Fingerprints of exceptional points in the survival probability of resonances in atomic spectra

The unique time signature of the survival probability exactly at the exceptional point parameters is studied here for the hydrogen atom in strong static magnetic and electric fields. We show that indeed the survival probability $S\left(t\right)={\left|\langle \psi \left(0\right)|\psi \left(t\right)\rangle \right|}^2$ decays exactly as ${|1-at|}^2{e}^{-{\Gamma }_{\mathrm{EP}}t/\hslash }$, where ${\Gamma }_{\mathrm{EP}}$ is associated with the decay rate at the exceptional point and $a$ is a complex constant depending solely on the initial wave packet that populates exclusively the two almost degenerate states of the non-Hermitian Hamiltonian. This may open the possibility for a first experimental detection of exceptional points in a quantum system.

Figure 1

Survival probability for the resonances at the exceptional point labeled 8 in Ref. [20] with an offset of $\delta ={10}^{-5}$. Shown are the direct evaluation $S_{\mathrm{d}}\left(t\right)$ according to Eq. (13) and the expected form $S_{\mathrm{m}}\left(t\right)$ of Eq. (8a) including a linear term in $t$ in addition to the exponential decay. Note that on the scale of the figure the survival probabilities calculated by the two methods are not distinguishable. Since we use here the c-product rather than the regular scalar product, the survival probability can assume values larger than 1. This nonphysical behavior results from the way we normalize ${\Psi }_0$ [see Eq. (12)]. As we will show later this problem disappears when the initial wave packet is prepared by using a laser excitation of the field-free ground state. On the c-product, read Refs. [21, 25].

Figure 2

(a) Survival probability $S_{\mathrm{d}}\left(t\right)$ for the same exceptional point as in Fig. 1 but for several different offsets $\delta ={10}^{-8},\cdots ,{10}^{-2}$. Small deviations appear only for the largest $\delta ={10}^{-2}$. In this case also a small difference from the shape $S_{\mathrm{m}}\left(t\right)$ at an exceptional point becomes observable. (b) The modulus $d=\langle {\Psi }_1+i{\Psi }_2|{\Psi }_1+i{\Psi }_2\rangle$ vanishes at the exceptional point as is expected due to the phase relation (10).

Figure 3

(a) Occupation probabilities $|A_i{|}^2$ in arbitrary units for resonances in the energy vicinity of the exceptional point. The two resonances connected with the exceptional point are marked with blue (filled) circles. On the scale of our plot the two almost degenerate states are not distinguishable. (b) Survival probability for the excited state ${\Psi }_{\mathrm{F}}$, which is prepared by using a laser to excite the field-free ground state [see Eq. (16)]. The division by the exponential part in the inset demonstrates the presence of the polynomial contribution. The behavior of the survival probability as presented here is exactly as in the analytical expression given in Eq. (8a).