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arXiv:2211.16832 · Journal of Physics A Mathematical and Theoretical

Simple determination of dark states in a general multi-level system

Kaixuan Zhou, June Wu, Junheng Shi, Tim Byrnes

Published Feb 18, 2025• 0 citations
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Abstract

Abstract In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.

DOI
10.1088/1751-8121/adb7a8

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Abstract In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.
Optical dark resonance in systems with arbitrary multilevels and multiphotons is studied theoretically. When the system configuration is treelike (the term of graph theory), all the coherence terms in the system can be categorized into several groups on dark resonance. It is proved for generic system configurations with arbitrary multilevels that the resonance conditions can be obtained successively by setting the determinant of the submatrix of the Hamiltonian with respect to each group to zero. We calculate, as one of the practical examples, all the conditions of dark resonance of potassium atoms with two laser fields and one rf field mixing one excited state and eight ground states in the hyperfine structure. Although the system configuration is very complicated, the condition is easily obtained in an analytic form and verified by the result of a numerical simulation. Furthermore, our theory can reveal the fine structure of dark lines with power broadening in the observed light-induced-fluorescence spectra and numerical calculations. In addition, the dark lines with the complicated splitting pattern observed in experiments can be assigned to the relevant level configuration and transitions by the present theory. \textcopyright{} 1996 The American Physical Society.
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 applications 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. Furthermore, 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 atomic physics and quantum optics.
Figure 2.1 shows a schematic diagram of a two-level system, called a two-level atom, with the ground state |1〉 and the upper (excited) state |2〉. In fact, the electron can make transitions between many energy states, but we can limit the transitions to only between two states. In practice, it is done by a suitable choice of the frequency of an external field that will force the electron to oscillate only between these two selected states [3, 4]. Multi-level transitions involving more than two energy states are much more complex and we are not intend to consider them here, but the formalism presented here can be extended to multi-level cases.a The energy of the electron in the selected states |2〉 and |1〉 is determined by the stationary Schro¨dinger equation HˆA |1〉 = E1 |1〉 , HˆA |2〉 = E2 |2〉 , (2.10) where HˆA is the Hamiltonian of the atom whose explicit form is to be determine.
Abstract Quantum mechanics allows for light-matter setups that hold excitations without releasing them as light. Arising from destructive interference processes, they are best seen in a Tavis-Cummings-like setup where two-level atoms (or qubits) are placed within a lossy cavity. If the system is initialized with some qubits excited and some in the ground state, there is a non-zero likelihood of no photons being emitted. This can be framed as a Stern-Gerlach measurement, with a detector to measure if one or more photons leave the cavity. If no photons are detected, the qubits collapse onto a dark state. This can be viewed as heralding of a dark state based on zero photon detection. Building upon this idea, we propose a protocol to measure the number of independent dark states. Remarkably, we show that this quantity is robust to arbitrary levels of disorder in the qubit-photon coupling constants. This provides a rare example of an observable that remains unchanged with any amount of disorder. We then discuss a phase transition where the number of dark states plays the role of an order parameter. This provides a rare example of a phase transition that is completely insensitive to disorder.
Quantum mechanics allows for light-matter setups that hold excitations without releasing them as light. Arising from destructive interference processes, they are best seen in a Tavis-Cummings-like setup where two-level atoms (or qubits) are placed within a lossy cavity. If the system is initialized with some qubits excited and some in the ground state, there is a non-zero probability that no photons will be emitted. This can be framed as a Stern-Gerlach measurement, with a detector to measure if one or more photons leave the cavity. If no photons are detected, the qubits collapse onto a dark state. This can be viewed as heralding of a dark state based on zero photon detection. Building upon this idea, we propose a protocol to measure the number of independent dark states. Moreover, we show that this quantity is robust to arbitrary levels of disorder in the qubit-photon coupling constants. We then discuss a phase transition where the number of dark states plays the role of an order parameter. This provides an exciting example of a phase transition that is completely insensitive to disorder.