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 nth atom has non-degenerate excited states |±n⟩ with frequency shifted by ±μBn/ℏ. Blue lines indicate a linear trend of Bn as in Eq. (2). (c) Band structures with different constant magnetic fields Bc with values Bc=0,4ℏγ0/μ,8ℏγ0/μ,12ℏγ0/μ, respectively. Black arrows indicate values of Bc 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 Bc. Here, a=0.1λ.

Fig. 2. Bloch oscillations for Gaussian excitations initially centered at nc=0 with kc=1.5k0 on band I (left) and kc=4k0 on band II (right), shown by temporal evolution for excitation probabilities of (a1), (a2) |C+,n|2; (b1), (b2) |C−,n|2; (c1), (c2) PI(ky); (d1), (d2) PII(ky); (e1), (e2) P+, P−, and Pt. Here, a=0.1λ and μB0/ℏ=0.2γ0.

Fig. 3. Bloch oscillations for Gaussian excitations initially centered at nc=0 and kc=k0 on band I with a static magnetic field (left) and a controllable magnetic field (right), whose zero point is shifted to y=30a over a time period from 0 to 20γ0−1 and back to y=0 over a time period from 180γ0−1 to 200γ0−1, shown by temporal evolution of (a1), (a2) Pn; (b1), (b2) PI+PII; (c1), (c2) P+, P−, and Pt. Other parameters are the same as those in Fig. 2.

Fig. 4. Bloch oscillations for a Gaussian excitation initially centered at nc=0 with kc=1.5k0 (left) and kc=3k0 (right) on band I. Temporal evolution of (a1), (a2) Pn=|C+,n|2+|C−,n|2 and (b1), (b2) PI(ky)+PII(ky) in an array with atomic number N=201 within a time period of 300 γ0−1 after t=108 γ0−1 (left) and after t=1013 γ0−1 (right). (c1), (c2) The total probability Pt 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=0.1λ and μB0/ℏ=0.2γ0.

Fig. 5. Bloch oscillations started from nc=0 for Gaussian excitations initially centered at kc=4k0 on band I (left), kc=2k0 on band II (middle), and kc=−0.5k0 on band II (right), shown by temporal evolutions of (a1)–(a3) Pn, (b1)–(b3) PI(ky)+PII(ky), (c1), (c2) total excitation probabilities P+ (red solid line), P− (black dash line), and Pt (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=0.1λ, and μB0/ℏ=0.2γ0.

Fig. 6. Bloch oscillations started from nc=50 for a Gaussian excitation initially centered at kc=1.5k0 on band I, shown by the temporal evolution of (a) Pn, (b) PI+PII, (c) P+ (red solid line), P− (black dash line), and Pt (green dash-dotted line). Here N=201, a=0.1λ, and μB0/ℏ=0.2γ0.

Fig. 7. Numerically calculated decay rates in descending order of single-excitation eigenstates in an atomic array under a linear magnetic field with varying μB0/ℏγ0 indicated by different colors for (a) a=0.1λ, N=201; (b) a=0.1λ, N=101; and (c) a=0.2λ, N=201.

Fig. 8. Decay rates of the most subradiant single-excitation modes with (a) increasing spatial disorder characterized by the deviation width δa and (b) increasing Doppler broadening ΔD for B0=0 (blue dash line) and B0=0.2ℏγ0/μ (red solid line). a=0.1λ and N=201.