State flip at exceptional points in atomic spectra

We study the behavior of nonadiabatic population transfer between resonances at an exceptional point in the spectrum of the hydrogen atom. It is known that, when the exceptional point is encircled, the system always ends up in the same state, independent of the initial occupation within the two-dimensional subspace spanned by the states coalescing at the exceptional point. We verify this behavior for a realistic quantum system, viz., the hydrogen atom in crossed electric and magnetic fields. It is also shown that the nonadiabatic hypothesis can be violated when resonances in the vicinity are taken into account. In addition, we study nonadiabatic population transfer in the case of a third-order exceptional point, in which three resonances are involved.

Figure 1

(a) Two resonances interchanging their positions in the complex energy plane for a parameter-space circle (inset) around the exceptional point listed in the first row of data in Table 1. The center of the parameter-space circle is identical to the exceptional point and $\delta ={10}^{-2}$ was chosen. The initial point in the parameter plane and the corresponding energy values of the resonances are represented by symbols on the lines. (b, c) Dynamics of two nearby resonances with small imaginary parts, which are visible only as filled symbols in (a) and are marked by the labels 3 and 4.

Figure 2

Temporal evolution of the populations. Only the two resonances connected with the exceptional point were taken into account. (a, b) The actual populations are plotted for an initial population $a_1\left(0\right)=1$ and $a_2\left(0\right)=1$, respectively. The overall evolution is dominated by the decay emergent from the nonzero imaginary parts of the eigenvalues. (c) and (d) Weighted coefficients according to Eq. (7) are shown (c) for the initial condition of (a) and (d) for the initial condition of (b). Results of the adiabatic approximation ($a_{\mathrm{ad}}$) are shown for comparison.

Figure 3

Temporal evolution of the populations with all four resonances taken into account. (a) The initial population was exclusively in the resonance described by the coefficient $a_1$; (b) only $a_2$ was initially populated. In (a) and (b) it is already visible that the resonances with smaller moduli of the imaginary parts decay much more slowly. Eventually the resonance with a smaller modulus of the imaginary part dominates. (c, d) This is especially clearly visiblehere, where the weighted coefficients for the cases in (a) and (b), respectively, are shown. The whole surviving population transfers into the resonance represented by $a_4$.

Figure 4

The center of the circle in parameter space is shifted in the positive $\gamma$ direction to study the transition from an interchange scenario to a noninterchange case. (a) Circles for different values of $s$ as introduced in Eq. (10). The time evolution of the absolute (b) and weighted (c) coefficients demonstrate a significant qualitative change as soon as the exceptional point is no longer located within the parameter-space loop. The line styles of the circles correspond to those used in the temporal evolution. For nonweighted coefficients the adiabatic expectation is plotted alongside.

Figure 5

Final values of the coefficients after a full traversal of the circle in parameter space as a function of the shift parameter $s$, where the absolute coefficients (a) and their weighted counterparts (b) are shown. The (red) plus symbols represent $a_1$; (green) $\mathsf{X}$'s, $a_2$; and black circles, the adiabatic approximation. The shaded area (left half) denotes that the circle for this value of $s$ encircles the exceptional point.

Figure 6

Weighted coefficients $|a_i{|}^2$ from Fig. 5 as a function of the shift parameter $s$ and the evolved time $t$. (a, b) Two views of the ($s,t$) plane. One clearly recognizes the dramatic change as soon as the exceptional point is within the parameter-space circle, indicating the switch from one Riemann surface to the other.

Figure 7

Map of the energy space of a structure identical to that of a third-order exceptional point for a parameter-space circle around the two exceptional points from the last two rows of data in Table 1 (see inset). The plot shows three resonances interchanging their positions and some nearby resonances in the energy space. Energy eigenvalues for the initial point in the parameter-space circle are represented by symbols on the lines.

Figure 8

Temporal evolution of the populations for a circle around a third-order exceptional point. Initial populations were set up in (a, d) $a_1$, (b, e) $a_2$, and (c, f) $a_3$. In the case of the initial population being prepared in $a_3$ the system evolves adiabatically. In all cases the nondissipated population ends up in $a_3$. This is clearly shown by the weighted coefficients ${\overline{a}}_i$.

Figure 9

Temporal evolution of the populations for the same parameter-space circle as in Fig. 8 but with six resonances taken into account in the calculation. Again, initial populations were set up in (a, d) $a_1$, (b, e) $a_2$, and (c, f) $a_3$. The population always ends up in the resonance with the least modulus of the imaginary part, here $a_5$.