Spin-domain formation in antiferromagnetic condensates

Antiferromagnetic condensates are generally believed not to display modulational instability and subsequent spin-domain formation. Here we demonstrate that in the presence of a homogeneous magnetic field antiferromagnetic spin-1 Bose-Einstein condensates can undergo spatial modulational instability followed by the subsequent generation of spin domains. Employing numerical simulations for realistic conditions, we show how this effect can be observed in sodium condensates confined in an optical trap. Finally, we link this instability and spin-domain formation with stationary modes of the condensate.

Figure 1

(Color online) Diagram of existence of stationary states in spin-1 condensates. In addition to the three-component phase-matched and anti-phase-matched solutions shown in the diagram, two-component solutions with ${\rho }_0=0$ exist for arbitrary $M$ and one-component solutions ${\rho }_j=1$ exist with $j=-,0,+$. Dark shading (top), anti-phase-matched state; light shading (bottom), phase-matched state; cross hatching, both states. The dotted and dashed lines at $M=0$ indicate the absence of a phase-matched or anti-phase-matched state, respectively.

Figure 2

(Color online) Modulational instability growth rate in a phase-matched steady state of a sodium condensate (circles) versus the quadratic Zeeman shift in the case $M=0$. The solid line is a square fit to the numerical data. The upper shaded area shows the range of $k$ vectors corresponding to the unstable (imaginary) frequencies.

Figure 3

(Color online) Spin-domain formation in a ${}^{23}\mathrm{Na}$ condensate confined in an optical harmonic trap. Top panels: evolution of $n_0$ (darker shading higher density). Bottom panels: densities at given times. (a), (b), (c) Phase-matched initial state, ${\rho }_{+,0,-}=0.351,0.3,0.349$; (d), (e), (f) anti-phase-matched initial state, ${\rho }_{+,0,-}=0.4,0.01,0.59$. The thin dotted line corresponds to the total condensate density $n$, and the dashed, solid, and dotted lines correspond to $n_{+}$, $n_0$, and $n_{-}$, respectively. Parameters are $N=3.7\times {10}^3$, $B=175\mathrm{mG}$, ${\omega }_{\perp }=2\pi \times {10}^3\mathrm{Hz}$, and ${\omega }_{\parallel }=2\pi \times 32\mathrm{Hz}$.

Figure 4

(Color online) Examples of (a) phase-matched and (b) anti-phase-matched stationary states in a ${}^{23}\mathrm{Na}$ condensate. Shown are the total density $n$ (thin dotted line) and component densities $n_{+,0,-}$ (dashed, solid, and dotted lines, respectively).