Information Geometry of Complex Hamiltonians and Exceptional Points

Information geometry provides a tool to systematically investigate the parameter sensitivity of the state of a system. If a physical system is described by a linear combination of eigenstates of a complex (that is, non-Hermitian) Hamiltonian, then there can be phase transitions where dynamical properties of the system change abruptly. In the vicinities of the transition points, the state of the system becomes highly sensitive to the changes of the parameters in the Hamiltonian. The parameter sensitivity can then be measured in terms of the Fisher-Rao metric and the associated curvature of the parameter-space manifold. A general scheme for the geometric study of parameter-space manifolds of eigenstates of complex Hamiltonians is outlined here, leading to generic expressions for the metric.

[1]  Y. Fyodorov,et al.  Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality. , 2012, Physical review letters.

[2]  Biorthogonal quantum systems , 2005, quant-ph/0507015.

[3]  Tsampikos Kottos,et al.  Experimental study of active LRC circuits with PT symmetries , 2011, 1109.2913.

[4]  S. Pancharatnam The propagation of light in absorbing biaxial crystals , 1955 .

[5]  J. D. Cloizeaux Extension d'une formule de Lagrange à des problèmes de valeurs propres , 1960 .

[6]  Shachar Klaiman,et al.  Visualization of branch points in PT-symmetric waveguides. , 2008, Physical review letters.

[7]  R. More THEORY OF DECAYING STATES. , 1971 .

[8]  Daniel W. Hook,et al.  Quantum phase transitions without thermodynamic limits , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  E. Graefe,et al.  Signatures of three coalescing eigenfunctions , 2011, 1110.1489.

[10]  Dorje C. Brody,et al.  Geometrisation of Statistical Mechanics , 1997 .

[11]  M. Sternheim,et al.  Non-Hermitian Hamiltonians, Decaying States, and Perturbation Theory , 1972 .

[12]  I. Rotter,et al.  Projective Hilbert space structures at exceptional points , 2007, 0704.1291.

[13]  Yujun Zheng,et al.  Geometric phases in non-Hermitian quantum mechanics , 2012 .

[14]  N. Sinitsyn,et al.  Sensitivity field for nonautonomous molecular rotors. , 2011, The Journal of chemical physics.

[15]  Y. Lacasse,et al.  From the authors , 2005, European Respiratory Journal.

[16]  Bypassing the bandwidth theorem with PT symmetry , 2012, 1205.1847.

[17]  W. Heiss,et al.  The physics of exceptional points , 2012, 1210.7536.

[18]  Li Ge,et al.  Pump-induced exceptional points in lasers. , 2011, Physical review letters.

[19]  Eva-Maria Graefe,et al.  Mixed-state evolution in the presence of gain and loss. , 2012, Physical Review Letters.

[20]  T. D. Lee,et al.  Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation , 1952 .

[21]  J. Main,et al.  Eigenvalue structure of a Bose–Einstein condensate in a PT?>-symmetric double well , 2013, 1306.3871.

[22]  T. D. Lee,et al.  Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model , 1952 .

[23]  Songky Moon,et al.  Observation of an exceptional point in a chaotic optical microcavity. , 2009, Physical review letters.

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

[25]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[26]  Oleg N. Kirillov,et al.  Geometric phase around exceptional points , 2005, Physical Review A.

[27]  A. Mostafazadeh Spectral singularities of complex scattering potentials and infinite reflection and transmission coefficients at real energies. , 2009, Physical review letters.

[28]  Rivier,et al.  Geometrical aspects of statistical mechanics. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  S. Heinze,et al.  Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions. , 2007, Physical review letters.

[30]  Torre,et al.  Non-Hermitian evolution of two-level quantum systems. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[31]  Berry Phases and Quantum Phase Transitions , 2006, quant-ph/0602091.

[32]  Dorje C. Brody,et al.  Geometrization of statistical mechanics , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[33]  R. A. Blythe,et al.  The Lee-Yang theory of equilibrium and nonequilibrium phase transitions , 2003, cond-mat/0304120.

[34]  J. Main,et al.  Bifurcations and exceptional points in dipolar Bose–Einstein condensates , 2013, 1302.5615.

[35]  H. Korsch,et al.  A non-Hermitian symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points , 2008, 0802.3164.

[36]  E. Brändas Non-hermitian quantum mechanics , 2012 .

[37]  Alexei A. Mailybaev,et al.  Multiparameter Stability Theory with Mechanical Applications , 2004 .

[38]  Aharonov,et al.  Geometry of quantum evolution. , 1990, Physical review letters.

[39]  Claude Mahaux,et al.  Shell-model approach to nuclear reactions , 1969 .

[40]  D. Brody,et al.  Coherent states and rational surfaces , 2010, 1001.1754.

[41]  Rory A. Fisher,et al.  Theory of Statistical Estimation , 1925, Mathematical Proceedings of the Cambridge Philosophical Society.

[42]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

[43]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[44]  Dorje C. Brody,et al.  Statistical geometry in quantum mechanics , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[45]  Z. Musslimani,et al.  PT -symmetric optical lattices , 2010 .

[46]  C. Bender,et al.  Observation of PT phase transition in a simple mechanical system , 2012, 1206.4972.

[47]  B. Mandelbrot The Role of Sufficiency and of Estimation in Thermodynamics , 1962 .

[48]  Tosio Kato Perturbation theory for linear operators , 1966 .

[49]  Alan Edelman,et al.  Nongeneric Eigenvalue Perturbations of Jordan Blocks , 1998 .

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

[51]  Brody,et al.  Geometry of Quantum Statistical Inference. , 1996, Physical review letters.

[52]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

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

[54]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[55]  A. Mostafazadeh,et al.  Geometric Phase for Non-Hermitian Hamiltonians and Its Holonomy Interpretation , 2008, 0807.3405.

[56]  Scaling of geometric phases close to the quantum phase transition in the XY spin chain. , 2005, Physical review letters.

[57]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[58]  E. Wright,et al.  Complex geometrical phases for dissipative systems , 1988 .

[59]  Li Ge,et al.  Unconventional modes in lasers with spatially varying gain and loss , 2011, 1106.3051.

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

[61]  H. Harney,et al.  PT symmetry and spontaneous symmetry breaking in a microwave billiard. , 2011, Physical review letters.

[62]  Paolo Zanardi,et al.  Information-theoretic differential geometry of quantum phase transitions. , 2007, Physical review letters.

[63]  Dorje C. Brody,et al.  Information geometry in vapour–liquid equilibrium , 2008, 0809.1166.

[64]  V. Arnold ON MATRICES DEPENDING ON PARAMETERS , 1971 .

[65]  A. Pell Biorthogonal systems of functions , 1911 .