Stability diagram of the collective atomic recoil laser with thermal atoms

We experimentally investigate cold thermal atoms in a single sidedly pumped optical ring resonator for temperatures between 0.4 and $9\mu \mathrm{K}$. The threshold for collective atomic recoil lasing (CARL) is recorded for various pump-cavity detunings. The resulting stability diagram is interpreted by simulating the classical CARL equations. We find that the stability diagram for thermal atoms shows the same asymmetry as observed for Bose-Einstein condensates in previous experiments. Whereas for condensates the asymmetry is well explained by a Dicke-type quantum model we here discuss a simplified classical model. It complements the quantum model and provides an intuitive explanation based on the change in the long-range atomic interaction with pump-cavity detuning.

Figure 1

Data for a condensate (black triangles), 200.000 atoms at 400 nK (red circles), and for a classical theory for 200.000 atoms at 400 nK (blue squares). The threshold definition for the BEC measurements is chosen as $30\%\pm 10\%$ atoms being in the first momentum state. For clouds at 400 nK, momentum states cannot be clearly separated anymore. An aspect ratio of the cloud of $1.4\pm 0.2$ was chosen as threshold both in experiment and in theory.

Figure 2

Left: For $\mathrm{\Delta }>0$ with increasing pump power, a part of the atoms reaches high velocities, whereas another part stays at zero velocity. As shown here for a situation with $\mathrm{\Delta }=1.25\mathrm{MHz}$ and a pump power of 40 mW, this happens in both (a) experiment and (b) theory. For $\mathrm{\Delta }<0$, this behavior does not develop. Instead, the atomic cloud is mainly broadened with increasing pump power [(c) and (d) for $\mathrm{\Delta }=-1.75\mathrm{MHz}$ and pump power of 95 mW]. Right: The experimental velocity values at the end of the pump period for $\mathrm{\Delta }>0$ are plotted against pump power [(e) blue squares] and compared to theoretical values (red circles). The data correspond to $2.8\times {10}^6$ atoms ($1.4\times {10}^6$ in the theory) at $9\mu \mathrm{K}$ for a detuning of 1.25 MHz. Error bars show the standard deviation for averaging between 4 and 15 data points and 4 simulations, respectively.

Figure 3

(a) Velocity distribution of the atoms for positive detuning after $50\text{−}\mu \mathrm{s}$ interaction time. Atoms slower than 0.2 m/s are marked in blue; faster atoms are marked in red. (b) The position distribution of the atoms in (a) at the beginning of the instability is shown (blue and red or dark and light gray) and compared to the optical potential (black). Atoms near the potential maxima are those that remain at zero velocity in the end.

Figure 4

(a) Dipole potential of a single atom in a ring resonator. If the laser frequency matches to a cavity resonance ($\mathrm{\Delta }=0$), the dipole potential has its maximum slope at the position of the atom. In the case of positive or negative detuning, the atom is located close to a minimum or maximum of its self-generated dipole potential. (b) A minimization of the potential energy occurs if a second atom could align at distance $\frac{\lambda }{2}$ in the case of positive detuning and $\frac{\lambda }{4}$ in the case of negative detuning. For these situations each atom has minimal energy in the dipole potential of the other (see also Fig. 5).

Figure 5

(a) Total potential energy for two atoms vs distance $\delta z=z_2-z_1$ for negative detuning (red or upper line) and positive detuning (green or lower line). For positive detuning, the optimal distances are $q(\lambda /2),q\in {\mathbb{N}}_0$. For these distances the scattered light fields interfere constructively. For negative detuning, the optimal distances are given by $(2q+1)(\lambda /4)$. This leads to destructive interference between the scattered light fields. (b) Time evolution of positions $z_n(n=1,2)$ for $\mathrm{\Delta }>0$. Inset: The relative distance $\delta z$ oscillates around zero with a time-independent amplitude.

Figure 6

(a) Time evolution of positions $z_1$ (green), $z_2$ (red), and $z_3$ (black) for $\mathrm{\Delta }>0$. At time $t_1\approx 70$, one atom separates from the others. (b) The relative distance $\delta z=z_3-z_2$ stabilizes after $t_1\approx 70$ time units. Two atoms can bunch using the third one to take away their potential energy.

Figure 7

c.m. position in units of the optical wavelength $\lambda$ for $\mathrm{\Delta }=2\pi (\pm 3\mathrm{MHz})$. For negative detuning the pump field amplitude has to be larger to move the c.m. over distance $D$ than for negative detuning. The simulations were performed with three atoms.

Figure 8

Stability diagram of three atoms in the simplified model: The white dots mark the pump powers at which the c.m. moved over distance $D$ during a fixed interaction time and given detuning (see also Fig. 7).