Population transfer at exceptional points in the spectra of the hydrogen atom in parallel electric and magnetic fields

We study the population transfer between resonance states for a time-dependent loop around the exceptional points in the spectra of the hydrogen atom in parallel electric and magnetic fields. The exceptional points are well suited for population transfer mechanisms, since a closed loop around these in parameter space commutes eigenstates. We address the question of how shape and duration of the dynamical parameter loop affect the transferred population, in order to optimize the latter. Since the full quantum dynamics of the expansion coefficients is time consuming, we furthermore present an approximation method, based on a $2\times 2$ matrix.

Figure 1

Eigenvalue trajectories for an elliptical field strength loop according to Eq. (17) with parameters $r={10}^{-3}$ and ${\varphi }_0=0$. The insets schematically show the trajectories of the isolated resonances. Start positions are marked with points. The EP resonances (1 and 2) commute, whereas the neighboring four side resonances perform closed loops. Population is transferred from the eigenstate corresponding to the start position of resonance 1 to the eigenstate of the start position at resonance 2.

Figure 2

Post-loop resonance populations $|{\alpha }_i{(t=T)|}^2$ for different encircling durations $T$ with $r=1\times {10}^{-3}$ and ${\varphi }_0=0$. The same colors as in Fig. 1 are used for labeling the resonances. Since an adiabatic basis is used the states 1 and 2 commute after one parameter loop. The maximum transferred population is $|{\alpha }_1\left(T_{\mathrm{opt}}\right){|}^2=0.0889\%$ at $T_{\mathrm{opt}}=2.39\times {10}^{-3}$. The dynamics of the EP resonances calculated with the $2\times 2$ matrix model (dash-dotted lines) coincide with the exact dynamics (solid lines).

Figure 3

Post-loop populations $|{\alpha }_i{(t=T)|}^2$ for different ellipse radii $r$, while $T=2.5\times {10}^3$ and ${\varphi }_0=0$. Since an adiabatic basis is used the population transfer is given by $|{\alpha }_1{\left(T\right)|}^2$. The latter strongly increases with $r$ for small $r$ values, goes through a global maximum at $r=0.12$ with $4.57\%$ transferred population, and decreases slightly for large $r$ values. For large radii $r$ the approximate solution (dash-dotted lines) shows deviations when compared to the exact solution (solid lines).

Figure 4

Complex eigenvalue trajectories for elliptical parameter loops according to Eq. (17) for six different ellipse radii $r$. For small $r$, EP resonances are strongly coupled. For large $r$, couplings to side resonances increase and the initially occupied resonance 1 (black solid line) drifts deeper into the complex energy plane.

Figure 5

Population transfer $|{\alpha }_1{\left(T\right)|}^2$ as a function of the ellipse radius $r$ and encircling duration $T$. The green curve is the optimal encircling duration for given $r$. The optimal encircling duration decreases with increasing $r$, since the decay rate of resonance 1 increases (see Fig. 4). For the global optimum $T_{\mathrm{opt}}=2.001\times {10}^3$, $r_{\mathrm{opt}}=0.1368$, the transfer amounts to $4.783\%$.

Figure 6

(a) Post-loop populations $|{\alpha }_i{\left(T\right)|}^2$ as a function of the starting angle ${\varphi }_0$ of the parameter ellipse. The $4\pi$-periodic transfer reaches the global maximum $|{\alpha }_1{\left(T\right)|}^2=12.67\%$ at ${\varphi }_{0,\mathrm{opt}}=2.55276\pi$. In (b) the transfer $|{\alpha }_1{\left(T\right)|}^2$ (color bar) is related to the corresponding start position of the initially populated resonance 1 in energy space. The arrows indicate the direction of rotation of the eigenvalue for growing $t$ in Eq. (17). Transfer is maximal for the eigenvalue path of resonance 1, which exhibits the smallest global decay.

Figure 7

Population dynamics $|{\alpha }_i{\left(t\right)|}^2$ for the optimal loop parameters $T_{\mathrm{opt}}=2.001\times {10}^3$, $r_{\mathrm{opt}}=0.1368$, and ${\varphi }_{0,\mathrm{opt}}=2.55276\pi$. The amount of transferred population denotes $12.67\%$.