It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalizing the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalizes the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalizing the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalizes the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.

arXiv:0803.3721Journal of Physics A Mathematical and TheoreticalMay 22, 200823 citations

Time evolution of quantum systems

Series in atomic and molecular physics/Series in atomic molecular physicsDec 1, 20030 citations

Generalized Floquet theory for open quantum systems

C. M. Dai, Hong Li, W. Wang +1 more

For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator $Λ(t,0)$ of the system can be factorized as $Λ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}$ with micro-motion operator $\mathcal{D}(t)$ possessing the same period as the external driving, and time-independent operator $\mathcal{L}_{eff}$ . In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator $Λ(t,0)$ of the system can be factorized as $Λ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}$ with micro-motion operator $\mathcal{D}(t)$ possessing the same period as the external driving, and time-independent operator $\mathcal{L}_{eff}$. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.

arXiv:1707.05030Jul 17, 20174 citations

Time evolution of quantum systems

Institute of Physics eBooksNov 26, 20040 citations

Response of a quantum system to a time-dependent external field anddynamical symmetry of the system

Wang, SJTypel, SWeiguny +1 more

The response of a quantum system to a time-dependent periodic external The response of a quantum system to a time-dependent periodic external

Jan 1, 19980 citations

Towards Time-Discrete Quantum Mechanics

Springer eBooksJan 1, 20020 citations

For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator

Λ(t,0)

of the system can be factorized as

Λ(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}

with micro-motion operator

\mathcal{D}(t)

possessing the same period as the external driving, and time-independent operator

\mathcal{L}_{eff}

. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.