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arXiv:1601.03499 · Physical Review A

Non-Hermitian tight-binding network engineering

Stefano Longhi

Published Feb 1, 2016• 41 citations
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Abstract

We suggest a simple method to engineer a tight-binding quantum network based on proper coupling to an auxiliary non-Hermitian cluster. In particular, it is shown that effective complex non-Hermitian hopping rates can be realized with only complex onsite energies in the network. Three applications of the Hamiltonian engineering method are presented: the synthesis of a nearly transparent defect in an Hermitian linear lattice; the realization of the Fano-Anderson model with complex coupling; and the synthesis of a $\mathcal{PT}$-symmetric tight-binding lattice with a bound state in the continuum.

DOI
10.1103/physreva.93.022102

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We suggest a simple method to engineer a tight-binding quantum network based on proper coupling to an auxiliary non-Hermitian cluster. In particular, it is shown that effective complex non-Hermitian hopping rates can be realized with only complex onsite energies in the network. Three applications of the Hamiltonian engineering method are presented: the synthesis of a nearly transparent defect in an Hermitian linear lattice; the realization of the Fano-Anderson model with complex coupling; and the synthesis of a PT\mathcal{PT}PT-symmetric tight-binding lattice with a bound state in the continuum.
Reflectionless defects in Hermitian tight-binding lattices, synthesized by the intertwining operator technique of supersymmetric quantum mechanics, are generally not invisible and time-of-flight measurements could reveal the existence of the defects. Here it is shown that, in a certain class of non-Hermitian tight-binding lattices with complex hopping amplitudes, defects in the lattice can appear fully invisible to an outside observer. The synthesized non-Hermitian lattices with invisible defects possess a real-valued energy spectrum; however, they lack parity-time (PT\mathcal{PT}PT) symmetry, which does not play any role in the present work.
The interaction of a discrete state coupled to a continuum is a longstanding problem of major interest in different areas of quantum and classical physics. In Hermitian models, several dynamical decoupling schemes have been suggested, in which the discrete-continuum interaction can be substantially reduced and even suppressed. In this work, we consider a discrete state interacting with a continuum via a time-dependent non-Hermitian coupling with finite (albeit arbitrarily long) duration, and show rather generally that for a wide class of coupling temporal shapes, in which the real and imaginary parts of the coupling are related each other by a Hilbert transform, the discrete state returns to its initial condition after the interaction with the continuum, while the continuum keeps trace of the interaction. Such a behavior, which does not have any counterpart in Hermitian dynamics, can be referred to as non-Hermitian pseudo decoupling. Non-Hermitian pseudo decoupling is illustrated by considering a non-Hermitian extension of the Fano–Anderson model in a one-dimensional tight-binding lattice. Such a non-Hermitian model can describe, for example, photonic hopping dynamics in a tight-binding chain of optical microrings or resonators, in which non-Hermitian coupling can be realized by fast modulation of the real and imaginary (gain/loss) parts of the refractive index of the edge microring.
Engineering quantum bath networks through non-Hermitian subsystem Hamiltonians has recently emerged as a promising strategy for qubit cooling, state stabilization, and fault-tolerant quantum computation. However, scaling these systems while maintaining precise control over their complex interconnections, especially in the optical domain, poses significant challenges in both theoretical modeling and physical implementation. In this work, drawing on principles from quantum and mathematical physics, we introduce a systematic framework for constructing non-Hermitian subsystems within entirely Hermitian photonic platforms. In particular, controlled exponential decay without actual absorption loss is realized in finite 1-D waveguide chains through discrete-to-continuum coupling and Lanczos transformations. Using this new methodology, we implement parity-time symmetric quantum systems and experimentally demonstrate that these artificial bath environments accurately replicate the dynamics of non-Hermitian arrangements in both single- and multi-photon excitation regimes. Since the non-Hermitian subsystem response deterministically arises from an artificially built Hermitian bath, the quantum evolution can be monitored via post-selection in this fully conservative configuration. This approach bridges the gap between theoretical models and experimental realizations, thus paving the way for exploiting quantum bath engineering in advanced information processing and emerging quantum technologies.
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians, it is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wavefunction with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter whether the non-Hermitian is broken symmetry or even non- symmetry. Exactly solvable models are presented to demonstrate the main points of this paper.
A method for high-fidelity quantum state transfer in a quantum network coupled to a continuum, based on time reversal in the continuum after decay, is theoretically suggested. Provided that the energy spectrum of the network is symmetric around a reference energy and symmetric energy states are coupled the same way to the common continuum, ideal perfect state transfer can be obtained after time reversal. In particular, it is shown that in a linear tight-binding chain a quantum state can be transformed into its mirror image with respect to the center of the chain after a controllable time. As compared to a quantum mirror image based on coherent transport in a static chain with properly tailored inhomogeneous hopping rates, our method does not require hopping rate engineering and is less sensitive to disorder for long transfer times.
The decay of a bound state weakly coupled to a non-Hermitian tight-binding unstable continuum, i.e., a continuum of states comprising energies with positive imaginary part, is theoretically investigated. As compared to quantum decay in a Hermitian continuum, in the non-Hermitian case a richer scenario can be found as a result of nonunitary dynamics. Different behaviors are observed depending on the kind of instability of the continuum. These include complete or fractional decay in convectively unstable continua, the absence of quantum decay for a bound state with energy embedded in the continuum loop, and unstable (secular) growth with pseudoexponential amplification in the absolutely unstable regime. Analytical results are presented for a nearest-neighboring tight-binding continuum with asymmetric hopping rates ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$, which shows a transition from convective to absolute instability when the sign ${\ensuremath{\kappa}}_{1}{\ensuremath{\kappa}}_{2}$ changes from positive to negative. In the convectively unstable regime, the model describes the decay of a bound state coupled to a tight-binding lattice with an imaginary gauge field, which shows a pseudo-Hermitian dynamics. In the absolutely unstable regime, pseudo-Hermitian dynamics is broken and a pseudoexponential secular growth is observed.