Wave-particle duality of photons with losses in the Mach-Zehnder interferometer (MZI) is investigated experimentally and theoretically. The experiment is done with the standard MZI with the beam splitter or the beam merger being continuously varied.
The losses are deliberately introduced either inside the MZI (the two arms between the beam splitter and beam mergers) or outside the MZI (after the beam merger). It is proved that the unbalanced losses have great influence on the predictability $P$ (particle nature) and visibility $V$ (wave nature). For the former case the duality inequality holds while for the later the duality inequality is ``violated. We get $P^2+V^2>1$. This ``violation could be eliminated in principle by switching the two paths and detectors and then averaging the results. The observed results can be exactly explained theoretically. The experiment is done with coherent beam, instead of single photons, and we have proved that they are exactly equivalent in duality experiment with MZI.
Based on $(106.41 pm 0.86)times 10^{6}$ $psi(3686)$ events collected with the BESIII detector at the BEPCII collider, the branching fractions of $psi(3686) to pi^+pi^- J/psi$, $J/psi to e^+e^- $, and $J/psi to mu^+mu^-$ are measured. We obtain ${cal
B}[psi(3686) to pi^+pi^-J/psi]=(34.98pm 0.02pm 0.45)%$, ${cal B}[J/psi to e^+e^-] = (5.983 pm 0.007 pm 0.037)%$ and ${cal B}[J/psi to mu^+mu^-] = (5.973 pm 0.007 pm 0.038)%$. The measurement of ${cal B}[psi(3686) to pi^{+}pi^{-}J/psi]$ confirms the CLEO-c measurement, and is apparently larger than the others. The measured $J/psi$ leptonic decay branching fractions agree with previous experiments within one standard deviation. These results lead to ${cal B}[J/psi to l^+l^-] = (5.978 pm 0.005 pm 0.040)%$ by averaging over the $e^{+}e^{-}$ and $mu^{+}mu^{-}$ channels and a ratio of ${cal B}[J/psi to e^+e^-] / {cal B}[J/psi to mu^+mu^-] = 1.0017 pm 0.0017 pm 0.0033$, which tests $e$-$mu$ universality at the four tenths of a percent level. All the measurements presented in this paper are the most precise in the world to date.
The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $beta$ and the per
iodic solution, and a conclusion that the critical value of $beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics.
