Theoretical study of a cold-atom beam splitter

A theoretical model is presented for the study of the dynamics of a cold atomic cloud falling in the gravity field in the presence of two crossing dipole guides. The cloud is split between the two branches of this laser guide, and we compare experimental measurements of the splitting efficiency with semiclassical simulations. We then explore the possibilities of optimization of this beam splitter. Our numerical study also gives access to detailed information, such as the atom temperature after the splitting.

Figure 1

Schematic view of the guiding setup. The initial ${}^{87}\mathrm{Rb}$ cloud is shown at $z=0$. When the magneto-optic trap is switched off, a vertical laser beam is switched on and the cloud, partially trapped by the associated dipole force, falls in the gravity field. At time $t_0$, a second oblique guide is switched on. The two guides cross at the height $z_c$ and form an angle $\gamma$. The trapping potential seen by the atoms at two different heights is shown in the insets. Finally, the atoms are probed $1\mathrm{cm}$ below the initial position.

Figure 2

(Color online) (a) Classical trajectories of a cold atom falling in the gravity field in the presence of the two trapping potentials with $U_0=30\mu \mathrm{K}$, $U_1=10\mu \mathrm{K}$, $w_0=0.2\mathrm{mm}$, and $w_1=0.3\mathrm{mm}$. The three trajectories correspond to an initial height $z_0=0$ and initials positions $x_0=-0.2\mathrm{mm}$ (black solid line), $x_0=0$ (red dashed line), and $x_0=+0.2\mathrm{mm}$ (blue dash-dotted line) with zero initial momentum (${\dot{z}}_0=0$ and ${\dot{x}}_0=0$). The two guides cross at the height $z_c=-4\mathrm{mm}$ with an angle $\gamma =0.12\mathrm{rad}$, and the oblique guide is switched on at $t_0=28.6\mathrm{ms}$. At this time, a free fall dynamics would give the position $z_{\mathrm{ff}}\left(t_0\right)=z_0-{\dot{z}}_{0}t-\frac{1}{2}g{t}_0^2=-4\mathrm{mm}$ corresponding exactly to $z_c$. Unless specified, these laser parameters remain fixed throughout the paper. The thin vertical and oblique dotted lines reveal the geometry of the laser beams. (b) Deviation of the trajectories from a free fall dynamics in the $z$ direction for the same initial conditions as in (a).

Figure 3

(Color online) One-dimensional (black solid line) and two-dimensional (red dashed line) trapping probability as a function of the dimensionless ratio $U_0∕k_B{T}_0$ (logarithmic scale) for ${\sigma }_0∕w_0=1.5$. The blue point located at the abscissa $U_0∕k_B{T}_0=1.3$ with the error bar is an experimental measurement extracted from Ref. 38. The one-dimensional trapping probabilities for ${\sigma }_0∕w_0=0.5$ and 2.5 are also shown as blue and green dotted lines with the labels (1, 2), respectively. All other parameters are as in Fig. 2.

Figure 4

(Color online) Initial population of the various vibrational levels $P\left(v_0\right)$ normalized with respect to the population of the ground state $P\left(0\right)$ as a function of the ratio of their energy ${\epsilon }_0$ to $U_0$. The black solid, red dashed, and blue dash-dotted lines correspond to ${\sigma }_0∕w_0=1.5$, 0.5, and 2.5, respectively. All other parameters are as in Fig. 2. The energies of the levels $v_0=0$, 2000, 4000, 6000, 8000, 10 000, and 11 704 are indicated by the thin vertical dotted lines.

Figure 5

(Color online) Contour plot depicting the time evolution of the envelope of ${\mid \phi (x,t)\mid }^2$ on a logarithmic scale as a function of $x$ and of $z\left(t\right)=z_0-{\dot{z}}_{0}t-\frac{1}{2}g{t}^2$. The atom is falling in the gravity field in the presence of the two trapping potentials with $U_0=30\mu \mathrm{K}$, $U_1=10\mu \mathrm{K}$, $w_0=0.2\mathrm{mm}$, and $w_1=0.3\mathrm{mm}$. The two guides cross at the height $z_c=-4\mathrm{mm}$ with an angle $\gamma =0.12\mathrm{rad}$, and the oblique guide is switched on at $t_0=28.6\mathrm{ms}$. The vertical and oblique dotted white lines reveal the directions of propagation of the laser beams. The two white arrows point to the location of the maximum probability density at the height $z=-10\mathrm{mm}$. The initial level here is $v_0=6000$.

Figure 6

(Color online) State-dependent splitting efficiency as a function of the initial level $v_0$. Upper graph (a): the probability $P_R(v_0,0,0)$ of finding the atom in the oblique guide at the end of the propagation is shown as a black solid line for the initial condition $z_0={\dot{z}}_0=0$. The same probability averaged over $z_0$ and ${\dot{z}}_0$, $⟨P_R⟩\left(v_0\right)$, is shown as a red dashed line. Lower graph (b): The averaged splitting probability $P_s\left(v_0\right)$ scaled by the same factor $P\left(0\right)$ as in Fig. 4 is shown as a function of $v_0$ for various sizes of the atomic cloud ${\sigma }_0$ and for $T_0=14\mu \mathrm{K}$. The cloud sizes ${\sigma }_0=0.15$, 0.30, and $0.45\mathrm{mm}$ correspond to the solid black, dashed green, and dash-dotted blue curves, respectively. These probabilities have been averaged over the initial classical conditions chosen for $z_0$ and ${\dot{z}}_0$. The guide parameters are as in Fig. 5. The two vertical lines indicate the average value of $v_0$ in the oblique guide for ${\sigma }_0=0.15\mathrm{mm}$ (black solid line), $⟨v_0⟩=7750$, and ${\sigma }_0=0.45\mathrm{mm}$ (blue dash-dotted line), $⟨v_0⟩=8140$.

Figure 7

Total splitting efficiency $P_s$ [Eq. (22)] as a function of the crossing height $z_c$ of the laser beams. This probability has been averaged over the initial conditions chosen for $z_0$ and ${\dot{z}}_0$, and summed over $v_0$. The size of the initial atomic cloud is ${\sigma }_0=0.30\mathrm{mm}$. All other parameters are as in Fig. 6.

Figure 8

(Color online) Total splitting efficiency $P_s$ [Eq. (22)] for the initial conditions $z_0={\dot{z}}_0=0$ as a function of the ratio of the oblique to vertical potential depths $U_1∕U_0$, and therefore as a function of the ratio of the beam intensities. For this calculation, $U_0$ is fixed $\left(30\mu \mathrm{K}\right)$ and $U_1$ is varied. The crossing height between the two guides is $z_c=-4\mathrm{mm}$. The waist of the vertical beam is $w_0=0.2\mathrm{mm}$, and the splitting efficiencies calculated for an oblique waist of $w_1=0.1$, 0.2, and $0.3\mathrm{mm}$ are shown as black solid, red dashed, and blue dash-dotted lines, respectively. All other parameters are as in Fig. 7.

Figure 9

(Color online) Average energies $⟨E_0⟩$ (black solid line) and $⟨E_1⟩$ (red dashed line) in the vertical and oblique guides as a function of $U_1∕U_0$ for the initial conditions $z_0={\dot{z}}_0=0$. For this calculation, $U_0$ is fixed $\left(30\mu \mathrm{K}\right)$ and $U_1$ is varied. The crossing height between the two guides is $z_c=-4\mathrm{mm}$. The waists of the vertical and oblique beams are $w_0=0.2\mathrm{mm}$ and $w_1=0.3\mathrm{mm}$. All other parameters are as in Fig. 7. The total average energy $⟨E⟩$ is also shown as a green dash-dotted line.