Encircling an exceptional point.

We calculate analytically the geometric phases that the eigenvectors of a parametric dissipative two-state system described by a complex symmetric Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a parameter setting where the two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce. We show that it can be encircled on a path along which the eigenvectors remain approximately real and discuss a microwave cavity experiment, where such an encircling of an EP was realized. Since the wave functions remain approximately real, they could be reconstructed from the nodal lines of the recorded spatial intensity distributions of the electric fields inside the resonator. We measured the geometric phases that occur when an EP is encircled four times and thus confirmed that for our system an EP is a branch point of fourth order.

[1]  M. Berry,et al.  The optical singularities of birefringent dichroic chiral crystals , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  Phases of wave functions and level repulsion , 1999, quant-ph/9901023.

[3]  Lauber,et al.  Geometric phases and hidden symmetries in simple resonators. , 1994, Physical review letters.

[4]  M. Berry.  Quantal phase factors accompanying adiabatic changes , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5]  Off-diagonal geometric phases. , 1999, Physical review letters.

[6]  S. Pancharatnam.  The propagation of light in absorbing biaxial crystals — I. Theoretical , 1955 .

[7]  F. Wilczek,et al.  Geometric Phases in Physics , 1989 .

[8]  Ingrid Rotter,et al.  Dynamics of quantum systems embedded in a continuum , 2003 .

[9]  M. Berry,et al.  Diabolical points in the spectra of triangles , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  I. Rotter,et al.  Collectivity, phase transitions and exceptional points in open quantum systems , 1998, quant-ph/9805038.

[11]  H. L. Harney,et al.  The chirality of exceptional points , 2001 .

[12]  I Rotter.  Exceptional points and double poles of the S matrix. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  H. Harney,et al.  Experimental observation of the topological structure of exceptional points. , 2001, Physical review letters.

[14]  H. Stöckmann,et al.  Quantum Chaos: An Introduction , 1999 .

[15]  Geometric phase in open systems. , 2003, Physical review letters.

[16]  W. Voigt.  Beiträge zur Aufklärung der Eigenschaften pleochroitischer Krystalle , 1902 .

[17]  G. Nenciu,et al.  On the adiabatic theorem for nonself-adjoint Hamiltonians , 1992 .

[18]  S. Sridhar,et al.  Experimental observation of scarred eigenfunctions of chaotic microwave cavities. , 1991, Physical review letters.

[19]  Berry phase in a nonisolated system. , 2002, Physical review letters.

[20]  F. Keck,et al.  Unfolding a diabolic point: a generalized crossing scenario , 2003 .

[21]  Heiss.  Repulsion of resonance states and exceptional points , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  Richter,et al.  Frequency and width crossing of two interacting resonances in a microwave cavity , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  Heidelberg,et al.  Observation of a chiral state in a microwave cavity. , 2002, Physical review letters.

[24]  Heine,et al.  First experimental evidence for chaos-assisted tunneling in a microwave annular billiard , 1999, Physical review letters.