Theory of Subradiant States of a One-Dimensional Two-Level Atom Chain

Recently, the subradiant states of one-dimensional two-level atom chains coupled to light modes were found to have decay rates obeying a universal scaling, and an unexpected fermionic character of the multiply excited subradiant states was discovered. In this Letter, we theoretically obtain the singly excited subradiant states, and by eliminating the superradiant modes, we demonstrate a relation between the multiply excited subradiant states and the Tonks-Girardeau limit of the Lieb-Liniger model which explains the fermionic behavior. In addition, we identify a new family of subradiant states with correlations different from the fermionic ansatz.

Figure 1

Outline of the theory: the imaginary part of the effective atomic Hamiltonian, $H_{\text{eff}}^I$, identifies the superradiant ($|\pm k_{1\mathrm{D}}⟩$) and dark subspaces of states (left panel), coupled perturbatively by $H_{\text{eff}}^R$ to produce the subradiant modes with decay rates ${\gamma }_{\xi }\propto {\xi }^2/N^3(\xi \ll N)$. The Holstein-Primakoff transformation bosonizes the super- and subradiant modes and introduces a coupling ($Q$ and $Q^{†}$) between them (right panel). The coupling effectively yields a strong interaction $V_{\mathrm{sub}}$ among the multiply excited subradiant modes leading to the formation of states with fermionic character.

Figure 2

(a) The two-excitation eigenstates of the system with $k_{1\mathrm{D}}=0.2\pi /d$ and $N=20$ are sorted by increasing decay rates. The bars show the maximal fidelity that a fermionic ansatz can achieve for each eigenstate. The fermionic ansatz fits a broad range of the most subradiant states while a few exceptional states (the dips in the fidelity, e.g., state No. 7) show distinct nonfermionic behavior. (b) Position distributions of the atomic excitations, $|⟨{\psi }_{1(7)}|e_m,e_n⟩{|}^2$ with $m(n)=0,1,\dots ,19$ of a typical fermionic subradiant state, $|{\psi }_1⟩$ (upper panel) and the nonfermionic state $|{\psi }_7⟩$ (lower panel). The lower panel feature at $|z_m-z_n|\approx 2d$ indicates that $|{\psi }_7⟩$ supports dimerlike bound excitations.