Foundations of quantum mechanics (QM) are designed by the physical interpretation: each observable must be a Hermitian operator to guarantee the reality of its spectrum, and wave functions must belong to a Hilbert space with a conserved probability measure. Recently, extended quantum theories, which do not require such a restrictive mathematical structure, have been developed.
In the late 90s, a broader set of Hamiltonian operators was considered by replacing the Hermiticity condition with the weaker requirement of a space-time reflection symmetry (PT-symmetry). This allows to obtain complex Hamiltonians whose spectra are still real and positive. In the 80s, QM in rigged Hilbert spaces showed that even the reality of the Hamiltonian spectrum is not necessary: the imaginary part of a complex energy takes into account the presence of a time asymmetric (TA) evolution, and describes damping.
Surprisingly, even if these two theories seem unrelated and were conceived independently, their link becomes deeply relevant when one considers the spontaneous PT-symmetry breaking phenomena in optical applications.
The present research aims at an exhaustive first quantization theory able to commit together non-Hermitian and PT-symmetric Hamiltonians with TA-QM. This goal is not only fundamentally important but has a relevant counterpart in studying optical propagation, where experimental tests and applications can be conceived and realized. We formerly showed the way TA-QM describes dispersive shock waves in nonlinear optics, and other scientists elaborated optical analogues of PT-symmetric systems. Our goal is to study the propagation of light in more complex systems, where PT-symmetry and TA evolution are simultaneously present.
This project presents several applications in optics and condensed matter physics, including the study of optical intrinsically irreversible propagation of ultra-short laser pulses interacting with matter and decay of phonons in disordered lattices.
The purpose of this project is to include PT-symmetric and time asymmetric (TA) QM in a more extended but fictive quantum theory. The possibility to create a QM not constrained to consider only Hermitian operators with real spectra as observables, or probability measures that are conserved during the evolution, opens novel and promising perspectives to the study of physics and science in general. A growing community of scientists [1-4] has been moving towards this direction, and developing a complete and extended QM, which does not require a restrictive mathematical structure, is a challenging and timely goal.
Actually no one has tried to put together PT-symmetry and TA-QM, because apparently their aims are different: the first one attains a QM whose wave function evolution is governed by a broader class of Hamiltonian operators, replacing the Hermiticity condition by the weaker and more physical space-time reflection symmetry, whereas the second one enlarges the Hilbert space of the wave functions to a rigged Hilbert space (RHS), where the Hamiltonian cannot be Hermitian for construction, and finds that complex spectra are able to express the time asymmetry in a theory, the QM, where the evolution was always defined unitary. Summarizing, it seems that PT-symmetry leads to a QM tied to Hamiltonians with real-valued spectra, while the TA-QM does the opposite, but this is not what recent studies have discovered [4-6]: above a certain threshold of specific parameters, a PT-symmetric Hamiltonian experiences a ¿phase transition¿ that moves the spectrum from the real axis to the whole complex plane, the so-called spontaneous PT-symmetry breaking. After this breaking, exponentially enhancing or decaying phenomena appear, with evolution similar to the one described by Gamow vectors (GVs) in TA-QM. This was proved in optics [4,5] by several physicists, but without any reference to GVs.
The unified theory we want to developed can be simulated in optics, by considering light propagation ruled by a Hamiltonian composed by an overlapping of reversed harmonic oscillators [7], defined in the RHS, and a gain-loss potential [4], characteristic of PT-symmetric dynamics. This is another relevant objective our research: the theoretical and numerical study of an extended QM optical analogue, able to take into account both PT-symmetric and TA quantum theories.
Such a treatment may be also experimentally tested, and might provides new insight not only in optics but in condensed matter physics too. In fact, the coexistence of PT-symmetry and exponential decays in lattices can unveil a rigorous theoretical model for important physical phenomena, such as the phonon decays in disordered systems and ultrashort laser propagation.
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[4] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev and D. Kip, Nat. Phys. 6, 192 (2010).
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[6] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[7] S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe and C. Conti, Phys. Rev. A 92, 023801 (2015).