Theory of a single-atom laser including light forces

We study a single incoherently pumped atom moving within an optical high-$Q$ resonator in the strong-coupling regime. Using a semiclassical description for the atom and field dynamics, we derive a closed system of differential equations to describe this coupled atom-field dynamics. For sufficiently strong pumping, the system starts lasing when the atom gets close to a field antinode, and the associated light forces provide for self-trapping of the atom. For a cavity mode blue detuned with respect to the atomic transition frequency, this is combined with cavity-induced motional cooling, allowing for long-term steady-state operation of such a laser. The analytical results for temperature and field statistics agree well with our earlier predictions based on quantum Monte Carlo simulations. We find sub-Doppler temperatures that decrease with gain and coupling strength, and can even go beyond the limit of passive cavity cooling. Besides demonstrating the importance of light forces in single-atom lasers, this result also gives strong evidence to enhance laser cooling through stimulated emission in resonators.

Figure 1

(a) Photon number $N$ (solid line) and population inversion $z$ (dashed line) for the steady state as a function of the particle’s position $x$. The parameters are $(\gamma ,\nu ,g,\Delta )=(10,20,100,200)\kappa$. (b) Stationary force acting on the atom $F$ (solid line) and corresponding potential $U$ (dashed line) for the same parameters. (c) Time evolution of the photon number $N$, particle position $x$, and particle momentum $p$ for the same parameters. (d) Cutout of (c) demonstrating the cooling mechanism (see text).

Figure 2

(a) Stationary photon number obtained from the rate equations as a function of the atomic position $x$ and the detuning $\Delta$. The parameters are $(\gamma ,\nu ,g)=(20,25,20)\kappa$. (b) Atomic upper-state population for the same parameters. (c) The dipole force $F$ is an odd function of $\Delta$.

Figure 3

(a) Optical potential depth for $(\gamma ,g)=(10,50)\kappa$. In each curve, the pumping rate $\nu$ is adjusted resulting in constant maximum photon number $N$, as indicated. (b) Position-averaged friction coefficient in arbitrary units for the same parameters as in (a). (c) Equilibrium temperature associated with the atomic center-of-mass motion. It can be lower than for free-space Doppler cooling, where $T_D=\hslash \gamma$. (d) Ratio of atomic kinetic energy and the light potential. For $N>1$, it drops well below unity when the cavity is far detuned from the atom indicating strong particle localization.

Figure 4

(a) Atomic temperature as a function of the maximum cavity photon number $N$ for different values of the pumping rate $\nu$. The other parameters are $(\gamma ,g)=(10,50)\kappa$ while the atom-field detuning $\Delta$ was continuously adjusted. (b) Corresponding ratio $E∕V$.

Figure 5

Atomic temperature as a function of the atom-field coupling constant $g$ for different values of the pumping rate where $(\gamma ,\Delta )=(5,250)\kappa$. The marks show the results obtained from Monte Carlo wave function simulations.

Figure 6

(a) Atomic temperature in units of $\hslash \kappa ∕k_B$. For $y>1∕2$, it can fall below one which is the limit of passive cavity cooling. (b) Minimum temperature (solid line) as a function of $y=\nu ∕\Delta$ and the parameter $a$ (dashed line). The latter determines the operating point of the system.