Resonances in open quantum systems

The Hamilton operator of an open quantum system is non-Hermitian. Its eigenvalues are generally complex and provide not only the energies but also the lifetimes of the states of the system. The states may couple via the common environment of scattering wave functions into which the system is embedded. This causes an external mixing (EM) of the states. Mathematically, EM is related to the existence of singular (the so-called exceptional) points. The eigenfunctions of a non-Hermitian operator are biorthogonal, in contrast to the orthogonal eigenfunctions of a Hermitian operator. A quantitative measure for the ratio between biorthogonality and orthogonality is the phase rigidity of the wave functions. At and near an exceptional point (EP), the phase rigidity takes its minimum value. The lifetimes of two nearby eigenstates of a quantum system bifurcate under the influence of an EP. At the parameter value of maximum width bifurcation, the phase rigidity approaches the value one, meaning that the two eigenfunctions become orthogonal. However, the eigenfunctions are externally mixed at this parameter value. The $S$ matrix and therewith the cross section do contain, in the one-channel case, almost no information on the EM of the states. The situation is completely different in the case with two (or more) channels where the resonance structure is strongly influenced by the EM of the states and interesting features of non-Hermitian quantum physics are revealed. We provide numerical results for two and three nearby eigenstates of a non-Hermitian Hamilton operator that are embedded in one common continuum and are influenced by two adjoining EPs. The results are discussed. They are of interest for an experimental test of the non-Hermitian quantum physics as well as for applications.

Figure 1

Plot of eigenvalues ${\mathcal{E}}_i\equiv E_i+\frac{1}{2}{\mathrm{\Gamma }}_i$, phase rigidity $r_i$ and $1-r_i$, and mixing $|b_{ij}|$ of the eigenfunctions ${\mathrm{\Phi }}_i$ of the Hamiltonian ${\mathcal{H}}^{\left(2\right)}$ as a function of $a$. The value $\omega$ is independent of $a$: (a)–(e) $\omega =0.01(i+\frac{1}{10})$ and (f)–(j) $\omega =0.5(i+\frac{1}{10})$. The parameters are (a)–(e) $e_1=1-a/2$, $e_2=a$, ${\gamma }_1/2=-0.495$, and ${\gamma }_2/2=-0.493$ and (f)–(j) $e_1=1-a/2$, $e_2=a$, ${\gamma }_1/2=-0.495$, and ${\gamma }_2/2=-0.595$. The dotted lines show $e_i$ and ${\gamma }_i/2$ in (a,b,f,g).

Figure 2

Plot of eigenvalues ${\mathcal{E}}_i\equiv E_i+\frac{1}{2}{\mathrm{\Gamma }}_i$, phase rigidity $r_i$ and $1-r_i$, and mixing $|b_{ij}|$ of the eigenfunctions ${\mathrm{\Phi }}_i$ of the Hamiltonian ${\mathcal{H}}^{\left(3\right)}$ as a function of $a$. The value $\omega$ is independent of $a$: (a)–(e) $\omega =0.01(i+\frac{1}{10})$ and (f)–(j) $\omega =0.5(i+\frac{1}{10})$. The parameters are (a)–(e) $e_1=1-a/2$, $e_2=a$, $e_3=-1/3+1.5a$, ${\gamma }_1/2=-0.495$, ${\gamma }_2/2=-0.493$, and ${\gamma }_3/2=-0.49$ and (f)–(j) $e_1=1-a/2$, $e_2=a$, $e_3=-1/3+1.5a$, ${\gamma }_1/2=-0.495$, ${\gamma }_2/2=-0.595$, and ${\gamma }_3/2=-0.545$. The dotted lines show $e_i$ and ${\gamma }_i/2$ in (a,b,f,g).

Figure 3

Cross section with two resonance states. The parameters are the same as in Figs. 1–1, but $\omega =0$ in (b). In (a) and (b) $a=0$ (dashed red line), $a=1$ (dotted blue line), and $a=0.653333$ (solid black line). (c) A 2D contour plot with $\omega =0.01(i+\frac{1}{10})$.

Figure 4

Cross section with two resonance states. The parameters are the same as in Figs. 1–1, but $\omega =0$ in (b). In (a) and (b), $a=0$ (dashed red line), $a=1$ (dotted blue line), and $a=0.6502$ (solid black line). (c) A 2D contour plot with $\omega =0.5(i+\frac{1}{10})$.

Figure 5

Cross section with three resonance states. The parameters are the same as in Figs. 2–2, but $\omega =0$ in (b). In (a) and (b) $a=0$ (dashed red line), $a=1$ (dotted blue line), and $a=0.6502$ (solid black line). (c) A 2D contour plot with $\omega =0$.