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Communications in Theoretical Physics

Universal Pseudo- P T -Antisymmetry on One-Dimensional Atomic Optical Lattices*

Xin Wang, Chang-Pu Sun

Published Sep 1, 2018• 0 citations
Publisher

Abstract

Abstract We present the interesting result that under sinusoidal field detuning setting along the propagation direction of 1D atomic lattices, the probe susceptibility response of the lattices, regardless of atomic configuration, uniformly demonstrates pseudo - P T - antisymmetry, which by our definition corresponds to n(z) = −n * (−z), the complex refractive index antisymmetry along propagation axis, and when being cast back to quantum mechanical side, corresponds to V(x,t) = −V * (x,−t), the conjugate time-reversal antisymmetry of complex potential. We define this as the pseudo - P T - antisymmetry, and prove the reason for this phenomenon to be the quantum-mechanical nature described by master equation under weak field approximation for any configuration of 1D atomic lattices. This work will help to deepen the understanding of origin of optical response features of atomic lattices, and will certainly open up the gate to a more rigorous, durable and flexible method of atomic optical lattice design .

DOI
10.1088/0253-6102/70/3/303

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arXiv:2010.04342•Physical Review A•Oct 22, 2020•7 citations
Abstract We present the interesting result that under sinusoidal field detuning setting along the propagation direction of 1D atomic lattices, the probe susceptibility response of the lattices, regardless of atomic configuration, uniformly demonstrates pseudo - P T - antisymmetry, which by our definition corresponds to n(z) = −n * (−z), the complex refractive index antisymmetry along propagation axis, and when being cast back to quantum mechanical side, corresponds to V(x,t) = −V * (x,−t), the conjugate time-reversal antisymmetry of complex potential. We define this as the pseudo - P T - antisymmetry, and prove the reason for this phenomenon to be the quantum-mechanical nature described by master equation under weak field approximation for any configuration of 1D atomic lattices. This work will help to deepen the understanding of origin of optical response features of atomic lattices, and will certainly open up the gate to a more rigorous, durable and flexible method of atomic optical lattice design .
We study an ensemble of ultracold atoms trapped in the one-dimensional optical lattices under the N level configuration to examine the absorption and dispersion properties modulated into different ways along the lattice direction z. We find these trapped atoms with a Gaussian density distribution in each period may exhibit either symmetry χp(z)=χp*(-z) or antisymmetry χp(z)=-χp*(-z) of parity-time (PT) in terms of the probe susceptibility. Such intriguing twofold modulations of real (χ'p) and imaginary (χ″p) susceptibilities with a π/2 phase shift are attained by spatially modulating intensities or frequencies of one driving field in a suitable way. The PT-symmetric or PT-antisymmetric atomic lattices correspond in fact to complex photonic crystals and may be extended to develop functional devices like photonic diodes and transistors, a task impossible for real photonic crystals.
A collection of cold rubidium atoms in three-level configuration trapped in two dimensional (2D) optical lattices is revisited. The trapped atoms are considered in the Gaussian density distribution and we study the realization of P T -, non- P T -, and P T -antisymmetry in 2D optical lattices. Such a fascinating modulation is achieved by spatially modulating the intensity of the driving field. Interestingly, control over P T - to non- P T -symmetry and vice versa in 2D optical lattices is achieved via a single knob such as microwave field, probe field and relative phase of optical and microwave fields. In addition, control over P T -antisymmetry to non- P T -symmetry and vice versa is also achieved via relative phase. The coherent control of P T - non- P T - and P T -antisymmetry in optical susceptibility of 2D atomic lattices can be extended to 2D optical devices including modulators, detectors, and the 2D atomic lattices can also be extended to photonic transistors and diodes.
Abstract We propose a scheme for realizing exact parity-time ( P T ) antisymmetry of complex susceptibility in a wide range of probe detuning by considering a one-dimensional lattice of cold atoms driven into the simplest three-level Λ configuration. This is attained by modulating the intensity of a standing-wave coupling field with a proper phase shift to counteract the product of a single-photon detuning and a two-photon detuning. Such a dynamically controlled P T -antisymmetric lattice supports the integration of a few nontrivial scattering behaviors including unidirectional light reflectionless, asymmetric perfect absorption, and directional signal quenching. These behaviors, depending in particular on atomic densities and lattice lengths, facilitate the on-demand realization of unidirectional or bidirectional photon transport blockade.
The probe-transmission spectrum of a one-dimensional optical lattice is calculated. The optical lattice consists of atoms that are cooled by a pair of counterpropagating, linearly polarized fields, whose frequency \ensuremath{\omega} is detuned to the blue side of an atomic transition. A weak probe field of frequency ${\mathrm{\ensuremath{\omega}}}^{\ensuremath{'}}$ is assumed to be propagating in the same direction as one of the cooling fields. It is shown that, owing to the existence of dark and gray states in the lattice, the probe-transmission spectrum exhibits interesting features that are qualitatively different from those of a ``normal'' optical lattice with red-detuned cooling fields. Examples of these features include subrecoil linewidths and reversed absorption-amplification sidebands. It is shown that by monitoring the magnitudes of resonance signals in the spectrum, one can gain information on the evolution of the dark-state population in the lattice.
A collection of cold rubidium atoms in three-level configuration trapped in one dimensional (1D) optical lattice is revisited. The trapped atoms are considered in the Gaussian density distribution and study the realization of P T -, non- P T - and P T anti-symmetry in optical susceptibility in 1D atomic lattices in a periodic structure. Such a fascinating modulation is achieved by spatially modulating the intensity of the driving field. Interestingly, a nonreciprocal optical propagation phenomenon is investigated. In this system, we have introduced a microwave that couples to the two ground states, spatial modulation of the coupling field, and the atomic density with Gaussian distribution in practice. With a proper detuning and coupling field Rabi frequencies, we can find the condition of P T -symmetry along with field propagation direction, and the novel properties of transmission and reflections have been discussed. The large difference of field reflections from the two ends of the atomic lattice medium shows strong evidence that the nonreciprocal behavior can be greatly enhanced by increasing the spatial modulation amplitude.
Bogoliubov excitations of Bose-Einstein condensates in optical lattices may possess band topology in analogous to topological insulators in class AII of fermions. Using the language of the Krein-space theory, this topological property is shown to be protected by a pseudo-time-reversal symmetry that is pseudo-antiunitary and squares to $\ensuremath{-}1$, with the associated bulk topological invariant also being a Z2{\mathbb{Z}}_{2}Z2​ index. We construct three equivalent expressions for it, relating to the Pfaffian, the pseudo-time-reversal polarization, and most practically the Wannier center flow, all adopted from the fermionic case, defined here with respect to the pseudo inner product. In the presence of an additional pseudo-unitary and pseudo-Hermitian inversion symmetry, a simpler expression is derived. We then study two toy models feasible on cold atom platforms to numerically confirm the bulk-boundary correspondence. The Krein-space approach developed in this work is a universal formalism to study all kinds of symmetry-protected topological bosonic Bogoliubov bands.