Stability of a flattened dipolar binary condensate: Emergence of the spin roton

We develop a theory for a two-component miscible dipolar condensate in a planar trap. Using numerical solutions and a variational theory we solve for the excitation spectrum and identify regimes where density- and spin-roton excitations are favored. We characterize the various instabilities that can emerge in this system over a wide parameter regime and present results for the stability phase diagram. Importantly this allows us to identify the parameter regimes where a novel roton-immiscibility transition can occur, driven by the softening of the spin-roton excitation.

Figure 1

Schematic showing a miscible binary dipolar Bose-Einstein condensate where the components have (upper) parallel and (lower) antiparallel polarization of dipole moments. The red and blue arrows represent the dipole moments of components 1 and 2, respectively.

Figure 2

(a1)–(a3) Stability phase diagram for a uniform balanced system with parallel dipoles. Results for $\alpha =0$ and (a1) $n{g}_{12}^s/\hslash {\omega }_z{l}_z=5$, (a2) 0, and (a3) $-5$. The shaded regions identify where the uniform state is either stable as a miscible mixture, or has various instabilities as described in the main text. (b) Shows the excitation spectrum for $n(g_{ii}^s,g_{ii}^{dd})/\hslash {\omega }_z{l}_z=(10,10.8)$ [shown with a “$+$” symbol in subplot (a1)]. The density $\lambda =+1$ (spin $\lambda =-1$) excitation branches are shown as solid (dashed) lines. The roton minimum is indicated. (c) The roton wave vector at the point that the roton excitation softens to zero energy. (d) The interaction parameters for the roton boundary. In subplots (a)–(d) blue lines indicate the results from the full numerical calculation of the GPE and BdG equations, whereas the orange lines are from the variational theory.

Figure 3

(a1)–(a3) Stability phase diagram for a uniform balanced system with antiparallel dipoles. Results for $\alpha =0$ and (a1) $n{g}_{12}^s/\hslash {\omega }_z{l}_z=5$, (a2) 0, and (a3) $-5$. The shaded regions identify where the uniform state is either stable as a miscible mixture, or has various instabilities as described in the main text. (b) Shows the excitation spectrum for $n(g_{ii}^s,g_{ii}^{dd})/\hslash {\omega }_z{l}_z=(10,5.1)$ [shown with a “$+$” symbol in subplot (a1)]. The density $\lambda =+1$ (spin $\lambda =-1$) excitation branches are shown as solid (dashed) lines. The roton minimum is indicated. (c) The roton wave vector at the point that the roton excitation softens to zero energy. (d) The interaction parameters for the roton boundary. In subplots (a)–(d) blue lines indicate the results from the full numerical calculation of the GPE and BdG equations, whereas the orange lines are from the variational theory.

Figure 4

The density and spin-density dynamic and static structure factors for a balanced system with parallel (a1)–(a3) and antiparallel (b1)–(b3) dipole orientations. Results in (a1)–(a3) are for the parallel case considered in Fig. 2 and (b1)–(b3) are for the antiparallel case in Fig. 3. White solid (dashed) lines show the in-phase/density (out-of-phase/spin) excitation spectrum for reference. The dynamic structure factors are frequency broadened by a frequency width of ${\omega }_B=0.1{\omega }_z$.

Figure 5

The density and spin-density dynamic and static structure factors for an unbalanced system with $n(g_{11}^{dd},g_{22}^{dd})/\hslash {\omega }_z{l}_z=(14,7)$ (a1)–(a3), (13.4,0) (b1)–(b3), and other parameters as in Fig. 2. White solid lines show the excitation spectrum for reference. The insets to (a3) and (b3) show the axial density profiles $|{\psi }_1{|}^2$ (blue) and $|{\psi }_2{|}^2$ (red). The dynamic structure factors are frequency broadened by a frequency width of ${\omega }_B=0.1{\omega }_z$.

Figure 6

Stability phase diagrams for an unbalanced system with axes ${\mu }_2^m/{\mu }_1^m$, (a) $g_{12}^s/\sqrt{g_{11}^s{g}_{22}^s}$, and (b) $g_{12}/\sqrt{g_{11}{g}_{22}}$. Color of the stable miscible region reveals the dominant character of the roton when present (see text). Blue (purple) boundaries identify when a roton (phonon) mode goes soft leading to an instability. (c)–(e) Wave functions of the components at various points in the stable region as indicated in the main plot. Parameters: $n{g}_{ii}^s/\hslash {\omega }_z{l}_z=5$, $n{g}_{11}^{dd}/\hslash {\omega }_z{l}_z=4.5$, and $\alpha =0$.