Classification of dark states in multilevel dissipative systems

Dark states are eigenstates or steady states of a system that are decoupled from the radiation. Their use, along with associated techniques such as stimulated Raman adiabatic passage, has extended from atomic physics, where it is an essential cooling mechanism, to more recent versions in the condensed phase where it can increase the coherence times of qubits. These states are often discussed in the context of unitary evolution and found with elegant methods exploiting symmetries, or via the Morris-Shore transformation. However, the link with dissipative systems is not always transparent, and distinctions between classes of coherent population trapping are not always clear. We present a detailed overview of the arguments to find stationary dark states in dissipative systems, and examine their dependence on the Hamiltonian parameters, their multiplicity, and purity. We evidence the class of dark states that depends not only on the detunings of the lasers but also on their relative intensities and phases. We illustrate the criteria with the more complex physical system of the hyperfine transitions of ${}^{87}\mathrm{Rb}$ and show how a knowledge of the dark-state manifold can guide the preparation of pure states.

Figure 1

$N$-level system consisting of a ground-state manifold coupled radiatively to an excited-state manifold. The excited state can also relax dissipatively to the ground state.

Figure 2

Populations of the $M$ system as a function of time in the (a) diagram of the $M$ system connectivity and probability amplitudes of the pure, dark, stationary state (red is positive probability amplitude, blue is negative). The dynamics of the system is plotted in the site basis and in the eigenstate basis. The system evolves towards a pure state occupying only the ground states. Parameters are $V_{21}=0.56$, $V_{22}=0.23$, $V_{32}=0.45$, ${\gamma }_{11}=0.04$, ${\gamma }_{12}=0.01$, ${\gamma }_{13}=0.09$, ${\gamma }_{21}=0.14$, ${\gamma }_{22}=0.02$, and ${\gamma }_{23}=0.04$. (b) Same as in (a) in the presence of the coupling $V_{31}=0.57$. In both cases the dark state is ${\varphi }_5$ (all values in units of $V_{11}$).

Figure 3

Tripod system with four ground-state levels in different detuning configurations. Below the systems are shown the wave functions that span the dark-state subspace (blue means negative coefficient, red positive coefficient, and the size of the circle is proportional to the coefficient). On the left side are shown the evolution of the populations for an initially excited state. The purity of the steady state is evaluated by calculating $\mathrm{Tr}\left({\rho }_{\text{SS}}^2\right)$. Parameters of the simulations are $V_{21}=V_{31}=V_{41}=1$, and ${\gamma }_{11}=0.1$, ${\gamma }_{12}=0.2$, ${\gamma }_{13}=0.3$, ${\gamma }_{14}=0.4$ (all values in units of $V_{11}$).

Figure 4

Excited-state population as a function of detuning and field intensities for a pair of two-level systems. Parameters of the simulations are $V_{12}=2$, $V_{12}=1$, and ${\gamma }_{11}=0.2$, ${\gamma }_{12}=0.5$, ${\gamma }_{21}=0.44$, ${\gamma }_{22}=0.7$ (all values in units of $V_{22}$).

Figure 5

Excited-state population for a pair of two-level systems in the case of (a) only coupling to one excited state ($V_{21}=1$), (b) coupling to both excited states in the dark-state condition ($V_{21}=1$, $V_{12}=1$, and $V_{22}=1$), and (c) coupling to both excited states not fulfilling the dark-state condition ($V_{21}=1$, $V_{12}=1$, and $V_{22}=2$). The relaxation rates are ${\gamma }_{11}=0.2$, ${\gamma }_{12}=0.5$, ${\gamma }_{21}=0.44$, ${\gamma }_{22}=0.7$ (all values in units of $V_{11}$) and all detunings have been set equal.

Figure 6

Possible connectivities for the hyperfine levels of ${}^{87}\mathrm{Rb}$ atoms. In the case of multiple manifold of dark states these are labeled by different colors (blue (solid) corresponds to $d_1$ and red (dashed) corresponds to $d_2$ of Table 1). The case investigated in Ref. [50] corresponds to case 3.