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arXiv:1211.0819 · Physica B Condensed Matter

Excitation of time-dependent quantum systems: An application of time–energy uncertainty relations

Tarek Khalil, J. Richert

Published Sep 28, 2014• 6 citations
arXivHTMLPDF
DOI
10.1016/j.physb.2014.09.027

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We introduce a general framework for defining context-dependent time distributions in quantum systems using projective measurements. The time-of-flow (TF) distribution, derived from population transfer rates into a measurement subspace, yields a time-energy uncertainty relation of the form $\mathrm{\ensuremath{\Delta}}\mathcal{T}\mathrm{\ensuremath{\Delta}}H\ensuremath{\ge}\ensuremath{\hbar}/(6\sqrt{3})\ensuremath{\delta}\ensuremath{\theta}$, where $\ensuremath{\delta}\ensuremath{\theta}$ quantifies the net population transfer measured by the projector. This bound applies to arbitrary projectors under unitary dynamics and reveals that time uncertainty is inherently measurement dependent. We demonstrate the framework with two applications: a general time-of-arrival--energy uncertainty relation and a driven three-level system under detuned coherent driving. The TF framework unifies timing observables across spin, atomic, and matter-wave systems and offers an experimentally accessible route to probing quantum timing in controlled measurements.
Abstract Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.
We study the temporal formation of quantum-mechanical bound states within a one-dimensional attractive square-well potential by first solving the time-independent Schr\"odinger equation and then studying a time-dependent system with an external time-dependent potential. For this we introduce Gaussian potentials with different spatial and temporal extensions and generalize this description also for subsequent pulses and for random noisy potentials. Our main goal is to study the timescales in which the bound state is populated and depopulated. Particularly, we clarify a likely connection between the uncertainty relation for energy and time and the transition time between different energy eigenstates. We demonstrate that the formation of states is not delayed due to the uncertainty relation but follows the pulse shape of the perturbation. In addition we investigate the (non-)applicability of first-order perturbation theory on the considered quantum system.