining to the complete expression in Eq. (22). However, in section 4 we shall apply all results of the present and the following section to a simple graph and derive analytical results for both, the classical and the quantum delay distribution.
3 The classical delay-time distribution
In the present section we provide a classical description of the delay-time distribution. It is a valid approximation when quantum interference effects are negligible, either because of decoherence mechanisms in the scattering process or for short times, when the contributing trajectories do not have isometric partners. For long times and coherent dynamics a comparison to the reference provided by the classical description can highlight the features of the delay distribution which are due to genuine quantum (wave) properties of the scattering process, e.g. an enhancement of long delay times (algebraic vs. exponential decay) in Section 4.
In the classical analogue of the scattering process described above, one considers a classical particle which moves with a constant speed on the incoming lead , and its probability to enter the graph through an edge is . Reaching the next vertex after traversing a distance , it scatters into any of the connected edges with probability (10) and so on until it leaves the graph from the edge to the lead after being scattered on intermediate vertices. The length of the traversed trajectory between the entrance and exit vertices is . Thus, the delay-time distribution is
This expression could be further reduced by grouping together trajectories which share the same lengths, and the result reproduces the expression for given in Eq. (26).
Again, it is convenient to define the cumulative probability,
in complete analogy to Eq. (23). Again the cumulative probability is monotonically decreasing as a function of the edge lengths since all the factors multiplying the Heavyside function in (29) are positive. Hence one can bound in a similar way as in (25).
We will now derive the leading asymptotic behavior of for large time. To this end we consider the Laplace transform of Eq. (28)
and is a diagonal matrix with entries . Note that according to Eq. (31) the poles of are related to the zeroes of and the residues at these poles can be computed with Jacobi’s formula (adj = adjugate):
The idea is now to use this information about the the analytic properties of in order to invert the Laplace transform by a complex contour integral. This procedure can be put on a solid basis by applying the Wiener-Ikehara theorem to Eq. (31). Using the results of  one gets
where is the largest real zero of . It depends on both the graph connectivity and the set of edge lengths . Eq. (35) is the main result of the present section.
4.1 The T-junction model.
As an example we choose a graph which is simple enough to allow for an analytical treatment and still rich enough to exhibit all aspects of the theory outlined above. In particular the model demonstrates the influence of quantum interferences on the delay distribution, from Eq. (27). The graph consists of two edges (, ) which are connected at a central vertex. Moreover, at this vertex a single scattering lead is attached. Thus the central vertex has the total degree three. Both internal edges end in vertices of degree one with Neumann b.c. The graph can be depicted as shown in Fig. 2 and we refer to it as a T-junction. In order to specify the model completely we need to define the lengths of the two edges and the 33 scattering matrix of the central vertex. For the lengths we choose two rationally independent values such that the total length is . This is no restriction of generality as the delay time scales proportionally to this quantity. Our choice for is motivated by analytical simplicity,
Here the lower right 22 block describes the scattering within the interior of the graph. Our calculations are simplified by the fact that in this block no phases must be considered. The first column and the first row contain the transition amplitudes from the scattering lead into the graph and back. The amplitude at the central vertex for a direct back scattering into the lead is zero, .
Note that according to  any choice of a unitary scattering matrix at some fixed wave number is compatible with a self-adjoint Laplacian. However, this choice also fixes the variation of with wave number which depends on the parameter . As we consider here an envelope function with a width we can approximate and ignore the energy dependence of the vertex scattering matrix.
4.2 The S-matrix.
(see A for details). The first line of is a compact representation which is suitable for numerical calculations and clearly highlights the resonance structure of the scattering matrix. The second line is an expansion of in terms of families of isometric trajectories starting and ending on the scattering lead. These families are labelled by pairs counting the number of reflections from the the first and second outer vertex, respectively. Trajectories which are restricted to a single edge are accounted for by the second sum. In the notation of Eq. (21) the numbers q defining a family count the traversals of directed bonds. However, in our simple model, an edge is always traversed outward and inward successively, thus and . We will refer to the integer value as the topological time of a path on the T-junction graph. As in Eq. (13) the oscillating phase factors in Eq. (38) depend on the total length of the trajectories within a family,
while the rational prefactors represent the sum of amplitudes from all trajectories within a family, as in Eq. (21).
Fig. 3 shows the time delay density computed with Eqs. (16), (17), (37) by a Fourier transform of the scattering matrix . The two curves correspond to two different envelope widths . As predicted above in Eqs. (19), (20) a series of sharp peaks centered at the lengths of scattering trajectories develops as grows. For example, the first two peaks at and each correspond to a single scattering trajectory which enters the graph, visits one of the outer vertices 1 or 2 and returns to the lead. However, to most of the peaks more than one trajectory contributes and their interference, expressed by the rational prefactors in Eq. (38), determines the height of the peak. For growing time , an increasing fraction of peaks have a separation of the order of or smaller and overlap. This is a limitation to Eq. (20) and the subsequent theory. An example at is magnified and compared to the prediction of Eq. (19) in the inset Fig. 3(b).
4.3 The topological delay time distribution.
Within the asymptotic approximation for broad envelope functions (short pulses), Eq. (20), we can evaluate the (cumulative) distribution of delay times (23) for the T-junction. According to Eqs. (21)-(23) the squared coefficients from Eq. (38) provide the weigth of a family and we obtain
As in Eq. (25), this function can be bound from below and above by a variation of the edge lengths. Define to denote the r.h.s of Eq. (40) with both edge lengths , replaced by some value such that is . Then the Heavyside functions in Eq. (40) are and select all terms with topological times up to (the largest integer below ). Thus, if denotes the sum of coefficients of all terms with some fixed topological time , is the cumulant sum
Starting with the substitution we can evaluate as
while and . Eq. (43) can be found with the help of standard computer algebra, and a formal proof can be based on the methods outlined in . is a normalized discrete probability distribution (the distribution of topological time delays) and its cumulant sum is