Stabilizing an atom laser using spatially selective pumping and feedback

We perform a comprehensive study of stability of a pumped atom laser in the presence of pumping, damping, and outcoupling. We also introduce a realistic feedback scheme to improve stability by extracting energy from the condensate and determine its effectiveness. We find that while the feedback scheme is highly efficient in reducing condensate fluctuations, it usually does not alter the stability class of a particular set of pumping, damping, and outcoupling parameters.

Figure 1

General behavior of the condensate and outcoupled fields. Shown are the condensate density over a $2\mathrm{s}$ period (top) and the outcoupled field density over the same period (bottom). Note that the fluctuations of the untrapped field follow those of the trapped field. System parameters were $a=1.0\times {10}^{-10}$, $\sigma =9.0\times {10}^{-6}\mathrm{m}$, $r=3.7\times {10}^8{\mathrm{s}}^{-1}$.

Figure 2

Increasing the nonlinear interaction leads to stability. Left, trapped field density; right, untrapped field density. Darker regions indicate higher densities. From top to bottom the scattering lengths are $a=0$, $1.0\times {10}^{-10}$, and $a=4.65\times {10}^{-10}\mathrm{m}$. Other parameters were $\sigma =9.0\times {10}^{-6}\mathrm{m}$, $r=3.7\times {10}^8{\mathrm{s}}^{-1}$.

Figure 3

Top: Central density of the condensate over a $2\mathrm{s}$ period. Bottom: The associated frequency power spectrum over the periods $1.0–1.5\mathrm{s}$ (solid line) and $1.5–2.0\mathrm{s}$ (dashed line).

Figure 4

(Color online) The effect of the pumping rate on the stability of the laser when feedback is not present. Above the upper line in each plot the laser is absolutely unstable; below the lower line it is absolutely stable. Between the lines only certain modes are stable. Shown are pumping rates of $r=3.0\times {10}^7$ (top left), $1.0\times {10}^8$ (top right), $6.0\times {10}^8$ (bottom left), and $3.0\times {10}^9{\mathrm{s}}^{-1}$ (bottom right).

Figure 5

An example of a set of parameters where the attractor for the condensate is not the ground state, but rather the first excited state with energy $3∕2\hslash \omega$ per particle. The top figure shows the density distribution of the condensate over time, and the lower figure shows the energy per particle of the condensate in units of $\hslash \omega$. Parameters are $a=0$, $\sigma =10.0\times {10}^6$, and $r=3.7\times {10}^8$.

Figure 6

(Color online) The effect of feedback. Top row: central density of the condensate and its energy when no feedback is present. Bottom row: central density of the condensate and its energy when feedback is turned on at $t=0.3\mathrm{s}$. Energies are in units of $\hslash \omega$ per particle. Note that the ground-state energy for this system is $0.839\hslash \omega$ per particle after the particle number has reached steady state (calculated using an imaginary-time algorithm). Parameters used are $r=3.7\times {10}^8{\mathrm{s}}^{-1}$, $\sigma =9.0\times {10}^{-6}\mathrm{m}$, $a=4.65\times {10}^{-11}\mathrm{m}$.

Figure 7

(Color online) How adding feedback improves stability. Dashed line is the boundary of stability with feedback; solid line the boundary of stability without feedback. Pumping rate was $r=6.0\times {10}^7{\mathrm{s}}^{-1}$. Note that the stable-unstable boundary indicates where at least one higher mode begins to grow without limit, and therefore corresponds to the lower lines of the plots in Fig. 4.