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arXiv:1711.05806 · Physical Review A

Chiral state conversion without encircling an exceptional point

Absar U. Hassan, G. López-Galmiche, Gal Harari, Patrick LiKamWa, Mercedeh Khajavikhan, Mordechai Segev, Demetrios N. Christodoulides

Published Nov 27, 2017• 89 citations
arXivHTMLPDF

Abstract

Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a sufficiently slow variation of parameters, even away from the system's exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed-form solutions are obtained when the amplification or attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.

DOI
10.1103/physreva.96.052129

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Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a sufficiently slow variation of parameters, even away from the system's exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed-form solutions are obtained when the amplification or attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.
Physical systems with gain and loss can be described by a non-Hermitian Hamiltonian, which is degenerated at the exceptional points (EPs). Many new and unexpected features have been explored in the non-Hermitian systems with a great deal of recent interest. One of the most fascinating features is that chiral state conversion appears when one EP is encircled dynamically. Here, we propose an easy-controllable levitated microparticle system that carries a pair of EPs and realize slow evolution of the Hamiltonian along loops in the parameter plane. Utilizing the controllable rotation angle, gain and loss coefficients, we can control the structure, size and location of the loops in situ. We demonstrate that, under the joint action of topological structure of energy surfaces and nonadiabatic transitions, the chiral behavior emerges both along a loop encircling an EP and even along a straight path away from the EP. This work broadens the range of parameter space for the chiral state conversion, and proposes a useful platform to explore the interesting properties of exceptional points physics.
In the context of dynamical control, non-Hermitian systems offer unique opportunities and challenges. In the simplest case of two interacting levels with gain and/or loss, the existence of an exceptional point (EP) of degeneracy in the chiral complex eigenvalue landscape fundamentally changes how the system responds to adiabatic changes in the coupling $\ensuremath{\gamma}$ and energy separation $\ensuremath{\delta}$ of the bare levels. In particular, previous studies have shown that selective population transfer can be achieved by adiabatically steering the system around closed paths that encircle the EP with the proper helicity in the $\ensuremath{\gamma}$ vs $\ensuremath{\delta}$ parameter space. Here we show that efficient state-selective population transfer can, in some cases, be achieved even with control loops that do not enclose the EP.
Non-Hermitian systems have attracted much interest in recent decades, driven partly by the existence of exotic spectral singularities, known as exceptional points (EPs), where the dimensionality of the system evolution operator is reduced. Among various intriguing applications, the discovery of EPs has suggested the potential for implementing a symmetric mode switch, when encircling them in a system parameter space. However, subsequent theoretical and experimental works have revealed that dynamical encirclement of EPs invariably results in asymmetric mode conversion; namely, the mode switching depends only on the winding direction but not on the initial state. This chirality arises from the failure of adiabaticity due to the complex spectrum of non-Hermitian systems. Although the chirality revealed has undoubtedly made a significant impact in the field, a realization of the originally sought symmetric adiabatic passage in non-Hermitian systems with EPs has since been elusive. In this work, we bridge this gap and theoretically demonstrate that adiabaticity, and therefore a symmetric state transfer, is achievable when dynamically winding around an EP. This becomes feasible by specifically choosing a trajectory in the system parameter space along which the corresponding evolution operator attains a real spectrum. Our findings, thus, offer a promise for advancing various wave manipulation protocols in both quantum and classical domains.
One intriguing property of non-Hermitian systems is the breakdown of adiabatic theorem and chiral state conversion as the system dynamically encircles exceptional points. However, the subtle dependence of the chiral dynamics on the loop geometry, the starting point, the encircling speed and especially the noise has not been studied systematically. Here we propose a non-chirality degree χcχ_cχc​ to measure the chirality quantitatively and analyze it in dynamics without noise by exact solution and dynamics with noise by numerical integration. The exact dynamics starting from the broken phase show chirality oscillations, which are extremely sensitive to noise when the speed is small. The encircling speed and the noise strength are found to compete with each other in determining χcχ_cχc​, resulting in two distinguished limits, namely the noisy limit and the clean limit. The critical boundary between the two limits satisfies a simple scaling law, which could be explained in terms of first-order perturbation theory and the condition number of the transfer matrix. Our findings reveal the essential role played by noise in non-Hermitian dynamics and are relevant for both theoretical and experimental investigations.
In this paper, we formally prove how, by cyclically varying the parameters of a generalized two-level discrete and non-Hermitian Hamiltonian, the respective state vector converts to the instantaneous eigenstate of the system in the adiabatic limit, irrespective of how the state was initially prepared or of the form of the cyclic trajectory followed within the parameter space $(\mathsc{R})$ or of whether a singularity/exceptional point is encircled or not. Our proof also applies to continuous configurations in the limit of infinitesimally small discrete time steps. The observed mode switching behavior, which is a clear signature of the irreversible nature of non-Hermitian arrangements, can be either of symmetric (clockwise [CW] and counterclockwise [CCW] encirclements in $\mathsc{R}$ space lead to state conversion to the same instantaneous eigenstate) or asymmetric (CW and CCW encirclements in $\mathsc{R}$ space lead to state conversion to different instantaneous eigenstates). As a specific example, we investigate the case of rhombic parametric trajectories for a discrete parity-time (PT)(\mathcal{P}\phantom{\rule{0.16em}{0ex}}\mathcal{T}\phantom{\rule{0.28em}{0ex}})(PT)-symmetric-like setting. Exact analytical solutions are retrieved in terms of Weber/parabolic cylinder (large rhombic loops) and Airy functions (small rhombic loops), with their asymptotic behavior dictated by the Stokes phenomenon. Both analytical derivations and numerical computations indicate that the observed mode conversion is primarily an artifact of the adiabatic character of the state vector evolution, while the encirclement or not of a singularity/exceptional point affects only the magnitude of the required adiabaticity rate for mode switching to take place. Overall, our results provide a deeper theoretical insight on slowly varying discrete non-Hermitian Hamiltonian systems, and pave the way towards exploring the dynamics underpinning the traversal of higher-dimensional cyclic parametric trajectories in the vicinity (or not) of higher-order spectral singularities for both continuous and discrete settings under linear or nonlinear conditions.
The exceptional point, known as the non-Hermitian degeneracy, has special topological structure, leading to various counterintuitive phenomena and novel applications, which are refreshing our cognition of quantum physics. One particularly intriguing behavior is the mode switch phenomenon induced by dynamically encircling an exceptional point in the parameter space. While these mode switches have been explored in classical systems, the experimental investigation in the quantum regime remains elusive due to the difficulty of constructing time-dependent non-Hermitian Hamiltonians in a real quantum system. Here we experimentally demonstrate dynamically encircling the exceptional point with a single nitrogen-vacancy center in diamond. The time-dependent non-Hermitian Hamiltonians are realized utilizing a dilation method. Both the asymmetric and symmetric mode switches have been observed. Our Letter reveals the topological structure of the exceptional point and paves the way to comprehensively explore the exotic properties of non-Hermitian Hamiltonians in the quantum regime.