One of the fundamental axioms of quantum mechanics is associated with the Hermiticity of physical observables 1 . In the case of the Hamiltonian operator, this requirement not only implies real eigenenergies but also guarantees probability conservation. Interestingly, a wide class of non-Hermitian Hamiltonians can still show entirely real spectra. Among these are Hamiltonians respecting parity‐time (PT) symmetry 2‐7 . Even though the Hermiticity of quantum observables was never in doubt, such concepts have motivated discussions on several fronts in physics, including quantum field theories 8 , nonHermitian Anderson models 9 and open quantum systems 10,11 , to mention a few. Although the impact of PT symmetry in these fields is still debated, it has been recently realized that optics can provide a fertile ground where PT-related notions can be implemented and experimentally investigated 12‐15 . In this letter we report the first observation of the behaviour of a PT optical coupled system that judiciously involves a complex index potential. We observe both spontaneous PT symmetry breaking and power oscillations violating left‐right symmetry. Our results may pave the way towards a new class of PT-synthetic materials with intriguing and unexpected properties that rely on non-reciprocal light propagation and tailored transverse energy flow. Before we introduce the concept of spacetime reflection in optics, we first briefly outline some of the basic aspects of this symmetry within the context of quantum mechanics. In general, a Hamiltonian HD p 2 =2mCV(x

/pdf/observation-of-parity-time-symmetry-in-optics-2qb17qrvql.pdf

Observation of parity–time symmetry in optics

In 1998, Bender and Boettcher found that a wide class of Hamiltonians, even though non-Hermitian, can still exhibit entirely real spectra provided that they obey parity-time requirements or PT symmetry. Here we demonstrate experimentally passive PT-symmetry breaking within the realm of optics. This phase transition leads to a loss induced optical transparency in specially designed pseudo-Hermitian guiding potentials.

Observation of PT-Symmetry Breaking in Complex Optical Potentials

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose +complex conjugate) is replaced by the physically transparent condition of space?time reflection ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive!Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.The Lee model provides an excellent example of a -symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm 'ghost' state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The operator for the Lee model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

https://iopscience.iop.org/article/10.1088/0034-4885/70/6/R03/pdf

Making sense of non-Hermitian Hamiltonians

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

Making Sense of Non-Hermitian Hamiltonians

We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.

/pdf/pseudo-hermiticity-versus-pt-symmetry-the-necessary-24944d6gro.pdf

Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian

To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.

/pdf/quasi-hermitian-formulation-of-quantum-mechanics-using-two-3cuwwyi2.pdf

Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations

Paper “Asymptotic iteration method for eigenvalue problems” by Hakan Ciftci, Richard L. Hall and Nasser Saad [1] represents one of the really remarkable contributions to the 50 years of history of publishing of the results in mathematical physics in Journal of Physics A. The paper successfully combined the choice of a sufficiently popular subject (viz., the constructive approach to ODEs) with a sufficiently transparent formulation of the problem. Briefly, the problem may be identified as the necessity of compatibility between the tools of mathematics (yielding various systematic polynomial approximations to general wave functions ψ(x)) with the needs of quantum physics requiring the matching of the general solutions to the specific, i.e., in our case, bound-state (BS) boundary conditions. The second reason of success (measured, e.g., by the degree of inspiration by the proposal, i.e., by the number of citations) may be seen in the fact that the underlying idea was fresh and neatly presented. Its essence was transparent the recipe generated all derivatives of the general solution and subsequently, a clever “asymptotic aspect” (AA) assumption (cf. eq. (2.8) in loc. cit.) reduced the general (i.e., two-parametric) solution to the mere double quadrature (cf. eq. (2.12) in loc. cit.). The third, independent source of appeal of the asymptotic iteration method (AIM) may be seen in its illustrative applications to Schrödinger equations with various potentials. In particular, for their two or three exactly solvable samples, a serendipitious aspect of the method has been discovered in a mysterious emergence of a strict equivalence between the above, purely formal AA assumption and the specific physical BS boundary conditions. Last but not least, the authors emphasized that a hypothesis of an AABS correspondence seems to work even beyond the exactly solvable class, in

http://iopscience.iop.org/article/10.1088/1751-8113/49/45/451003/pdf

A method of constructive quantum mechanics of remarkable hidden beauty

In the framework of quantum mechanics using quasi-Hermitian operators the standard unitary evolution of a non-stationary but still closed quantum system is only properly described in the non-Hermitian interaction picture (NIP). In this formulation of the theory both the states and the observables vary with time. A few aspects of implementation of this picture are illustrated via the “wrong-sign” quartic oscillators. It is shown that in contrast to the widespread belief, both of the related Schrödinger-equation generators G(t) and the Heisenberg-equation generators Σ(t) are just auxiliary concepts. Their spectra are phenomenologically irrelevant and, in general, complex. It is argued that only the sum H(t)=G(t)+Σ(t) of the latter operators retains the standard physical meaning of the instantaneous energy of the unitary quantum system in question.

https://www.mdpi.com/1099-4300/25/4/692/pdf?version=1681987414

Non-Stationary Non-Hermitian “Wrong-Sign” Quantum Oscillators and Their Meaningful Physical Interpretation

Discrete multiparametric 1D quantum well with PT-symmetric long-range boundary conditions is proposed and studied. As a nonlocal descendant of the square well families endowed with Dirac (i.e., Hermitian) and with complex Robin (i.e., non-Hermitian but still local) boundary conditions, the model is shown characterized by the survival of solvability in combination with an enhanced spectral-design flexibility. The solvability incorporates also the feasibility of closed-form constructions of the physical Hilbert-space inner products rendering the time-evolution unitary.

Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions

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Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations

A method of constructive quantum mechanics of remarkable hidden beauty

Non-Stationary Non-Hermitian “Wrong-Sign” Quantum Oscillators and Their Meaningful Physical Interpretation

Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions

Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions