## Abstract

In this review article, we will demonstrate the power of microwave experiments in the realm of fidelity also known as Loschmidt echoes. As the determination of the fidelity itself is experimentally tedious and error prone, we will introduce the scattering fidelity which under the conditions of chaotic systems and weak coupling approaches the fidelity itself. The main ingredient in fidelity investigations is the type and strength of a perturbation. The perturbations presented here will be both global and local boundary perturbations, as well as local perturber movements but also the change of coupling to the environment. All these perturbations will produce their own fidelity decay as a function of the perturbation strength, which will be discussed in this article.

## 1. Introduction

The sensitivity of quantum dynamics to perturbations is an important issue in many fields of physics and chemistry, e.g. for quantum computing or chemical reactions. To quantify this, Peres suggested to look at the overlap between time-evolved perturbed (*H*_{1}) and unperturbed Hamiltonians (*H*_{0}) with same initial state |*ψ*_{0}〉 [1], in contrast with investigation of classical chaos, where the sensitivity on initial conditions is considered. The stability of quantum time evolution has been studied from various viewpoints and under different names in the past. In the following equation
1.1

the basic idea of interest for this review is written explicitly. The equation can be described by two possible interpretations: either the ‘fidelity’ *F*(*t*), where *f*(*t*) is the fidelity amplitude, or the ‘Loschmidt echo’ (LE) *M*(*t*).

The fidelity considers the overlap of the time evolved quantum state under the unperturbed Hamiltonian *H*_{0} with the state that results from the same initial state by evolving the latter for the same time, but under a perturbed Hamiltonian *H*_{1}. The LE compares the overlap of the initial state |*ψ*_{0}〉 and the state |*ψ*_{12}(*t*)〉 obtained by first propagating |*ψ*_{0}〉 for a time *t* under the Hamiltonian *H*_{0}, and then in reversed time under *H*_{1}. The considered overlap equals unity at *t*=0 and typically decays further in time.

The behaviour of the decay on different kinds of Hamiltonians *H*, e.g. describing not only chaotic, mixed or integrable systems but also disordered systems, and perturbations, e.g. spatial perturbations ranging from global to local or non-diagonal perturbations, have been intensely investigated (for a review, see [2] and references therein). Apart from the overall behaviour which often looks at system averages, individual systems are of interest also in the field of quantum information and quantum computation [3], where fidelity during the time of calculations needs to stay of the order *F*≈0.9999 [4] even though 0.999 might be sufficient after error correction.

Experimentally, the fidelity itself is difficult to handle as it implies an integration over the entire space. In two-dimensional microwave billiards, upon which this review will concentrate, the antenna always represents a perturbation, and thus moving the antenna defeats the purpose of a fidelity measurement, as the wave function taken at any point is that of a slightly different system. In contrast with wave function measurements, in fidelity experiments one is precisely interested in such differences, and thus wave functions measured with movable antennas [5–7] or movable perturbation bodies [8–10] are not appropriate. In elastodynamic experiments on solid bodies, eigenmodes inside the volume seem to be inaccessible anyway [11–13]. Thus, different approaches have been suggested to measure quantities that resemble the fidelity. For microwave experiments, the quantity used is the scattering fidelity [14], which will be introduced in detail later. It tests the sensitivity of *S*-matrix elements to perturbations and reduces to the ordinary fidelity for fully chaotic systems in the limit of weak coupling. But it is also of intrinsic interest, as the scattering matrix may be considered as the basic building block at least in the case of quantum theory [15,16].

Microwave experiments with flat cavities are meanwhile a well-known paradigm in the field of quantum chaos [17], since approximately 25 years ago with a seminal paper on the spectral properties of a stadium shaped metallic resonator with top and bottom plate parallel to each other [18]. The system is described by a two-dimensional electromagnetic wave equation (Helmholtz equation) with Dirichlet boundary conditions, where the *z*-direction can be separated. This review concentrates on the first transverse magnetic (TM_{0}) mode, where the electric field is pointing in the *z*-direction and is also independent of *z*. As long as a maximum frequency , where *h* is the height of the resonator and *c* denotes the speed of the light, is not surpassed only the TM_{0} mode exists and a one-to-one correspondence to the stationary problem in quantum mechanics is given. The correspondence of the two-dimensional Helmholtz equation to the two-dimensional Schrödinger equation of a particle in a box with the same boundary conditions allows one to study predictions for quantum billiards by means of flat microwave resonators. This correspondence also holds for open scattering systems, thus giving the possibility to verify predictions from quantum-mechanical scattering theory by means of open microwave resonators with a number of attached open channels, either antennas or waveguides. All experimental studies presented in this work are based on this correspondence.

This paper reviews shortly the effective Hamiltonian approach (§2), and introduces the relation to scattering theory, thereby defining the scattering fidelity (§3). Thereafter, the perturbations involving only the internal ‘closed’ Hamiltonian are presented in §4. Finally, perturbations in the coupling with the environment are investigated, which naturally enters the concept of scattering fidelity. This review is mainly revisiting microwave experiments presented already in [14,19–24] apart from some additional comparisons in §5.

## 2. Effective Hamiltonian and eigenvalue dynamics

Any measurement couples a wave system with the continuum, thus drastically changing the system properties by converting the discrete energy levels of the closed system into decaying resonance states of the open system, thus inducing currents that do not exist in the closed system. A standard description is the effective Hamiltonian (see [25,26] for overviews). Thus the fidelity, defined via (1.1), will have an additional decay due to its norm leakage, i.e. the subunitary evolution [27]. This probability decay can also be interpreted as a fidelity, where the imaginary part of the effective Hamiltonian is the perturbation. The coupling with the environment also gives rise to additional perturbation possibilities, e.g. the coupling strength with the environment, and leads to new contributions as perturbations will vary the decay rate. We assume here the general case of *M* scattering channels attached to *N* levels of the closed Hermitian Hamiltonian *H*_{0} including an arbitrary perturbation *V* with perturbation strength λ
2.1where *W*_{c} are *M* vectors of length *N* containing the information on the coupling of the levels with the continuum through *M* coupling channels. *W*_{c} are assumed to be normalized to one, , and *κ*_{c} is the coupling constant of channel *c*. Note that both *κ*_{c}=*κ*_{cr}+i*κ*_{ci} and the Hamiltonian have an internal scale, namely the mean level spacing Δ. The effective Hamiltonian approach initially developed in nuclear physics [28–30] has been successfully applied to study various aspects of open systems, including wave billiards [17,31–33]. Usually, *κ*_{c} are considered to be real which enter the final expressions via the so-called transmission coefficients *T*_{c}. Generally, in the case of an experiment it might be difficult to vary only the coupling of a channel without perturbing the system otherwise, thus leading to complex coupling coefficients *κ*_{c}. As the coupling constant *κ*_{c} will also be varied (see §5), the imaginary part needs to be taken into account [23]. Non-zero Im(*κ*_{c}) arises also in shell-model calculations due to the principal value term of the self-energy operator (cf. [28,29]). The only constraint is Re(*κ*_{c})=*κ*_{cr}≥0, due to the causality condition on the *S*-matrix. Typically, the real perturbation induced by *κ*_{ci} is absorbed in the Hamiltonian *H*_{0}, even though the term might not have the same symmetries as the Hamiltonian *H*_{0} itself. Note that needs to take into account also evanescent channels which can be neglected for the real part *κ*_{cr}.

The non-Hermiticity of *H*_{eff} results in a set of complex eigenvalues associated with two distinct sets of eigenvectors called left {〈*ψ*^{L}_{n}|} and right {|*ψ*^{R}_{n}〉} eigenvectors and , where the complex eigenvalue gives the energy *E*_{n} and the resonance width *Γ*_{n}>0 of the *n*th resonance, respectively. The left and right eigenvectors, which describe the resonance states, satisfy the condition of bi-orthogonality, , and completeness, In the context of microwave cavities, the validity of model (2.1) for real *κ*_{c} has been established in previous works [33,34]. More general models of non-Hermitian Hamiltonian operators may be used to study open quantum systems (e.g. the topical review [25]). This formalism has been successfully applied to wave billiards for which antennas and absorption are respectively described by physical and fictitious coupling channels [34–36].

In figure 2, the eigenvalue dynamics measured in microwave cavities are shown for three types of perturbation, where the perturbation strength is indicate by λ. In figure 2*a*, the perturbation is introduced by the shift of a wall in a stadium billiard (for details see [18,37]) corresponding to a full perturbation matrix *V* . It is the same kind of perturbation as sketched in figure 1*a*. As this perturbation acts on all levels in the same way for already small parameter variations, thus the dynamics seen in figure 2*a* is quite homogeneous. For this kind of perturbation the distribution of level velocities *v*=d*E*/dλ is Gaussian [17,18,37,42]. In figure 2*b,* a local perturbation is performed by moving a scatterer with a radius much smaller than the wavelength (for details see [38,39]). This perturbation can be described by a rank 2 perturbation, i.e. , where *W*_{λ} is a 2×*N* matrix and *H*_{0} is *N*×*N*. Note that the *W*_{λ} can be interpreted as two fully closed channels *W*_{c}, i.e. *κ*_{cr}=0 and λ=*κ*_{ci}. Here the level dynamics shows partly nearly linear behaviour and on other parts strong variations (figure 2*b*). All this leads to a velocity distribution which is described by a modified Bessel function *K*_{0}. Generally, the type of perturbations depends on the ratio between radius and wavelength leading to a transition from local to global perturbation [38,39,43]. Note that the effect of opening the system has been so far neglected for the level dynamics and their velocity distributions. In figure 2*a*,*b,* the coupling of the measuring antenna is small (*κ*≤0.3), thus the non-Hermiticity here leads only to slight changes, which still can be recognized in the level-spacing distribution [44]. A perturbation via *λV* also changes the eigenfunctions, leading to changes of the individual coupling *W*_{c}. This induces the variation of resonance width, i.e. the imaginary part of the eigenenergy which can be determined and related to the non-orthogonality of the wave function [45,46].

All perturbations presented before deal with the real part of the effective Hamiltonian, but in microwave cavities also the coupling can be varied, either by varying a diaphragm of an attached waveguide (figure 1*e*) or by varying the antenna coupling (figure 1*f*). In figure 2*c*, the eigenvalue dependence as a function of the coupling strength *κ*_{c} is presented, realized via a changing diaphragm of a lead (for details see [40,41]). In the case of small openings the coupling is weak and the resonance width is small *Γ*≈10^{−3} and increasing with the opening. Additionally, a slight shift to lower eigenfrequencies is seen, which mainly results from the change of the boundary conditions at the diaphragm from Dirichlet to Neumann, i.e. corresponding to a variation of the imaginary part of *κ*_{c}. Additionally, there will be contributions to *κ*_{ci} from diffraction at the corners of the diaphragm and coupling with evanescent modes of the waveguide behind the diaphragm. When increasing the diaphragm width more and more, the behaviour changes and finally the resonance width is reducing. This effect is called resonance trapping and is related to the generation of so-called doorway states or collective modes [30,47–50]. Owing to the strong coupling the eigenstates can be reshuffled leading to broad resonances, which strongly couple with the environment via the existing channels, and sharper resonances, which avoid the region of the channels. In the experiment shown in figure 2*c* the broad state could not be identified, as it cannot be distinguished from the background.

All these parametric variations will of course also show up in the stability of the wave system, i.e. the fidelity. As this review is concerned with the microwave experiments in the realm of fidelity, we will not go any further in the details of investigating the eigenvalues or eigenfunctions of the effective Hamiltonian. In the next section, the scattering fidelity is defined using scattering theory in the framework of the effective Hamiltonian.

## 3. Scattering theory and scattering fidelity

In nuclear physics much insight is gained by performing nuclear reaction experiments using particle accelerators. The principle of these experiments can be described as a three-step process. In a first step, an accelerated particle is moving towards the reaction target. Ideally all quantum numbers (spin, parity, momentum, etc.) are known. This set of numbers labels the incident channel. In a second step, the particle hits the target, that is, it interacts locally with some potential which might cause some of the quantum numbers to change. In the third and final step, a particle leaves the interaction region to be registered by some detector system that determines the new set of quantum numbers which now labels the final channel. This whole process defines a scattering problem where the fundamental challenge is to determine the transition probability from a given initial channel to a given final channel. In the presented experiments microwaves are fed into the resonator using attached waveguides or antennas, which are introduced into the cavity through a small hole. The reflection and transmission properties between different antennas are described by the scattering matrix *S*. It is defined by
3.1where *a*=(*a*_{1},*a*_{2},⋯ ) is the vector of amplitudes of the waves entering through the different channels and *b*=(*b*_{1},*b*_{2},⋯ ) is the amplitude vector of the outgoing waves. The diagonal element *S*_{ii} of *S* corresponds to the reflection amplitude at antenna *i*, whereas the non-diagonal elements *S*_{ij} are related to the transmission amplitudes between antennas *i* and *j* (figure 3). Microwave experiments allow measuring of the complete scattering matrix including the phases in frequency space using a vector network analyser.

According to the general scattering formalism [28,29,51], the resonance part of the *S*-matrix at the scattering energy *E* can be expressed in terms of *H*_{eff} (equation (2.1)) as follows:
3.2

Here, we have followed Köber *et al.* [23] to take into account the possible case of complex coupling constants, *κ*_{c}, as mentioned earlier in §2. Usually, the coupling amplitudes change slowly with the energy when *E* is far from the channel opening thresholds. Then the complex eigenvalues of *H*_{eff} are the only singularities of the *S*-matrix in the complex energy plane.

The coupling strength of a particular channel is characterized by
3.3where 〈.〉 denotes average over ensemble and/or energy. Thus, *T*_{c} is accessible directly from a reflection measurement. Using random matrix theory (RMT) for large matrices at *E*=0, the transmission coefficients are given in terms of the average *S*-matrix diagonal elements by [23,29,52]
3.4*κ*_{c} can be determined uniquely if Im(*κ*_{c})=0. Note that for real *κ*_{c}, one gets *T*_{c}=(4*κ*_{c}/(1+*κ*_{c})^{2})∈[0,1], so that *T*_{c}≪1 and *T*_{c}=1 correspond to an almost closed and a perfectly open channel *c*, respectively. In the case of purely imaginary *κ*_{c} corresponding to perfect reflection, the channel is closed, *T*_{c}=0. We will use this in the investigation of the coupling fidelity (see §5a and [23]), where the effect of adding a channel (i.e. *T*_{c}=0→*T*_{c}=1) is investigated in the scattering fidelity.

As *κ*_{c} have only a weak energy dependence within the resonance width, the elements of the scattering matrix can be written as a sum of Lorentzians
3.5where *a*_{n} is the complex amplitude, *E*_{n} the eigenenergy and *Γ*_{n} the width of the resonance. In the time domain, *Γ*_{n} describes the exponential decay of the *n*th eigenstate. In the weak coupling regime, the resonances do not overlap and the non-Hermitian part of *H*_{eff} can be treated as a perturbation of the Hermitian part *H* and one can relate the parameters of the Lorentzians to the closed system.

Fidelity, defined by (1.1), can be also applied to scattering systems. A wave packet can be evolved with two slightly different scattering Hamiltonians. This would be the standard fidelity of a scattering system. However, it is instructive to consider the so-called ‘scattering fidelity’ which can be obtained from simple scattering data, though under certain conditions it agrees with the standard fidelity.

We introduce the concept of scattering fidelity. In microwave experiments, one can measure scattering matrix elements for unperturbed () and perturbed () systems independently in frequency space. By taking the Fourier transform 3.6of any scattering matrix element one obtains , the scattering matrix in the time domain. This leads to the definition of the scattering fidelity amplitude [2,14]: 3.7By this definition, the overall decay of the correlation functions due to absorption or other open channels drops out, provided the decay is the same for the parametric cross-correlation functions in the numerator and the autocorrelation functions in the denominator. The scattering fidelity itself is 3.8For chaotic systems and weak coupling of the measuring antenna, the scattering fidelity approaches the ordinary fidelity (equation (1.1)) [2,14].

For a suitable description of the experiments (see §5), which have been performed for varying the coupling with the system, in general, one has to use complex numbers for the coupling constant of the varied (perturbing) channel. So all coupling constants *κ*_{c} will be treated as complex numbers, where Re(*κ*_{c})≥0.

In the next two sections, we will look at different types of perturbations (global, local, local boundary and coupling), and two different Hamiltonians (ballistic and disordered) as indicated in figure 1.

## 4. Scattering fidelity by varying the ‘closed’ Hamiltonian

### (a) Global perturbation in a ballistic chaotic system

We now study the dependence of the fidelity decay on a global perturbation as a function of the perturbation strength. For this the side wall of a chaotic cavity without bouncing ball components or stable islands is moved (figure 1*a*).

For the random matrix model, this corresponds to a perturbation with a full rank perturbation matrix *V* . In this case, one expects a transition from linear to quadratic decay near the Heisenberg time *t*_{H}. The time *τ*=*t*/*t*_{H} is given in units of the Heisenberg time. For small perturbations, in the so-called perturbative regime, the linear term in the exponential is still negligible and we observe a Gaussian decay of the fidelity amplitude approaching an exponential in the Fermi Golden Rule regime [53–55]. An exact solution for the random matrix model has been obtained using supersymmetry techniques [56]. For systems with time reversal symmetry (GOE) the fidelity is given by
4.1In the limit of a small perturbation, (4.1) can be simplified to
4.2where *C*(*t*) is given by
4.3for the Gaussian orthogonal ensemble, *b*_{2}(*t*) being the 2-point form factor. This has been derived before in the linear-response approximation [57]. Equation (4.2) will be used to overlay all experimental results to a single curve in §4e.

In the experiment, λ varies from λ=0.01 for *n*=1 and *ν*=3–4 GHz up to λ=0.5 for *n*=10 and *ν*=17–18 GHz. Figure 4*a* shows the fidelity amplitude for three different frequency windows. The perturbation parameter λ has been fitted to the experimental curves, as its determination from the level dynamics is time consuming and for strong perturbations not always feasible. To improve statistics, experimental results from the different *S*-matrix elements have been superimposed. The exact solutions (4.1) are shown as dashed lines in figure 4*a* and a good correspondence in all three cases is found, where λ was a free fit parameter.

### (b) Local perturbation in a ballistic chaotic system

Now we investigate a local perturbation performed by a scatterer whose diameter is smaller than the wavelength, as shown in figure 1*b*. We have already seen in figure 2 that the level dynamics and the velocity distribution between local and global perturbation are quite different. Starting from a weak, point-like perturbation, which produces a small eigenenergy shift proportional to the intensity of the unperturbed wave function at the perturber position, one derives the final expression for the fidelity amplitude [20]
4.4where
4.5where *A* is the area of the billiard and *α* describes the scattering strength of the scatterer, which can be determined by the level dynamics [58]. This result is valid for finite *λt* where λ→0 and . An exact result was obtained showing first a linear decay and then approaching (4.4) [59]. Have in mind that the perturbation *V* is here of rank 2 as we are looking into a shift of a perturber. A rank 1 perturbation would be realized if we would compare the system without perturber to the one with one perturber.

Figure 4*b* shows the scattering fidelity for three different perturber shifts. We find a good agreement to the theoretical prediction from (4.4) (solid lines) in all cases. There are only a few percent deviations for the coupling parameter *α* between the values obtained by fitting the scattering fidelity from those obtained via the level velocities. As the perturbation and the coupling of the antenna are weak, i.e. the experiment was performed in the isolated resonance regime, it was possible to calculate directly the ordinary fidelity as given by (1.1). The measurement has been performed at a fixed antenna position *r*_{0}, leading to a delta-like initial state localized at *r*_{0}. The ordinary fidelity can be expressed by the eigenenergy shifts and the eigenfunctions at the antenna position which can be obtained via the resonance depths [60]. A good agreement between the ordinary fidelity and the scattering fidelity was found. For more details, refer to [20]. We emphasize that the fidelity decay here is not monotonic, but oscillates once the shift is of the order of half the wavelength, which have not been explored experimentally. In the next subsection, a non-monotonic behaviour is also found in the case of local boundary perturbations.

### (c) Local boundary perturbation in a ballistic chaotic system

We now go on for a local boundary perturbation by pushing out a piston-like part of the boundary as indicated in figure 1*c*. In contrast with the previous perturbation, the length *w* of the boundary that is changed is larger than the wavelength but in contrast with the global one the classical dynamics will be affected not on a very short time scale. This piston-like boundary perturbation was studied in the semiclassical context and the fidelity decay is approximately given by [21,61,62]
4.6with the effective decay rate *κ* given in the limit of *h*≪*w* by
4.7with **H**_{1} being the Struve H-function of first order. *k* denotes the wavenumber. In figure 4*c*, the fidelity decay rate is plotted for three different perturbations *h*. On the one hand, the decay is well described by the exponential decay (4.6), but the decay is not monotonically increasing with the perturbation *h*, as already indicated by the Struve H-function in (4.7). We will revisit this behaviour in §4e.

### (d) Boundary perturbation in a strongly disordered system

Now the results of a boundary perturbation in a disordered system are presented (figure 1*d*). The idea is to measure the Anderson localization length in quasi-one-dimensional waveguides by means of the scattering fidelity due to small perturbations. Again a piston boundary perturbation is applied, but this time the system is not chaotic but disordered. Thus, the time scale is not defined by the classical Lyapunov exponent but is related to the localization length. The fidelity decay of the microwave radiation in the localized regime for small perturbations λ is shown in figure 4*d*. The perturbation is realized by moving one side wall, i.e. corresponding in a ballistic or diffusive system to a global perturbation. One observes deviations from a Gaussian decay of the fidelity, given by the dotted line, expected in a frequency interval associated with extended waves, i.e. diffusive or ballistic chaotic regime [2,14,63]. Based on a banded RMT the fidelity amplitude decay in the localized regime was found to be [64]
4.8where is the inverse participation number, inversely proportional to the localization length. It is assumed that *ψ*(**r**) is normalized. Equation (4.8) is shown in figure 4*d* as a solid line, showing a better agreement than a Gaussian decay.

Using λ as a fitting parameter, we have attempted to fit the experimental data with (4.2). We found that the overall agreement is poor (figure 4*d*). Further analysis (see below) confirms that (4.2) is inapplicable in the localized regime. On the contrary, when fitting the experimental data with (4.8) a good agreement is found.

### (e) Fidelity versus rescaled parameters

To obtain the fidelity one needs to average over ensembles and/or energy. As the fidelity is typically highly fluctuating, to increase statistics and verify universal behaviours, it is attractive to look at it on a rescaled parameter axis.

Equation (4.2) is valid for global perturbations and weak perturbation strengths allowing one to go to a rescaled parameter axis 4*π*^{2}λ^{2}*C*(*t*). Thus, the fidelity is given by an exponential decay, which is seen in figure 5*a*. Note also the small fluctuations due to the increased statistics.

In the case of the local perturbation by a small scatterer (rank 2), (4.6) exhibits a scaling behaviour on a rescaled time axis *λt*. Thus, all experimental results for the scattering fidelity fall onto one single curve which is indeed demonstrated in figure 5*b*. Again due to better statistics, the fluctuations are suppressed.

### (f) Perturbation strength versus fidelity decay

Figure 6*a* shows the fidelity decay rate as a function of the perturbation strength. For small perturbations, we find first a monotonic increase but then an oscillating behaviour of the fidelity decay rate, thus showing a non-monotonic crossover from the Fermi Golden Rule to the escape-rate regime of the fidelity decay with the perturbation *h*, where the limit corresponds to the classical decay rate, where the system is opened at the perturbed boundary. In this limit, the fidelity is independent of the perturbation strength and is called the Lyapunov regime [2,63,65,66].

In the case of localization the scaling is different. By fitting the experimental data using (4.8) and the Gaussian decay, where *α*,λ are used as fitting parameters. We have extracted *α*,λ, for various wall shifts *w* and plotted them versus a rescaled shift, *w*/*δw*. The results are summarized in figure 6*b*. We find that in the frequency window 6–7.5 GHz (where Anderson localization is present), the best fit with a Gaussian gives , with *γ*=1.9±0.05. This result violates the theoretical expectation *γ*≈1 and constitutes a direct confirmation that the RMT result (4.2) is not applicable in the localized regime. At the same time, the best fit of the experimental scattering fidelity with the prediction of the banded RMT modelling (see (4.8)) gives with *γ*∼0.92±0.05, which is in agreement with our theory. The extracted slope can be used as an estimation for the localization properties of our sample. Note that this works also in the presence of absorption, where other techniques often fail to extract the localization length.

Within the delocalized frequency window 10.5–12 GHz, a fit with a Gaussian decay works perfectly well with *γ*≈1.0±0.05, in good agreement with the theory (see inset of figure 6*b*). Here we meet the situation found for the fidelity decay observed in chaotic billiards when moving one wall [19] (see §4a).

## 5. Scattering fidelity by varying the coupling

### (a) Varying the channel coupling

The diaphragm of an attached waveguide can be varied to change the channel coupling strength *κ*_{c} as depicted in figure 1*d*. The coupling constant *κ*_{c} could then be determined directly from a reflection measurement at antenna *c* using (3.4). As figure 7*a* shows, *κ*_{c} changes from *κ*_{c}=0 (no coupling) to *κ*_{c}=1 (perfect coupling) by increasing the opening *d* of the slit. In figure 7*b,* the fidelity for two different diaphragm openings *d* is shown. The dotted line shows the theoretical prediction of the coupling fidelity using the experimentally determined *κ*_{c} obtained by a reflection measurement in the attached lead via (3.3) (for details see [23]). The two dotted curves do not agree with the experimentally found fidelity decay, showing that a description of the fidelity decay purely by a variation of coupling is sufficient. As a next step, we fitted the decay of the fidelity |*f*(*t*)|^{2}, which gave approximately a purely imaginary coupling *κ*_{c,fit}. These curves are shown as solid lines figure 7*b* and a good agreement with the data is found. Thus, the fidelity decay is mainly governed by the change of boundary condition at the diaphragm, i.e. by the imaginary part *κ*_{ci} of the coupling *κ*_{c} (see (2.1)), and not due to the coupling with the environment via the channel, i.e. the real part *κ*_{cr} of the coupling *κ*_{c}. Finally, we can state that the variable diaphragm works essentially as a scattering centre leading to partial masking of the change of coupling by the change of scattering properties in the fidelity decay, even though it shows that the ansatz using a complex coupling constant *κ*_{c} is sufficient to describe the results. To isolate the effect of coupling, in the next two subsections, we treat the case of an additional attached antenna inside the cavity with three different closures.

### (b) Additional channel with 50 Ω termination

One can single out a single measurement channel ‘*a*’ of the scattering matrix and describe the effective Hamiltonian
5.1where *κ*_{W} is the channel coupling strength, i.e. the coupling strength of the antenna, whereas *κ*_{T}=(1−*r*_{T})/(1+*r*_{T}) is defined by the reflection properties of the termination , which is described by its absorption *α*_{T} and phase *φ*_{T}. Here *κ*_{c}=*κ*_{T}*κ*_{W} is our perturbation parameter for the coupling fidelity. Thus, the 2×2 scattering matrix including the measuring antenna and the variable channel has been reduced to a single scattering matrix element for the measuring antenna only:
5.2Therefore, the influence of the variable channel with different terminations can be taken into account by an appropriate modification of the Hamiltonian. For details on the derivation of the coupling fidelity refer to [23].

In this subsection, the case of complete absorption is treated, which is realized experimentally by 50 Ω termination and described by a purely real *κ*_{T}=1
5.3

In figure 8*a*, the transmission of the measuring channel *T*_{a} and the varying channel *T*_{c} is presented. From *T*_{c} the coupling strength is calculated via (3.4) and displayed in figure 8*b*. According to (5.3) the coupling is purely imaginary with *κ*_{50Ω}=*κ*_{c}. In figure 9*a*, the fidelity is plotted for two frequency ranges corresponding to two different coupling strengths *κ*_{c} and in figure 9*c*,*e* the corresponding real and imaginary parts of the fidelity amplitude. Using the experimentally determined *κ*_{c} here denoted as coupling parameter *κ*^{expt}_{50Ω}, one gets already a very good agreement between experimental results (filled symbols) and theoretical curves (dotted line) without any fit. A fit of *κ*_{c} to the experimental curves (the corresponding values are denoted by *κ*^{fit}_{50Ω}), which are plotted as dashed lines, shows only a minor improvement for the correspondence between experiment and theory.

As a second example, figure 9*b* presents the fidelity decay for the frequency range 7.2–7.7 GHz and the corresponding real and imaginary parts of the fidelity amplitude. An obvious deviation between the experimental results (filled circles) and the theoretical curve based on the experimental parameter (dotted line) is found. To determine the decay of the experimental fidelity decay, the data are fitted in two possible ways. The first way uses a real-valued fitting parameter *κ*^{fit,re}_{50Ω} (black dashed line), whereas in the second way the experimental parameter has an additional imaginary part *κ*^{fit,im}_{50Ω}. In both cases, the fidelity is nicely fitted, whereas only the complex valued fit properly fits the individual real and imaginary parts of the fidelity amplitude.

### (c) Coupling of an additional ‘closed’ channel

For the two cases, where the antenna is terminated by a reflecting hard wall or an open end, we may assume *α*=0, resulting in , and
5.4Here the coupling is purely real and the antenna does not correspond any longer to an open channel but only to a scattering centre. In fact, this interpretation is true only as long as the absorption in the antenna can really be neglected. This becomes questionable, as soon as *φ*_{T} approaches *π*, corresponding to the excitation of a resonance within the antenna. For this singular situation, the perturbative treatment of the antenna coupling applied in the derivation loses its justification. In figure 10, the fidelity and the real and imaginary parts of the fidelity amplitude are plotted for the two different terminations, i.e. open and short. The experimental data and also the relation between the two fidelities are well described. The value of the total phase shift *φ*_{T} depends on the length of the antenna in units of the wavelength and thus on frequency *ν*. But independently of frequency the difference of the phase shift *φ*_{T} for the reflection at the open end (oe) and the hard wall (hw), respectively, is always *π*, as already mentioned. A phase difference of *π* means a replacement of the tangent by the cotangent in (5.4), i.e. the coupling constants *κ*_{T} for the two situations are related via
5.5With the above-introduced total coupling constant *κ*=*κ*_{T}*κ*_{W} this may be alternatively written as
5.6as *κ*_{W} is the coupling constant for the 50 Ω load (see (5.3)). *κ*_{hw} and *κ*_{oe} denote the total coupling constants for the hard-wall and the open-end reflections. These relations allow for explicit tests of the theory. In figure 8*b*, is plotted and is close to the *κ*_{50Ω} apart from a few frequency ranges. In these ranges, the antenna connector with the terminator corresponds to multiples of half the wavelength.

## 6. Conclusion

We have seen that microwave experiments can be used to investigate the fidelity by means of the scattering fidelity for various kinds of systems (chaotic, diffusive and localized). A quantitative understanding of the scattering fidelity behaviour is also found for different kinds of perturbations. The main ingredient is scattering theory involving the effective Hamiltonian approach. All investigations until now were performed on flat cavities, thus reducing the problem to scalar equations. From the experimental side, going to the vector equation is straightforward and it has been shown that many concepts of RMT also hold in three-dimensional cavities [67,68] like in reverberation chambers [69] used for electromagnetic compatibility. In this case, the fidelity or scattering fidelity needs to be extended to the vectorial case which forms a promising direction of future research.

## Competing interests

The author declares that he has no competing interests.

## Funding

I received no funding for this study.

## Acknowledgements

I thank M. Barth, J.D. Bodyfelt, T. Gorin, A. Goussev, R. Höhmann, B. Köber, T. Kottos, O. Legrand, F. Mortessagne, K. Richter, D. Savin, R. Schäfer, T. Seligman, H.-J. Stöckmann and M.C. Zheng who contributed to the work by performing either measurements, data evaluation, ideas and theoretical background.

## Footnotes

One contribution of 12 to a theme issue ‘Loschmidt echo and time reversal in complex systems’.

- Accepted March 4, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.