Theory of the light potential

An atom in a near-resonant light field of frequency $\omega$ experiences both a conservative force (due to stimulated photon exchange) and a dissipative force (due to spontaneous photon emission)[82,50]. The conservative force is the gradient of a spatially-dependent potential $U_{\rm dip}{({\mathbf r})}$ which can be viewed as the time-averaged induced dipole interaction energy (proportional to the real part of the classical polarizability) in the electric field, or equally well as the `light shift' (that is, energy level shift due to the ac Stark effect) of the atomic ground state [2,85].

We assume that we apply a monochromatic light field of detuning $\Delta \equiv \omega - \omega_0$ to an alkali atom (with the $n\,s \rightarrow n\,p$ transition resonant at $\omega_0$), in the far-detuned regime ($\Delta$ greater than the excited state hyperfine splitting, but much less than $\omega_0$) and the low saturation regime ( $\Omega_{\rm rabi}\ll \Delta$, where the Rabi flopping rate $\Omega_{\rm rabi}$ is defined ^8.1 by $\hbar \Omega_{\rm rabi}\equiv \mu E_0$, the dipole matrix element multiplied by the electric field amplitude). The dipole potential has both a scalar and a magnetic part:

$$U_{\rm dip}{({\mathbf r})} = \beta_s \, \frac{\hbar \Gamma}{8} \, \frac{\Gamma}{\Delta} \, s{({\mathbf r})}\$$

$$+ \mu_{\rm Bohr}\, g_{\rm nuc}(L,S,J,i,F) m_F \vert{\bf H}{({\mathbf r})}\vert \,,$$ (8.1)

where $\Gamma$ is the spontaneous decay rate, $s{({\mathbf r})}$ is the spatially-dependent saturation parameter, and the potential is taken to be much less than the ground state hyperfine splitting. Only the magnetic part is affected by $m_F$, which is defined as the projection of the total angular momentum $F$ on the direction of the local effective magnetic field ${\bf H}{({\mathbf r})}$. The constant $g_{\rm nuc}$ is the nuclear Landé g-factor appropriate for the $F$ of the ground state. The scalar potential is identical to the case of a two-level atom, apart from the strength factor $\beta_s$ which is ${\textstyle\frac{1}{3}}$ for detuning from the D1 line, ${\textstyle\frac{2}{3}}$ for the D2 [85].

It is important to be precise with the definition of the saturation parameter. We write

$$s{({\mathbf r})}\equiv \frac{E_0^2{({\mathbf r})}}{\Es2} \,,$$ (8.2)

where $\Es2 \equiv 2 I_{\rm sat} / \epsilon_0 c$ (in the MKSA system) is the squared electric field amplitude in a plane wave of intensity $I_{\rm sat}$ (following definition in [85]). This avoids the ambiguities present with the usual definition $s{({\mathbf r})}\equiv I{({\mathbf r})}/I_{\rm sat}$ in the case of a general monochromatic light field (composed of an arbitrary coherent sum of travelling and evanescent waves), and emphasizes that it is the local electric field alone that causes the dipole potential.

The effective magnetic field has a strength and direction given by the circularly polarized component of the electric field[85], which can be written thus:

$$\mu_{\rm Bohr} {\bf H}{({\mathbf r})}\, = \, \beta_m \, \fra... ...{\mathbf r})}] \times \Im[{\bf E}_0{({\mathbf r})}]}{\Es2} \,,$$ (8.3)

where the strength factor $\beta_m$ is $-{\textstyle\frac{2}{3}}$ for D1, ${\textstyle\frac{2}{3}}$ for D2, and we take the physical electric field (with amplitude $E_0{({\mathbf r})}\equiv \vert{\bf E}_0{({\mathbf r})}\vert$) to be the real part of a complex field ${\bf E}({\bf r},t) \equiv {\bf E}_0{({\mathbf r})}\exp(-i \omega t)$. The reason for the `dummy' constant $\mu_{\rm Bohr}$ is aesthetic, so that (8.1) can be expressed in a standard magnetic form. Note that for nonzero $m_F$ the magnetic and scalar contributions to the potential are of similar order, if the fields have significant circular polarization (this will be true for our trapping fields, for the reason that the optical guide is close to cut-off).

The fact that $U_{\rm dip}{({\mathbf r})}$ has its sign controlled by the sign of the detuning allows both attractive (red-detuned) and repulsive (blue-detuned) potentials to be created. The potential scales as $I/\Delta$ but the spontaneous emission rate scales as $I/\Delta^2$; from this follows the well-known result that, if coherence time is an important factor, it is best to be far off-resonance and use high intensities in order to achieve the desired trap depth [85,2].

For simplicity, in this chapter we will restrict our further analysis and simulations to $m_F = 0$, although our initial calculations suggest that the effect of the magnetic part of our potential when trapping in other $m_F$ states will not pose major problems (assuming the spin axis adiabatically follows the ${\bf H}{({\mathbf r})}$ field direction), and can even be used to our advantage by increasing the depth and the transverse oscillation frequency $\omega_x$ in the case $m_F > 0$. Also, we will consider the effect of only a single resonance (choosing D2 because it has a larger $\beta_s$ than D1), which is a valid approximation when the detunings from this resonance are much less than the alkali atom fine structure splitting $\Delta_{\rm fs}$. Even when it becomes advantageous to use a large detuning of the order of $\Delta_{\rm fs}$, it is possible to cancel the effect of the other resonance by a simple shift in the two detunings (as we will see at the end of Section 8.3.1).

If we now have two light fields of differing frequency, the atomic potentials add[82,156], as long as we assume that the timescale of atomic motion is much slower than the beating period (that is, the inverse of the frequency difference). In our case, atomic motion occurs at $\sim 10^{5}$Hz and our light field frequency difference is $\sim 10^{15}$Hz, so this assumption is valid. Choosing equal but opposite detunings $\pm \Delta$ about the D2 line, the trapping potential for $m_F = 0$ is

$$U_{\rm dip}{({\mathbf r})}= \frac{2}{3} \, \frac{\hbar \Ga... ... [s_{\rm blue}{({\mathbf r})}-s_{\rm red}{({\mathbf r})}] \,,$$ (8.4)

written in terms of the saturation parameters for the two colors. The spontaneous scattering rate[2] is a factor $\Delta / \Gamma$ smaller than $U_{\rm dip}{({\mathbf r})}/ \hbar$ (in fact this relation applies for any $m_F$ state), but it depends on the sum of the saturation parameters rather than the difference. It is also spatially dependent, and has the form

$$\Gamma_{\rm scatt}{({\mathbf r})}= \frac{2}{3} \, \frac{\G... ... [s_{\rm blue}{({\mathbf r})}+ s_{\rm red}{({\mathbf r})}] \,.$$ (8.5)