Gain and loss in open quantum systems

Photosynthesis is the basic process used by plants to convert light energy in reaction centers into chemical energy. The high efficiency of this process is not yet understood today. Using the formalism for the description of open quantum systems by means of a non-Hermitian Hamilton operator, we consider initially the interplay of gain (acceptor) and loss (donor). Near singular points it causes fluctuations of the cross section which appear without any excitation of internal degrees of freedom of the system. This process occurs therefore very quickly and with high efficiency. We then consider the excitation of resonance states of the system by means of these fluctuations. This second step of the whole process takes place much slower than the first one, because it involves the excitation of internal degrees of freedom of the system. The two-step process as a whole is highly efficient, and the decay is biexponential. We provide, if possible, the results of analytical studies, otherwise characteristic numerical results. The similarities of the obtained results to light harvesting in photosynthetic organisms are discussed.

Figure 1

Eigenvalues ${\mathcal{E}}_i^{\left(1\right)}\equiv E_i^{\left(1\right)}+\frac{i}{2}{\mathrm{\Gamma }}_i^{\left(1\right)}$ (a, b) and eigenfunctions ${\mathrm{\Phi }}_i^{\left(1\right)}$ (c–e) of the Hamiltonian ${\tilde{\mathcal{H}}}^{(2,1)}$ as a function of $a$ and contour plot (f) of the cross section; ${\omega }^{\left(1\right)}=0.5025$. Parameters: $e_1^{\left(1\right)}=1-a/2;e_2^{\left(1\right)}=a;{\gamma }_1^{\left(1\right)}/2=0.55;{\gamma }_2^{\left(1\right)}/2=-0.5$ (dashed lines in a, b).

Figure 2

The same as Fig. 1 but ${\omega }^{\left(1\right)}=0.45+0.05i$ and $e_1^{\left(1\right)}=0.5;e_2^{\left(1\right)}=0.5+0.1a;{\gamma }_1^{\left(1\right)}/2=0.5;{\gamma }_2^{\left(1\right)}/2=-0.4$.

Figure 3

The same as Fig. 1 but ${\omega }^{\left(1\right)}=0.9+0.1i$ and $e_1^{\left(1\right)}=0.5;e_2^{\left(1\right)}=0.5+0.1a;{\gamma }_1^{\left(1\right)}/2=0.5;{\gamma }_2^{\left(1\right)}/2=-1.3$.

Figure 4

The same as Fig. 1 but ${\omega }^{\left(1\right)}=0.9+0.1i$ and $e_1^{\left(1\right)}=0.5;e_2^{\left(1\right)}=0.3;{\gamma }_1^{\left(1\right)}/2=0.5a;{\gamma }_2^{\left(1\right)}/2=-0.5a$.

Figure 5

Eigenvalues ${\mathcal{E}}_i^{(1,2)}\equiv E_i^{(1,2)}+\frac{i}{2}{\mathrm{\Gamma }}_i^{(1,2)}$ (a, b) and eigenfunctions ${\mathrm{\Phi }}_i^{(1,2)}$ (c–e) of the Hamiltonian ${\tilde{\mathcal{H}}}^{(2,2)}$ and contour plot (f) of the cross section as function of $a$ with two merging states in the first channel, ${\omega }^{\left(2\right)}=0.005+0.045i$. $e_1^{\left(2\right)}=0.5;e_2^{\left(2\right)}=0.3;{\gamma }_1^{\left(2\right)}/2=-0.05;{\gamma }_2^{\left(2\right)}/2=-0.05$ (dashed lines in a, b). The parameters of the first channel are the same as those in Fig. 2.

Figure 6

Left: Eigenvalues ${\mathcal{E}}_i^{(1,2)}\equiv E_i^{(1,2)}+\frac{i}{2}{\mathrm{\Gamma }}_i^{(1,2)}$ (a, b) and eigenfunctions ${\mathrm{\Phi }}_i^{(1,2)}$ (c–e) of the Hamiltonian ${\tilde{\mathcal{H}}}^{(2,2)}$ as function of $a$ with one state in the second channel, ${\omega }^{\left(2\right)}=0$. $e_1^{\left(2\right)}=0.5;{\gamma }_1^{\left(2\right)}/2=-0.05$ (dashed lines in a, b). The parameters of the first channel are the same as those in Fig. 4. Right: Contour plots of the cross section with the same parameters of the first channel as in Fig. 4; ${\omega }^{\left(2\right)}=0$; and (f) $e_1^{\left(2\right)}=0.5;{\gamma }_1^{\left(2\right)}/2=-0.05$; (g) $e_1^{\left(2\right)}=0.5;{\gamma }_1^{\left(2\right)}/2=-0.5$; (h) $e_1^{\left(2\right)}=-0.5;{\gamma }_1^{\left(2\right)}/2=-0.5$.

Figure 7

Eigenvalues ${\mathcal{E}}_i^{(1,2)}\equiv E_i^{(1,2)}+\frac{i}{2}{\mathrm{\Gamma }}_i^{(1,2)}$ (a, b) and eigenfunctions ${\mathrm{\Phi }}_i^{(1,2)}$ (c–e) of the Hamiltonian ${\tilde{\mathcal{H}}}^{(2,2)}$ and contour plot (f) of the cross section as function of $a$ with two states in the second channel, ${\omega }^{\left(2\right)}=0.01+0.09i$. $e_1^{\left(2\right)}=0.4;e_2^{\left(2\right)}=0.5;{\gamma }_1^{\left(2\right)}/2=-0.06;{\gamma }_2^{\left(2\right)}/2=-0.06$ (dashed lines in a, b). The parameters of the first channel are the same as those in Fig. 4.