Probing Anderson Localization via Fidelity via Fidelity Probing Anderson Localization via Fidelity via Fidelity Joshua D. Bodyfelt in collaboration with T. Kottos, M.C. Zheng, U. Kuhl, & H.-J. Stöckmann TSICS09 – Dresden – Wesleyan University Complex Quantum Dynamics & Mesoscopic Physics Group in cooperation with Philipps-Universität Marburg AG Quantenchaos References: ● Physical Review Letters, 102, (2009). ● Acta Physica Polonica A, submitted for Proceedings of the 4th Workshop on Quantum Chaos and Localization Phenomena.

Synopsis Synopsis Introduction What is Anderson Localization? Fidelity – A Measure of Irreversibility Fidelity in Diffusive/Localized Systems Comin' In & Goin' Out – The Scattering Matrix Drin tatsächlichen Erde - Scattering Fidelity Experimental Setup & Theoretical Modeling Theory versus Experiment Conclusions

Diffusive: Critical: Localized: Determining Anderson Localization Determining Anderson Localization

A. Aspect, et al., Nature, 453, 891 (2008) BECs in Optical Lattices Ultrasound Waves in 3D Elastic Networks B.A. van Tiggelen, et al., Nature Phys. 4, 945 (2008) Light Waves in Planar Lattices T. Schwartz, et al., Nature 446, 52 (2007) Dynamical Localization in Rough Cavities H. Cao, et al., PNAS, 101, (2004) Anderson Localization in Experiments Anderson Localization in Experiments

F.M. Cucchietti, et al., PRE, 65, (2002) r p r p H 1 H 2 Fidelity and the Loschmidt Echo Fidelity and the Loschmidt Echo

Scattering System Weidenmüller, Ann. Phys., 158, 120 (1984) Scattering Theory – Weidenmüller Approach Scattering Theory – Weidenmüller Approach

R. Schäfer, H.J. Stöckmann, T. Gorin, & T.H. Seligman, PRL, 95, (2005) Theoretical: Experimental: From Scattering to Fidelity From Scattering to Fidelity

Borland Delphi User Interface Agilent 8720ES Network Analyzer w #1 #2 Quasi-1D Experimental Apparatus Quasi-1D Experimental Apparatus

Localized Extended A. Genack & A. Chabanov, J. Phys A, 38, (2005) Signatures of Localization Signatures of Localization

Theoretical Modeling - RMT Theoretical Modeling - RMT

R. Schäfer, T. Gorin, et al., PRL 95, (2005) Delocalized Supersymmetry (RMT) Result Supersymmetry (RMT) Result

Localized RMT Localized Result Localized Result

where Y. Fyodorov, PRL, 73, 2688 (1994) Perturbative Regime – Theory Perturbative Regime – Theory

Perturbative Regime – Theory & Numerics Perturbative Regime – Theory & Numerics

Perturbative Regime – Experimental Perturbative Regime – Experimental

Delocalized

Perturbative Regime – Experimental Perturbative Regime – Experimental Localized Delocalized

Conclusion Conclusion Localized Wavefunctions ● Fidelity Decay Rate in the Perturbative Regime for Localized Wavefunctions is i.e., localization length found from Fidelity Decay Rate. Delocalized Wavefunctions ● Fidelity Decay Rate in the Perturbative Regime for Delocalized Wavefunctions....reproduces known RMT results.

Wigner Regime – Theory Wigner Regime – Theory L mn

Wigner Regime – Theory & Numerics Wigner Regime – Theory & Numerics

Wigner Regime – Experimental Wigner Regime – Experimental