Use of vector polarizability to manipulate alkali-metal atoms

Abstract

We review a few ideas and experiments that our laboratory in Korea University has proposed and carried out to use vector polarizability $\beta$ to manipulate alkali-metal atoms. $\beta$ comes from spin-orbit coupling, and it produces an ac Stark shift that resembles a Zeeman shift. When a circularly polarized laser field is properly detuned between the $D_1$ and $D_2$ transitions, an ac Stark shift of a ground-state atom takes the form of a pure Zeeman shift. We call it analogous Zeeman effect, and experimentally demonstrated optical Stern–Gerlach effect and an optical trap that behaves exactly like a magnetic trap. By tuning polarization of a trapping beam, and thereby controlling a shift proportional to $\beta$, we demonstrated elimination of an inhomogeneous broadening of a ground hyperfine transition in an optical trap. We call it magic polarization. We also showed significant narrowing of a Raman sideband transition at a specific well depth. A Raman sideband in an optical trap is broadened owing to anharmonicity of the trap potential, and the broadening can be eliminated by $\beta$-induced differential ac Stark shift at a magic well depth. Finally, we proposed and experimentally demonstrated a cooling scheme that incorporated the idea of velocity-selective coherent population trapping to Raman sideband cooling to enhance cooling efficiency of the latter outside the Lamb–Dicke regime. We call it motion-selective coherent population trapping, and $\beta$ is responsible for the selectivity. We include as a Supplementary Material a program file that calculates both scalar and vector polarizabilities of a given alkali-metal atom when wavelength of an applied field is specified. It also calculates depth of a potential well and photon-scattering rate of a trapped atom in a specific ground state when power, minimum spot size, and polarization of a trap beam are given.

Appendix A: Numerical calculation of scalar and vector polarizabilities

In the Excel file of the Supplementary Material, to obtain numerical values of $\alpha$ and $\beta$, we use the definitions in Eqs. (8) and (9), respectively, including the off-resonant terms. Those terms are not negligible when a trap beam is far detuned. When the principal quantum number of the ground state of an alkali-metal atom is n, we include couplings to only the $nP_{J^\prime }$ and $(n+1)P_{J^\prime }$ states with $J^\prime =1/2,3/2$. To obtain numerical values of a and b, we use Eqs. (19) and (20), respectively, and restrict the sums to $n^\prime =n$. Since we consider a far-detuned trap beam, either ground or excited state hyperfine splitting is not important, and effect of an isotope shift is negligible too.

There are two definitions for a reduced matrix element commonly used in literature. We follow that of Racah [11], according to which the Wigner–Eckart theorem is written as [12]

$$\begin{aligned} \langle n^\prime , J^\prime , m_J^\prime | d_q | n, J, m_J \rangle = (-1)^{J^\prime -m^\prime } \left( \begin{array}{ccc} J^\prime &{} 1 &{} J \\ -m_J^\prime &{} q &{} m_J \end{array} \right) \langle n^\prime J^\prime || d || n J \rangle , \end{aligned}$$

(A1)

and the reduced matrix element for an electric dipole moment between the $|n^\prime P_{J^\prime } \rangle$ and $|nS_{1/2} \rangle$ states of an alkali-metal atom is related to the lifetime $\tau (n^\prime P_{J^\prime })$ by

$$\begin{aligned} |\langle n^\prime P_{J^\prime } || d || n S_{1/2} \rangle |^2 = \frac{1}{\tau (n^\prime P_{J^\prime })} \frac{3\pi \epsilon _0\hbar c^3}{\omega _0^3}(2J^\prime +1), \end{aligned}$$

(A2)

where $\omega _0$ is the $|nS_{1/2} \rangle$ $\rightarrow$ $|n^\prime P_{J^\prime } \rangle$ transition frequency. For spectroscopic data, we use NIST Atomic Spectra Database Lines Data [13], where Einstein coefficient $A(n^\prime P_{J^\prime }) = 1/\tau (n^\prime P_{J^\prime })$ of the upper state $|n^\prime P_{J^\prime } \rangle$ and wavelength $\lambda _0 = 2\pi c /\omega _0$ are tabulated.

In the Excel file, for a given wavelength $\lambda$ of a trap beam, $\alpha , \beta$ and a, b are calculated in atomic units. For given power P, minimum spot size $w_0$ ($e^{-2}$ intensity radius), and degree of circularity $\eta$ of the trap beam, well depth $U_\textrm{AC}(F,m_F)$ for the ground hyperfine state $|nS_{1/2}, F, m_F \rangle$ according to Eq. (10) is given in mK:

$$\begin{aligned} U_{\mathrm{{AC}}} (F,m_F)= (\alpha -\eta \beta g_F m_F)(c\mu _0 I_0/2). \end{aligned}$$

(A3)

A photon-scattering rate $R_\gamma (F,m_F)$ according to Eq. (18) is given in s$^{-1}$:

$$\begin{aligned} R_\gamma (F,m_F)= (a-\eta b g_F m_F)(c\mu _0 I_0/2)\Gamma , \end{aligned}$$

(A4)

where we use $\Gamma = A(nP_{1/2})$ of the $D_1$ transition.