Near-resonant optical forces beyond the two-level approximation for a continuous source of spin-polarized cold atoms

According to the notations in Fig. 2, the Hamiltonian of the atom in the rotating frame is, in the $\left\{\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle\right\}$ basis,

By introducing the jump operators $\sigma_{12}=\left|1\right\rangle\left\langle 2\right|$ and $\sigma_{23}=\left|2\right\rangle\left\langle 3\right|$, the atom-light interaction can be modeled with the following Hamiltonian:

where the optical phases are assumed to be constant in time so that the Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$ are real numbers. The Hamiltonian that describes the internal energy of an atom coupled to the two optical fields is thus $V=H_{\rm at}+H_{\rm int}$.

The internal state evolution of the system is governed by the following master equation:

where the Lindblad superoperators that model the spontaneous emission processes are

and $\left\{A,B\right\}=AB+BA$ is the anticommutator. Using this master equation, the optical Bloch equations for a $\Xi$-system are derived in App. A.1.

In the end, we are interested in the external motion of the atom which occurs at timescales much larger than the internal state evolution. We thus assume that the internal atomic state is always in the steady state $\rho_{\infty}\equiv\lim_{t\rightarrow\infty}\rho$. The steady state satisfies $\partial_{t}\rho_{\infty}=0$ and its value is derived in App. A.2.

The dressed states are defined as the eigenstates of the Hamiltonian $V$, which admits the following eigendecomposition:

where $\widetilde{V}=\mathrm{diag}(\widetilde{E}_{\alpha},\widetilde{E}_{\beta},\widetilde{E}_{\gamma})$ is diagonal in the basis $\left\{\left|\alpha\right\rangle,\left|\beta\right\rangle,\left|\gamma\right\rangle\right\}$ called the dressed states basis. $U$ is the unitary transformation between the bare and dressed states basis $\left\{\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle\right\}\leftrightarrow\left\{\left|\alpha\right\rangle,\left|\beta\right\rangle,\left|\gamma\right\rangle\right\}$ Whitley and Stroud (1976).

A knowledge of the unitary transformation $U$ allows us to define the jump operators for the dressed states as $\widetilde{\sigma}^{\dagger}=U\sigma^{\dagger}U^{\dagger}$. Similarly, the density matrix in the dressed basis can be obtained from the solution $\rho$ of the Bloch equations in the bare basis: $\widetilde{\rho}=U\rho U^{\dagger}$.

The main advantage of the dressed-atom description is that when the system is strongly driven, the secular approximation applies and the equation of motion for the internal degrees of freedom reduce to rate equations, as shown in App. B.1. The decay rate between the dressed states $\left|k\right\rangle$ and $\left|k^{\prime}\right\rangle$ is

The rate equations are

where $\widetilde{\bm{\rho}}=\left(\widetilde{\rho}_{\alpha\alpha},\widetilde{\rho}_{\beta\beta},\widetilde{\rho}_{\gamma\gamma}\right)^{T}$ is the vector of the dressed state populations and the rate matrix is

The stationary populations $\widetilde{\bm{\rho}}^{\infty}$ are obtained by solving the system $\bm{\Gamma}\widetilde{\bm{\rho}}^{\infty}=\mathbf{0}$ with $\left\|\widetilde{\bm{\rho}}^{\infty}\right\|_{1}=1$.

If the Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$ are functions of the spatial position coordinates $\mathbf{r}$, as expected for Gaussian optical beams, then the overall Hamiltonian of the atom, including both the internal and external degrees of freedom, is

where $\mathbf{p}$ is the atomic momentum operator and $m$ is the mass of a single atom.

The dipole force operator can be defined from the Heisenberg equation of motion for the momentum operator Dalibard and Cohen-Tannoudji (1985):

Over a time interval long compared to the lifetimes $1/\Gamma_{1}$ and $1/\Gamma_{2}$, the mean force experienced by an atom at position $\mathbf{r}$ is obtained from the average for the steady internal state,

Note that $\left\langle\mathbf{F}(\mathbf{r})\right\rangle\neq-\nabla\left\langle V(\mathbf{r})\right\rangle$ since the density operator $\rho_{\infty}$ depends on the position $\mathbf{r}$.

Since the trace of an operator is independent of the basis choice, the mean dipole force can be written using the steady state density matrix in the dressed states basis: $\left\langle\mathbf{F}\right\rangle=\mathrm{Tr}\left(\mathbf{F}\rho_{\infty}\right)=\mathrm{Tr}\left(U\mathbf{F}U^{\dagger}U\rho_{\infty}U^{\dagger}\right)=\mathrm{Tr}(\widetilde{\mathbf{F}}\widetilde{\rho}_{\infty})=\langle\widetilde{\mathbf{F}}\rangle$. In that case, a dipole force can be associated with each dressed state,

where $\left\langle\widetilde{\mathbf{F}}_{k}\right\rangle=-\widetilde{\rho}^{\infty}_{kk}\nabla\widetilde{E}_{k}$. The potential can then be obtained by integrating the force.

The diffusion coefficient associated with the radiation pressure at the wavelength $\lambda_{k}$ ($k=1,2$) is

where $k_{R}$ is the single photon recoil momentum at $\lambda_{k}$, $m$ is the atomic mass, and the scattering rate is proportional to the decay rate and to the population in the bare state $\left|k\right\rangle$: $\Gamma_{S}^{(k)}(\mathbf{r})=\Gamma_{k}\rho^{\infty}_{kk}(\mathbf{r})$.

If the diffusion coefficient in a given spatial direction is desired, a prefactor must be applied depending on the polarization of the considered transition. For example, for the isotropic diffusion due to $s$-wave scattering the prefactor is $1/3$ for all directions, whereas for a $\sigma^{+/-}$ transition a factor of $1/4$ must be applied for the two directions orthogonal to the dipole axis and a factor of $1/2$ for the last direction.

Since the internal atomic state changes at random times, the dipole force fluctuates around its mean value [Eq. (12)]. From the rate equations of the dressed states populations [Eq. (7)], we see that the stochastic evolution of the dipole force $\{\widetilde{\mathbf{F}}_{t},t\geq 0\}$ is a continuous time Markovian process with infinitesimal generator $\mathbf{\Gamma}$ given by Eq. (8).

The diffusion coefficient is given by Ref. Dalibard and Cohen-Tannoudji (1985):

Since the process is Markovian, we show in App. B.2 that the diffusion coefficient along a direction $\varepsilon\in\{x,y,z\}$ satisfies the following relation:

where $\widetilde{\mathbf{F}}^{(\varepsilon)}=(\widetilde{\mathbf{F}}^{(\varepsilon)}_{\alpha},\widetilde{\mathbf{F}}^{(\varepsilon)}_{\beta},\widetilde{\mathbf{F}}^{(\varepsilon)}_{\gamma})^{T}$ is the vector of the forces in the different dressed states with $\widetilde{\mathbf{F}}^{(\varepsilon)}_{k}=-(\nabla\widetilde{E}_{k})_{\varepsilon}$, $\mathbf{\Gamma}^{\sharp}$ is the group inverse of $\mathbf{\Gamma}$, and the scalar product is defined as $\left\langle\mathbf{x},\mathbf{y}\right\rangle=\sum_{i}\widetilde{\rho}^{\infty}_{ii}x_{i}y_{i}$. This formula allows us to calculate the diffusion coefficient of any system described by a set of rate equations (Markovian process). Moreover, it provides an efficient way to perform a numerical estimation.

A closed $\Xi$-system within the ⁸⁷Rb structure is realized using the $5^{2}\mathrm{S}_{1/2}\rightarrow 5^{2}\mathrm{P}_{3/2}$ transition at 780.24 nm and the $5^{2}\mathrm{P}_{3/2}\rightarrow 4^{2}\mathrm{D}_{5/2}$ transition at 1529.37 nm Lee et al. (2007). Therefore, it realizes a very good approximation of the theoretical model developed above. Note that this idea can be generalized to other alkali-metal atoms since they have analogous structures. Furthermore, the choice of the 1529 nm wavelength is appealing since laser technologies developed for optical telecommunications can be used.

The use of $\sigma^{+}$ polarized electromagnetic fields both optically pumps the sample and drives closed transitions between the Zeeman substates $\left|F=2,m_{F}=2\right\rangle\rightarrow\left|F^{\prime}=3,m^{\prime}_{F}=3\right\rangle$ and $\left|F^{\prime}=3,m^{\prime}_{F}=3\right\rangle\rightarrow\left|F^{\prime\prime}=4,m^{\prime\prime}_{F}=4\right\rangle$, as depicted in Fig. 3 (a). Note also that, with this choice of hyperfine states, a decay into the $5^{2}\mathrm{P}_{1/2}$ state is forbidden ($\Delta F>1$), as shown in Fig. 3 (b). Such a configuration is thus equivalent to the $\Xi$ system sought for. Moreover, since an atom in $\left|F=2\right\rangle$ loaded inside the guide cannot decay into $\left|F=1\right\rangle$ after a spontaneous emission event the atom will remain in the guided state, thus avoiding its loss during propagation.

We analyze the properties of the guide obtained in the following specific configuration. The 780 nm beam has a waist of 1 mm and an intensity of $1000$ mW/cm² (the optical power is thus 16 mW), blue detuned by $\Delta_{1}=-2\pi\times 485$ MHz from the $\left|F=2\right\rangle\rightarrow\left|F^{\prime}=3\right\rangle$ transition and $\Gamma_{1}=2\pi\times 6.065$ MHz Steck (2001).

The beam at 1529 nm has a waist of 300 $\mu$m and thus a Rayleigh range of 18 cm. Its intensity is $3\times 10^{5}$ mW/cm² (a power of 420 mW), and the detuning is $\Delta_{2}=+2\pi\times 133$ MHz to the red of the $\left|F^{\prime}=3\right\rangle\rightarrow\left|F^{\prime\prime}=4\right\rangle$ transition. The transition linewidth is $\Gamma_{2}=2\pi\times 2$ MHz Heavens (1961); Moon et al. (2007).

From these parameters, we determine the steady state density matrix versus spatial position using the results in App. A.2, and estimate the dipole potential after integrating the dipole force [Eq. (12)]. The results are presented in Fig. 4.

We see in Fig. 4 (a) that, for an atom in $\left|F=2\right\rangle$, the potential has a double barrier shape, creating a pipe-like potential due to the cylindrical symmetry of the problem. The guide depth is 365 $\mu$K. Conversely, for an atom in $\left|F=1\right\rangle$, the potential is flat at the barrier position. The proposed configuration thus implements the state-dependent potential necessary to obtain the atom-diode effect.

Nevertheless, for the atom-diode effect to occur an atom in $\left|F=1\right\rangle$ must first pass through the barrier and then be repumped into $\left|F=2\right\rangle$, after passing the barrier maximum. In other words, an atom in $\left|F=1\right\rangle$ should not scatter more than a few photons between the moment when it enters the 780 nm beam and the moment when it reaches the position of the barrier maximum. From Fig. 4 (b), we see that the barrier width is about 0.1 mm wide and the scattering rate is below 1 kHz before the barrier maximum. As a consequence, an atom with a velocity of $\sim$ 10 cm/s scatters less than one photon while crossing the barrier, resulting in a low probability for the atom to be repumped. Conversely, the scattering rate of an atom in $\left|F=1\right\rangle$ strongly increases at the guide center and the atom will be pumped into the trap sensitive state, thus realizing the atom-diode based loading. This effect is studied in details using a numerical simulation in Sec. V.6.2.

By creating a differential light-shift, the 1529 nm radiation not only creates the guide barriers but also reduces the scattering rates, and thus the diffusion coefficients due to photon recoil, at 780 nm and 1529 nm by about two orders of magnitude [Fig. 4 (c)]. This effect strongly lowers the heating rate of the guided sample.

Nevertheless, the heating rate due to the recoils cannot be made arbitrarily small. The reason is that the diffusion due to the dipole force fluctuation increases with the intensity at 1529 nm because of the enhanced light-shift. A compromise must thus be found between the spontaneous emission rate and the dipole force diffusion. In the present configuration, the parameters are adjusted so that the diffusion due to the recoil is of the same order as the one induced by the dipole force fluctuation.

To transport the atoms, an extra force must be applied along the guide axis. By counterpropagating the beams at 780 nm and at 1529 nm, the scattering imbalance between the two beams pushes the atoms in a given direction with a force

where $k_{R}^{(k)}$ is the single-photon recoil impulsion at the wavelength $\lambda_{k}$ and $\mathbf{u}_{z}$ is the unit vector in the $z$ direction. The spatial distribution of the pushing force is depicted in Fig. 5. The velocity of an atom inside the guide is almost constant whereas the atom accelerates near the barrier.

Since the heating processes limit the lifetime of the guided sample, a compromise must be found between the transport velocity and the guided length: the atomic sample at the guide output should be cold – meaning that the output velocity and its dispersion should be of the order of a few meters per second – while the sample should be guided along a few centimeters. This distance is sufficient, for instance, to deliver atoms inside the magnetic shield required for any kind of high-performance atom interferometer, particularly since the MOT magnetic field is continuously operated.

To validate that the proposed configuration satisfies these requirements we numerically simulate the trajectories of atoms along the guide, solving a set of Langevin equations. The statistical analysis of the behavior over a large number of trajectories allows us to estimate the expected performances of the system in terms of guided distance and output velocity distribution. Moreover, the loading process is included in the simulation to consider its influence on the guided sample.