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arXiv:1809.07175 · Physical Review A

Non-Hermitian adiabatic transport in spaces of exceptional points

Judith Hoeller, N. Read, Jack Harris

Published Sep 16, 2020• 39 citations
arXivHTMLPDF

Abstract

We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians ($n=2$, 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 $n\mathrm{th}$ 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.

DOI
10.1103/physreva.102.032216

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A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.
We consider a two-state model which extends the exponential time-dependent Demkov model of non-adiabatic coupling to complex energies and interactions. Using the method of steepest descent, exact and practicable asymptotic expansions of the Hankel functions H v j (1) (λ), H v j (2) (λ) are obtained when | v j | and | λ | are large and of the same order. The semiclassical scattering two channel matrix within the strong-coupling Demkov model and embracing non-adiabatic parameters is derived. Closed form expressions for the transition probabilities, which incorporate the effects of the classical turning point, are also derived. The matrix elements are shown to reduce correctly in known limiting cases. These include the case of exact symmetric resonance, the Landau-Zener and Callaway-Bartling model and the Stueckelberg noncrossing formula. In the case of a Hamiltonian matrix which is complex Hermitian on the real t axis, Berry has shown that the phenomenon of anholonomy arises as the physical manifestation of a non-integrable real geometric phase in adiabatic parallel transport round a closed circuit in real space. More recently Berry has reinterpreted this theory as arising from transport around a circuit in complex time. The geometric amplitude factor, which is simply the anholonomic phase corresponding to the connection on a complex fibre bundle, occurs naturally in the Demkov model which is an example of a closed system in parameter space driven by a complex Hermitian Hamiltonian. It is shown that the Berry phase phenomenon also involves contributions from both real and complex circuit adiabatic phases.
This work presents a comprehensive study of geometric transport in non-Hermitian Floquet lattices. We derive the non-Hermitian Berry curvature using the biorthogonal formalism and prove an exact algebraic no-go theorem: perfectly anti-correlated driving (φ=0) yields zero net displacement for any wavepacket. When driving symmetry is broken (φ=π/2), a Gaussian wavepacket acquires finite directional displacement per cycle. Using adaptive peak-centered quadrature with analytic derivatives and direct time-dependent wavepacket simulations on finite chains, we demonstrate transport enhancement near the exceptional-point manifold, accompanied by non-monotonic behavior due to moving curvature hotspots. The paper includes full analytical characterization of the EP manifold topology, explicit convergence tests, finite-size scaling, and clear separation between geometric pumping and non-Hermitian skin effect. All Python scripts, raw outputs, and high-resolution figures are provided for full reproducibility. Keywords: non-Hermitian Floquet systems, exceptional points, geometric transport, Berry curvature, no-go theorem, adiabaticity, wavepacket dynamics
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.
We develop a Rayleigh--Schrödinger-like perturbation theory for non-Hermitian quantum systems at an exceptional point of order NNN. Working in the Jordan basis of the unperturbed Hamiltonian and employing a Puiseux expansion of the perturbed eigenvalues and eigenstates, we derive explicit recursion relations for the expansion coefficients. The corrections to the unperturbed eigenvalue in the Puiseux expansion govern the splitting near the exceptional point; the first two are obtained iteratively in two equivalent forms. One is given in terms of the perturbation Hamiltonian in the Jordan basis, and the other in terms of the generator that drives the eigenvalue evolution with respect to the perturbation. The latter constitutes the exceptional-point counterpart of a geometric perturbation method recently developed for the non-exceptional-point regime. Both representations are verified explicitly for the N=2N = 2N=2 and N=3N = 3N=3 cases.
Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.