Bound states in the continuum in spin-orbit-coupled atomic systems

We show that the interplay between spin-orbit coupling and Zeeman splitting in atomic systems can lead to the existence of bound states in the continuum (BICs) supported by trapping potentials. Such states have energies falling well within the continuum spectrum, but nevertheless they are localized and fully radiationless. We report the existence of BICs, in some cases in exact analytical form, in systems with tunable spin-orbit coupling and show that the phenomenon is physically robust. We also found that BIC states may be excited in spin-orbit-coupled Bose-Einstein condensates, where under suitable conditions they may be metastable with remarkably long lifetimes.

Figure 1

(a) BIC energy (red dots) and the discrete spectrum ${\mu }_m$ of (3) with $\nu =0.7$ vs SOC strength. The gray region corresponds to the continuous spectrum. (b) Integral widths of the BIC $w_{\mathrm{bic}}$ and of the states of the discrete spectrum $w_m$ vs SOC strength. The vertical dashed lines indicate the value $\gamma =1/\nu$, beyond which the potential (3) ceases to exist. Note that even though at $\gamma =0$ the localized mode of the system can be found, it is not a BIC anymore, because in this limit the two components of the spinor become decoupled and have independent spectra. All quantities in this and subsequent figures are plotted in dimensionless units.

Figure 2

(a) The potential (3) and the BIC (4) at $\gamma =0.5$ and $\nu =0.7$. (b) State with oscillating tails in $V_{\mathrm{pert}}$ with $v_0=0.1$ at $\gamma =0.5$ and $\nu =0.7$. (c) BIC in a perturbed potential obtained for $\gamma =0.723$. (d) Usual guided mode supported by the designed potential at $\gamma =1$. The dashed lines in all panels show the inverted potential $-V\left(x\right)$.

Figure 3

(a) Lifetime of dynamically excited states vs SOC strength for the potential (3) where the BIC exists at $\gamma =0.5$ and $\nu =0.7$ (black circles), and for the perturbed potential $V_{\mathrm{pert}}\left(x\right)$ with $v_0=0.05$ (red circles) and $v_0=0.1$ (green circles) in the linear case. (b) Lifetime in the potential (3) with $\gamma =0.5$ and $\nu =0.7$ vs nonlinearity strength $g$.

Figure 4

(a) Excitation of a BIC in the potential designed to support it at $\gamma =0.5$ and $\nu =0.7$. (b) Slow decay of a leaky state in the potential used in (a) when $\gamma =0.7$. (c) Simultaneous excitation of a coexisting BIC and standard guided modes in the potential (3) with $\gamma =1.3$ and $\nu =0.7$, leading to beatings. The evolution is shown within the window $x\in [-20,20]$ for propagation up to $t={10}^4$ in panels (a) and (b) and up to $t=200$ in panel (c).

Figure 5

(a) Evolution of the peak amplitude of the ${\psi }_1$ component when the input is a Gaussian wave packet in the potential (3) for $g=-2$, $\gamma =0.5$, and $\nu =0.7$. Notice the logarithmic scale of the vertical axis. Evolution of the ${\psi }_1$ component of the perturbed nonlinear BIC in the potential (7) for $\gamma =1.4$ (b) and $\gamma =0.2$ (c) for $g=1$ and $\nu =0.7$