Quantum Zeno effect by incomplete measurements

Abstract

A measurement process needs a time duration in many actual cases, such as the measurement on atomic system by the electron shelving technique. If this timescale is deficient, the measurement would be incomplete. Here, we investigate the quantum Zeno effect by incomplete measurements. We show that an efficient freeze of the quantum state by incomplete measurements is available. And interestingly, this state freeze can be more significant than that obtained in the complete measurement case if the parameters of incomplete measurements are properly set.

Appendix

It is instructive to consider how the parameter $\varDelta$ affects the QZE induced by incomplete measurements. After an incomplete measurement, we assume the system S evolves to a state expressed by the density matrix form $\rho ^\mathrm{S} = \mathrm{Tr_A}\rho$, where $\rho$ is the density matrix describing product state Eq. (5). Define two phase terms of the Bloch vector by $u = \rho _{ge}^\mathrm{S} + \rho _{eg}^\mathrm{S} = \langle \sigma _x\rangle$ and $v = (\rho _{ge}^\mathrm{S} - \rho _{eg}^\mathrm{S})/\mathrm{i} = \langle \sigma _y\rangle$ where $\rho _{ge}^\mathrm{S}$ and $\rho _{eg}^\mathrm{S}$ are nondiagonal elements of $\rho ^\mathrm{S}$; thus, one can find the evolution of them satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} u = u(0) \cdot \mathrm{Re}[a] - v(0) \cdot \mathrm{Im}[a], \nonumber \\ v = v(0) \cdot \mathrm{Re}[a] + u(0) \cdot \mathrm{Im}[a]. \end{array}\right. } \end{aligned}$$

$\mathrm{Re}[a]=e^{-\gamma t_{\mathrm{m}}}\cos {(t_{\mathrm{m}}\varDelta )}$, $\mathrm{Im}[a] = e^{-\gamma t_{\mathrm{m}}}\sin {(t_{\mathrm{m}}\varDelta )}$ are real and imaginary parts of a. If $\varDelta \ne 0$, it implies that the Bloch vector of system S will rotate by $t_{\mathrm{m}}\varDelta$ around $\sigma _z$ axis.