Spin structure of harmonically trapped one-dimensional atoms with spin-orbit coupling

We introduce a theoretical approach to determine the spin structure of harmonically trapped atoms with two-body zero-range interactions subject to an equal mixture of Rashba and Dresselhaus spin-orbit coupling created through Raman coupling of atomic hyperfine states. The spin structure of bosonic and fermionic two-particle systems with finite and infinite two-body interaction strength $g$ is calculated. Taking advantage of the fact that the $N$-boson and $N$-fermion systems with infinitely large coupling strength $g$ are analytically solvable for vanishing spin-orbit coupling strength $k_{\mathrm{so}}$ and vanishing Raman coupling strength $\mathrm{\Omega }$, we develop an effective spin model that is accurate to second order in $\mathrm{\Omega }$ for any $k_{\mathrm{so}}$ and infinite $g$. The three- and four-particle systems are considered explicitly. It is shown that the effective spin Hamiltonian, which contains a Heisenberg exchange term and an anisotropic Dzyaloshinskii-Moriya exchange term, describes the transitions that these systems undergo with the change of $k_{\mathrm{so}}$ as a competition between independent spin dynamics and nearest-neighbor spin interactions.

Figure 1

Single-particle momentum-space potentials ${(p\pm \hslash k_{\mathrm{so}})}^2/\left(2m\right)$ (dashed lines) and noninteracting single-particle momentum-space wave functions (solid lines) for (a) $k_{\mathrm{so}}{a}_{\mathrm{ho}}=2$ and (b) $k_{\mathrm{so}}{a}_{\mathrm{ho}}=4$, respectively. The potential and wave functions centered at $p=-\hslash k_{\mathrm{so}}$ are for the spin-up component and those centered at $p=\hslash k_{\mathrm{so}}$ are for the spin-down component. For small $k_{\mathrm{so}}$, the spin-up and spin-down single-particle wave functions in the Hilbert space $H_L$ (here, states with $E\le 3.5\hslash \omega$) have overlap, indicating that the first-order effective Hamiltonian dominates. For large $k_{\mathrm{so}}$, the spin-up and spin-down single-particle wave functions in the Hilbert space $H_L$ have essentially zero overlap, indicating that coupling to highly excited states is dominant, leading to a dominant second-order effective Hamiltonian.

Figure 2

Spin structure densities for the $N=2$ ground state for $\mathrm{\Omega }=\hslash \omega /2$ and varying $g$. Panels (a) and (b) show the spin structure densities $n_x^b\left(x\right)$ and $n_z^b\left(x\right)$, respectively. Dashed, dotted-dashed, dotted-dotted-dashed, and solid lines show the spin densities for $g=\sqrt{2}\hslash \omega a_{\mathrm{ho}},3\sqrt{2}\hslash \omega a_{\mathrm{ho}},15\sqrt{2}\hslash \omega a_{\mathrm{ho}},$ and $\infty$, respectively. The spin structure density is independent of $k_{\mathrm{so}}$. The spin structure density $n_x^f\left(x\right)$ is independent of $g$ and shown by the circles in (a). The spin structure density $n_z^f\left(x\right)$ coincides with $n_z^b\left(x\right)$ for all $g$.

Figure 3

Spin structure coefficients (a) $C_x^b$, (b) $C_z^b$, (c) $C_x^f$, and (d) $C_z^f$ for the $N=2$ ground state obtained within the effective Hamiltonian approach for $\mathrm{\Omega }=\hslash \omega /2$. The spin structure coefficients in (a) and (b) are for the system consisting of two identical bosons. The spin structure coefficients in (c) and (d) are for the system consisting of two identical fermions. Dashed, dotted-dashed, dotted-dotted-dashed, and solid lines show the spin structure coefficients for $g=\sqrt{2}\hslash \omega a_{\mathrm{ho}},3\sqrt{2}\hslash \omega a_{\mathrm{ho}},15\sqrt{2}\hslash \omega a_{\mathrm{ho}}$, and $\infty$, respectively. For infinitely large $g$ (solid lines), the spin structure coefficients display a “sharp spike” at $k_{\mathrm{so}}{a}_{\mathrm{ho}}\approx 3.4$; this is the result of an avoided crossing between the ground and first excited states.

Figure 4

Benchmarking the effective Hamiltonian approach for the $N=2$ ground state for $\mathrm{\Omega }=\hslash \omega /2$ and $g=\infty$. Panels (a) and (b) show $\langle S_x\left(x\right)\rangle$ and $\langle S_z\left(x\right)\rangle$ for $k_{\mathrm{so}}{a}_{\mathrm{ho}}=\frac{1}{5}$; panels (c) and (d) show $\langle S_x\left(x\right)\rangle$ and $\langle S_z\left(x\right)\rangle$ for $k_{\mathrm{so}}{a}_{\mathrm{ho}}=4$. The solid lines show the spin structure obtained from the exact diagonalization, while the dashed lines show the spin structure obtained within the effective Hamiltonian approach. On the scale shown, the solid and dashed lines nearly coincide.

Figure 5

Expectation value of $P_{|M_s|=m}$ for (a) $N=2$, (b) $N=3$, and (c) $N=4$ with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$ as a function of $k_{\mathrm{so}}$. (a) The solid and dashed lines show $P_{|M_s|=0}$ and $P_{|M_s|=2}$, respectively. (b) The solid and dashed lines show $P_{|M_s|=1}$ and $P_{|M_s|=3}$, respectively. (c) The solid, dashed, and dotted-dashed lines show $P_{|M_s|=0},P_{|M_s|=2}$, and $P_{|M_s|=4}$, respectively.

Figure 6

Spin structure for the three-particle system with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$ as a function of $k_{\mathrm{so}}$. (a) The solid and dashed lines show the spin structure coefficients $C_{1x}$ and $C_{2x}$, respectively. (b) The solid and dashed lines show the spin structure coefficient $C_z$. (c) The solid and dashed lines show the spin structure densities $n_{1x}\left(x\right)$ and $n_{2x}\left(x\right)$, respectively. (d) The solid line shows the spin structure density $n_z\left(x\right)$.

Figure 7

Spin structure for the four-particle system with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$ as a function of $k_{\mathrm{so}}$. (a) The solid and dashed lines show the spin structure coefficients $C_{1x}$ and $C_{2x}$. (b) The solid and dashed lines show the spin structure coefficients $C_{1z}$ and $C_{2z}$. (c) The solid and dashed lines show the spin structure densities $n_{1x}\left(x\right)$ and $n_{2x}\left(x\right)$. (d) The solid and dashed lines show the spin structure densities $n_{1z}\left(x\right)$ and $n_{2z}\left(x\right)$.

Figure 8

The spin structure for two identical particles with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$. Panel (a) shows the spin vector at the leftmost slot while panel (b) shows the spin vector at the rightmost slot. For $k_{\mathrm{so}}\lesssim k_{\mathrm{so}}^{\mathrm{cr}}\approx 3.4/a_{\mathrm{ho}}$, the spin vector at each slot follows the local effective $\vec{B}$ field. The spin vector rotates with increasing $k_{\mathrm{so}}$. For $k_{\mathrm{so}}\gtrsim k_{\mathrm{so}}^{\mathrm{cr}}$, the spin-spin interaction is dominant, resulting in an approximately vanishing spin vector at each slot.

Figure 9

The spin structure for three identical particles with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$. Panel (a) shows the spin vector at the leftmost slot, panel (b) shows the spin vector at the middle slot, and panel (c) shows the spin vector at the rightmost slot. For $k_{\mathrm{so}}\lesssim k_{\mathrm{so}}^{\mathrm{cr}}\approx 3.7/a_{\mathrm{ho}}$, the spin vector at each slot follows the local effective $\vec{B}$ field. The spin vector rotates with increasing $k_{\mathrm{so}}$. For $k_{\mathrm{so}}\gtrsim k_{\mathrm{so}}^{\mathrm{cr}}$, the spin-spin interaction is dominant, resulting in a spin vector with approximately constant magnitude and orientation at each slot. The inset in (b) replots the spin vector using a different orientation of the coordinate system.

Figure 10

The spin structure for four identical particles with $g=\infty$ and $\mathrm{\Omega }=\hslash \omega /2$. Panel (a) shows the spin vector at the leftmost slot. Panel (b) shows the spin vector at the second slot from the left. Panel (c) shows the spin vector at the third slot from the left. Panel (d) shows the spin vector at the rightmost slot. For $k_{\mathrm{so}}\lesssim k_{\mathrm{so}}^{\mathrm{cr}}\approx 4.3/a_{\mathrm{ho}}$, the spin vector at each slot follows the local effective $\vec{B}$ field. The spin vector rotates with increasing $k_{\mathrm{so}}$. For $k_{\mathrm{so}}\gtrsim k_{\mathrm{so}}^{\mathrm{cr}}$, the spin-spin interaction is dominant, resulting in an approximately vanishing spin vector at each slot.