A general dark-state theory for arbitrary multilevel quantum systems

TL;DR

This work presents a general arrowhead-matrix framework to determine both the number and explicit form of dark states in arbitrary multilevel quantum systems. By splitting the Hilbert space into upper and lower subspaces, diagonalizing each subspace to obtain dressed states, and mapping the system to a thick arrowhead Hamiltonian, the authors derive clear criteria based on the rank of the coupling submatrix to identify dark states within degenerate lower-state subspaces. They apply the method to three-, four-, and five-level systems across multiple configurations, providing concrete expressions for dark states and conditions under which they exist, including degenerate and nondegenerate scenarios. The approach unifies and extends prior results on dark-state polaritons and offers a practical route to engineer and control dark states in complex quantum platforms. Overall, this arrowhead-matrix theory supplies a versatile tool for dark-state manipulation in quantum information, coherent control, and quantum technologies.

Abstract

The dark-state effect, caused by destructive quantum interference, is an important physical effect in atomic physics and quantum optics. It not only deepens the understanding of light-atom interactions, but also has wide application in quantum physics and quantum information. Therefore, how to efficiently and conveniently determine the number and form of the dark states in multilevel quantum systems with complex transitions is an important and interesting topic in this field. In this work, we present a general theory for determining the dark states in multilevel quantum systems with any coupling configuration using the arrowhead-matrix method. To confirm the dark states in a multilevel system, we first define the upper- and lower-state subspaces, and then diagonalize the Hamiltonians restricted within the two subspaces to obtain the dressed upper and lower states. By further expressing the transitions between the dressed upper and lower states, we can map the multilevel system to a bipartite-graph network, in which the nodes and links are acted by the dressed states and transitions, respectively. Based on the coupling configurations of the network, we can determine the lower dark states with respect to the upper-state subspace. As examples, we analyze the dark states in three-, four-, and five-level quantum systems, for all possible configurations through the classification of the numbers of upper and lower states. Further, we extend the framework to multilevel quantum systems and discuss the existence of dark states in some typical configurations. We also recover the results of the dark-state polaritons in driven three-level systems with the arrowhead-matrix method. Our theory paves the way for manipulating and utilizing the dark states of multilevel quantum systems in modern quantum science and technology.

Figures (6)

• Figure 1: (a) The energy-level diagram of a general multilevel quantum system with all possible transitions among these energy levels (some transitions are omitted for concision), which are divided into two components: $N_{u}$ upper states marked with the red line ($\left\vert u_{1}\right\rangle=\left\vert N\right\rangle$, $\left\vert u_{2}\right\rangle=\left\vert N-2\right\rangle$, $...$, and $\vert u_{N_{u}}\rangle=\left\vert j\right\rangle$) and $N_{l}$ lower states marked with the blue line ($\left\vert l_{1}\right\rangle=\left\vert N-1\right\rangle$, $...$, $\vert l_{N_{l}-1}\rangle=\left\vert 2\right\rangle$, and $\vert l_{N_{l}}\rangle=\left\vert 1\right\rangle$). (b) The bipartite-graph presentation of the $N$-level system with the dressed upper states ($\left\vert U_{1}\right\rangle$, $\left\vert U_{2}\right\rangle$, ..., and $\vert U_{N_{u}}\rangle$) and the dressed lower states ($\left\vert L_{1}\right\rangle$, $\left\vert L_{2}\right\rangle$, ..., and $\vert L_{N_{l}}\rangle$), where the couplings only exist between the dressed upper states and the dressed lower states.
• Figure 2: (a) A general $\Delta$-type three-level system with all transitions among three energy levels expressed in the bare-state representation. (b) The single configuration of the three-level system, according to the numbers of the upper and lower states, expressed in the dressed upper- and lower-state representation. Based on the $\Delta$-type three-level system, we can further obtain three specific configurations by cutting one coupling channel (here we only cut one coupling such that these three levels are still connected): (c) $\Lambda$-type three-level system, (d) $\Xi$-type three-level system, and (e) $\mathrm{V}$-type three-level system. The red (blue) levels denote the dressed upper (lower) states of the system. We point out that the selection of the upper and lower states does not depend on the specific high- and low-energy levels, but depends on the specific research topic.
• Figure 3: (a) Schematic of a general four-level quantum system with all possible transitions among four energy levels expressed in the bare-state representation. According to the numbers of the upper and lower states, it can be divided into two configurations expressed in the dressed upper- and lower-state representation. (b) Configuration 1: one upper state and three lower states with real symmetric couplings $\Omega _{12}=\Omega _{13}=\Omega _{23}=\Omega$ and under the resonance condition $\Delta _{14}=\Delta _{24}=\Delta _{34}=\Delta$. (c) Configuration 2: two upper states and two lower states with $\Omega _{34}=0$ and under the resonance condition $\Delta _{14}=\Delta _{24}=\Delta$. The red (blue) levels denote the dressed upper (lower) states of the system. Note that the upper and lower states can be chosen on demand in different configurations, and we only present one representative case as an example.
• Figure 4: (a) Schematic of a general five-level quantum system expressed in the bare-state representation. According to the numbers of the upper and lower states, it can be divided into three configurations expressed in the dressed upper- and lower-state representation. (b) Configuration 1: one upper state and four lower states with real symmetric couplings $\Omega _{34}=\Omega _{24}=\Omega _{12}=\Omega _{13}=\Omega _{1}$ and $\Omega _{23}=\Omega_{14}=\Omega _{2}$ and under the resonance condition $\Delta _{45}=\Delta _{35}=\Delta _{25}=\Delta _{15}=\Delta$. (c) Configuration 2: two upper states and three lower states with real symmetric couplings $\Omega _{23}=\Omega _{12}=\Omega _{13}=\Omega$ and the resonance condition $\Delta _{35}=\Delta _{25}=\Delta _{15}=\Delta$. (d) Configuration 3: three upper states and two lower states with real symmetric couplings $\Omega _{45}=\Omega _{35}=\Omega _{34}=\Omega$ and the resonance conditions $\Delta_{25}=\Delta _{15}=\Delta$ and $\Delta _{45}=\Delta _{35}=0$. The red (blue) levels denote the dressed upper (lower) states. Similarly, the upper and lower states can be chosen in different configurations on demand, and we only present one case as an example.
• Figure 5: Schematic of five typical coupling configurations of the $N$-level quantum systems. (a) Configuration 1: multipod quantum system with one upper state and $N-1$ lower states. (b) Configuration 2: shared-lower-state multiple-$\Lambda$ system with $N-2$ upper states and two lower states. (c) Configuration 3: $\Lambda$-chain system with a zigzag coupling satisfying $N_{l}=N_{u}+1$. (d) Configuration 4: shared-edge $\mathrm{N}$-chain system with a zigzag coupling satisfying $N_{l}=N_{u}$. (e) Configuration 5: $\mathrm{V}$-chain system with a zigzag coupling satisfying $N_{l}=N_{u}-1$. The red (blue) lines denote the upper (lower) states, and the detunings are omitted for concision.