Multicomponent gap solitons in spinor Bose-Einstein condensates

We model the nonlinear behavior of spin-1 Bose-Einstein condensates (BECs) with repulsive spin-independent interactions and either ferromagnetic or antiferromagnetic (polar) spin-dependent interactions, loaded into a one-dimensional optical lattice potential. We show that both types of BECs exhibit dynamical instabilities and may form spatially localized multicomponent structures. The localized states of the spinor matter waves take the form of vector gap solitons and self-trapped waves that exist only within gaps of the linear Bloch-wave band-gap spectrum. Of special interest are the nonlinear localized states that do not exhibit a common spatial density profile shared by all condensate components, and consequently cannot be described by the single mode approximation (SMA) frequently employed within the framework of the mean-field treatment. We show that the non-SMA states can exhibits Josephson-like internal oscillations and self-magnetization, i.e., intrinsic precession of the local spin. Finally, we demonstrate that nonstationary states of a spinor BEC in a lattice exhibit coherent undamped spin-mixing dynamics, and that their controlled conversion into a stationary state can be achieved by the application of an external magnetic field.

Figure 1

Band-gap structure of the linear Bloch-wave spectrum for a periodic potential of a one-dimensional optical lattice $V_{\mathrm{OL}}=V_0{\sin }^2\left(x\right)$. (a) Variation of bands (shaded) and gaps (clear areas) with the potential depth $\mu \left(V_0\right)$ and (b) eigenvalue spectrum $\mu \left(k\right)$ reduced to the first Brillouin zone shown for $V_0=2$. Points (c) and (d) correspond to the ground and excited state linear Bloch waves with the typical spatial wave functions in a single hyperfine component of the condensate shown in panels (c) and (d), respectively.

Figure 2

Development of modulation instability for $F=1$ spinor Bloch states in (a) ferromagnetic $\left({}^{87}\mathrm{Rb}\right)$ and (c) polar $\left({}^{23}\mathrm{Na}\right)$ BECs at the top edge of the first band ($V_0=4$, $\mu =1.4$). Shown are the peak densities of the $m_F=+1$ (dashed line), $m_F=0$ (dash-dotted line), and $m_F=-1$ (dotted line) condensate components at $x=100$. For comparison, the behavior of the modulationally stable ground state nonlinear Bloch waves at $\mu =1.28$ are shown for a (b) ferromagnetic $\left({}^{87}\mathrm{Rb}\right)$ and (d) polar $\left({}^{23}\mathrm{Na}\right)$ BECs.

Figure 3

Instability-induced dynamics for a ${}^{23}\mathrm{Na}$ condensate at the top edge of the ground band ($V_0=2$, $\mu =1.12$). Shown are snapshots of the total BEC density (solid lines) with the periodic OL potential (gray shading, magnified by a factor of 20) at the times (a) $t=0$, (b) $t=20$, (c) $t=40$, (d) $t=100$, (e) $t=120$, and (f) $t=200$.

Figure 4

(Color online) Examples of spatial structure of the (a), (c), (d) ferromagnetic-type (in-phase) and (b) polar-type (out-of-phase) gap solitons of a polar ${}^{23}\mathrm{Na}$ condensate in an optical lattice ($V_0=4$, $\mu =2.8$). Lines of different shade/color (red, green, and blue) show the ${\varphi }_{-1}$, ${\varphi }_0$, and ${\varphi }_{+1}$ components, respectively. The corresponding values of the local spin vector $\vec{\mathcal{S}}=({\mathcal{S}}_x,{\mathcal{S}}_y,{\mathcal{S}}_z)$ are plotted in the cross-sectional plane ${\mathcal{S}}_y=0$ of the 3D spin sphere (shaded); $\beta$ is the Euler rotation angle.

Figure 5

Comparison of the values of gap soliton peak amplitudes ${\varphi }_{+1}$ and ${\varphi }_{-1}$ at $x=0$ (dashed lines), with the amplitudes of a homogeneous condensate (solid lines) for (a) out-of-phase and (b) in-phase stationary states of a ${}^{87}\mathrm{Rb}$ condensate in a lattice with $V_0=4$, plotted as a function of ${\varphi }_0$. Gap soliton is taken at $\mu =2.8$, and the homogeneous condensate at ${\mu }_{\mathrm{eff}}=2.0$, to account for the shift of the spectrum from zero (see text). Analytically obtained parameters of the semiaxes of the ellipses and the ellipse displacement are (a) $A=11.59$, $B=8.19$ and (b) $A=8.21$, $B=5.8$ and ${\Delta }_{+1}=B$.

Figure 6

(Color online) Self-trapped gap states at $\mu =2.8$ corresponding to the nonlinear Bloch waves of a ${}^{87}\mathrm{Rb}$ condensate (a) in the ground state and (b) at the edge of the first band of the optical lattice potential (shaded) with $V_0=4$. Lines of differet shade/color (red, green, and blue) show ${\varphi }_{-1}$, ${\varphi }_0$ and ${\varphi }_{+1}$ hyperfine states, respectively.

Figure 7

(Color online) Non-SMA stationary states for a ${}^{23}\mathrm{Na}$ condensate: (a) centered on the potential maximum (i.e., off-site) and (b) centred on the potential minimum (i.e., on-site). Both stationary states are found for the values of the chemical potentials ${\mu }_{+1}=2.8$, ${\mu }_{-1}=2.7$, and ${\mu }_0=\frac{1}{2}({\mu }_{+1}+{\mu }_{-1})=2.75$. Lines of different shade/color (red, blue, and green) mark ${\varphi }_{+1}$, ${\varphi }_{-1}$, and ${\varphi }_0$ components, respectively. Shaded areas show the corresponding optical lattice potential $(V_0=4)$.

Figure 8

(Color online) Spin dynamics of unstable off-site non-SMA state of ${}^{23}\mathrm{Na}$ for parameters given in Fig. 9. The magnitude of ${\mathcal{S}}_x$ is taken at $x=3\pi ∕2$. Initially, the local spin oscillates with period $T=\frac{2\pi }{\delta \mu }\simeq 104.7$. For a time interval of nine oscillations, which correspond to $t=942.5$, the dynamics is stable. Afterwards the unstable evolution commences.

Figure 9

Non-SMA stationary states of ${}^{23}\mathrm{Na}$ for parameters ${\mu }_{+1}=1.28$, ${\mu }_{-1}=1.4$, ${\mu }_0=\frac{1}{2}({\mu }_{+1}+{\mu }_{-1})=1.34$, and $V_0=2$. All hyperfine components of the unstable off-site soliton at $t=0$ (a), (b), (c) and at $t=2000$ (d), (e), (f) are compared against the corresponding profiles of a stable on-site non-SMA state (g), (h), (i).

Figure 10

Projections of local spin of the non-SMA states of ${}^{23}\mathrm{Na}$ (shown in Fig. 9) onto the planes $S_y=0$ (a), (b), (c) and $S_x=0$ (d), (e), (f) of the spin sphere (shaded). Distribution of the local spin vector of the unstable off-site non-SMA state in its (a), (d) initial and (b), (e) final, i.e., at $t=2000$ configurations. For comparison, panels (c), (f) show the spin distribution for the stable on-site state.

Figure 11

(Color online) Dynamics of nonequilibrium states in ${}^{87}\mathrm{Rb}$ ($\mu =2.8$, $V_0=4$). (a) Spin mixing oscillations of the populations between $m_F=-1$ (marked as $N_{-1}$), $m_F=+1$ (marked as $N_{+1}$), and $m_F=0$ (marked as $N_0$) hyperfine components. (b) The dynamical trajectories in the $n_0\left(\Delta \theta \right)$ phase space for the scenario presented in panel (a). Initial and final positions in the phase space are marked by the respective points $t_{\text{initial}}$ and $t_{\text{final}}$. The arrow indicates the direction of the spin mixing oscillations in the phase space.

Figure 12

(Color online) Arrest of spin-mixing oscillations shown in Fig. 11 due to quadratic Zeeman effect. The magnetic field of magnitude $B=1$ Gauss is applied over the period of $t_B=\left(\Delta \theta \right)∕\left(\Delta E\right)\simeq 2$ $\left(1.4\mathrm{ms}\right)$.

Figure 13

Density profiles of the hyperfine components of the vector gap soliton corresponding to $t_{\text{final}}=1000$ in Fig. 12. Periodic lattice potential (shaded) has been increased by a factor of 5 for better visualization.