Following an idea by Joyner et al. [Europhys. Lett. 107, 50004 (2014)], a microwave graph with an antiunitary symmetry T obeying T^{2}=-1 is realized. The Kramers doublets expected for such systems are clearly identified and can be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level spacings distribution of the Kramers doublets is in agreement with the predictions from the Gaussian symplectic ensemble expected for chaotic systems with such a symmetry.

Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartan's notation) appear. A microwave study of the chiral ensembles is presented using a linear chain of evanescently coupled dielectric cylindrical resonators. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero could be verified. PACS numbers: 05.45.MtRandom matrix theory originally had been developed by Wigner, Dyson, Mehta [1, 2] and others as a tool to describe the spectral properties of chaotic systems. For time-reversal symmetric systems the Hamiltonian H commutes with the time-reversal operator T , HT = T H, where T 2 = 1 for systems without spin 1/2, and T 2 = −1 in the presence of a spin 1/2 [3]. The three options (T 2 = 1, no T , T 2 = −1) give rise to the three classical random matrix ensembles, the Gaussian orthogonal (GOE), unitary (GUE) and symplectic (GSE) one, respectively. Further there may be a chiral symmetry, i. e., an operator C anticommuting with H, HC = −CH, again with the two options C 2 = 1 and C 2 = −1. Such a symmetry exists, e. g., for the Dirac equation [4]. All possible combinations of (T 2 = 1, no T , T 2 = −1) and (C 2 = 1, no C, C 2 = −1) yield a total of nine random matrix ensembles. Together with the last remaining option (no T , no C, but CT ) one finally ends up with the ten-fold way [5,6].For the GOE there is an abundant number of realizations, see Sec. 3.2 of Ref. 7, but for the GUE the number of experiments is still small [8][9][10]. The GSE has been realized only recently by us in a peculiarly designed microwave network mimicking a spin 1/2 [11].For the new ensembles systematic random matrix studies are still missing though there are a lot of studies of systems showing chiral symmetry [12]. In the present work a microwave study of the chiral relatives of the classical ensembles shall be presented, the chiral orthogonal (chOE), the chiral unitary (chUE), and the chiral symplectic (chSE) ensemble (the BDI, the AIII, and the CII in Cartan's notation). We omit the 'G' in the notation, since the ensembles studied by us are partly not Gaussian.For a chiral symmetry particles and anti-particles are not really needed. Sufficient is a system consisting of two subsystems I and II with interactions only between I and II, but no internal interactions within I or II. The Hamiltonian for such a situation may be written aswhere the diagonal blocks belong to the two subsystems, and the off-diagonal blocks describe the interaction. Hamiltonian (1) is chiral symmetric, since it anticommutes with C = diag(1 n , −1 m ). The characteristic polynomial of ...

Following an idea by Joyner et al. [Europhys. Lett. 107, 50004 (2014)EULEEJ0295-507510.1209/0295-5075/107/50004], a microwave graph with antiunitary symmetry T obeying T^{2}=-1 has been realized, thus mimicking a spin-1/2 system. The Kramers doublets expected for such systems have been clearly identified and could be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level-spacing distribution of the Kramers doublets agreed with the predictions from the Gaussian symplectic ensemble (GSE), expected for chaotic systems with such a symmetry. In addition, we studied the random matrix equivalents of the used graphs both analytically and numerically. Here small deviations from the GSE level-spacing distribution were found, too small to be seen in the experiment but clearly visible in the simulations. Furthermore, results on the two-point correlation function, the spectral form factor, the number variance, and the spectral rigidity are presented, as well as on the transition from Gaussian symplectic to Gaussian orthogonal statistics by continuously changing T from T^{2}=-1 to T^{2}=1.

The Landauer-Büttiker formalism establishes an equivalence between the electrical conduction through a device, e. g. a quantum dot, and the transmission. Guided by this analogy we perform transmission measurements through three-port microwave graphs with orthogonal, unitary, and symplectic symmetry thus mimicking three-terminal voltage drop devices. One of the ports is placed as input and a second one as output, while a third port is used as a probe. Analytical predictions show good agreement with the measurements in the presence of orthogonal and unitary symmetries, provided that the absorption and the influence of the coupling port are taken into account. The symplectic symmetry is realized in specifically designed graphs mimicking spin 1/2 systems. Again a good agreement between experiment and theory is found. For the symplectic case the results are marginally sensitive to absorption and coupling strength of the port, in contrast to the orthogonal and unitary case. 73.21.Hb, 72.10.Fk Wave transport and wave scattering phenomena have been of great interest in the last decades, both from experimental and theoretical points of view (see for instance Ref.[1]). Apart from the intrinsic importance in the complex scattering in a particular medium, the interest also comes from the equivalence between physical systems belonging to completely different areas, in which the dimensions of the systems may differ by several orders of magnitude [2]. One of these equivalences occurs in mesoscopic quantum systems, where the electrical conduction reduces to a scattering problem through the Landauer-Büttiker formalism [3][4][5]. Following this line, classical analogies of quantum systems have been used as auxiliary tools to understand the properties of the conductance of electronic devices in two-terminal configurations [6][7][8][9][10]. A plethora of chaotic scattering experiments in presence of time reversal invariance (TRI) and no spin 1/2 have been performed [7,8,[10][11][12][13][14][15][16], while very few experimental studies regarding absence of TRI are reported [7,8,17,18]. Furthermore, due to its intrinsic complexity, there are no scattering experiments up to now for systems with TRI and spin 1/2, where the signatures of the symplectic ensemble are expected, though there is one study of the spectral statistics in Au nanoparticles obeying this symmetry [19]. Moreover, very recently the appearance of a microwave experiment showing the signatures of the symplectic symmetry [20,21] for eigenvalue statistics has opened the possibility to study transport in the presence of this symmetry.Multiterminal devices are good candidates to provide experimental realizations for the three symmetry classes: Device Terminal 1 Terminal 2 Terminal 3 Junction FIG. 1. Sketch of a three-terminal setting that allows the measurement of the voltage along a device. The device carries a current while the vertical wire measures the voltage drop. Thin lines represent perfect conductors connected to sources of voltages V1, V2, and V3.orthogonal, un...

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