Macroscopic random Paschen-Back effect in ultracold atomic gases

We consider spin- and density-related properties of single-particle states in a one-dimensional system with random spin-orbit coupling. We show that the presence of an additional Zeeman field $\mathrm{\Delta }$ induces both nonlinear spin polarization and delocalization of states localized at $\mathrm{\Delta }=0$, corresponding to a random macroscopic analog of the Paschen-Back effect. While the conventional Paschen-Back effect corresponds to a saturated $\mathrm{\Delta }$ dependence of the spin polarization, here the gradual suppression of the spin-orbit coupling effects by the Zeeman field is responsible both for the spin saturation and delocalization of the particles.

Figure 1

Model distribution of SOC impurities corresponding to the disorder in Eq. (4).

Figure 2

(a) Spin component ${\langle {\sigma }_x\rangle }_n$ and (b) log scale of the IPR as a function of the state number for different Zeeman fields [the legend is shown in (b)]. To avoid degeneracy, we use here $0^{+}={10}^{-3}.$ The horizontal dash-dotted line corresponds to ${\zeta }_L=3/4L$ value. (c) Actual realization of the random potential (solid line) and three densities corresponding to the energies $E_n=E_0,-1$, and 0. Here ${\alpha }_0=4,d=\xi =0.5$, and $L=40$.

Figure 3

Disorder-averaged quantities as a function of the state energy for different Zeeman $\mathrm{\Delta }$'s [the legend is shown in (a)]. (a) Expectation value ${\langle {\sigma }_x\rangle }_n,$ (b) the IPR, where the inset shows the fraction $f$ of localized states (out of 300 lowest eigenstates) with the ${\zeta }_n>2{\zeta }_L$ (dash-dot horizontal line), and (c) the density of states. The averaging is performed over ${10}^3\alpha \left(x\right)$ realizations with the parameters same as in Fig. 2.

Figure 4

Schematic illustration of the spin-conserving backscattering caused by the random SOC. Lower and upper parabolas correspond to $k^2/2-\mathrm{\Delta }$ and $k^2/2+\mathrm{\Delta }$ branches, respectively, with the virtual transitions shown by dashed lines.

Figure 5

Dependence of the ground state ${\langle {\sigma }_x\left(\mathrm{\Delta }\right)\rangle }_0$ (main plot) and the IPR (inset) for two typical realizations of the random potential (see Appendix pp3 for more details). As expected for low purity spin states, $\left|{\langle {\sigma }_x\left(0\right)\rangle }_0\right|\ll 1,$ corresponding to typical $l_0\left|\alpha \left(x_0\right)\right|\ge 1$ for the chosen parameters of disorder, here the same as in Fig. 2.

Figure 6

Correlator ${\mathcal{K}}_{mm}(0,x)$ (averaged over ${10}^3$ realizations) for two random potentials, both with $d=0.5$. (a) ${\alpha }_0=1,\xi =d/4$, and best-fitting parameter $\beta =0.03$; (b) ${\alpha }_0=0.06125,\xi =4d$, and best-fitting parameter $\beta =0.026.$

Figure 7

Dependence of the ground-state spin on the Zeeman $\mathrm{\Delta }$ for random (solid line) and regular [as in Eq. (C11), dashed line] SOC. These dependences are very similar for both types of coupling. Inset shows the qualitative difference between the IPR for the random and the regular realizations. While at small $\mathrm{\Delta }$ the behavior of the IPR is the same, their large $\mathrm{\Delta }$ dependences are different: the IPR rapidly decreases for the random SOC and returns to its value at $\mathrm{\Delta }=0$ for the regular one.