Dynamical Phase Transitions in Quantum Systems

Author(s)
Ingrid Rotter

ABSTRACT

Many years ago Bohr characterized the fundamental differences between the two extreme cases of quantum mechanical many-body problems known at that time: between the compound states in nuclei at extremely high level density and the shell-model states in atoms at low level density. It is shown in the present paper that the compound nucleus states at high level density are the result of a dynamical phase transition due to which they have lost any spectroscopic relation to the individual states of the nucleus. The last ones are shell-model states which are of the same type as the shell-model states in atoms. Mathematically, dynamical phase transitions are caused by singular (exceptional) points at which the trajectories of the eigenvalues of the non-Hermitian Hamilton operator cross. In the neighborhood of these singular points, the phases of the eigenfunctions are not rigid. It is possible therefore that some eigenfunctions of the system align to the scattering wavefunctions of the environment by decoupling (trapping) the remaining ones from the environment. In the Schrödinger equation, nonlinear terms appear in the neighborhood of the singular points.

Many years ago Bohr characterized the fundamental differences between the two extreme cases of quantum mechanical many-body problems known at that time: between the compound states in nuclei at extremely high level density and the shell-model states in atoms at low level density. It is shown in the present paper that the compound nucleus states at high level density are the result of a dynamical phase transition due to which they have lost any spectroscopic relation to the individual states of the nucleus. The last ones are shell-model states which are of the same type as the shell-model states in atoms. Mathematically, dynamical phase transitions are caused by singular (exceptional) points at which the trajectories of the eigenvalues of the non-Hermitian Hamilton operator cross. In the neighborhood of these singular points, the phases of the eigenfunctions are not rigid. It is possible therefore that some eigenfunctions of the system align to the scattering wavefunctions of the environment by decoupling (trapping) the remaining ones from the environment. In the Schrödinger equation, nonlinear terms appear in the neighborhood of the singular points.

KEYWORDS

Non-Hermitian Quantum Physics, Dynamical Phase Transitions, Phase Rigidity of Eigen-functions, Nonlinear Schrodinger Equation, Exceptional Points

Non-Hermitian Quantum Physics, Dynamical Phase Transitions, Phase Rigidity of Eigen-functions, Nonlinear Schrodinger Equation, Exceptional Points

Cite this paper

nullI. Rotter, "Dynamical Phase Transitions in Quantum Systems,"*Journal of Modern Physics*, Vol. 1 No. 5, 2010, pp. 303-311. doi: 10.4236/jmp.2010.15043.

nullI. Rotter, "Dynamical Phase Transitions in Quantum Systems,"

References

[1] N. Bohr, “Neutron Capture and Nuclear Constitution,” Nature, Vol. 137, No. 3461, 1936, pp. 344-348.

[2] N. Bohr, “Neutroneneinfang und Bau der Atomkerne,” Die Naturwissenschaften, Vol. 24, 1936, pp. 241-245.

[3] The Nobel Prize in Physics 1963: Maria Goeppert-Mayer and J. Hans D. Jensen for their discoveries concerning nuclear shell structure.

[4] F. M. Dittes, H. L. Harney and A. Muller, “Nonexponential Decay of a Stochastic One-Channel System,” Physical Review A, Vol. 45, No. 2, 1992, pp. 701-705.

[5] H. L. Harney, F. M. Dittes and A. Muller, “Time Evolution of Chaotic Quantum Systems,” Annals of Physics, Vol. 220, No. 2, 1992, pp. 159-187.

[6] R. U. Haq, A. Pandey and O. Bohigas, “Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree?” Physical Review Letters, Vol. 48, No. 16, 1982, pp. 1086-1089(51 pages), and references therein.

[7] I. Rotter, “A Non-Hermitian Hamilton Operator and the Physics of Open Quantum Systems,” Journal of Physisc A, Vol. 42, No. 15, 2009, p. 153001.

[8] H. Feshbach, “Unified Theory of Nuclear Reactions,” Annals of Physics, Vol. 5, No. 4, 1958, pp. 357-390.

[9] H. Feshbach, “Unified Theory of Nuclear Reactions II,” Annals of Physics, Vol. 19, No. 2, 1962, pp. 287-313.

[10] N. Auerbach and V. Zelevinsky, “Doorway States in Nuclear Reactions as a Manifestation of the SuperRadiant Mechanism,” Nuclear Physics A, Vol. 781, No. 1-2, 2007, pp. 67-80.

[11] E. P. Kanter, D. Kollewe, K. Komaki, I. Leuca, G. M. Temmer and W. M. Gibson, “The Time Evolution of Compound Elastic Scattering Measured by Crystal Blocking,” Nuclear Physics A, Vol. 299, No. 2, 1978, pp. 230- 254.

[12] I. Rotter, “A Continuum Shell Model for the Open Quantum Mechanical Nuclear System,” Reports on Progress of Physics, Vol. 54, 1991, pp. 635-682, and references therein.

[13] T. Kato, “Peturbation Theory for Linear Operators,” Springer, Berlin, 1966.

[14] J. Okolowicz and M. Ploscajczak, “Exceptional Points in the Scattering Continuum,” Physical Review C, Vol. 80, No. 3, 2009, p. 034619(7 pages).

[15] L. D. Landau, “Zeitschrift fur Physik (Sowjetunion),” Physics, Vol. 2, 1932, p. 46.

[16] C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proceedings of Royal Society, Series A, Vol. 137, No. 833, London, 1932, pp. 696-702.

[17] I. Rotter, “Dynamics of Quantum Systems,” Physical Review E, Vol. 64, No. 3, 2001, p. 036213 (12 pages).

[18] C. Jung, M. Muller and I. Rotter, “Phase Transitions in Open Quantum Systems,” Physical Review E, Vol. 60, No. 1, 1999, pp. 114-131.

[19] W. D. Heiss, M. Muller and I. Rotter, “Collectivity, Phase Transitions, and Exceptional Points in Open Quantum Systems,” Physical Review E, Vol. 58, No. 3, 1998, pp. 2894-2901.

[20] R. G. Nazmitdinov, K. N. Pichugin, I. Rotter and P. Seba, “Whispering Gallery Modes in Open Quantum Billiards,” Physical Review E, Vol. 64, No. 5, 2001, p. 056214 (5 pages).

[21] R. G. Nazmitdinov, K. N. Pichugin, I. Rotter and P. Seba, “Conductance of Open Quantum Billiards and Classical Trajectories,” Physical Review B, Vol. 66, No. 8, 2002, p. 085322 (13 pages).

[22] R. G. Nazmitdinov, H. S. Sim, H. Schomerus and I. Rotter, “Shot Noise and Transport in Small Quantum Cavities with Large Openings,” Physical Review B, Vol. 66, No. 24, 2002, p. 241302(R) (4 pages).

[23] I. Rotter, “Environmentally Induced Effects and Dynamical Phase Transitions in Quantum Systems,” Journal of Optics, Vol. 12, No. 6, 2010, p. 065701 (9 pages).

[24] E. N. Bulgakov, I. Rotter and A. F. Sadreev, “Phase Rigidity and Avoided Level Crossings in the Complex Energy Plane,” Physical Review E, Vol. 74, No. 5, 2006, p. 056204 (6 pages).

[25] E. N. Bulgakov, I. Rotter and A. F. Sadreev, “Correlated Behavior of Conductance and Phase Rigidity in the Transition from the Weak-Coupling to the Strong-Coupling Regime,” Physical Review B, Vol. 76, No. 21, 2007, p. 214302 (13 pages).

[26] E. Persson, I. Rotter, H. J. Stockmann and M. Barth, “Observation of Resonance Trapping in an Open Microwave Cavity,” Physical Review Letters, Vol. 85, No. 12, 2000, pp. 2478-2481.

[27] A. Yacoby, M. Heiblum, D. Mahalu and H. Shtrikman, “Coherence and Phase Sensitive Measurements in a Quantum Dot,” Physical Review Letters, Vol. 74, No. 20, 1995, pp. 4047-4050.

[28] R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky and H. Shtrikman, “Phase Measurement in a Quantum Dot via a Double Slit Interference Experiment,” Nature, Vol. 385, 1997, pp. 417-420.

[29] M. Avinun-Kalish, M. Heiblum, O. Zarchin, D. Mahalu and V. Umansky, “Crossover from ‘Mesoscopic’ to ‘Universal’ Phase for Electron Transmission in Quantum Dots,” Nature, Vol. 436, 2005, pp. 529-533.

[30] M. Muller and I. Rotter, “Phase Lapses in Open Quantum Systems and the Non-Hermitian Hamilton Operator,” Physical Review A, Vol. 80, No. 4, 2009, p. 042705 (14 pages).

[31] G. A. Alvarez, E. P. Danieli, P. R. Levstein and H. M. Pastawski, “Environmentally Induced Quantum Dynamical Phase Transition in the Spin Swapping Operation,” Journal of Chemical Physics, Vol. 124, No. 19, 2006, p. 194507 (8 pages).

[32] E. P. Danieli, G. A. Alvarez, P. R. Levstein and H. M. Pastawski, “Quantum Dynamical Phase transition in a System with Many-Body Interactions,” Solid State Communications, Vol. 141, No. 7, 2007, pp. 422-426.

[33] G. A. Alvarez, P. R. Levstein and H. M. Pastawski, “Signatures of a Quantum Dynamical Phase Transition in a Three- Spin System in Presence of a Spin Environment,” Physica B, Vol. 398, No. 2, 2007, pp. 438-441.

[34] H. M. Pastawski, “Revisiting the Fermi Golden Rule: Quantum Dynamical Phase Transition as a Paradigm Shift,” Physica B, Vol. 398, No. 2, 2007, pp. 278-286.

[35] A. D. Dente, R. A. Bustos-Marun and H. M. Pastawski, “Dynamical Regimes of a Quantum SWAP Gate Beyond the Fermi Golden Rule,” Physical Review A, Vol. 78, No. 6, 2008, p. 062116 (10 pages).

[36] R. El-Ganainy, K. G. Makris, D. N. Christodoulides and Z. H. Musslimani, “Theory of Coupled Optical PT-Symmetric Structures,” Optics Letters, Vol. 32, No. 17, 2007, pp. 2632-2634.

[37] K. G. Makris, R. El-Ganainy, D. N. Christodoulides and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Physical Review Letters, Vol. 100, No. 10, 2008, p. 103904 (4 pages).

[38] Z. H. Musslimani, K. G. Makris, R. El-Ganainy and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Physical Review Letters, Vol. 100, No. 3, 2008, p. 030402 (4 pages).

[39] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Physical Review Letters, Vol. 103, No. 9, 2009, p. 093902 (4 pages).

[40] C. E. Ruter, G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, “Observation of Parity-Time Symmetry in Optics,” Nature Physics Letters, Vol. 6, 2010, pp. 192-195.

[41] T. Kottos, “Broken Symmetry Makes Light Work,” Nature Physics, Vol. 6, 2010, pp. 166-167.

[42] K. G. Makris, R. El-Ganainy, D. N. Christodoulides and Z. H. Musslimani, “PT-Symmetric Optical Lattices,” Physical Review A, Vol. 81, No. 6, 2010, p. 063807 (10 pages).

[43] S. Longhi, “Optical Realization of Relativistic Non-Hermitian Quantum Mechanics,” Physical Review Letters, Vol. 105, No. 1, 2010, p. 013903 (4 pages).

[1] N. Bohr, “Neutron Capture and Nuclear Constitution,” Nature, Vol. 137, No. 3461, 1936, pp. 344-348.

[2] N. Bohr, “Neutroneneinfang und Bau der Atomkerne,” Die Naturwissenschaften, Vol. 24, 1936, pp. 241-245.

[3] The Nobel Prize in Physics 1963: Maria Goeppert-Mayer and J. Hans D. Jensen for their discoveries concerning nuclear shell structure.

[4] F. M. Dittes, H. L. Harney and A. Muller, “Nonexponential Decay of a Stochastic One-Channel System,” Physical Review A, Vol. 45, No. 2, 1992, pp. 701-705.

[5] H. L. Harney, F. M. Dittes and A. Muller, “Time Evolution of Chaotic Quantum Systems,” Annals of Physics, Vol. 220, No. 2, 1992, pp. 159-187.

[6] R. U. Haq, A. Pandey and O. Bohigas, “Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree?” Physical Review Letters, Vol. 48, No. 16, 1982, pp. 1086-1089(51 pages), and references therein.

[7] I. Rotter, “A Non-Hermitian Hamilton Operator and the Physics of Open Quantum Systems,” Journal of Physisc A, Vol. 42, No. 15, 2009, p. 153001.

[8] H. Feshbach, “Unified Theory of Nuclear Reactions,” Annals of Physics, Vol. 5, No. 4, 1958, pp. 357-390.

[9] H. Feshbach, “Unified Theory of Nuclear Reactions II,” Annals of Physics, Vol. 19, No. 2, 1962, pp. 287-313.

[10] N. Auerbach and V. Zelevinsky, “Doorway States in Nuclear Reactions as a Manifestation of the SuperRadiant Mechanism,” Nuclear Physics A, Vol. 781, No. 1-2, 2007, pp. 67-80.

[11] E. P. Kanter, D. Kollewe, K. Komaki, I. Leuca, G. M. Temmer and W. M. Gibson, “The Time Evolution of Compound Elastic Scattering Measured by Crystal Blocking,” Nuclear Physics A, Vol. 299, No. 2, 1978, pp. 230- 254.

[12] I. Rotter, “A Continuum Shell Model for the Open Quantum Mechanical Nuclear System,” Reports on Progress of Physics, Vol. 54, 1991, pp. 635-682, and references therein.

[13] T. Kato, “Peturbation Theory for Linear Operators,” Springer, Berlin, 1966.

[14] J. Okolowicz and M. Ploscajczak, “Exceptional Points in the Scattering Continuum,” Physical Review C, Vol. 80, No. 3, 2009, p. 034619(7 pages).

[15] L. D. Landau, “Zeitschrift fur Physik (Sowjetunion),” Physics, Vol. 2, 1932, p. 46.

[16] C. Zener, “Non-Adiabatic Crossing of Energy Levels,” Proceedings of Royal Society, Series A, Vol. 137, No. 833, London, 1932, pp. 696-702.

[17] I. Rotter, “Dynamics of Quantum Systems,” Physical Review E, Vol. 64, No. 3, 2001, p. 036213 (12 pages).

[18] C. Jung, M. Muller and I. Rotter, “Phase Transitions in Open Quantum Systems,” Physical Review E, Vol. 60, No. 1, 1999, pp. 114-131.

[19] W. D. Heiss, M. Muller and I. Rotter, “Collectivity, Phase Transitions, and Exceptional Points in Open Quantum Systems,” Physical Review E, Vol. 58, No. 3, 1998, pp. 2894-2901.

[20] R. G. Nazmitdinov, K. N. Pichugin, I. Rotter and P. Seba, “Whispering Gallery Modes in Open Quantum Billiards,” Physical Review E, Vol. 64, No. 5, 2001, p. 056214 (5 pages).

[21] R. G. Nazmitdinov, K. N. Pichugin, I. Rotter and P. Seba, “Conductance of Open Quantum Billiards and Classical Trajectories,” Physical Review B, Vol. 66, No. 8, 2002, p. 085322 (13 pages).

[22] R. G. Nazmitdinov, H. S. Sim, H. Schomerus and I. Rotter, “Shot Noise and Transport in Small Quantum Cavities with Large Openings,” Physical Review B, Vol. 66, No. 24, 2002, p. 241302(R) (4 pages).

[23] I. Rotter, “Environmentally Induced Effects and Dynamical Phase Transitions in Quantum Systems,” Journal of Optics, Vol. 12, No. 6, 2010, p. 065701 (9 pages).

[24] E. N. Bulgakov, I. Rotter and A. F. Sadreev, “Phase Rigidity and Avoided Level Crossings in the Complex Energy Plane,” Physical Review E, Vol. 74, No. 5, 2006, p. 056204 (6 pages).

[25] E. N. Bulgakov, I. Rotter and A. F. Sadreev, “Correlated Behavior of Conductance and Phase Rigidity in the Transition from the Weak-Coupling to the Strong-Coupling Regime,” Physical Review B, Vol. 76, No. 21, 2007, p. 214302 (13 pages).

[26] E. Persson, I. Rotter, H. J. Stockmann and M. Barth, “Observation of Resonance Trapping in an Open Microwave Cavity,” Physical Review Letters, Vol. 85, No. 12, 2000, pp. 2478-2481.

[27] A. Yacoby, M. Heiblum, D. Mahalu and H. Shtrikman, “Coherence and Phase Sensitive Measurements in a Quantum Dot,” Physical Review Letters, Vol. 74, No. 20, 1995, pp. 4047-4050.

[28] R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky and H. Shtrikman, “Phase Measurement in a Quantum Dot via a Double Slit Interference Experiment,” Nature, Vol. 385, 1997, pp. 417-420.

[29] M. Avinun-Kalish, M. Heiblum, O. Zarchin, D. Mahalu and V. Umansky, “Crossover from ‘Mesoscopic’ to ‘Universal’ Phase for Electron Transmission in Quantum Dots,” Nature, Vol. 436, 2005, pp. 529-533.

[30] M. Muller and I. Rotter, “Phase Lapses in Open Quantum Systems and the Non-Hermitian Hamilton Operator,” Physical Review A, Vol. 80, No. 4, 2009, p. 042705 (14 pages).

[31] G. A. Alvarez, E. P. Danieli, P. R. Levstein and H. M. Pastawski, “Environmentally Induced Quantum Dynamical Phase Transition in the Spin Swapping Operation,” Journal of Chemical Physics, Vol. 124, No. 19, 2006, p. 194507 (8 pages).

[32] E. P. Danieli, G. A. Alvarez, P. R. Levstein and H. M. Pastawski, “Quantum Dynamical Phase transition in a System with Many-Body Interactions,” Solid State Communications, Vol. 141, No. 7, 2007, pp. 422-426.

[33] G. A. Alvarez, P. R. Levstein and H. M. Pastawski, “Signatures of a Quantum Dynamical Phase Transition in a Three- Spin System in Presence of a Spin Environment,” Physica B, Vol. 398, No. 2, 2007, pp. 438-441.

[34] H. M. Pastawski, “Revisiting the Fermi Golden Rule: Quantum Dynamical Phase Transition as a Paradigm Shift,” Physica B, Vol. 398, No. 2, 2007, pp. 278-286.

[35] A. D. Dente, R. A. Bustos-Marun and H. M. Pastawski, “Dynamical Regimes of a Quantum SWAP Gate Beyond the Fermi Golden Rule,” Physical Review A, Vol. 78, No. 6, 2008, p. 062116 (10 pages).

[36] R. El-Ganainy, K. G. Makris, D. N. Christodoulides and Z. H. Musslimani, “Theory of Coupled Optical PT-Symmetric Structures,” Optics Letters, Vol. 32, No. 17, 2007, pp. 2632-2634.

[37] K. G. Makris, R. El-Ganainy, D. N. Christodoulides and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Physical Review Letters, Vol. 100, No. 10, 2008, p. 103904 (4 pages).

[38] Z. H. Musslimani, K. G. Makris, R. El-Ganainy and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Physical Review Letters, Vol. 100, No. 3, 2008, p. 030402 (4 pages).

[39] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Physical Review Letters, Vol. 103, No. 9, 2009, p. 093902 (4 pages).

[40] C. E. Ruter, G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, “Observation of Parity-Time Symmetry in Optics,” Nature Physics Letters, Vol. 6, 2010, pp. 192-195.

[41] T. Kottos, “Broken Symmetry Makes Light Work,” Nature Physics, Vol. 6, 2010, pp. 166-167.

[42] K. G. Makris, R. El-Ganainy, D. N. Christodoulides and Z. H. Musslimani, “PT-Symmetric Optical Lattices,” Physical Review A, Vol. 81, No. 6, 2010, p. 063807 (10 pages).

[43] S. Longhi, “Optical Realization of Relativistic Non-Hermitian Quantum Mechanics,” Physical Review Letters, Vol. 105, No. 1, 2010, p. 013903 (4 pages).