Spin-orbit coupling manipulating composite topological spin textures in atomic-molecular Bose-Einstein condensates

Atomic-molecular Bose-Einstein condensates (BECs) offer brand new opportunities to revolutionize quantum gases and probe the variation of fundamental constants with unprecedented sensitivity. The recent realization of spin-orbit coupling (SOC) in BECs provides a new platform for exploring completely new phenomena unrealizable elsewhere. In this study, we find a way of creating a Rashba-Dresselhaus SOC in atomic-molecular BECs by combining the spin-dependent photoassociation and Raman coupling, which can control the formation and distribution of a different type of topological excitation—carbon-dioxide-like skyrmion. This skyrmion is formed by two half-skyrmions of molecular BECs coupling with one skyrmion of atomic BECs, where the two half-skyrmions locate at both sides of one skyrmion. Carbon-dioxide-like skyrmion can be detected by measuring the vortices structures using the time-of-flight absorption imaging technique in real experiments. Furthermore, we find that SOC can effectively change the occurrence of the Chern number in $k$ space, which causes the creation of topological spin textures from some separated carbon-dioxide-like monomers each with topological charge $-2$ to a polymer chain of the skyrmions. This work helps in creating dual SOC atomic-molecular BECs and opens avenues to manipulate topological excitations.

Figure 1

Experimental setup for creating the spin-orbit coupled atomic-molecular BECs of ${}^{87}\mathrm{Rb}$. (a) The experimental geometry. The formation of spin molecular BECs: Laser field $L1$ finishes the spin-dependent photoassociation [39], laser field $LT$ performs the $\mathrm{\Delta }F=-1, \mathrm{\Delta }{m}_F=+1$ is the transiting process of the excited molecules, and laser field $L2$ induces the excited molecules to emit a photon and forms the stable spin-1 molecular BECs. The creation of dual SOC: The laser fields $L3$ and $L4$ ($L5, L6$, and $L7$) are used to create the SOC of atomic (molecular) BECs along the $x$ axis. $B$ denotes a weak magnetic field along the $z$ axis, which is used to induce the Zeeman shift. (b) Level diagram of Raman coupling within the atomic BECs. Two lasers ($L3$ and $L4$) are used to couple states of $|\uparrow \rangle$ (i.e., $|1,-1\rangle$) and $|\downarrow \rangle$ (i.e., $|1,0\rangle$). (c) Level diagram of Raman coupling within the $F=1$ molecular BECs. The three lasers ($L5, L6$, and $L7$) couple states of ${\varphi }_{-1}, {\varphi }_0$, and ${\varphi }_{+1}$. (d) The procedure of creating $|F=1,m_F=-1\rangle$ state molecular BEC. A close pair of atoms from the condensate in a state $|\downarrow \rangle$ absorb a photon from laser field $L1$ and then transit to a bound excited molecular state $|F=2,m_F=-2\rangle$, i.e., the spin-dependent photoassociation process $|1,-1\rangle \otimes |1,-1\rangle ⟶|2,-2\rangle$ [39]. Via the laser field $LT$, the excited molecule transits to $|F=1,m_F=-1\rangle$ state. Finally, laser field $L2$ induces the excited molecular state to emit a photon into molecular state ${\varphi }_{-1}$ which keeps $F=1$ and $m_F=-1$. Panels (e) and (f) are the procedure of creating $|F=1,m_F=0\rangle$ and $|F=1,m_F=+1\rangle$ state molecular BECs, respectively.

Figure 2

Creating a carbon-dioxide-like skyrmion in the dual SOC atomic-molecular BECs. (a) The densities of each component when the system reaches the equilibrium state. The particles number $(N_{\uparrow },N_{\downarrow },N_{-1},N_0,N_{+1})$ is $(2.56\times {10}^3,2.98\times {10}^3,1.72\times {10}^3,3.22\times {10}^3,1.59\times {10}^3)$. (b) The corresponding phases of each component in (a). [(c) and (d)] The spin texture of the atomic BECs and molecular BECs, respectively. The color of each arrow indicates the magnitude of $S_z$. [(e) and (f)] The corresponding spin texture under the transformation: $({\mathbf{S}}_x^{{}^{\prime }},{\mathbf{S}}_y^{{}^{\prime }},{\mathbf{S}}_z^{{}^{\prime }})=({\mathbf{S}}_z,{\mathbf{S}}_y,-{\mathbf{S}}_x)$. (g) A combined scheme of the carbon-dioxide-like skyrmion. A skyrmion (center) couples with two half-skyrmions, which have to be arranged at the two sides of the skyrmion. (h) The difference of spin vectors between the atomic BECs and the molecular BECs, where $y=0$. [(i) and (j)] The topological charge density of the atomic BECs and molecular BECs, respectively. The ellipse and rectangles mark out the regions of a skyrmion and two half-skyrmions, respectively. The symbol $\circ , ☆, ▹, *$, and $◃$ are the position of vortices formed by the $\uparrow , \downarrow , m_F=-1, m_F=0$, and $m_F=+1$ components, respectively. (k) The sum of the topological charge densities. It denotes the creation of carbon-dioxide-like skyrmion that the ellipse overlaps with the two rectangles. The unit of length, strength of $\chi , {\mathrm{\Omega }}_A$ and dual SOC are $0.76\mu \mathrm{m}, \hslash \omega , \omega$, and $0.96\times {10}^{-3}\mathrm{m}/\mathrm{s}$, respectively.

Figure 3

The effect of dual SOC on the vortices in rotating atomic-molecular BECs. The rotation frequency is ${\mathrm{\Omega }}_{\mathrm{rotation}}=0.5\omega$, where $\omega =200\times 2\pi \mathrm{Hz}$. We set the parameters $g_{\uparrow ,\uparrow }=g_{\uparrow ,\downarrow }=g_A$ with the scattering length $a_A=101.8a_B, g_{\downarrow ,\downarrow }=0.95g_A, g_{AM}=0.5g_A, g_n=2g_A, g_s=0.03g_A, \chi =0.08, \varepsilon =0, {\mathrm{\Omega }}_A=0.8$, and ${\mu }_m=2{\mu }_a=3.5{\mu }_{m,0}=7{\mu }_{a,0}=42\hslash \omega$. (a) Densities of each components under the equilibrium state, the strength of dual SOC $\lambda =0$. Note that the components are pointed out by the title: $|\uparrow \rangle , |\downarrow \rangle , m_F=-1, m_F=0$, and $m_F=+1$, respectively. (b) The corresponding phases of each components in (a). [(c) and (d)] The corresponding densities and phases with $\lambda =1$. [(e) and (f)] The corresponding densities and phases with $\lambda =2$. [(g) and (i)] The position of vortices in (a), (c), and (e), respectively. In (g), the rectangle points out a vortex cluster. In (h), the rectangle points out the region where the vortices form a lattice. In (i), the red (blue) rectangle marks out a carbon dioxide vortex, and the black rectangle points out the vortices lattice. The particles number $(N_{\uparrow },N_{\downarrow },N_{-1},N_0,N_{+1})$ for the three cases are $(5.82\times {10}^3,8.01\times {10}^3,4.96\times {10}^3,8.23\times {10}^3,4.93\times {10}^3), (7.86\times {10}^3,8.80\times {10}^3,7.95\times {10}^3,3.93\times {10}^3,7.85\times {10}^3)$, and $(11.42\times {10}^3,12.29\times {10}^3,10.03\times {10}^3,2.00\times {10}^3,10.04\times {10}^3)$, respectively. The unit of length, strength of $\chi , {\mathrm{\Omega }}_A$, and dual SOC are $0.76\mu \mathrm{m}, \hslash \omega , \omega$, and $0.96\times {10}^{-3}\mathrm{m}/\mathrm{s}$, respectively.

Figure 4

The effect of SO coupling on the spin texture in the atomic-molecular BECs. [(a)–(c)] The corresponding spin textures of atomic BECs in Figs. 3, 3, and 3, respectively. [(d)–(f)] The corresponding spin textures of molecular BECs in Fig. 3, 3, and 3, respectively. Note that the spin textures are under the transformation: $({\mathbf{S}}_x^{{}^{\prime }},{\mathbf{S}}_y^{{}^{\prime }},{\mathbf{S}}_z^{{}^{\prime }})=({\mathbf{S}}_z,{\mathbf{S}}_y,-{\mathbf{S}}_x)$. The color of each arrow indicates the magnitude of $S_z^{{}^{\prime }}$. The blue ellipses point out the regions of the carbon-dioxide-type skyrmion that skyrmion of atomic BECs couples two half-skyrmion of molecular BECs. In addition, we mark the position of vortices in order to illuminate the relationship between spin texture and position of vortices clearly. The meanings of the marks are the same as that in Figs. 3–3. The unit of length, strength of $\chi , {\mathrm{\Omega }}_A$, and SO coupling are $0.76\mu \mathrm{m}, \hslash \omega , \omega$, and $0.96\times {10}^{-3}\mathrm{m}/\mathrm{s}$, respectively.

Figure 5

The topological charge density of spin textures in SO coupled atomic-molecular BECs. [(a)–(c)] The corresponding topological charge density in Fig. 4–4, respectively. [(d)–(f)] The corresponding topological charge density in Figs. 4–4, respectively. In addition, we mark the position of vortices in $x<0$ region in order to illuminate the relationship between spin texture and position of vortices clearly. The meanings of the marks are the same as that in Figs. 3–3. [(g)–(i)] The total topological charge density of (a) and (d), (b) and (e), and (c) and (f), respectively. We use the blue ellipses to point out the regions of the carbon-dioxide-type skyrmion. The unit of length and SO coupling are $0.76\mu \mathrm{m}$ and $0.96\times {10}^{-3}\mathrm{m}/\mathrm{s}$, respectively.

Figure 6

Chern number verifying the high-$\left|k\right|$ topological excitations in the dual SOC atomic-molecular BECs. [(a)–(c)] Chern number in $k$ space of atomic-molecular BECs in Figs. 3, 3, and 3, respectively. The top inset in Fig. 6 is a local enlargement. The red dotted line and the red arrow indicate the minimum value $K_{\min }$ for the first hop as $\left|k\right|$ increases. The black dotted line and black arrow show the first hop of average position $K_a$ in atomic BECs of ${\psi }_{\uparrow }$ and ${\psi }_{\downarrow }$. Similarly, the blue dotted line and blue arrow show the first hop of average value $K_m$ in molecular BECs of ${\varphi }_{+1}$ and ${\varphi }_{-1}$. The bottom inset in Fig. 6 is $A_{\uparrow }/i$ under $\lambda =0$, where $A_{\uparrow }$ is the Berry curvature of ${\psi }_{\uparrow }$. Similarly, the inset in Fig. 6 is $A_{\uparrow }/i$ under $\lambda =1$. The inset in Fig. 6 is $A_{-1}/i$ under $\lambda =2$. (d) Chern number as function of strength of SOC $\lambda$. (e) $\langle C\left(k\right)\times k^2\rangle$ as function of strength of SOC $\lambda$. (f) $\langle C\left(k\right)\times k^2\rangle /\langle C\left(k\right)\rangle$ as function of strength of SOC $\lambda$.

Figure 7

Phase diagram of topological spin textures in the dual SOC atomic-molecular BECs. (a) Ration frequency-SOC (${\mathrm{\Omega }}_{\mathrm{rotation}}-\lambda$) phase diagram for the products. We set the parameters $\omega =200\times 2\pi \mathrm{Hz}, \chi =0.03, {\mathrm{\Omega }}_A=0.8, g_{\uparrow ,\uparrow }=g_{\uparrow ,\downarrow }=g_A$ with the scattering length $a_A=101.8a_B, g_{\downarrow ,\downarrow }=0.95g_A, g_n=2g_A, g_s=0.03g_A, g_{AM}=0.5g_A, \varepsilon =0$ and ${\mu }_m=2{\mu }_a=3.5{\mu }_{m,0}=7{\mu }_{a,0}=28\hslash \omega$. (b) Atom-molecule interaction-SOC [$(g_{AM}/g_A)-\lambda$] phase diagram for the products. We set the parameters $\omega =200\times 2\pi \mathrm{Hz}, \chi =0.03, {\mathrm{\Omega }}_A=0.8, g_{\uparrow ,\uparrow }=g_{\uparrow ,\downarrow }=g_A$ with the scattering length $a_A=101.8a_B, g_{\downarrow ,\downarrow }=0.95g_A, g_n=2g_A, g_s=0.03g_A, \varepsilon =0$, and ${\mu }_m=2{\mu }_a=3.5{\mu }_{m,0}=7{\mu }_{a,0}=28\hslash \omega$. The unit of length, strength of $\chi , {\mathrm{\Omega }}_A$, and SOC are $0.76\mu \mathrm{m}, \hslash \omega , \omega$, and $0.96\times {10}^{-3}\mathrm{m}/\mathrm{s}$, respectively.