Analysis of resonant population transfer in time-dependent elliptical quantum billiards

A Fermi golden rule for population transfer between instantaneous eigenstates of elliptical quantum billiards with oscillating boundaries is derived. Thereby the occurrence of both the recently observed resonant population transfer between instantaneous eigenstates and the empirical criterion stating that these transitions occur when the driving frequency matches the mean difference of the latter [Lenz et al., New J. Phys. 13, 103019 (2011)] is explained. As a second main result a criterion judging which resonances are resolvable in a corresponding experiment of certain duration is provided. Our analysis is complemented by numerical simulations for three different driving laws. The corresponding resonance spectra are in agreement with the predictions of both criteria.

Figure 1

Energy eigenvalues $E_n\left(\tau \right)$ of eigenstates $\left|n;r\left(\tau \right)\right.〉$ with even $\eta$- and $\xi$-parity eigenvalues for the axis-ratio-preserving driving law. The instantaneous shape of the ellipse at five different values of $\tau$ is drawn below the energy eigenvalue curves. The parameters are $a_0=1$, $b_0=\sqrt{0.51}$, and $A=0.1$.

Figure 2

Dependence of the maximal and minimal energies on the driving frequency $\omega$ for the axis-ratio-preserving driving law. The vertical lines show all predicted resonance frequencies (31) with an interaction time ${\tau }_{\mathrm{int}}$ of less than 2000. The darker the lines are, the longer the corresponding interaction time is. Numerical values can be taken from Table . The parameters are the same as in Fig. 1.

Figure 3

Population dynamics $p_n\left(\tau \right)$ for the axis-ratio-preserving driving law at $\omega =10.81$. The parameters are the same as in Fig. 1.

Figure 4

Population dynamics $p_n\left(\tau \right)$ for the axis-ratio-preserving driving law at $\omega =15.17$. The parameters are the same as in Fig. 1.

Figure 5

Energy eigenvalues $E_n\left(\tau \right)$ of eigenstates $\left|n;r\left(\tau \right)\right.〉$ with even $\eta$ and $\xi$ parity for the breathing driving law. The parameters are the same as in Fig. 1.

Figure 6

Analog of Fig. 2 for the breathing driving law. Numerical values can be taken from Table . The parameters are the same as in Fig. 1.

Figure 7

Energy eigenvalues $E_n\left(\tau \right)$ of eigenstates $\left|n;r\left(\tau \right)\right.〉$ with even $\eta$- and $\xi$-parity eigenvalues for the volume-preserving driving law. The parameters are the same as in Fig. 1.

Figure 8

Analog of Fig. 2 for the volume-preserving driving law. Numerical values can be taken from Table . The parameters are the same as in Fig. 1.

Figure 9

Population dynamics $p_n\left(\tau \right)$ for the volume-preserving driving law at $\omega =3.32$. The parameters are the same as in Fig. 1.