3 FIG. 4: (Color online) The dependence of the wave vector k$_{−}$ for the most unstable mode on the modulation frequency Ω. The inset illustrates the imaginary part of a typical Bo- goliubov spectrum under modulation (blue double contours). For comparison, black dashed lines denote the results without modulation. FIG. 3: (Color online) (a) Time-dependent fractional pop- ulation ρ 0 and (b) time-dependent relative phase θ for the periodic square modulation Eq. (6) with τ = w = 1 / 4 and d = − 1. 10 −1 10 0 10 1 10 2 0 0.1 0.2 0.3 0.4  /  0 k −  / k 0 0 0.05 0.1 −2 0 2 x 10 −3 k / k$_{0}$ Im(F) / F 0 k$_{−}$ 0 1 2 3 −0.2 0 0.2 Time ( )  ( ) (b) 0 1 2 3 0.3 0.4 0.5 0.6 0.7 Time ( ) 0 (a) features as shown in Figs. 3(a) and 3(b). The modulation dynamics considered above offers many interesting possibilities. A direct application is to remove a dynamical instability observed in a ferromag- netically interacting spin-1 condensate [7–9, 31, 32]. This instability is removed whenever the imaginary part of the eigenfrequency for the corresponding Bogoliubov excita- tion becomes zero in a modulation cycle. We show this instability is indeed suppressed in the following for the cosine modulation c$_{2}$ ( t ) = d cos(Ω t ). This suppression inhibits the spontaneous formation of spin domains. to the modulation free results (in dashed lines), we find the most unstable mode (in double contours) is not only suppressed in amplitude but also shifted to larger wave vector. The dependence of k$_{−}$ on the modulation fre- quency Ω is illustrated in Fig. 4. The almost indepen- dence of k$_{−}$ on Ω at small values contrasts with a strong monotonic increase at large values of Ω. The emergence of spin domains is prohibited due to the modulation. On one hand, the suppression of F implies a smaller effective spin exchange interaction thus a longer effective spin healing length ξ ; On the other hand, the up- shift of k$_{−}$ means shorter wave length λ = 2 π/k$_{−}$ . If the unstable mode wave length (potentially domain width) is smaller than the spin healing length, the condensate is able to heal by itself. The domain structure would never appear if ξ exceeds λ . We now consider our system of a homogeneous $^{87}$Rb spin-1 condensate starting from an off-equilibrium initial state [7]. The averaged spin ⃗ f = 〈 F$_{x}$ 〉 ˆ x + 〈 F$_{y}$ 〉 ˆ y + 〈 F$_{z}$ 〉 ˆ z is conserved where F$_{x,y,z}$ are spin-1 matrices. Starting from any stationary point, the evolution of the collective excitations takes a compact form Because of the modulation, however, the resulting dy- namics becomes completely different from the case of c$_{2}$ = 0 or no spin dynamics at all. The modulation does not stop spin mixing dynamics, i.e. , as we continue to observe spin waves which is nominally disguised in exper- iments by the spontaneously formed spin domains [8, 9]. Furthermore, we expect the modulated spin dynamics to be robust against various experimental noises because the periodic modulation effectively cancels uncorrelated noises from alternating modulation periods. Finally we confirm the above conclusions for a trapped spin-1 condensate with full numerical simulations. We adopt experimental parameters as in Ref. [7]: The initial conditions are as in the experiment [8], with $^{87}$Rb con- densates [ ρ$_{0}$ (0) = 0 . 744, θ (0) = 0, N = 2 . 0 × 10 $^{5}$, and m = 0], in a trap V$_{ext}$ ( ⃗ r ) = ( M/ 2)( ω 2 $_{x}$x 2 + ω 2 $_{y}$y 2 + ω 2 $_{z}$z $^{2}$) where ⃗ x = ( δ Ψ$_{+}$ , δ Ψ$_{0}$ , δ Ψ$_{−}$ , δ Ψ ∗ $_{+}$, δ Ψ ∗ $_{0}$, δ Ψ ∗ $_{−}$) T with devi- ations δ Ψ$_{j}$ and δ Ψ ∗ j from the stationary solution [7]. The general solution to Eq. (7) is ⃗ x ( t ) = U ( t, 0) ⃗ x (0) where U ( t$_{2}$, t$_{1}$ ) = T exp[ − ( i/ ℏ ) ∫ t$_{2}$ t$_{1}$ ds M ( s )] with T the time ordering operator. For periodic modulation, the so- lution during t ∈ [ pT, ( p + 1) T ] is further simplified to ⃗ x ( t ) = U ( t, pT )( U$_{T}$ ) $^{p}$⃗ x (0), where p = 0 , 1 , 2 , · · · indexes the number of periods. T = 2 π/ Ω is the period of modu- lation, and U$_{T}$ = U ( T, 0) = T exp[ − ( i/ ℏ ) ∫ T 0 ds M ( s )] is the evolution operator for a complete period. | ⃗ x ( t ) | will grow (decay) exponentially with p if the modulus of U$_{T}$ are larger (smaller) than unity. Diagonalizing U$_{T}$ and rewriting it as U$_{T}$ = V $^{†}$exp( − iT F ) V , where the Fluo- quet operator F is a 6-by-6 diagonal matrix, the crite- ria for stable dynamics reduces to a vanishing imaginary part of F$_{q}$ ( q = 1 , 2 , · · · , 6). Unstable dynamics arises if the imaginary part is not zero, while stable dynamics emerges if the imaginary part of all F$_{q}$ is exactly zero. The inset of Fig. 4 shows the imaginary part of a typ- ical spectrum for the system under a cosine modulation of c$_{2}$ . We focus on the most unstable mode which in principle dominates the unstable dynamics. Compared i $_{ℏ}$∂⃗ x ∂t = M ( t ) · ⃗ x, (7)