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Journal of Physics A Mathematical and Theoretical

On the observability and asymmetry of adiabatic state flips generated by exceptional points

Raam Uzdin, Alexei A. Mailybaev, Nimrod Moiseyev

Published Oct 5, 2011• 238 citations
Publisher

Abstract

In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.

DOI
10.1088/1751-8113/44/43/435302

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In open quantum systems where the effective Hamiltonian is not Hermitian, it is known that the adiabatic (or instantaneous) basis can be multivalued: by adiabatically transporting an eigenstate along a closed loop in the parameter space of the Hamiltonian, it is possible to end up in an eigenstate different from the initial eigenstate. This ‘adiabatic flip’ effect is an outcome of the appearance of a degeneracy known as an ‘exceptional point’ inside the loop. We show that contrary to what is expected of the transport properties of the eigenstate basis, the interplay between gain/loss and non-adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect.
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.
We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians (n=2n=2n=2, 3, ...) that are equivalent to a single $n\ifmmode\times\else\texttimes\fi{}n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\ensuremath{\varepsilon}$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}$ times an expansion in powers of ${\ensuremath{\varepsilon}}^{1/n}$. The exponential of the term of nthn\mathrm{th}nth order (which is equivalent to the ``geometric'' or Berry phase modulo $2\ensuremath{\pi}$) is thus independent of $\ensuremath{\varepsilon}$ as $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$; it depends only on the homotopy class of the loop and is an integer power of ${e}^{2\ensuremath{\pi}i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.
Abstract We study state conversion in a parity-time (PT) symmetry broken non-Hermitian two level system. We construct a theory and explain the underlying mechanism for state conversion and define adiabatic evolutions in non-Hermitian systems. The adiabatic theorem can be used if the initial state is an instantaneous eigenstate in Hermitian systems. We show that the system can adiabatically follow the modulationally stable instantaneous eigenstate sooner or later, regardless of the initial state if the system parameters are varied slowly in non-Hermitian systems. We discuss the topological feature of dynamical circling in the parameter space.
Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues Ei≡Ei+iΓi/2 of the non-Hermitian Hamiltonian H coalesce (where Ei is the energy and Γi is the inverse lifetime of the state i ). We show that also the eigenfunctions Φi of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points a=acr in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.
We investigate the theory of exceptional points (EPs); degeneracies of non-hermitian systems famous for the swaps of eigenstates that arise when encircling the EP according to the adiabatic approximation. We review various characterizations of EPs that are used in the literature. We observe that they are not equivalent and consequently a unanimous definition is lacking. The theory of PT symmetric systems is studied as an aid in the study of EPs. In particular, we deal explicitly with a three-channel waveguide system originating from PT theory and supporting various EPs. We show how one can properly compose the eigenvalue permutations arising from EPs and how this generates a non-abelian group which we call the Λ-group. We finish with the needed geometry as induced by the adiabatic approximation. The adiabatic change as measured in experiment can be modelled naturally on a principal C× o Sn-bundle consisting of the eigenframes of the operator family. To the best of our knowledge this bundle is not found in the literature to date. The adiabatic approximation induces a canonical connection on this bundle. The special shape of the holonomy group induces three groups, one of which is the Λ-group, that fit together in a short exact sequence (SES). This SES can be used to describe and distinguish degeneracy structures. In particular, the Λ-group allows one to phrase a new definition of exceptional point that specializes existing definitions. Explicitly, one can define an EP by demanding that swaps of eigenvalues occur arbitrarily close to the EP. This definition is widely applicable, yet guarantees that the eigenvalue and eigenstates sheets are connected in a non-trivial way.
Exceptional points (EPs) have attracted extensive research interest due to their intriguing properties. One of the hallmarks of EP physics is that dynamically encircling the EPs induces chiral mode switching, arising from the breakdown of adiabaticity due to the presence of a complex spectrum in the system's Hamiltonian. While such chiral mode behavior has been widely observed experimentally, achieving truly adiabatic, and thus symmetric, state transfer, regardless of the winding direction, in time-modulated non-Hermitian systems has remained elusive. In this work, we demonstrate that this long-sought adiabatic state dynamics can indeed be restored. By steering a two-mode photonic setup along specifically designed trajectories in parameter space, we realize conditions where the associated non-Hermitian evolution operator acquires a purely real spectrum. Moreover, our experimental platform enables controlled switching between symmetric (adiabatic) and chiral (non-adiabatic) state-transfer regimes for the same set of initial modes, thus effectively implementing a universal symmetric-asymmetric two-mode switch. Our results therefore open new avenues for harnessing unique topological spectral properties of non-Hermitian systems, paving the way for the practical design of versatile optical wave-manipulation devices and for advancing both classical and quantum information technologies.