Quantum ground-state phase diagram of a noninteracting spin-orbit-coupled Floquet spinor BEC in the space of the Rabi frequency Ω and the quadratic Zeeman field ε. The driving is α/ω = 2 (J0(α/ω) = 0.224). The background corresponds to values of the tensor magnetization F 2 z . The black and white dotted lines represent first-order and secondorder phase transitions, respectively. Below the dotted lines is the plane-wave phase, and beyond is the zero momentum phase. The red star denotes a tricritical point. Insets show the lowest bands of the single-particle dispersion. The black dashed lines separate different regions where the lowest band of the dispersion has three, two or one local energy minima.

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... the quasimomentum can be determined by solving ∂E k /∂k = 0. The occupation of k = 0 is the zero momentum (ZM) state, and the occupation of a nonzero quasimomentum is the plane-wave (PW) state. Fig. 1 shows ground-state phase diagram in the (Ω, ε) plane, in which the tensor magnetization F 2 z = cos 2 θ is chosen as the order parameter. The dotted lines are the transition line between PW and ZM phases, above which is the ZM phase and below is the PW phase. We also show the lowest band of H SOC in Fig. 1. The dashed line in the ZM ...

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... is the plane-wave (PW) state. Fig. 1 shows ground-state phase diagram in the (Ω, ε) plane, in which the tensor magnetization F 2 z = cos 2 θ is chosen as the order parameter. The dotted lines are the transition line between PW and ZM phases, above which is the ZM phase and below is the PW phase. We also show the lowest band of H SOC in Fig. 1. The dashed line in the ZM regime is a separation, above which the lowest band has only one minimum at k = 0 and below which it has three local minima but the lowest one at k = 0. In the PW regime, the lowest band may have three local minima or two. The separation between these two cases is demonstrated by the blakc dashed lines. Two ...

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... which it has three local minima but the lowest one at k = 0. In the PW regime, the lowest band may have three local minima or two. The separation between these two cases is demonstrated by the blakc dashed lines. Two dashed lines merge together with the phase transition line at the so-called tricritical point, which is labeled by the red star in Fig. 1. The location of the tricritical point can be analytically calculated from ∂ 2 E k /∂k 2 = 0 and the equal energy between the PW and ZM states [30,31]. The calculated value for the tricritical point is (Ω * , ε * ) = (30.14, −1.66). When Ω < Ω * the PW-ZM transition is first-order and when Ω > Ω * the phase transition is ...

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... interaction is c 2 > 0, which is typical for the 23 Na BEC. Fig. 2 demonstrates the phase diagram for antiferromagnetic interactions with a driving α/ω = 2 in the space of the quadratic Zeeman field ε and the Rabi frequency Ω. When ε is negative, the single-particle dispersion has two lowest minima locating at ±k m [see the inset in Fig. 1], the antiferromagnetic interaction allows atoms to simultaneously occupy these two minima to form a stripe for a low Ω. This stripe phase labeled as S1 in Fig. 2 has |C + | = |C − | = 1/ √ 2 and C 0 = 0. Using the wave functions in Eq. (10) with C 0 = 0 and considering the single-particle spinors at ±k m having ϕ = π/2, we get the ...

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... first glance, the phase diagram shown in Fig. 2 is similar to that of a usual spin-orbit-coupled BEC demonstrated in Refs. [31,32] (i.e., Fig. 1(a) in [31] and Fig. 1 in [32]). There are two tricritical points represented by stars in Fig. 2. The first (second) order phase transitions between different phases are shown by black-dotted (white-dotted) lines. However, there are two new features in our system. (i) All phases exist in a broadened region of the Rabi frequency. This is a ...

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... first glance, the phase diagram shown in Fig. 2 is similar to that of a usual spin-orbit-coupled BEC demonstrated in Refs. [31,32] (i.e., Fig. 1(a) in [31] and Fig. 1 in [32]). There are two tricritical points represented by stars in Fig. 2. The first (second) order phase transitions between different phases are shown by black-dotted (white-dotted) lines. However, there are two new features in our system. (i) All phases exist in a broadened region of the Rabi frequency. This is a straightforward ...

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... in our system. (i) All phases exist in a broadened region of the Rabi frequency. This is a straightforward consequence of the Bessel-function modulation ΩJ 0 . (ii) The existence of the S2 phase is also extended in the ε domain. In the usual spin-orbit-coupled antiferromagnetic BEC the S2 phase exists in an extremely narrow region of ε (see Fig. 1(a) in [31] and Fig. 1 in [32]). Our Floquet system has a large extension, which benefits from the Floquet-induced ...

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... All phases exist in a broadened region of the Rabi frequency. This is a straightforward consequence of the Bessel-function modulation ΩJ 0 . (ii) The existence of the S2 phase is also extended in the ε domain. In the usual spin-orbit-coupled antiferromagnetic BEC the S2 phase exists in an extremely narrow region of ε (see Fig. 1(a) in [31] and Fig. 1 in [32]). Our Floquet system has a large extension, which benefits from the Floquet-induced ...

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... (c 2 < 0) and it cannot minimize the first term c 2 ¯ n 2 /2(|C − | 2 −|C + | 2 ) 2 which is minimized by the PW phase. With the effect of the quadratic Zeeman field, the S3, PW and ZM phases are distributed in the way shown in Fig. 4. These three phases are similar to the previous studies [31,32] (i.e., Fig. 1(b) in [31] and Fig. 2 in [32]), but with an outstanding feature that every phase exists in a broaden region of Ω due to the Bessel-function ...