Variational functions in degenerate open quantum systems

We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven $\Lambda$-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.

Figure 1

Level structure of the degenerate $n$-level system. The upper level has $n-1$ decay channels denoted with ${\Gamma }_1,\dots ,{\Gamma }_{n-1}$ into different stationary states $\mid 1⟩,\dots ,\mid n-1⟩$, which can be additionally degenerate. Since there are $n-1$ stationary states, the eigenvalues of the Lindblad operator have at least a $(n-1)$ algebraic multiplicity of the eigenvalue 0.

Figure 2

Level structure of the driven three-level system. The Rabi frequencies are denoted by ${\Omega }_1$ and ${\Omega }_2$ describing the coupling between the atomic transitions $\mid 1⟩\leftrightarrow \mid 3⟩$ and $\mid 2⟩\leftrightarrow \mid 3⟩$, respectively. The decay rates are given by ${\Gamma }_1$ and ${\Gamma }_2$. In addition, we assume both light fields having the same frequency. However, there can be a detuning $\Delta$ between the frequencies of the atomic transitions and the light fields.

Figure 3

Entropy functional $S_{\mathbf{\Omega }}\left(t\right)$ as a function of time with ${\Gamma }_1={\Gamma }_2=2\gamma$ and, consequently, ${\gamma }_1={\gamma }_2=\gamma$. The initial conditions are ${\rho }_{33}\left(0\right)=1$ for the solid line, ${\rho }_{ij}\left(0\right)=1∕3$ for the dashed line (representing a pure state with equally populated upper and lower levels), and ${\rho }_{11}\left(0\right)={\rho }_{22}\left(0\right)={\rho }_{33}\left(0\right)=1∕3$ for the dotted line (representing a mixed state of equally populated upper and lower levels). The steady-state values of the entropy functional are identical for the first (solid line) and the third (dotted line) initial condition as a result of the particular choice of decay rates. The quantities plotted are dimensionless.

Figure 4

Entropy functional $S_{\mathbf{\Omega }}^{\text{mod}}\left(t\right)$ as a function of time with the same initial conditions as in Fig. 3. Since the Fano entropy of the system in steady state is added, the modified entropy functional is positive definite. The quantities plotted are dimensionless.

Figure 5

Entropy functional $S_{\mathbf{\Omega }}\left(t\right)$ as a function of time and different initial conditions $\mid \rho \left(0\right)⟩⟩$, represented by the solid, dashed, and dotted lines. The initial conditions are the same as in Figs. 3, 4. The interaction strengths are given by ${\Omega }_1=4\gamma$ and ${\Omega }_2=2\gamma$. The detuning is $\Delta =\gamma$ while the decay rates are ${\Gamma }_1={\Gamma }_2=2\gamma$ and accordingly ${\gamma }_1={\gamma }_2=\gamma$. The quantities plotted are dimensionless.

Figure 6

Entropy functional $S_{\mathbf{\Omega }}\left(t\right)$ as a function of time and different initial conditions $\mid \rho \left(0\right)⟩⟩$ represented by the solid, dashed, and dotted lines and being the same as in the previous figures. The interaction strengths are ${\Omega }_1=2\gamma$ and ${\Omega }_2=6\gamma$ while the detuning is relatively small and given by $\Delta =0.2\gamma$. The decay rates are identical with those of the previous figure. The quantities plotted are dimensionless.