Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results
Scattering is a ubiquitous phenomenon which is observed in a variety of physical systems which span a wide range of length scales. The scattering matrix is the key quantity which provides a complete description of the scattering process. The universal features of scattering in chaotic systems is most generally modeled by the Heidelberg approach which introduces stochasticity to the scattering matrix at the level of the Hamiltonian describing the scattering center. The statistics of the scattering matrix is obtained by averaging over the ensemble of random Hamiltonians of appropriate symmetry. We derive exact results for the distributions of the real and imaginary parts of the off-diagonal scattering matrix elements applicable to orthogonally-invariant and unitarily-invariant Hamiltonians, thereby solving a long standing problem.
Distributions of Off-Diagonal Scattering Matrix Elements: Exact Results
Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany
03.65.Nk, 11.55.-m, 05.45.Mt, 24.30.-v
Scattering is a truly fundamental issue in physics Scattering (); FKS2005 (). A major part of our information about quantum systems stems from scattering experiments. Rutherford’s gold-foil experiment RGM1911 () is a classic example which led us towards the understanding of the atomic structure. Even in modern times, powerful particle accelerators rely on scattering experiments to probe deeper and deeper into the structure of matter. Moreover, scattering plays a crucial role in classical wave systems as well and one can often relate the relevant observables to the scattering parameters. Along with the atomic nuclei FPW1954 (); R1991 (); BLR1993 (); DPS1996 (); MRW2010 (), atoms TNC1989 (); GDG1993 (); DZD1995 (); MG1994 () and molecules Gaspard1993 (); SKSD1995 (); LS1993 (), some of the other important examples where scattering phenomena have been of considerable interest are mesoscopic ballistic devices BM1994 (); Stone1995 (); SS1988 (); IWZ1990 (); Altland1991 (); Zirnbauer1992 (); MMZ1994 (); PWZLW1995 (); MB1995 (); BM1996 (); Beenakker1997 (); MK2004 (), microwave cavities SS1990 (); Sridhar1991 (); SSS1995 (); DSF1990 (); SAOO1995 (); ABGHPRS1993 (); AGHHLRSW1995 (); SKBLS2003 (); KMMS2005 (); HZAOA2005 (); HHZAOA2006 (); KHMS2008 (); D2009 (); YHBAOA2010 (); D2010a (); D2010b (), irregular graphs HTBKS2005 (); LHBSS2008 (), quantum graphs KS2000 (); GA2004 (); PW2013 (), elastomechanical billiards EGLNO1996 (); KSW2005 (); AEOGS2010 (), wireless communication YAOA2012 (); YOAA2012 (); BC2001 () etc.
The scattering process can be completely described in terms of the scattering matrix ( matrix). It relates the asymptotic initial and final Hilbert spaces spanned by a quantum system undergoing the scattering process. In simple words, it relates the incoming and outgoing waves. In a quantum mechanical context these are the wave functions, i.e. the probability amplitudes. However, in classical systems, the waves are the displacement vectors in elastomechanical systems or the electromagnetic field in microwave cavities. The flux conservation requirement constrains the matrix to be unitary, i.e., . As a consequence of the complicated dependence on the parameters of the incoming waves and the scattering center, scattering is quite often of chaotic nature. Accordingly one needs a statistical description of the scattering phenomenon and hence of the matrix, i.e., to describe the matrix and related observables in terms of correlations functions and distributions. Two standard approaches in this direction are the semiclassical approach BS1989 (); Smilansky1990 (); DS1992 (); Richter2000 () and the stochastic approach AWM1975 (); Weidenmuller1992 (); Mello1993 (); MPS1985 (). The former relies on representing the -matrix elements in terms of a sum over the classical periodic orbits, starting with the genuine microscopic Hamiltonian representing the system. The latter, in contrast, relies on introducing stochasticity to the scattering matrix or to the Hamiltonian describing the scattering center. Both of these have their advantages and drawbacks. For instance, the semiclassical approach suffers the restriction caused by an exponential proliferation of classical periodic unstable trajectories. It is further constrained by the formal condition which demands that the number of open channels be large and therefore does not cover all interesting cases. The stochastic approach, on the other hand, gets restricted by the very nature of the stochastic modeling. Moreover, in this case, one can expect only to explore the universal aspects, leaving aside the system specific properties. The comparison between these two approaches has been discussed in detail in LW1991 ().
As indicated above, within the stochastic approach, one can pursue one of the following two routes. In the first one, the matrix itself is regarded a stochastic quantity and is described by the Poisson kernel. Its derivation is based on imposing minimal information content along with the necessary conditions like unitarity, analyticity etc. This route was pioneered by Mello and coworkers and is often referred to as the Mexico approach Mello1993 (); MPS1985 (). The second path relies on introducing the stochasticity at the level of the Hamiltonian describing the scattering center. For this, one employs the random matrix universality conjecture and models the system Hamiltonian by one of the appropriate random matrix ensembles BFFMPW1981 (); Mehta2004 (); GMW1998 (). This path was laid by Weidenmüller and coworkers AWM1975 () and is referred to as the Heidelberg approach. Even though these two stochastic approaches appear very different in their formulation, they describe precisely the same quantity, the matrix. Naturally, one would expect that these two routes are equivalent. Indeed it was shown by Brouwer that the Poisson kernel can be derived using the Heidelberg approach by modeling the scattering-center Hamiltonian by a Lorentzian (or Cauchy) ensemble of random matrices Brouwer1995 (). Since the universal properties depend only on the invariance properties of the underlying Hamiltonian BFFMPW1981 (); Mehta2004 (); GMW1998 (), his result established the equivalence between the two approaches. Furthermore, very recently Fyodorov et al. have demonstrated this equivalence for a broad class of unitary-invariant ensembles of random matrices FKN2013 ().
In their pioneering work Verbaarschot et al. VWZ1985 () calculated the two-point energy correlation functions by implementing the supersymmetry technique Efetov1983 (); Berezin1987 (); Efetov1997 (); Guhr2010 () within the Heidelberg approach. Their result established the universality of the -matrix fluctuation properties in chaotic scattering. Further progress in characterizing the -matrix fluctuations was made in DB1988 (); DB1989 () where the authors derived up to the fourth-moment. In Refs. RFW2003 (); RFW2004 () a related problem of statistics of transmitted power in complex disordered and ray-chaotic structures was solved. The Landauer-Büttiker formalism Beenakker1997 (); MK2004 (); SS1988 () gives the quantum conductance of mesoscopic systems (quantum dots and quantum wires) in terms of the scattering matrix elements. In these systems the Heidelberg approach has been used to calculate the average and variance of conductance in Refs. IWZ1990 (); Altland1991 (); Zirnbauer1992 (); MMZ1994 (); PWZLW1995 (). These results served as important steps in our understanding of the nature of scattering in chaotic systems. However, a more stringent investigation of the universality at the level of individual -matrix elements requires information beyond that of a first few moments SKBLS2003 (); KMMS2005 (); HZAOA2005 (); HHZAOA2006 (); KHMS2008 (); D2009 (); YHBAOA2010 (); D2010a (); D2010b (). A complete description is provided only by the full distributions which is equivalent to having the knowledge of all the moments. In the limit of a large number of open channels and a vanishing average matrix, or equivalently, in the Ericson regime of strongly overlapping resonances MRW2010 (); AWM1975 (), the real and the imaginary parts of the -matrix elements exhibit Gaussian behavior. However, outside this regime the unitarity of the matrix results in significant deviations from the Gaussian distribution MRW2010 (); DB1988 (); DB1989 (); T1975 (); RSW1975 (). The available moments up to the fourth are insufficient to determine the exact behavior of these distributions in a general case. A significant progress in characterizing the behavior of diagonal -matrix elements in the general case was made in FSS2005 () where the authors succeeded in deriving the full distributions. The off-diagonal elements, however, could not be tackled by the well established methods and the problem of finding their distributions remained unsolved till very recently KNSGDMRS2013 (). Here, we derive an exact solution to this problem and present results which are valid in all regimes. We gave a brief presentation of the results in Ref. KNSGDMRS2013 (). In the present work we provide a full-fledged derivation with all the details, as well as some more new results concerning the statistics of off-diagonal -matrix elements. Our approach is based on a novel route to the nonlinear sigma model which involves obtaining the characteristic functions associated with the distributions. This is different from the usual approach where one formulates an appropriate generating function for the -matrix correlations. By contrast, the characteristic function is the moment generating function.
The paper is arranged as follows. In Section 2 we set up the model for the scattering process using the Hamiltonian formulation and implement the Heidelberg approach to introduce stochasticity. In Section 3 we define the quantities to be calculated, viz., the probability distributions for off-diagonal matrix elements and the associated characteristic functions. Section 4 deals with the exact results for unitarily-invariant Hamiltonians, which apply to systems with broken time-reversal invariance. Section 5 gives the exact results for orthogonally-invariant Hamiltonians, which apply to “spinless” time-reversal invariant systems. We conclude in Section 6 with a brief discussion. In the appendices, we collect some of the derivations.
2 Scattering Matrix
In the generic setting of the scattering problem the scattering event is assumed to take place inside only a certain part of the available space. Outside this “interaction region” the fragments exhibit a free motion which is characterized, along with the energy , by a set of quantum numbers. The states corresponding to these quantum numbers represent the states in which the fragments exist asymptotically before or after the scattering event and are referred to as channels of reaction.
We associate with the compact interaction region a discrete set of orthogonal states , which represent the bound states of the Hamiltonian describing the “closed” chaotic system. Moreover, we assume that at given energy there are exactly open channels, described by a continuous set of functions satisfying the orthogonality condition . The full Hamiltonian for the system can therefore be written as
Here describes the part of the Hamiltonian which is present without any interaction between the internal states of the system Hamiltonian and states of the open channels, viz.,
and represents the interaction part,
Here represents the threshold energy in a given channel , and thus integrals in the above two equations run over the energy region where the channel is open. are the -dimensional coupling vectors which encode the information about the interaction. In Eq. (2) any direct interaction between channels has been neglected for simplicity, thereby rendering the second term diagonal in . Furthermore, the dependence of the coupling vectors on energy has also been ignored as we are interested in a situation where the mean level spacing between the resonances is very small compared to the mean level spacing between the channel thresholds.
where the inverse of the resolvent is given by
The above -matrix ansatz provides the most general description of any scattering process in which an interaction zone and scattering channels can be identified. For the characterization of the matrix one has to specify properties of the coupling vectors . A convenient choice corresponds to the case when the average matrix is diagonal, viz., VWZ1985 (); LW1991 (). In this case the coupling vectors can be chosen to obey the following orthogonality relation VWZ1985 ():
This choice corresponds to the absence of any direct coupling between the channels VWZ1985 (); LW1991 (). An alternative choice which also fulfills the condition of being diagonal is of considering the elements of as zero-mean independent Gaussians with variances proportional to . It turns out that these two choices are equivalent as long as LSSS1995a (); LSSS1995b (), which is exactly the case we are interested in. We consider the former choice in our calculations , i.e., Eq. (6). We would like to remark here that for the case of a non-diagonal average matrix, a unitary transformation can always be found such that is diagonal on average and has the same fluctuation properties as the matrix without direct reactions VWZ1985 (); EW1973 (). Thus it suffices to consider a case which omits direct reactions.
We now evoke the random matrix universality conjecture, according to which the universal and generic properties of chaotic systems can be extracted by modeling the underlying Hamiltonian (or its analogue) by an ensemble of random matrices of appropriate symmetry class. We consider here the Gaussian ensemble of random matrices to model the interaction-region Hamiltonian . This particular choice of distribution is only for calculation convenience since it is known that for the universal properties, as long as one takes into account the proper invariance properties of the Hamiltonian to be modeled, the choice of distribution is immaterial. See for example MZ2010 () where the authors calculate the two-point correlation function by considering an arbitrary invariant Hamiltonian.
Depending on whether the system is time-reversal invariant or noninvariant, is chosen to belong to the Gaussian Orthogonal Ensemble (GOE) or the Gaussian Unitary Ensemble (GUE) BFFMPW1981 (); Mehta2004 (); GMW1998 (). These two ensembles are designated by the Dyson index and have the following probability distribution associated with them:
The GOE and GUE are described respectively by and . in the above equation represents the dimensionality of which is essentially the number of bound states, and is a free parameter which can be chosen to fix the energy scale. For , is a real-symmetric matrix and has independent parameters. On the other hand for , is Hermitian and involves independent parameters. For we obtain from Eq. (7), for both values of , the density of eigenvalues as the Wigner semicircle Mehta2004 (),
The level density is , and consequently the mean level spacing is , which for large behaves as in the bulk of the spectrum.
3 Distributions and Characteristic Functions
We are interested in the statistics of the off-diagonal elements of the matrix. The off-diagonal -matrix elements relate the amplitudes in different channels. Thus their statistical information is as important, if not more, as that of the diagonal -matrix elements which relate the amplitudes within the same channel. The -matrix elements being complex quantities, we need to investigate the behavior of their real and imaginary parts or equivalently that of their moduli and phases. We consider here the distributions of the real and imaginary parts and deal with them simultaneously. We introduce the notation , with to refer to the real and imaginary parts of respectively. For the off-diagonal case, by setting we obtain from Eq. (4),
To find the corresponding distributions, , we need to perform the following ensemble average:
Here the volume element represents the flat measure involving the product of the differentials of all independent variables occurring within . As mentioned in the introduction, we analyze the corresponding characteristic function, viz.
The characteristic function also serves as the moment generating function, i.e., all the moments of the real and imaginary parts of the -matrix elements can be obtained by expanding in powers of . The expression for can be retrieved from the Fourier transform of as
Thus our strategy is to calculate the characteristic function first and then obtain the distribution from it by taking the Fourier transform.
We introduce a -component vector involving the coupling vectors , and a -dimensional matrix composed of the resolvent as
The characteristic function can be expressed in terms of these quantities as
To evaluate we need to integrate over the ensemble of -matrices defined by Eq. (7). In general this is a nontrivial task, more so when the quantity to be averaged does not respect the invariance properties of , which is the case here. Further complications are caused here because of the extremely convoluted dependence of in the exponent in Eq. (14) – it appears in the denominator of the resolvent contained in the matrix . To overcome this problem we seek some trick which will invert the in Eq. (14), thereby rendering the exponent linear in . As we will see below, the supersymmetry formalism provides exactly such a solution Efetov1983 (); Berezin1987 (); Efetov1997 (); Guhr2010 ().
We introduce a -dimensional complex vector consisting of commuting (Bosonic) variables. Similarly we introduce a -dimensional vector consisting of anticommuting (Fermionic or Grassmann) variables. We note that the indices in these vectors are just dummy indices and do not have any direct dependence on the values of the indices signifying the -matrix element. We now consider the following multivariate Gaussian integral results for commuting and anticommuting variables:
In Eqs. (15) and (16), is an arbitrary normal matrix with complex entries. b, c in Eq. (15) are vectors consisting of commuting entries, while in Eq. (16) are vectors having anticommuting entries. The volume elements and in the above equation are given by and . The above two Gaussian-integral identities enable us to recast the characteristic function, Eq. (14), in the following form:
Eq. (17) can also be expressed in terms of an integral over a -dimensional supervector as:
Here and are -dimensional square-matrix and vector respectively. An ensemble average over an exponential of a bilinear form involving supervectors and a matrix, as in the equation above, is common in supersymmetry calculations. However, there is a difference here: is not in block-diagonal form. If we carry out the ensemble average using this form of it will result in problems incorporating the correct symmetry properties in the supermatrix which has to be introduced later. To resolve this problem we employ the trick of carrying out certain transformations in and , while leaving and as they are. This is allowed since and , being complex quantities, admit independent transformations.
To proceed further from this point, we have to take into account the appropriate symmetry of , i.e., whether it belongs to the GOE or GUE.
4 Unitarily Invariant Hamiltonians ()
We consider in this section the case when is modeled by the GUE, and thus is applicable to systems with completely broken time-reversal symmetry. In this case the Hamiltonian is complex-Hermitian BFFMPW1981 (); Mehta2004 (); GMW1998 (). The route to the final results will consist of three steps: (i) mapping the above result to a matrix integral in superspace, (ii) applying the large -limit and obtaining the nonlinear -model, and finally (iii) reducing the result to integrals over the radial coordinates.
4.1 Mapping to a matrix integral in superspace
As mentioned in the previous section we want to bring in Eq. (18) into a block-diagonal form. To accomplish this we consider the following transformations in the vectors:
The different transformations for and ensure proper symmetry and convergence properties of the supermatrix when we map the problem to a matrix integral in superspace. The Jacobian factor arising from the above transformations is . We therefore obtain
where and results from of Eq. (18) because of the rotation of the supervector . The new -dimensional matrix in the above equation is block diagonal as desired.
We now examine the -dependent part in the exponent in Eq. (21). It possesses the bilinear form, , which can also be written as , where
The Hermiticity of the matrix is evident. The GUE averaging in Eq. (21) therefore amounts to performing the following integral:
where the variable incorporates parameters of the problem as
where and and . The Boson-Boson and Fermion-Fermion blocks of the supermatrix are Hermitian, while the other two blocks are adjoints of each other, and therefore is Hermitian itself. We have
We have introduced here which is the -independent part of , viz.
and map the problem to a matrix integral in superspace by introducing a 4-dimensional supermatrix having same symmetry as . Also, observing that
we arrive at
with being the shifted matrix,
where and sdet represents the superdeterminant Efetov1997 (); Guhr2010 (). The supersymmetric representation given in Eq. (34) constitutes one of the key results of our paper. We have accomplished the difficult task of ensemble averaging and mapped the problem to a matrix integral in superspace. The parameter which occurred earlier implicitly also, as the dimension of matrix and supervectors, is now completely an explicit parameter in the integrand. To begin with we had independent integration variables (for ). We now have overall 16 independent integration variables. Thus we have achieved a considerable reduction in the number of degrees of freedom of our problem. This is one of the powerful aspects of the supersymmetry method.
4.2 Large- limit and nonlinear model
Our interest is in the limiting case of many resonances coupled with few open channels . Thus we analyze the characteristic function in a large- limit. From the above results we conclude that has the form
where the free energy and the perturbation around it are given respectively by
We observe that the term in the free energy is of order with respect to the term . The dominating part, , is invariant under the conjugation by which belongs to the Lie superspace . The sub-dominant part breaks this symmetry to . We now fix and apply the saddle point approximation to consider the limit. The saddle point equation is obtained by the first variation of the exponent as
which has the diagonal solution
where . We note that is the Wigner semicircle given in Eq. (8). The full solution to Eq. (42), which contributes to the integral in Eq. (39) in the limit, is a continuous manifold of the saddle point solutions described by
with . belongs to the coset superspace U(1,1/2)/[U(1/1)U(1/1)] and satisfies the conditions , .
We consider the solution as where represents the fluctuations around . The parts and may be referred to as the Goldstone and massive modes respectively SW1980 (). Substitution of this solution in Eq. (39) and expansion of the terms up to second order in leads to a separation of the Goldstone modes and the massive modes . The integrals over the massive modes are Gaussian ones, and therefore can readily be done and yield unity. We are therefore left with the expression of depending on the Goldstone modes only:
The as well as the in the above equation should now be interpreted as Eqs. (33) and (37) with in all the ingredients replaced by . Eq. (45) is the nonlinear-sigma model for our problem and constitutes another key result. We would like to emphasize that to arrive at this equation we followed a novel route which is based on the characteristic function. This is different from the earlier approaches where one starts with a generating function with source variables. It is also worth mentioning that the superdeterminant part in this equation is the same as those obtained in the earlier works FS1997 (). The exponential part, however, is new in our result and carries the information specific to the present problem.
We now use the parametrization of given in FS1997 (). It involves the pseudo eigenvalues , , angles and four Grassmann variables . To make the paper self-contained we present this parametrization in Appendix D. As suggested there we write , and since and commute, the product over the superdeterminant only depends on . Then it is straightforward to calculate it as
which depends only on the pseudo eigenvalues but no other integration variables. Here we defined
where is related to the transmission coefficient or the sticking probability as VWZ1985 (); FS1997 (). The transmission coefficient corresponding to a given channel signifies the portion of the flux which is not reflected back immediately, but penetrates the interaction region and participates in the formation of the long-living resonances LW1991 (). We will need later on in the case. can be referred to as the channel factor since the number of channels appears explicitly in this term only.
To evaluate the exponential contribution in Eq. (45) we have to explicitly calculate defined in Eq. (37) with replaced by . This has been done in Appendix E. On using the parametrization of Appendix D in this result we obtain,
We have introduced here
Thus we have the explicit expressions for all the terms in Eq. (45) in terms of four commuting and four anticommuting variables parametrizing the supermatrix . As we can see, in contrast to the channel factor, the exponential part depends on all integration variables.
4.3 Reduction to integrals over the radial coordinates
In this final step we perform the integral over the Grassmann variables and the angles to obtain the results in terms of radial coordinates only. Performing the Grassmann integrals amounts to expanding the exponential in the Grassmann variables and picking out the coefficient of . All terms with other combinations of Grassmann variables vanish. The nonvanishing term turns out to be
The factor comes from the convention followed in the definition of Grassmann integration. In the second line of the above equation we have recast the expression in front of the exponential as a differential operator acting on the exponential term. We note that there is no dependence in the integrand, the integral over it therefore just gives a factor of . The integral over can be performed using the following result GR2000 ():
Here represents the zeroth order Bessel function of the first kind. Application of the differential operator on this result then gives the desired expression as a 2-fold integral involving Bessel functions , which can be further simplified using their recurrence relations.
It is quite natural to expect a Rothstein or Efetov-Wegner contribution in this result Efetov1983 (); Efetov1997 (); Rothstein1987 (); KKG2009 (); Kieburg2011 (). It is a consequence of the particular choice of the parametrization of the supermatrix and comes from the term of zeroth order in the Grassmann variables, which is the product of a divergent result from the integration over the Bosonic variables and zero from the Grassmann integrations. It is known that there is no contribution from the second order Grassmannian term in accordance with the results due to Efetov and Zirnbauer Zirnbauer1968 (); Efetov1983 (). In our case the Efetov-Wegner contribution is ‘1’ in Eq. (52) below. It is essential to produce the correct value due to the normalization conditions for and ; see Eqs. (11), (12). We obtain for both real () and imaginary () parts identical expressions for the characteristic function,
Here is the Heaviside-theta function, assuming the value 0 for and 1 for . We obtain
The delta-function singularity at in the expression for gets canceled by another delta-function singularity hidden in the integrals. These singularities, however, do not create any problem if the evaluation of is carried out using Eq. (55).
The expression for the th moment can be easily obtained from Eq. (52) by using the series expansion of the Bessel function and examining the coefficients of . We have for ,
and for ,
We found above that the distributions of real and imaginary parts are equal in this case. It is therefore clear that the phase will have a uniform distribution, i.e.,
and that the joint density of real and imaginary parts will be a function of only. The last observation can be used to calculate the distribution of modulus also. As shown in the Appendix F we obtain for ,
with assuming the value,
Note that is normalized as
As we can see in this case the characteristic function and the distributions have dependence on the parameters of the problem via (or equivalently the
transmission coefficients) only. This is similar to the earlier results for -matrix element correlation functions, and distributions of diagonal
elements VWZ1985 (); FSS2005 (). We also note that the cross sections are given by the squared-moduli of the -matrix elements. Consequently we have
access to their distributions also. This is of particular relevance for the experiments where only cross sections are accessible.
All the above analytical results can be easily implemented in Mathematica Mathematica (). The corresponding Mathematica codes can be found as the supplemental material available with Ref. NKSG2014 (). To test these analytical results we also performed numerical simulations. These simulations were performed with an ensemble of 50000 random matrices of dimensions from the GUE. For each random matrix we obtain the matrix element . For the distributions we plot the histogram of such -matrix elements (real part, imaginary part or the modulus) obtained from the ensemble of -matrices. For the characteristic functions, instead of obtaining them from the distributions via an inverse Fourier transform, guided by Eq. (11), we use , where represents the number of matrices considered in the ensemble. In Figs. 1 and 2 we show the plots for (a) the characteristic function, (b) the distributions for the real and imaginary parts and (c) the distribution of the modulus of the scattering matrix element . The parameters used for the plots are indicated in the captions. The choice of parameters for Fig. 1 is such that the transmission coefficient is unity for all the channels (perfect coupling). In this case the matrix belongs to the Haar measure on the unitary group . In other words it is a member of Dyson’s circular unitary ensemble (CUE). In Fig. 2 we choose values of parameters which corresponds to an ensemble far from the CUE. As we can see the analytical predictions and the numerical simulation results are in excellent agreement in all cases.
It is known that in the case of strongly overlapping resonances (the Ericson regime) the distributions of real and imaginary parts can be well approximated by a Gaussian distribution MRW2010 (),
where the variance of the distribution , which is same as the second moment (the mean being zero), is determined using Eq. (57). In the same limit the the moduli become Rayleigh distributed, viz.
The corresponding characteristic function is given under this approximation by
We test these approximations in Fig. 3. Figs. 3 (a), (b), (c) show the characteristic function, distribution of the real (or imaginary) part and distribution of the modulus for channels which possess identical values for the transmission coefficient, . Similarly Figs. 3 (e), (f), (g) show these quantities for channels, each having the transmission coefficient . The solid lines are exact results while the dashed lines represent the approximations as in the above equations. In figures (b), (e) and (c), (f), for clarity, the plots have been shown respectively for and , instead of [-1,1] and [0,1]. Note that although we focus on the element , all off-diagonal elements will exhibit the same statistics because of identical choice of for all channels. Using the Weisskopf estimate BW1952 (); DRW2011 (),
we find that the ratio of the average resonance width () and average resonance spacing () is for the former choice and for the latter. The first case corresponds to that of weakly overlapping resonances, while the second one is closer to the Ericson regime of strongly overlapping resonances. As expected, we can see significant deviations from the approximate results in Figs. 3 (a), (b), (c), while in Figs. 3 (d), (e), (f) the approximations work quite well.
It is worth mentioning that Eq. (60) also provides the exact result for the Landauer conductance of a chaotic quantum dot with two non-ideal leads, each supporting a single mode. To see this we recall that the dimensionless Landauer conductance for a chaotic quantum dot supporting , (), modes in the two leads is given by Beenakker1997 (); MK2004 (); SS1988 (),