Fig. 2

Asymmetric evolution of the probability density \(|\psi _{\mathrm{s},n}^\epsilon ({q},t)|^2\) given in Eq. (41). The relative amplitude \(\epsilon\) associated to \(\psi _{\mathrm{s},n,-} ({q},t)\) is \(0.1\epsilon _0\) for (a) and \(10\epsilon _0\) for (b) where \(\epsilon _0=(1+i)/\sqrt{2}\). The conditions given in Eqs. (81)–(87) are used. The values of other parameters taken here are \(L=1\), \(C_0 = 1\), \(\omega _0=1\), \(Q_{\mathrm{t},c}(0)=1\), \(P_{\mathrm{t},c}(0)=1\), \(r=0.3\), \(\phi =1\), \(\hbar =1\), \(\beta =0.1\), \(n=3\), and \({\textsf {Q}}=0\), which, in fact, are the same as those of Fig. 1a