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A. Brandstötter, M. Kühmayer, P. Ambichl, S. Rotter:
"Coherent Wave Scattering in Billiards: Time-Delay and Beyond";
Talk: Dynamics Days Europe 2018, Loughborough (invited); 2018-09-03 - 2018-09-07.

We present a protocol for manipulating waves inside billiards or waveguides based on the Wigner-Smith time-delay operator. The concept of this operator emerged in scattering theory as a very useful tool to deduce the time associated with a scattering event from stationary measurements of the corresponding scattering amplitudes. Originally devised for nuclear scattering problems [1,2], this concept later resurfaced in mesoscopic physics [3]. The Wigner-Smith time-delay operator is constructed based on a system's scattering matrix by way of a frequency derivative, . The eigenvalues of , also called proper delay times, measure the time-delay caused by the scattering process at a given potential and their average value just depends on the outer shape of a billiard, but not its inner structure [4]. The corresponding eigenvectors of are states that can be associated with this well-defined time-delay. A slight variation of these states´ input frequency leaves the output profile invariant to first order - a very attractive property for light transmission through multi-mode fibers [5]. In the ballistic limit of vanishing disorder, this property leads to "particle-like" wave function patterns [6] even in billiards with chaotic classical dynamics [7]. In our recent work we took the concept of time-delay eigenstates to a new level such as to produce states that - instead of being insensitive with respect to a frequency variation - are invariant with respect to changes in the system configuration, like a local shift of a designated scatterer inside a billiard or a disordered medium [8]. In the same way as the frequency-insensitive principal modes are the eigenstates of the time-delay operator (involving a frequency derivative), the states we are looking for are eigenstates of a corresponding operator , where the parameter stands, e.g., for the position of a movable scatterer. Depending on the choice of the parameter , we can either focus onto a predetermined target inside a billiard or apply a well-defined torque onto it [9].

[1] E. P. Wigner, "Lower Limit for the Energy Derivative of the Scattering Phase Shift," Phys. Rev., vol. 98, no. 1, pp. 145-147, Apr. 1955.

[2] F. T. Smith, "Lifetime Matrix in Collision Theory," Phys. Rev., vol. 118, no. 1, pp. 349-356, Apr. 1960.

[3] P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, "Quantum Mechanical Time-Delay Matrix in Chaotic Scattering," Phys. Rev. Lett., vol. 78, no. 25, pp. 4737-4740, Jun. 1997.

[4] R. Savo, R. Pierrat, U. Najar, R. Carminati, S. Rotter, and S. Gigan, "Observation of mean path length invariance in light-scattering media," Science, vol. 358, pp. 765-768, Nov. 2017.

[5] W. Xiong, P. Ambichl, Y. Bromberg, B. Redding, S. Rotter, and H. Cao, "Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling," Phys. Rev. Lett., vol. 117, p. 053901, Jul. 2016.

[6] S. Rotter, P. Ambichl, and F. Libisch, "Generating Particlelike Scattering States in Wave Transport," Phys. Rev. Lett., vol. 106, no. 12, p. 120602, Mar. 2011.

[7] J. Böhm, A. Brandstötter, P. Ambichl, S. Rotter, and Ulrich Kuhl, "In situ realization of particlelike scattering states in a microwave cavity," Phys. Rev. A 97, p. 021801 (R), Feb. 2018.

[8] P. Ambichl, A. Brandstötter, J. Böhm, M. Kühmayer, U. Kuhl, and S. Rotter, "Focusing inside Disordered Media with the Generalized Wigner-Smith Operator," Phys. Rev. Lett., vol. 119, no. 3, p. 033903, Jul. 2017.

[9] M. Horodynski, M. Kühmayer, A. Brandstötter, and S. Rotter, Manuscript in preparation.

Wave front shaping, Time-Delay, Scattering