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arXiv:1605.02844 · Physical Review A

Quantum decay and amplification in a non-Hermitian unstable continuum

Stefano Longhi

Published Jun 29, 2016• 12 citations
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Abstract

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.

DOI
10.1103/physreva.93.062129

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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 \ensuremathκ1{\ensuremath{\kappa}}_{1}\ensuremathκ1​ and \ensuremathκ2{\ensuremath{\kappa}}_{2}\ensuremathκ2​, which shows a transition from convective to absolute instability when the sign \ensuremathκ1\ensuremathκ2{\ensuremath{\kappa}}_{1}{\ensuremath{\kappa}}_{2}\ensuremathκ1​\ensuremathκ2​ 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.
Controlling the decay dynamics of an unstable state coupled to a continuum is a longstanding problem of major interest in different areas of quantum and classical physics. Suppression of quan- tum mechanical decay and slow-down of the escape dynamics in macroscopic quantum and optical tunneling via dynamical decoupling methods have been proposed and experimentally demonstrated in different physical systems over the last two decades. So far most of dynamical decoupling schemes have been focused to Hermitian dynamics. In this work we consider the decay of an unstable state coupled to 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 metastable state returns to its initial condition after the interaction with the continuum. Re- markably, the absence of decay holds even beyond the weak-coupling (Markovian) limit and for arbitrarily slow (adiabatic) interactions. The method is illustrated by considering a paradigmatic example of decay dynamics of a metastable state coupled to a tight-binding continuum described by a Fano-Anderson model and realized by a semi-infinite tight-binding lattice. Non-Hermitian coupling can be implemented by fast modulation of the complex site energy of the edge site in the lattice.
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.
We consider non-Hermitian dynamics of a quantum particle hopping on a one-dimensional tight-binding lattice made of NNN sites with asymmetric hopping rates induced by a time-periodic oscillating imaginary gauge field. A deeply different behavior is found depending on the lattice topology. While in a linear chain (open boundary conditions) an oscillating field can lead to a complex quasienergy spectrum via a multiple parametric resonance; in a ring topology (Born--von Karman periodic boundary conditions) an entirely real quasienergy spectrum can be found and the dynamics is pseudo-Hermitian. In the large-NNN limit, parametric instability and pseudo-Hermitian dynamics in the two different lattice topologies are physically explained on the basis of a simple picture of wave-packet propagation.
A metastable quantum state coupled to a continuum undergoes an exponential decay in the Weisskopf-Wigner (Markovian) approximation. However, quantum theory strictly predicts deviations from an exponential decay law both at small and long time scales. In multilevel systems, even in the Markovian approximation strong deviations from an exponential decay can be observed in the intermediate time scale owing to interference of overlapping resonances. Such interference effects are simply described by an effective non-Hermitian Hamiltonian and are known to explain, for example, the existence of dark states and fractional decay. Here we show that the wide variety of anomalous behaviors observed in multilevel quantum decay are rooted in the non-normal nature of the effective non-Hermitian Hamiltonian, revealing strong similarities between anomalous quantum mechanical decay and non-normal dynamics found in hydrodynamic flows. Major signatures of non-normal dynamics include delayed decay, accelerated decay, and exponential-power-law decay at an exceptional point. Such signatures are exemplified by suggesting simple tight-binding lattice realizations of multilevel quantum decay, which could be emulated in integrated photonic experiments. Finally, it is shown that for noninteracting particles with fermionic statistics the usual exponential decay law is restored, i.e., all anomalous decay effects arising from single-particle non-normal dynamics are washed out.
Spectral singularities are predicted to occur in a non-Hermitian extension of the Friedrichs-Fano-Anderson model describing the decay of a discrete state ∣a⟩|a⟩∣a⟩ coupled to a continuum of modes. A physical realization of the model, based on electronic or photonic transport in a semi-infinite tight-binding lattice with an imaginary impurity site at the lattice boundary, is proposed. The occurrence of the spectral singularities is shown to correspond either to a diverging reflection probability (for an amplifying impurity) or to a vanishing reflection probability (for an absorbing impurity) from the lattice boundary. In the former case, the spectral singularity of the resolvent is also responsible for the nondecay of state ∣a⟩|a⟩∣a⟩ into the continuum, in spite of the absence of bound states.
In this paper, we investigate the open tight-binding model with NNN sites coupled to two reservoirs on its edges with the nonequilibrium Green function method to understand effects of open boundaries. As a result, we obtain an analytical expression of the electron density in the steady state and in the transient regime for N→∞N \rightarrow \inftyN→∞ and at absolute zero temperature. From the expression of the electron density in the steady state, we show that the open boundaries do not affect the qualitative behavior of the electron density and the phase transition which has been observed for the isolated tight-binding model also exists in our open case. Using the expression of the time-dependent electron density, we show that a two-step decay appears in the relaxation process purely caused by open boundaries. In addition, we show that the dependence of speed of convergence on boundary coupling strength qualitatively changes at a certain value of the boundary parameters. This is a result of special eigenvalues of a non-hermitian matrix whose imaginary parts do not vanish even in the limit N→∞N \rightarrow \inftyN→∞.