Clustering of exceptional points and dynamical phase transitions

The eigenvalues of a non-Hermitian Hamilton operator are complex and provide not only the energies but also the lifetimes of the states of the system. They show a nonanalytical behavior at singular (exceptional) points (EPs). The eigenfunctions 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 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. When the parameters are tuned to the point of maximum width bifurcation, the phase rigidity suddenly increases up to its maximum value. This means that the eigenfunctions become almost orthogonal at this point. This unexpected result is very robust as shown by numerical results for different classes of systems. Physically, it causes an irreversible stabilization of the system by creating local structures that can be described well by a Hermitian Hamilton operator. Interesting nontrivial features of open quantum systems appear in the parameter range in which a clustering of EPs causes a dynamical phase transition.

Figure 1

Energies $E_i$ (full lines) (a),(e), widths ${\mathrm{\Gamma }}_i/2$ (b),(f), phase rigidity $r_1=r_2$ (c),(g), and $1-r_k$ (d),(h) of the two eigenfunctions of ${\mathcal{H}}^{\left(2\right)}$ as a function of the distance $d$ between the two unperturbed energies $e_1$ and $e_2$. The parameters are given, respectively, in the first and second row of Table 1. The dashed lines in (a),(e) and (b),(f) show, respectively, the $e_i$ and ${\gamma }_i/2$ trajectories.

Figure 2

Same as Fig. 1 but with the parameters given, respectively, in the third and fourth row of Table 1.

Figure 3

Same as Fig. 1 but with the parameters given, respectively, in the fifth and sixth row of Table 1.

Figure 4

Same as Fig. 1 but with the parameters given, respectively, in the seventh and eighth row of Table 1. In (e) and (j), the mixing coefficients $|b_{ij}|$ of the wave functions, defined by (11), are shown.

Figure 5

Energies $E_i$ (full lines) (a),(e), widths ${\mathrm{\Gamma }}_i/2$ (b),(f), phase rigidity $r_i$ (c),(g), and $1-r_i$ (d),(h) of the two eigenfunctions of ${\mathcal{H}}^{\left(2\right)}$ (left panel) and of the three eigenfunctions of ${\mathcal{H}}^{\left(3\right)}$ (right panel), respectively, as a function of the parameter $a$. The dashed lines in (a),(e) and (b),(f) show, respectively, the $e_i$ and ${\gamma }_i/2$ trajectories. The parameters are as follows: $\omega =0.01i$, $e_1=1-1/2a,e_2=a$, $e_3=-1/3+3/2a$ (e)–(h); ${\gamma }_1/2={\gamma }_2/2=-0.495$, ${\gamma }_3/2=-0.4853$ (e)–(h).

Figure 6

Same as Fig. 5 but $\omega =0.005(1+i)$ and ${\gamma }_1/2=-0.5,{\gamma }_2/2=-0.51,e_1=0.5,e_2=a$ (left panel) and ${\gamma }_1/2=-0.5,{\gamma }_2/2=-0.505,{\gamma }_3/2=-0.51,e_1=0.5,e_2=a,e_3=2a-0.5$ (right panel).

Figure 7

Energies $E_i$ (full lines) (a),(f), widths ${\mathrm{\Gamma }}_i/2$ (b),(g), phase rigidity $r_i$ (c),(h), $1-r_i$ (d),(i), and mixing coefficients $|b_{ij}|$ of the wave functions, defined by (11), of the four eigenfunctions of ${\mathcal{H}}^{\left(4\right)}$ as a function of the parameter $a$. The dashed lines in (f),(g) show, respectively, the $e_i$ and ${\gamma }_i/2$ trajectories. The parameters are $\omega =0.01i$, $e_1=1-a/2,e_2=a,e_3=-1/3+3/2a,e_4=2/3,{\gamma }_1/2={\gamma }_2/2=-0.4950,{\gamma }_3/2=-0.4853,{\gamma }_4/2=-0.4950$ (left panel) and $\omega =0.005(1+i)$, $e_1=0.5,e_2=a,e_3=2a-0.5,e_4=1-a,{\gamma }_1/2=-0.5,{\gamma }_2/2=-0.505,{\gamma }_3/2=-0.51,{\gamma }_4/2=-0.505$ (right panel).