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

Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

A new form of the degenerate perturbation series is proposed. Its construction is inspired by the standard numerical algorithm of inverse iterations: As input, we only assume knowledge of an approximate non-diagonal propagator matrix at some trial zero-order energy. The convergence and a few other technicalities are illustrated by the anharmonic oscillator.

Numerically inspired new version of degenerate Rayleigh-Schrödinger perturbation theory

I propose a new version of the Rayleigh - Schr\"{o}dinger perturbation method. It admits a lower triangular matrix in place of the usual diagonal propagator. Illustrated on rational anharmonicities polynomial}(x)/polynomial}(x), treated as perturbations of (quasi-)exact anharmonic oscilators. In this sense the method works in an intermediate-coupling regime and bridges the gap between the weak- and strong-coupling expansions.

Perturbation method with triangular propagators and anharmonicities of intermediate strength

The conventional toy-model constructions of phase diagrams often use various versions of the standard Hermitian Bose-Hubbard Hamiltonians $H$. These studies were recently extended to cover several non-Hermitian PT-symmetric versions of the model. A key technical merit of the extensions has been found in the existence of Kato's phase-transition-related exceptional-point spectral degeneracies of the $N$th order (EPN). In this setting, an ill-conditioned nature of $H$ near its EPN singularity represented a decisive technical obstacle. In our paper we show how the obstacle can be circumvented, leading to a broader class of the eligible models with tridiagonal and symmetric complex Hamiltonians $H^{(N)}(z)$ exhibiting the EPN degeneracies. The implementation of the method (requiring the use of computer-assisted symbolic manipulations at the larger integers $N$) is illustrated by sampling the real or complex spectra of energies near as well as far from the EPN dynamical regime.

Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

We propose a new, very flexible version of the Rayleigh–Schrodinger perturbation method which admits a lower triangular matrix in place of the usual diagonal unperturbed propagator. The technique and its enhanced efficiency are illustrated on rational anharmonicities V(1)(x)=β×polynomial(x)/polynomial(x). They are shown tractable, in the intermediate coupling regime, as $\mathcal{O}(\beta {\text{ - }}\beta ^{{\text{(0)}}} )$ perturbations of exact states at non-vanishing β(0)≠0. In this sense our method bridges the gap between the current weak- and strong-coupling expansions.

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Perturbation method for non-square Hamiltonians and its application to polynomial oscillators

Numerically inspired new version of degenerate Rayleigh-Schrödinger perturbation theory

Perturbation method with triangular propagators and anharmonicities of intermediate strength

Generalized Bose-Hubbard Hamiltonians exhibiting a complete non-Hermitian degeneracy

Perturbation method with triangular propagators and anharmonicities of intermediate strength