Spin-orbit-induced resonances and threshold anomalies in a reduced-dimension Fermi gas

We calculate the reflection and transmission probabilities in a one-dimensional Fermi gas with an equal mixing of the Rashba and Dresselhaus spin-orbit coupling (RD-SOC) produced by an external Raman laser field. These probabilities are computed over multiple relevant energy ranges within the pseudopotential approximation. Strong scattering resonances are found whenever the incident energy approaches either a scattering threshold or a quasibound state attached to one of the energetically closed higher dispersion branches. A striking difference is demonstrated between two very different regimes set by the Raman laser intensity, namely, between scattering for the single-minimum dispersion versus the double-minimum dispersion at the lowest threshold. The presence of RD-SOC together with the Raman field fundamentally changes the scattering behavior and enables the realization of very different one-dimensional theoretical models in a single experimental setup when combined with a confinement-induced resonance.

Figure 1

(a) The total spectrum of states (the relative energy, $E$, vs the wave vector, $k$, where $k^{\prime }$ is the real part and $k^{\prime \prime }$ is the imaginary part of $k$) in the single-minimum regime of the center-of-mass frame, i.e., obtained by setting $K=0$. The parameters used are $\hslash \mathrm{\Omega }=5{\hslash }^2{\lambda }^2/\left(2\mu \right)$ and $\hslash \delta =0{\hslash }^2{\lambda }^2/\left(2\mu \right)$. The thick curves represent the real wave-vector solutions in different dispersion branches shown in thick red, thick green, and thick blue (with solid circles on top of it). When energy goes below the scattering thresholds, application of the analytical continuation gives us the thin curves, which stand for either the purely imaginary wave vectors (thin red, thin green, and thin blue with solid circles) or the complex ones (thin purple with diamonds). (b) A companion to plot (a). The solid (dashed) lines represent the real (imaginary) part of the wave vector, $k$.

Figure 2

(a) The total spectrum of states in the double-minimum regime in the center-of-mass frame. Parameters used are $\hslash \mathrm{\Omega }=1{\hslash }^2{\lambda }^2/\left(2\mu \right)$ and $\hslash \delta =0{\hslash }^2{\lambda }^2/\left(2\mu \right)$. The thick curves represent real wave-vector solutions (thick red, thick green, and thick blue with solid circles). Besides the thin green curve, which is identical to the one in Fig. 1, the purely imaginary solutions (thin red and thin blue with solid circles) now live between the real solutions. The complex solutions are represented by the thin purple lines with diamonds. (b) A companion to figure (a). The solid (dashed) lines represent the real (imaginary) part of the wave vector, $k$.

Figure 3

The reflection probability and occupation probability as a function of the relative energy, $E$, given the initial state in the blue dispersion branch ($i=6$). The gray dashed vertical lines label the scattering thresholds. When energy goes below a scattering threshold, one branch becomes a closed channel. Therefore, the state becomes evanescent and the amplitude modulus is interpreted as a measure of the occupation probability in the closed channel, which is represented by the dashed curve. Some curves have been rescaled to make them more clearly visible. The multiplication factors are labeled above those curves. The parameters for these calculations are $\hslash \mathrm{\Omega }=5{\hslash }^2{\lambda }^2/2\mu$, $\delta =0$, and $g_{\text{1D}}=-\hslash \lambda /m$.

Figure 4

The reflection probability and occupation probability are plotted as a function of the relative energy, $E$, given the initial state in the green dispersion branch ($i=2$). The gray dashed vertical lines label the scattering thresholds. When energy decreases below a scattering threshold, one branch becomes energetically closed. Therefore, the state becomes evanescent and the amplitude modulus is interpreted as the occupation probability in the closed channel, which is represented by the dashed curve. Some curves are scaled to be clearly visible. The multiplication factors are labeled above those curves. The parameters used here are $\hslash \mathrm{\Omega }=5{\hslash }^2{\lambda }^2/2\mu$, $\delta =0$, and $g_{\text{1D}}=-\hslash \lambda /m$.

Figure 5

The reflection probability is shown as a function of the relative energy, $E$, for an initial state indicated by the green dispersion branch ($i=4$). The parameters for this example are $\hslash \mathrm{\Omega }=5{\hslash }^2{\lambda }^2/2\mu$, $\delta =0$, and $g_{\text{1D}}=-\hslash \lambda /m$.

Figure 6

The reflection probability and occupation probability as a function of the relative energy, $E$, given the initial state in the blue dispersion branch ($i=6$). The occupation probabilities diverge at the branch point, which is labeled by the dot-dashed purple line. The gray dashed vertical lines label the scattering thresholds. When energy goes below a scattering threshold, one branch becomes closed. Therefore, the state becomes evanescent and the amplitude modulus is interpreted as the occupation probability, which is represented by the dashed curve. Some curves are scaled to be clearly visible. The multiplication factors are labeled near those curves. The parameters used here are $\hslash \mathrm{\Omega }=1{\hslash }^2{\lambda }^2/2\mu$, $\delta =0$, and $g_{\text{1D}}=-\hslash \lambda /m$.

Figure 7

Resonance position as a function of $g_{\text{1D}}$ in the energy range $[-\hslash \mathrm{\Omega },0]$, where only one channel is open. (a) The dashed orange, green, and blue lines label the lowest scattering threshold for each case. Parameters used are $\hslash \mathrm{\Omega }=2$ (orange, top), 4 (green, middle) and 8 (blue, bottom) in the unit of ${\hslash }^2{\lambda }^2/2\mu$. (b) Parameters used are $\hslash \mathrm{\Omega }=0.5$ (green, top) and 1.0 (blue, bottom) in the unit of ${\hslash }^2{\lambda }^2/2\mu$. (c) Bound-state spectra for $\hslash \mathrm{\Omega }=1$ (blue, top) and 4 (green, bottom) in the unit of ${\hslash }^2{\lambda }^2/2\mu$.

Figure 8

The reflection probability in the double-minimum regime when the incoming state has a negative wave vector but a positive flux current ($i=4$). The parameters used in this example are $\hslash \mathrm{\Omega }={\hslash }^2{\lambda }^2/2\mu$, $\delta =0$, and $g_{\text{1D}}=-\hslash \lambda /m$. Dashed gray lines are the scattering thresholds.