Fig. 1. (a) Schematic of the system. A Sr87 atom is trapped in a one-dimensional potential formed by two counter-propagating lasers of the “magic” wavelength. The S10−P30 transition of the atom is coupled by the clock laser at wavelength λC. The clock laser lies on the xy plane making an angle ϕ to the trap lasers. (b) Anharmonic energy spectrum of the Hamiltonian H′ with gk=gx and Ω=ω. Here |n,±⟩ denote the nth pair of dressed states with eigenenergies En,±=ℏ(nω±gxn).

Fig. 2. Distributions of (a) g1(2)(0) and (b) ns on the Ωgk parameter plane. (c) Time interval τ dependence of g1(2)(τ) for (Ω,gk)=(1,1)gx. (d) Distributions of g1(2)(0) (solid line) and ns (dashed line) along the line gk=Ω on the Ωgk parameter plane. In (a)–(d), the phonon frequency and the clock detuning are fixed at ω/gx=1 and δ/gx=0.005, respectively.

Fig. 3. (a) Distribution of ns on the ωδ parameter plane. The solid lines are the contour lines with ns=0.2 and the dashed lines show the δ dependence of ωn. (b) and (c) are, respectively, the distributions of g1(2)(0) and p˜(1) on the ωδ parameter plane around ω=ω1. All solid lines in (b) and (c) are contour lines.

Fig. 4. Statistic properties of motional two-phonon (left column) and three-phonon (right column) states. (a) and (b) show the distributions of g1(3)(0) and g1(4)(0), respectively, on the ωδ parameter plane. Solid lines in (a) and (b) are contour lines marking g1(3)(0)=1 and g1(4)(0)=1, respectively. The dashed lines in (a) and (b) represent the δ dependence of ω2 and ω3, respectively. (c) plots g1(2)(τ) (solid line) and g2(2)(τ) (dashed line) for (δ,ω)=(0.025,0.708)gx, i.e., the red square in (a). (d) shows g1(2)(τ) (solid line) and g3(2)(τ) (dashed line) for (δ,ω)=(0.1,0.58)gx, i.e., the red square in (b). (e) and (f) plot the steady-state phonon-number distribution p˜(q) corresponding to parameters marked by red squares in (a) and (b), respectively.

Fig. 5. Typical energy spectrum of Hamiltonian Eq. (B1) with the zero-clock shift δ=0. The red square shows the position of n-phonon resonance ωn. |n,±⟩=(|n,g⟩∓|n−1,e⟩)2 denotes the nth pair of dressed states of the system. The other parameters are gk/gx=1 and Ω/ω=1.

Fig. 6. (a) Energy spectrum of Hamiltonian Eq. (B1) with δ/gx=0.05. The vertical dashed lines denoting the positions of ωn are guides for the eyes. The δ dependence of (b) ωn and (c) Δn for the n-phonon states, respectively. The other parameters are gk/gx=1 and Ω/ω=1.

Fig. 7. ω dependence of g1(2)(0) in (a) and ns in (b) for different values of κe with δ/gx=0.005 and γd/κd=0.1.

Fig. 8. ω dependence of g1(2)(0) in (a) and ns in (b) for different values of γd with δ/gx=0.005 and κe/κd=2.2×10−6.