Numerical and theoretical analysis of electron survival probability is presented for the system where a periodically driven discrete state is coupled with continuum states of the tight-binding model. The stability of the discrete state is examined by analyzing the survival probability in the weak-coupling case. Stable nondecaying states as well as usual unstable decaying states appear in the systems where the oscillating energy of the discrete state and the energy band of the continuum states overlap. Dynamical stable states caused by the energy oscillation are qualitatively explained by an approximate solution of a periodically driven two-level model. The decay is suppressed as a result of vanishing reduced coupling constant at the resonance.
Numerical and theoretical analysis of electron survival probability is presented for the system where a periodically driven discrete state is coupled with continuum states of the tight-binding model. The stability of the discrete state is examined by analyzing the survival probability in the weak-coupling case. Stable nondecaying states as well as usual unstable decaying states appear in the systems where the oscillating energy of the discrete state and the energy band of the continuum states overlap. Dynamical stable states caused by the energy oscillation are qualitatively explained by an approximate solution of a periodically driven two-level model. The decay is suppressed as a result of vanishing reduced coupling constant at the resonance.
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 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.
We formulate a quantum mechanical system consisting of a single discrete state coupled to an infinite ladder of equally-spaced states, the coupling between the two being given by a Lorentzian profile. Various limits of this system correspond to well-known models from quantum optics, namely, the narrow resonance limit gives the Rabi system, the wide resonance limit gives the Bixon-Jortner system, the wide resonance, true continuum limit gives the Wigner-Weisskopf system, and the fixed resonance, true continuum limit gives a system that is typically studied by methods developed by Fano. We give a semi-analytical solution of the eigenvalue problem by reducing it to a transcendental equation, and demonstrate the aforementioned limiting behaviors. We then study the dynamics of the initial discrete state numerically, and show that it gives a wide range of behaviors in various limiting cases as predicted by our asymptotic theory including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of this system to interpolate between such a rich set of behaviors and existing model systems, and the accessibility of a semi-analytical solution, make it a useful model system in quantum optics and related fields.
The dynamics of a single quantum state embedded in one continuum or several (quasi)continua is one of the most studied phenomena in quantum mechanics. In this paper we investigate its discrete analog and consider short- and long-time dynamics based on numerical and analytical solutions of the Schr\"odinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [L. Guo, A. Grimsmo, A. F. Kockum, M. Pletyukhov, and G. Johansson, Phys. Rev. A 95, 053821 (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters, is identical to a qubit interacting with a finite number of parallel regularly spaced bands of states via constant couplings. As a consequence, the characteristic near periodic initial-state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon--Jortner model, the wide-resonance true continuum limit yields the Wigner--Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.
We formulate a quantum-mechanical system consisting of a single discrete state coupled to an infinite ladder of equally spaced states, with the coupling described by a Lorentzian profile. Various limiting cases of this system correspond to well-known models in quantum optics: the narrow-resonance limit yields the Rabi model, the wide-resonance limit yields the Bixon-Jortner model, the wide-resonance true continuum limit yields the Wigner-Weisskopf model, and the fixed-resonance true continuum limit yields a system typically studied using methods developed by Fano. We obtain a semianalytical solution to the eigenvalue problem by reducing it to a transcendental equation and demonstrate the limiting behaviors described above. We then investigate numerically the dynamics of the initial discrete state and show that, in various limiting regimes, the system exhibits a wide range of behaviors predicted by the asymptotic theory, including exponential decay, revivals, Rabi oscillations, and damped oscillations. The ability of the system to interpolate between such a rich variety of behaviors and established model systems, together with the accessibility of a semianalytical solution, makes it a useful model in quantum optics and related fields.