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arXiv:2111.11220

Accelerated Non-Reciprocal Transfer of Energy Around an Exceptional Point

Hugo Ribeiro, Florian Marquardt

Published Nov 22, 2021• 0 citations
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Abstract

We develop perturbative methods to study and control dynamical phenomena related to exceptional points in Non-Hermitian systems. In particular, we show how to find perturbative solutions based on the Magnus expansion that accurately describe the evolution of non-Hermitian systems when encircling an exceptional point. This allows us to use the recently proposed Magnus-based strategy for control to design fast non-reciprocal, topological operations whose fidelity error is orders of magnitude smaller than their much slower adiabatic counterparts.

DOI
10.48550/arxiv.2111.11220

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We develop perturbative methods to study and control dynamical phenomena related to exceptional points in Non-Hermitian systems. In particular, we show how to find perturbative solutions based on the Magnus expansion that accurately describe the evolution of non-Hermitian systems when encircling an exceptional point. This allows us to use the recently proposed Magnus-based strategy for control to design fast non-reciprocal, topological operations whose fidelity error is orders of magnitude smaller than their much slower adiabatic counterparts.
The existence of exceptional points (EPs) in the spectrum of non-Hermitian Hamiltonians leads to a nontrivial spectral topology that can be exploited to generate topological operations. However, their implementation has remained elusive due to the difficulty of generating a true adiabatic evolution. Here, we develop fast, robust control protocols that generate a desired topological operation. Our strategy relies on shortcuts to adiabaticity, but is not a trivial extension. The presence of EPs renders the strategy developed for Hermitian Hamiltonians impractical as it will lead to faulty control protocols. Moreover, due to the dynamics sensitivity to parameter uncertainties, not all shortcuts to adiabaticity can be used in a realistic setting. We illustrate our method in the context of a two-mode non-Hermitian Hamiltonian and discuss why in general celebrated shortcuts to adiabaticity like transitionless driving and superadiabatic transitionless are not appropriate control protocols for non-Hermitian systems.
We study the quantum evolution of a non-Hermitian qubit realized as a submanifold of a dissipative superconducting transmon circuit. Real-time tuning of the system parameters to encircle an exceptional point results in nonreciprocal quantum state transfer. We further observe chiral geometric phases accumulated under state transport, verifying the quantum coherent nature of the evolution in the complex energy landscape and distinguishing between coherent and incoherent effects associated with exceptional point encircling. Our work demonstrates an entirely new method for control over quantum state vectors, highlighting new facets of quantum bath engineering enabled through dynamical non-Hermitian control.
The exploration of the effect of non-Hermitian (NH) in quantum systems is gaining renewed momentum as a result of the recent progress in experiments. It has been indicated that the ability to engineer the orders of exceptional points (EPs) can be fundamental to its applications in quantum control, such as accelerating entanglement generation; however, an initial survey of the interactions between different orders of EPs is still missing. In this work, we try to partially address this issue by employing a non-reciprocally coupled cavity system which can be experimentally realizable with a two-dimensional reconfigurable NH gauged laser array. With deliberate design, our systems display both second and third orders of EPs. The sign of a novel, purely quantum continuous entanglement phase transition is observed, which reveals how multi-EP interactions alter entanglement dynamics across different regimes while maintaining accelerated generation. Our findings reveil the genuinely quantum NH physics of higher-order EPs, which has opened up a new avenue for future investigations of the higher-order degeneracy application and the dynamic phase transitions of quantum systems.
We study the quantum evolution of a non-Hermitian qubit realized as a sub-manifold of a dissipative superconducting transmon circuit. Real-time tuning of the system parameters results in non-reciprocal quantum state transfer associated with proximity to the exceptional points of the effective Floquet Hamiltonian. We observe chiral geometric phases accumulated under state transport, verifying the quantum coherent nature of the evolution in the complex energy landscape and distinguishing between coherent and incoherent effects associated with exceptional point encircling. Our work demonstrates an entirely new method for control over quantum state vectors, highlighting new facets of quantum bath engineering enabled through time-periodic (Floquet) non-Hermitian control.
Exceptional points define branch-exchange state transfers through holomorphic continuation of non-Hermitian eigenmodes, but realizing these transfers dynamically remains difficult. Slow encircling does not generally transport the full set of instantaneous eigenstates, while shortcuts to adiabaticity can require controls outside the native experimental manifold. Here, we introduce a constrained shortcut-to-adiabaticity principle for exceptional-point braiding using native controls. In a dressed instantaneous-eigenstate frame, the available controls cancel the accessible transition channels locally, while the residual channels remain active during the evolution but are constrained to have no net accumulated effect over the closed loop. The protocol therefore targets the endpoint state transfer selected by ideal adiabatic branch exchange, rather than enforcing complete local cancellation or adiabatic following throughout the trajectory. We demonstrate the construction in a minimal two-mode non-Hermitian model equivalent, up to a trace shift and a basis convention, to the effective Hamiltonian used in dissipative-transmon exceptional-point experiments, where detuning and drive amplitude provide real controls and the relative loss imbalance fixes the non-Hermitian scale. Smooth real waveforms reshape only these controls, reproduce the branch-exchange transfer, and remain accurate under calibration errors, exceptional-point uncertainty and finite-bandwidth filtering with modest overhead.
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.