Subwavelength-width optical tunnel junctions for ultracold atoms

We propose a method for creating far-field optical barrier potentials for ultracold atoms with widths that are narrower than the diffraction limit and can approach tens of nanometers. The reduced widths stem from the nonlinear atomic response to control fields that create spatially varying dark resonances. The subwavelength barrier is the result of the geometric scalar potential experienced by an atom prepared in such a spatially varying dark state. The performance of this technique, as well as its applications to the study of many-body physics and to the implementation of quantum-information protocols with ultracold atoms, are discussed, with a focus on the implementation of tunnel junctions.

Figure 1

Scheme for creating a subwavelength barrier. (a) Level diagram featuring a resonant probe field with a spatially uniform Rabi frequency $\mathrm{\Omega }$ and a resonant control field with spatially dependent Rabi frequency ${\mathrm{\Omega }}_{\mathrm{c}}\left(x\right)$. $\gamma$ is the linewidth of the excited state. (b) Spatial dependence of the probe and control Rabi frequencies. (c) Geometric scalar potential $U\left(x\right)$ of the dark state $|D\rangle$ features a potential barrier of height $E_w=1/\left(2m{w}^2\right)$ and subwavelength width $w$ determined by $\mathrm{\Omega }={\mathrm{\Omega }}_{\mathrm{c}}\left(w\right)$. The energies of the other two rotating-frame eigenstates $|+\rangle$ and $|-\rangle$ are shown schematically.

Figure 2

(a) Transmission ${\left|t\right|}^2$ and reflection ${\left|r\right|}^2$ probabilities for the geometric scalar potential $U\left(x\right)$ [Eq. (1)] as a function of the incoming energy $E$. (b) $|{\theta }^{\prime \prime }{\psi }_D+2{\theta }^{\prime }{\psi }_D^{\prime }|\sqrt{E_w/E}$ (red), its approximate form $1.27/{(1+x^2)}^{3/2}$ from Eq. (27) (black dashed), and $|{\psi }_{-}\left(x\right)|\mathrm{\Omega }/\sqrt{E{E}_w}$ (brown) for $\mathrm{\Omega }=20E_w$ and $E=E_w/50$. Excited state population can be determined via $P_e\sim {\left|{\psi }_{-}\left(0\right)\right|}^2\approx \frac{E{E}_w}{{\mathrm{\Omega }}^2}$, which agrees with Eq. (8), while the nonadiabaticity error is $P_{\mathrm{NA}}=\frac{k_2}{k}\left[|{\psi }_{-}{(-L)|}^2+{\left|{\psi }_{-}\left(L\right)\right|}^2\right]\approx 9\times {10}^{-5}$, which agrees with the approximate asymptotic formula [Eq. (29)] $p_{\mathrm{NA}}\approx 1.37\sqrt{E/E_w}{e}^{-1.75\sqrt{\mathrm{\Omega }/E_w}}\approx 8\times {10}^{-5}$.