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arXiv:2111.00378 · Physical Review A

Coupling-induced nonunitary and unitary scattering in anti- PT -symmetric non-Hermitian systems

H. S. Xu, L. Jin

Published Jul 26, 2021• 17 citations
arXivHTMLPDF

Abstract

We investigate the properties of two anti-parity-time (anti-$\mathcal{PT}$)-symmetric four-site scattering centers. The anti-$\mathcal{PT}$-symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-$\mathcal{PT}$ symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-$\mathcal{PT}$-symmetric scattering center possesses nonunitary scattering and the difference between the reflection and transmission is unity; for the real coupling, the odd-parity anti-$\mathcal{PT}$-symmetric scattering center possesses unitary scattering and the sum of the reflection and transmission is unity. The coupling-induced different scattering behaviors are verified in the numerical simulations. Our findings reveal that a significant difference in the dynamics can be caused by a slight difference between two similar anti-$\mathcal{PT}$-symmetric non-Hermitian scattering centers.

DOI
10.1103/physreva.104.012218

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We investigate the properties of two anti-parity-time (anti-PT\mathcal{PT}PT)-symmetric four-site scattering centers. The anti-PT\mathcal{PT}PT-symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-PT\mathcal{PT}PT symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-PT\mathcal{PT}PT-symmetric scattering center possesses nonunitary scattering and the difference between the reflection and transmission is unity; for the real coupling, the odd-parity anti-PT\mathcal{PT}PT-symmetric scattering center possesses unitary scattering and the sum of the reflection and transmission is unity. The coupling-induced different scattering behaviors are verified in the numerical simulations. Our findings reveal that a significant difference in the dynamics can be caused by a slight difference between two similar anti-PT\mathcal{PT}PT-symmetric non-Hermitian scattering centers.
We investigate the scattering properties of a parity-time-antisymmetric non-Hermitian system. The parity-time-antisymmetric scattering center possesses imaginary nearest-neighbor hoppings and real on-site potentials; it has been experimentally realized through dissipative coupling and frequency detuning between atomic spin waves. We find that such parity-time-antisymmetric system displays three salient features: First, the reflection and transmission are both reciprocal. Second, the reflection and transmission probabilities satisfy $R\ifmmode\pm\else\textpm\fi{}T=1$, which depends on the parity of the scattering center size. Third, the scattering matrix satisfies $\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right){\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right)}^{*}=I$ for a scattering center with an even site; for a scattering center with an odd site, the dynamics exhibits Hermitian scattering behavior, possessing unitary scattering matrix $S{S}^{\ifmmode\dagger\else\textdagger\fi{}}=I$.
We study the scattering problem for the non-Hermitian scattering center, which consists of two Hermitian clusters with anti-Hermitian couplings between them. Counterintuitively, it is shown that it acts as a Hermitian scattering center, that is, the SSS matrix is unitary, or the Dirac probability current is conserved, ∣r∣2+∣t∣2=1{|r|}^{2}+{|t|}^{2}=1∣r∣2+∣t∣2=1, when one of the two clusters is embedded in the waveguides. This conclusion can be applied to an arbitrary parity-symmetric real Hermitian graph with additional PT\mathcal{PT}PT-symmetric potentials, which is more feasible in the experiment. An exactly solvable model is presented to illustrate the theory. Bethe ansatz solution indicates that the transmission spectrum of such a cluster displays a peculiar feature arising from the non-Hermiticity of the scattering center.
PT\mathcal{PT}PT-symmetric electronics
We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time PT operator: typical normal modes, temporal evolution, and scattering processes. Utilizing a Liouvilian formulation, we can define an underlying PT-symmetric Hamiltonian, which provides important insight for understanding the behavior of the system. When the PT-dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the PT-symmetry of the scattering target. Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves. Having an exact theory, and due to its relative experimental simplicity, PT-symmetric electronics offers new insights into the properties of PT-symmetric systems which are at the forefront of the research in mathematical physics and related fields.
Abstract. We show both theoretically and experimentally that a pair of inductively coupled active LRC circuits (dimer), one with amplification and another with an equivalent amount of attenuation, display all the features which characterize a wide class of non-Hermitian systems which commute with the joint parity-time PT operator: typical normal modes, temporal evolution, and scattering processes. Utilizing a Liouvilian formulation, we can define an underlying PT-symmetric Hamiltonian, which provides important insight for understanding the behavior of the system. When the PT-dimer is coupled to transmission lines, the resulting scattering signal reveals novel features which reflect the PT-symmetry of the scattering target. Specifically we show that the device can show two different behaviors simultaneously, an amplifier or an absorber, depending on the direction and phase relation of the interrogating waves. Having an exact theory, and due to its relative experimental simplicity, PT-symmetric electronics offers new insights into the properties of PT-symmetric systems which are at the forefront of the research in mathematical physics and related fields. PACS numbers: 11.30.Er, 05.60.-k, 45.05.+x 1.
Abstract Parity-time-reversal (PT) symmetric quantum mechanics promotes the increasing research interest of non-Hermitian (NH) systems for the theoretical value, novel properties, and links to open and dissipative systems in various areas. Recently, anti-PT–symmetric systems and their featured properties have started to be investigated. In this work, we develop the PT and anti-PT symmetry to PT-arbitrary-phase symmetry (or PT- φ symmetry) for the first time, being analogous to bosons, fermions and anyons. It can also be seen as a complex extension of the PT symmetry, unifying the PT and anti-PT symmetries and having intermediate properties between them. Many of the established concepts and mathematics in the PT-symmetric system are still compatible. We mainly investigate quantum simulation of this novel NH system in two dimensions in detail and discuss for higher-dimensional cases in general using the linear combinations of unitaries in the scheme of duality quantum computing, enabling implementations and experimental investigations of novel properties on both small quantum devices and near-term quantum computers.
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.