We study the dynamics of a simple model for quantum decay, where a single state is coupled to a set of discrete states, the pseudocontinuum, each coupled to a real continuum of states. We find that for constant matrix elements between the single state and the pseudocontinuum the decay occurs via one state in a certain region of the parameters, involving the Dicke and quantum Zeno effects. When the matrix elements are random, several cases are identified. For a pseudocontinuum with small bandwidth there are weakly damped oscillations in the probability to be in the initial single state. For intermediate bandwidth one finds mesoscopic fluctuations in the probability with amplitude inversely proportional to the square root of the volume of the pseudocontinuum space. They last for a long time compared to the nonrandom case.
We study the dynamics of a simple model for quantum decay, where a single state is coupled to a set of discrete states, the pseudocontinuum, each coupled to a real continuum of states. We find that for constant matrix elements between the single state and the pseudocontinuum the decay occurs via one state in a certain region of the parameters, involving the Dicke and quantum Zeno effects. When the matrix elements are random, several cases are identified. For a pseudocontinuum with small bandwidth there are weakly damped oscillations in the probability to be in the initial single state. For intermediate bandwidth one finds mesoscopic fluctuations in the probability with amplitude inversely proportional to the square root of the volume of the pseudocontinuum space. They last for a long time compared to the nonrandom case.
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.
The dynamics of a single quantum state embedded in one or several (quasi-)continua is one of the most studied phenomena in quantum mechanics. In this work we investigate its discrete analogue and consider short and long time dynamics based on numerical and analytical solutions of the Schrödinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [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 band 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, the observation of polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.
This Letter addresses the dynamical quantum problem of a driven discrete energy level coupled to a semi-infinite continuum whose density of states has a square-root-type singularity, such as states of a free particle in one dimension or quasiparticle states in a BCS superconductor. The system dynamics is strongly affected by the quantum-mechanical repulsion between the discrete level and the singularity, which gives rise to a bound state, suppresses the decay into the continuum, and can produce Stueckelberg oscillations. This quantum coherence effect may limit the performance of mesoscopic superconducting devices, such as the quantum electron turnstile.
An isolated level coupled to a continuum of levels need not decay fully into them. Ordinarily one expects exponential decay, module finite-size effects, transients and long-time power-law tails. In the phenomenon we describe the amplitude for remaining in the initial state at first drops, but it then levels off and remains O(1) indefinitely. Because of the available continuum, this is different from certain quantum localization phenomena where there is an absence of on-shell levels. The origin of the effect we describe is the existence of thresholds and band edges. A condition relating the proximity to threshold with the strength of the coupling determines whether the limited decay occurs.
Small (but still containing many, about 10 2 -10 4 , atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins.Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior.The topic of our paper is in-between these limits.Nano-scale systems are characterized by small (but finite) mean inter-level spacing.In such a case with recurrence periods in pico-second range, Loschmidt echo and double resonance phenomena come into the game.System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics.We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period.We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig.It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states.In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system -reservoir coupling matrix elements are independent of the reservoir states.We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations.The coupling results to temperature dependent broadening and decay of the echo components.Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir.We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).
Abstract Small (but still containing many, about 102–104, atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrödinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in between these limits. Nano-scale systems are characterized by small (but finite) mean interlevel spacing. In such a case with recurrence periods in picosecond range, Loschmidt echo and double resonance phenomena come into the game. System behavior becomes nontrivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system—reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nanosystems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).