Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized ${ \mathcal P }{ \mathcal T }$-symmetric Scarf-II potential in the deep learning.     

${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics

A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Effects of Quantum Fluctuations on $\mathcal{PT}$-Symmetric Solitons of a Trapped Bose Gas *

Considering the quantum fluctuation effects, the existence and stabilityof solitons in a Bose-Einstein condensate subjected in a $\mathcal{PT}$-symmetric potentialare discussed. Using the variational approach, we investigate how the quantum fluctuationaffects the self-localization and stability of the condensate with attractivetwo-body interactions. The results show that the quantum fluctuation dramaticallyinfluences the shape, width, and chemical potential of the condensate.Analytical variational computation also predicts there exists a positive critical quantumfluctuation strength $q_{c}$ with each fixed attractive two-body interaction $g_{0}$, if thequantum fluctuation strength $q_{0}$ is bigger than $q_{c}$, there is no bright solitonsolution existence. We also study the effects of the quantum fluctuations on the stabilityof solitons using the Vakhitov-Kolokolov (VK) stability criterion. A robust stable brightsoliton will always exist when the quantum fluctuation strength $q_{0}$ belongs tothe parameter regimes $q_{c}\geq q_{0}＞0$.     

One-dimensional $\mathcal{PT}$-symmetric acoustic heterostructure

The explorations of parity-time ($\mathcal{PT}$)-symmetric acoustics have resided at the frontier in physics, and the pre-existing accessing of exceptional points typically depends on Fabry-Perot resonances of the coupling interlayer sandwiched between balanced gain and loss components. Nevertheless, the concise $\mathcal{PT}$-symmetric acoustic heterostructure, eliminating extra interactions caused by the interlayer, has not been researched in depth. Here we derive the generalized unitary relation for one-dimensional (1D) $\mathcal{PT}$-symmetric heterostructure of arbitrary complexity, and demonstrate four disparate patterns of anisotropic transmission resonances (ATRs) accompanied by corresponding spontaneous phase transitions. As a special case of ATR, the occasional bidirectional transmission resonance reconsolidates the ATR frequencies that split when waves incident from opposite directions, whose spatial profiles distinguish from a unitary structure. The derived theoretical relation can serve as a predominant signature for the presence of $\mathcal{PT}$ symmetry and $\mathcal{PT}$-symmetry-breaking transition, which may provide substantial support for the development of prototype devices with asymmetric acoustic responses.     

Gap solitons of spin-orbit-coupled Bose-Einstein condensates in $\mathcal{PT}$ periodic potential

We numerically investigate the gap solitons in Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in the parity-time ($\mathcal{PT}$)-symmetric periodic potential. We find that the depths and periods of the imaginary lattice have an important influence on the shape and stability of these single-peak gap solitons and double-peak gap solitons in the first band gap. The dynamics of these gap solitons are checked by the split-time-step Crank-Nicolson method. It is proved that the depths of the imaginary part of the $\mathcal{PT}$-symmetric periodic potential gradually increase, and the gap solitons become unstable. But the different periods of imaginary part hardly affect the stability of the gap solitons in the corresponding parameter interval.     

Effect of power-law nonlinearity on ${ \mathcal P }{ \mathcal T }$-symmetric optical system with fourth-order diffraction

Gaussian-type soliton solutions of the nonlinear Schrödinger (NLS) equation with fourth order dispersion, and power law nonlinearity in the novel parity-time (${ \mathcal P }{ \mathcal T }$)-symmetric quartic Gaussian potential are derived analytically and numerically. The exact analytical expressions of the solutions are obtained in the first two-dimensional (1D and 2D) power law NLS equations. By means of the linear stability analysis, the effect of power law nonlinearity on the stability of Gauss type solitons in different nonlinear media is carried out. Numerical investigations do confirm the stability of our soliton solutions in both focusing and defocusing cases, specially around the propagation parameters.     

Krein-space formulation of \mathcal{P}\mathcal{T} symmetry, \mathcal{C}\mathcal{P}\mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the $\mathcal{C}\mathcal{P}\mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics.     

Understanding Partial P T $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

International Journal of Theoretical Physics - A new kind of symmetry behaviour introduced as partial $\mathcal {P}\mathcal {T}$ -symmetry(henceforth $\partial _{\mathcal {P}\mathcal {T}}$ ) is...     

Oscillatory stability of quantum droplets in PT-symmetric optical lattice

We study stability and collisions of quantum droplets(QDs) forming in a binary bosonic condensate trapped in parity-time (PT)-symmetric optical lattices. It is found that the stability of QDs in the PT-symmetric system depends strongly on the values of the imaginary part W_0 of the PT-symmetric optical lattices, self-repulsion strength g, and the condensate norm N. As expected,the PT-symmetric QDs are entirely unstable in the broken PT-symmetric phase. However, the PT-symmetric QDs exhibit oscillatory stability with the increase of N and g in the unbroken PT-symmetric phase. Finally, collisions between PT-symmetric QDs are considered. The collisions of droplets with unequal norms are completely different from that in free space. Besides, a stable PT-symmetric QDs collides with an unstable ones tend to merge into breathers after the collision.     

Anti-$\mathcal{PT}$-symmetric Kerr gyroscope

Non-Hermitian systems can exhibit unconventional spectral singularities called exceptional points (EPs). Various EP sensors have been fabricated in recent years, showing strong spectral responses to external signals. Here we propose how to achieve a nonlinear anti-parity-time ($\mathcal{APT}$) gyroscope by spinning an optical resonator. We show that, in the absence of any nonlinearity, the sensitivity or optical mode splitting of the linear device can be magnified up to 3 orders compared to that of the conventional device without EPs. Remarkably, the $\mathcal{APT}$ symmetry can be broken when including the Kerr nonlinearity of the materials and, as a result, the detection threshold can be significantly lowered, i.e., much weaker rotations which are well beyond the ability of a linear gyroscope can now be detected with the nonlinear device. Our work shows the powerful ability of $\mathcal{APT}$ gyroscopes in practice to achieve ultrasensitive rotation measurement.     

P T $\mathcal {P}\mathcal {T}$ -symmetry in Compact Phase Space for a Linear Hamiltonian

International Journal of Theoretical Physics - We study the time evolution of a $\mathcal {P}\mathcal {T}$ -symmetric, non-Hermitian quantum system for which the associated phase space is compact....     

Impact of P T $\mathcal {P}\mathcal {T}$ -Symmetric Operation on Concurrence and the First-Order Coherence

International Journal of Theoretical Physics - In this paper, we lock the focus in effect of $\mathcal {P}\mathcal {T}$ -symmetric operation on the dynamics of concurrence and the first-order...     

To Quantify the Difference of η-Inner Products in P T $\mathcal {P}\mathcal {T}$ -Symmetric Theory

International Journal of Theoretical Physics - In this paper, we consider a typical continuous two dimensional $\mathcal {P}\mathcal {T}$ -symmetric Hamiltonian and propose two different approaches...     

\mathcal{PT}-symmetric nonlinear metamaterials and zero-dimensional systems

A one dimensional, parity-time ( $\mathcal{PT}$ )-symmetric magnetic metamaterial comprising split-ring resonators having both gain and loss is investigated. In the linear regime, the transition from the exact to the broken $\mathcal{PT}$ -phase is determined through the calculation of the eigenfrequency spectrum for two different configurations; the one with equidistant split-rings and the other with the split-rings forming a binary pattern ( $\mathcal{PT}$ dimer chain). The latter system features a two-band, gapped spectrum with its shape determined by the gain/loss coefficient as well as the interelement coupling. In the presence of nonlinearity, the $\mathcal{PT}$ dimer chain configuration with balanced gain and loss supports nonlinear localized modes in the form of a novel type of discrete breathers below the lower branch of the linear spectrum. These breathers that can be excited from a weak applied magnetic field by frequency chirping, can be subsequently driven solely by the gain for very long times. The effect of a small imbalance between gain and loss is also considered. Fundamental gain-driven breathers occupy both sites of a dimer, while their energy is almost equally partitioned between the two split-rings, the one with gain and the other with loss. We also introduce a model equation for the investigation of classical $\mathcal{PT}$ symmetry in zero dimensions, realized by a simple harmonic oscillator with matched time-dependent gain and loss that exhibits a transition from oscillatory to diverging motion. This behavior is similar to a transition from the exact to the broken $\mathcal{PT}$ phase in higher-dimensional $\mathcal{PT}$ -symmetric systems. A stability condition relating the parameters of the problem is obtained in the case of a piece-wise constant gain/loss function that allows the construction of a phase diagram with alternating stable and unstable regions.     

Singular $$ \mathcal{R}$$ -Matrices and Drinfeld's Comultiplication

We compute the $\mathcal{R}$ -matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for ${\text{U}}_q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ . This $\mathcal{R}$ -matrix contains terms proportional to the δ-function. We construct the algebra $A\left( \mathcal{R} \right)$ generated by the elements of the matrices L^±(z) with relations determined by $\mathcal{R}$ . In the category of highest-weight representations, there is a Hopf algebra isomorphism between $A\left( \mathcal{R} \right)$ and an extension $\overline {\text{U}} _q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ of Drinfeld's algebra.     

Analytic wave functions and energies for two-dimensional \mathcal{P}\mathcal{T} -symmetric quartic potentials

Analytic wave functions and the corresponding energies for a class of the $ \mathcal{P}\mathcal{T} $ -symmetric two-dimensional quartic potentials are found. The general form of the solutions is discussed.     

Valley-resolved transport in zigzag graphene nanoribbon junctions

Applying the transfer matrix and Green's function methods, we study the valley-resolved transport properties of zigzag graphene nanoribbon (ZGNR) junctions. The width of the left and right ZGNRs are N_L and N_R, and N_L ≥ N_R. The step/dip positions of the conductance G, the intravalley transmission coefficients (T_KK and ${T}_{{K}^{{\prime} }{K}^{{\prime} }}$), and the valley polarization efficiency ${P}_{{{KK}}^{{\prime} }}$ correspond to the subband edges of the right/left ZGNR that are controlled by N_R/N_L. The intervalley transmission coefficients (${T}_{{{KK}}^{{\prime} }}$ and ${T}_{{K}^{{\prime} }K}$) exhibit peaks at most of the subband edge of the left and right ZGNRs. In the bulk gap of the right ZGNR, ${T}_{{{KK}}^{{\prime} }}$ $={T}_{{K}^{{\prime} }K}$=0, and ${P}_{{{KK}}^{{\prime} }}$ = ±1, the valley polarization is well preserved. As N_R increases, the energy region for ${P}_{{{KK}}^{{\prime} }}$ = ±1 decreases. When N_L is fixed and N_R decreases, G, T_KK, ${T}_{{K}^{{\prime} }{K}^{{\prime} }}$ and ${P}_{{{KK}}^{{\prime} }}$ exhibit more and more dips, and the peaks of ${T}_{{{KK}}^{{\prime} }}$ (${T}_{{K}^{{\prime} }K}$) become more and more high, especially when (N_L − N_R)/2 is odd. These characters are quite useful for manipulating the valley dependent transport properties of carriers in ZGNR junctions by modulating N_L or N_R, and our results are helpful to the design of valleytronics based on ZGNR junctions.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of symmetry and the non- orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.