Tune-out wavelengths and landscape-modulated polarizabilities of alkali-metal Rydberg atoms in infrared optical lattices

Intensity-modulated optical lattice potentials can change sign for an alkali-metal Rydberg atom, and the atoms are not always attracted to intensity minima in optical lattices with wavelengths near the CO${}_2$ laser band. Here we demonstrate that such IR lattices can be tuned so that the trapping potential experienced by the Rydberg atom can be made to vanish for atoms in “targeted” Rydberg states. Such state-selective trapping of Rydberg atoms can be useful in controlled cold Rydberg collisions, cooling Rydberg states, and species-selective trapping and transport of Rydberg atoms in optical lattices. We tabulate wavelengths at which the trapping potential vanishes for the $ns$, $np$, and $nd$ Rydberg states of Na and Rb atoms and discuss advantages of using such optical lattices for state-selective trapping of Rydberg atoms. We also develop exact analytical expressions for the lattice-induced polarizability for the $m_z=0$ Rydberg states and derive an accurate formula predicting tune-out wavelengths at which the optical trapping potential becomes invisible to Rydberg atoms in targeted $l=0$ states.

Figure 1

Toy model of a Rydberg atom in one-dimensional lattice, where the vertical axis is the laser intensity $I\left(z\right)$. When the size of the Rydberg orbit $2z_e^0=\lambda /4$, the forces ${\mathbf{f}}_R$ and ${\mathbf{f}}_L$ acting on the localized “lumps” of electron density are equal in magnitude and the net force acting on the atom is zero: this is the tune-out scenario.

Figure 2

The modulation factor $\langle \cos \left(2k{z}_e\right)\rangle$ as a function of $n$ for $l=0$ states of Rb for various $\lambda$. At low $n$, $\langle \cos \left(2k{z}_e\right)\rangle \to 1$ and ${\alpha }_{nl{m}_z}^{\mathrm{lsc}}\left(\omega \right)\to {\alpha }_e\left(\omega \right)$. At higher $n$, ${\alpha }_r^{\mathrm{lsc}}$ oscillates around zero, and each of the wavelengths seen in the figure becomes a tune-out wavelength for an infinite number of $n$ states. The lower panel demonstrates the universal dependence of the factor $\langle \cos \left(2k{z}_e\right)\rangle$ from the upper panel on $n^2{a}_0/\lambda$.

Figure 3

Landscaping polarizabilities ${\alpha }_{ns}^{\mathrm{lsc}}\left(\omega \right)$ for $n=100$, 160, and 180 states of the Rb and Na atoms. The $\lambda$ axis is plotted in logarithmic scale to display the range and amplitude of the oscillations. Note that $\sim$10${}^4$ nm is the center of the CO${}_2$ laser band, and tune-out wavelengths can be found for all $n$ in optical lattices with $\lambda <{10}^4$ nm. Table tabulates the four largest tune-out wavelengths seen in this figure along with those for the $p$ and the $d$ states.

Figure 4

${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ for $l$ (rows), $m_z$ (columns) states for the Rb atom. Only $m_z>0$ are shown since ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ is identical for states with $+m_z$ and $-m_z$. For ($l,m_z)=(0,0)$, (1,0), (2,0), and (2,1) ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ oscillates and the angular coefficients in (11) come in with different signs. For the (1,1) and (2,2) states, it is always negative because the angular coefficients in (11) are all positive.

Figure 5

${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;X)$ for $l$ (rows), $m_z$ (columns) states for the Rb atom. Only $m_z>0$ are shown because ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;X)$ is a linear combination of ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ within the $l$ manifold when $z$ is chosen to be the quantization axis, and ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ are identical for states with $+m_z$ and $-m_z$ from Eq. (11). Note that ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;X)$ and ${\alpha }_{n,l,m_z}^{\mathrm{lsc}}(\omega ;Z)$ are the same for $l=0$.

Figure 6

Diagonal ${\alpha }_{100p_0}^{100p_0}(\omega ;X)$ (dashed orange) and off-diagonal ${\alpha }_{100p_{-1}}^{100p_1}(\omega ;X)$ (dashed blue) matrix elements of the landscaping polarizability matrix $\mathit{\alpha }(\omega ;X)$. The off-diagonal matrix element vanishes at several wavelengths.