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arXiv:1706.01162 · Physical Review A

Effect of non-Hermiticity on adiabatic elimination in coupled waveguides

Rahman Sharaf, Mojgan Dehghani, Hamidreza Ramezani

Published Jan 31, 2018• 5 citations
arXivHTMLPDF

Abstract

We investigate the influence of non-Hermiticity on the adiabatic elimination in coupled waveguides. We show that although the total norm of the system is not conserved in parity-time symmetric phase, the dark state intensity remains constant. However, in broken phase the eliminated waveguide loses its darkness, i.e., its field amplitude starts increasing. In both exact and broken phases, the effective Hamiltonian derived from the adiabatic elimination condition describes the dynamics of all states except the dark state. Specifically, it cannot predict the amplification in the intensity of the dark state. Our results can strengthen the control of the dynamics in cavities with active elements and improve the design of controllable absorbers.

DOI
10.1103/physreva.97.013854

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We investigate the influence of non-Hermiticity on the adiabatic elimination in coupled waveguides. We show that although the total norm of the system is not conserved in parity-time symmetric phase, the dark state intensity remains constant. However, in broken phase the eliminated waveguide loses its darkness, i.e., its field amplitude starts increasing. In both exact and broken phases, the effective Hamiltonian derived from the adiabatic elimination condition describes the dynamics of all states except the dark state. Specifically, it cannot predict the amplification in the intensity of the dark state. Our results can strengthen the control of the dynamics in cavities with active elements and improve the design of controllable absorbers.
We show how the dynamics of a specific subset of states can be separated from the dynamic of the total quantum state via a time-dependent projector-based formalism of adiabatic elimination. Within our formalism, we assume an explicit time dependency in the coupling between both subsystems. Additionally, we do not assume that the elements of the Hamiltonian commute, as in matter-wave optics, this is not given in general. Here, the center-of-mass degrees of freedom frequently need to be taken into account. Assuming non-commutativity also enables the application of this formalism to matrices of arbitrary dimension and the treatment of the adiabatic elimination in high dimensional systems, where the elimination may be needed to reduce the overall complexity. In particular, we apply our formalism to typical mechanisms in matter-wave optics, Raman, and Bragg diffraction.
Abstract Dark states, which are incapable of absorbing and emitting light, have been widely applied in multiple disciplines of physics. However, the existence of dark states relies on certain strict constraints on the system. For instance, in the fundamental Λ\LambdaΛ Λ system, a perturbation breaking the degeneracy between two energy levels may destroy the destructive interference and demolish the dark state. Here, we show that non-Hermiticity can be exploited as a constructive means to restore a dark state. By compensating for the undesired perturbations, non-Hermiticity produces unidirectional couplings such that the dark state remains decoupled from the rest of the system. Implementing this scheme in many-body systems, flat bands and edge states can be recovered by losses and gains. Further taking into account interactions, a range of novel quantum phases could arise in such non-Hermitian systems.
We show that the analog of the geometric phase for non-Hermitian coupled waveguides with PT\mathcal{PT}PT symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, PT\mathcal{PT}PT symmetry will be observed breaking earlier than theoretically predicted.
We propose an adiabatic method for the robust transfer of light between the two outer waveguides in a three-waveguide directional coupler. Unlike the established technique inherited from stimulated Raman adiabatic passage (STIRAP), the method proposed here is symmetric with respect to an exchange of the left and right waveguides in the structure and permits the transfer in both directions. The technique uses the adiabatic elimination of the middle waveguide together with level crossing and adiabatic passage in an effective two-state system involving only the external waveguides. It requires a strong detuning between the outer and the middle waveguide and does not rely on the adiabatic transfer state (dark state) underlying the STIRAP process. The suggested technique is generalized to an array of NNN waveguides and verified by numerical beam propagation calculations.
The generation of non-Abelian geometric phases from a system of evanescently coupled waveguides is considered within the framework of nonorthogonal coupled-mode theory. Here, we study an experimentally feasible tripod arrangement of waveguides that contain dark states from which a nontrivial U(2)U(2)U(2) mixing can be obtained by means of an adiabatic parameter variation. We investigate the influence of higher-order contributions beyond nearest-neighbor coupling as well as self-coupling on the stability of a non-Abelian U(3)U(3)U(3) phase generated from an optical tetrapod setup. Our results indicate that, despite the mode nonorthogonality, the symmetry of dark states protects the geometric evolution of light from distortion.
The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.