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arXiv:1809.10366 · Physical Review A

Scattering properties of a parity-time-antisymmetric non-Hermitian system

L. Jin

Published Aug 13, 2018• 18 citations
arXivHTMLPDF

Abstract

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 $S{S}^{\ifmmode\dagger\else\textdagger\fi{}}=I$.

DOI
10.1103/physreva.98.022117

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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±\else\textpm\fiT=1R\ifmmode\pm\else\textpm\fi{}T=1R\ifmmode±\else\textpm\fiT=1, which depends on the parity of the scattering center size. Third, the scattering matrix satisfies (S\ensuremathσz)(S\ensuremathσz)∗=I\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right){\left(S{\ensuremath{\sigma}}_{\mathrm{z}}\right)}^{*}=I(S\ensuremathσz​)(S\ensuremathσz​)∗=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 SS\ifmmode†\else\textdagger\fi=IS{S}^{\ifmmode\dagger\else\textdagger\fi{}}=ISS\ifmmode†\else\textdagger\fi=I.
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 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.
Lossy atomic photonic crystals can be suitably tailored so that the real and imaginary parts of the susceptibility are, respectively, an odd and an even function of position. Such a parity-time (PT)(PT)(PT) space antisymmetry in the susceptibility requires neither optical gain nor negative refraction, but is rather attained by a combined control of the spatial modulation of both the atomic density and their dynamic level shift. These passive photonic crystals made of dressed atoms are characterized by a tunable unidirectional reflectionlessness accompanied by an appreciable degree of transmission. Interestingly, such peculiar properties are associated with non-Hermitian degeneracies of the crystal scattering matrix, which can then be directly observed through reflectivity measurements via a straightforward phase modulation of the atomic dynamic level shift and even off resonance.
Abstract A wide class of non-Hermitian Hamiltonians can possess entirely real eigenvalues when they have parity-time (PT) symmetric potentials. Recently, this family of non-Hermitian systems has attracted considerable attention in diverse areas of physics due to their extraordinary properties, especially in optical systems based on solid-state materials, such as coupled gain–loss waveguides and microcavities. Considering the desired refractive index can be effectively manipulated through atomic coherence, it is important to realize such non-Hermitian optical potentials and further investigate their distinct properties in atomic systems. In this paper, we review the recent theoretical and experimental progress of non-Hermitian optics with coherently prepared multi-level atomic configurations. The realizations of (anti-) PT symmetry with different schemes have extensively demonstrated the special optical properties of non-Hermitian optical systems with atomic coherence.
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.
Non-Hermitian, one-dimensional potentials which are also nonlocal, allow for scattering asymmetries, namely, asymmetric transmission or reflection responses to the incidence of a particle from left or right. The symmetries of the potential imply selection rules for transmission and reflection. In particular, parity-time (PT) symmetry or the symmetry of any local potential do not allow for asymmetric transmission. We put forward a feasible quantum-optical implementation of non-Hermitian, nonlocal, non-PT potentials to implement different scattering asymmetries, including transmission asymmetries.