Luojia Wang, Da-Wei Wang, Luqi Yuan, Yaping Yang, Xianfeng Chen, "Extreme single-excitation subradiance from two-band Bloch oscillations in atomic arrays," Photonics Res. 12, 571 (2024)

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- Photonics Research
- Vol. 12, Issue 3, 571 (2024)

Fig. 1. (a) Schematic of a 1D V-type atomic array under a magnetic field B ( y ) . (b) Schematic of the corresponding lattice where the n th atom has non-degenerate excited states | ± n ⟩ with frequency shifted by ± μ B n / ℏ . Blue lines indicate a linear trend of B n as in Eq. (2 ). (c) Band structures with different constant magnetic fields B c with values B c = 0 , 4 ℏ γ 0 / μ , 8 ℏ γ 0 / μ , 12 ℏ γ 0 / μ , respectively. Black arrows indicate values of B c for corresponding bands. Decay rates of modes are color coded. Probabilities of eigenstates on (d) band II and (e) band I projected on | + ⟩ states for different B c . Here, a = 0.1 λ .

Fig. 2. Bloch oscillations for Gaussian excitations initially centered at n c = 0 with k c = 1.5 k 0 on band I (left) and k c = 4 k 0 on band II (right), shown by temporal evolution for excitation probabilities of (a1), (a2) | C + , n | 2 ; (b1), (b2) | C − , n | 2 ; (c1), (c2) P I ( k y ) ; (d1), (d2) P II ( k y ) ; (e1), (e2) P + , P − , and P t . Here, a = 0.1 λ and μ B 0 / ℏ = 0.2 γ 0 .

Fig. 3. Bloch oscillations for Gaussian excitations initially centered at n c = 0 and k c = k 0 on band I with a static magnetic field (left) and a controllable magnetic field (right), whose zero point is shifted to y = 30 a over a time period from 0 to 20 γ 0 − 1 and back to y = 0 over a time period from 180 γ 0 − 1 to 200 γ 0 − 1 , shown by temporal evolution of (a1), (a2) P n ; (b1), (b2) P I + P II ; (c1), (c2) P + , P − , and P t . Other parameters are the same as those in Fig. 2 .

Fig. 4. Bloch oscillations for a Gaussian excitation initially centered at n c = 0 with k c = 1.5 k 0 (left) and k c = 3 k 0 (right) on band I. Temporal evolution of (a1), (a2) P n = | C + , n | 2 + | C − , n | 2 and (b1), (b2) P I ( k y ) + P II ( k y ) in an array with atomic number N = 201 within a time period of 300 γ 0 − 1 after t = 10 8 γ 0 − 1 (left) and after t = 10 13 γ 0 − 1 (right). (c1), (c2) The total probability P t versus order of magnitude of time for N = 51 (blue solid line), N = 101 (green dash-dotted line), N = 151 (red dash line), and N = 201 (black dotted line). a = 0.1 λ and μ B 0 / ℏ = 0.2 γ 0 .

Fig. 5. Bloch oscillations started from n c = 0 for Gaussian excitations initially centered at k c = 4 k 0 on band I (left), k c = 2 k 0 on band II (middle), and k c = − 0.5 k 0 on band II (right), shown by temporal evolutions of (a1)–(a3) P n , (b1)–(b3) P I ( k y ) + P II ( k y ) , (c1), (c2) total excitation probabilities P + (red solid line), P − (black dash line), and P t (green dash-dotted line). An exponential function is used to fit the decay of the total excitation probability in (c3) (blue dotted line). Here, N = 201 , a = 0.1 λ , and μ B 0 / ℏ = 0.2 γ 0 .

Fig. 6. Bloch oscillations started from n c = 50 for a Gaussian excitation initially centered at k c = 1.5 k 0 on band I, shown by the temporal evolution of (a) P n , (b) P I + P II , (c) P + (red solid line), P − (black dash line), and P t (green dash-dotted line). Here N = 201 , a = 0.1 λ , and μ B 0 / ℏ = 0.2 γ 0 .

Fig. 7. Numerically calculated decay rates in descending order of single-excitation eigenstates in an atomic array under a linear magnetic field with varying μ B 0 / ℏ γ 0 indicated by different colors for (a) a = 0.1 λ , N = 201 ; (b) a = 0.1 λ , N = 101 ; and (c) a = 0.2 λ , N = 201 .

Fig. 8. Decay rates of the most subradiant single-excitation modes with (a) increasing spatial disorder characterized by the deviation width δ a and (b) increasing Doppler broadening Δ D for B 0 = 0 (blue dash line) and B 0 = 0.2 ℏ γ 0 / μ (red solid line). a = 0.1 λ and N = 201 .

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