Quantum Optics IX Optical Dipole Trap

In this article, we introduce the basic concepts of trapping neutral atoms using optical dipole potentials created by far-detuned laser light. In this regime, optical excitation is strongly suppressed, and the radiation-pressure force from photon scattering becomes negligible compared to the dipole (gradient) force.

Optical Dipole Trap

There are several common mechanisms that can trap cold atoms

1. Radiation-pressure trap (e.g. optical molasses)

   • Typical temperature: $\lesssim 10~\mu\mathrm{K}$ with sub-Doppler techniques
   • Limitations: the photon-recoil energy sets a lower bound on the temperature.

2. Magnetic trap

   • Principle: state-dependent force on the atomic magnetic dipole moment in an inhomogeneous magnetic field
   • Typical trap depth: on the order of $\sim 100~\mathrm{mK}$
   • Cooling: evaporative cooling
   • Limitation: only certain Zeeman sublevels (low-field seekers) can be trapped

3. Optical dipole trap

   • Principle: interaction of the induced electric dipole moment with far-detuned lightddd
   • Typical temperatures: $\lesssim 1~\mathrm{mK}$ when loaded directly from a MOT, and down to the $\mu\mathrm{K}$ regime after further cooling

Optical Tweezer

An optical tweezer is a tightly focused optical dipole trap, where the beam waist is on the order of the optical wavelength.

When an atom is illuminated by a laser light, the external electric field $\boldsymbol{E}$ induces an atomic dipole moment $\boldsymbol{d}$

$$\boldsymbol{d} = \alpha \boldsymbol{E} = -e\boldsymbol{R}$$

where $\boldsymbol{E}(\boldsymbol{x},t) = \boldsymbol{E}_0 \cos (\boldsymbol{k}\cdot \boldsymbol{x}-\omega t)$ is an oscillating field, $\alpha$ is the complex polarizability that may depend on the driving frequency $\omega$.

$$U = -\boldsymbol{d}\cdot \boldsymbol{E} = - \frac{1}{2} \mathrm{Re}(\alpha) |\boldsymbol{E}|^2 = - \frac{1}{c\varepsilon_0} \mathrm{Re}(\alpha) I^2$$

To evaluate the complex polarizability $\alpha$, we can use either semiclassical model or quantum model of light matter interaction.

Consider a two-level system without dissipation

$$H = \frac{1}{2} \hbar \omega_0 \sigma_z + \frac{1}{2}\hbar \Omega (\mathrm{e}^{\mathrm{i}\omega t} + \mathrm{e}^{-\mathrm{i}\omega t}) (\sigma^\dagger + \sigma)$$

The energy shift of ground state is

$$U=\frac{\Omega^2}{4\delta}$$

Plugging in $\Omega^2 = |\langle g | \boldsymbol{d} \cdot \boldsymbol{E} | e \rangle|^2 / \hbar^2 =\frac{|\boldsymbol{E}|^2}{\hbar^2} |\langle g | \hat{\boldsymbol{\epsilon}} \cdot \boldsymbol{d} | e \rangle|^2$, $I = \frac{1}{2} c \varepsilon_0 |\boldsymbol{E}|^2$ and $\Gamma = \frac{\omega_0^3}{3\pi\varepsilon_0 \hbar c^3}|\langle g | \hat{\boldsymbol{\epsilon}} \cdot \boldsymbol{d} | e \rangle|^2$, the dipole potential can be written as

$$U_\text{dip} = \frac{3\pi c^2 \Gamma}{2\omega_0^3} \frac{I}{\delta}$$

For a Gaussian beam trap

$$I = \frac{2P}{\pi w^2(z)} \exp\left[ -2 \frac{r^2}{w^2(z)} \right]$$

where $w(z) = w_0 \sqrt{1 + z^2/z_R^2}$ and $z_R$ is the Rayleigh length. At origin, the laser intensity can be approximated as

$$I \approx \frac{2P}{\pi w^2_0} \left(1 - \frac{z^2}{z_R^2} - 2 \frac{r^2}{w_0^2}\right)$$

Therefore, the dipole trap potential is

$$U_\text{dip} = \frac{3\pi c^2 \Gamma}{2\omega_0^3} \frac{1}{\delta} \frac{2P}{\pi w^2_0} \left(1 - \frac{z^2}{z_R^2} - 2 \frac{r^2}{w_0^2}\right) := U_0 \left(1 - \frac{z^2}{z_R^2} - 2 \frac{r^2}{w_0^2}\right)$$