To illustrate the concept of the experiment, let's consider an atom at rest near the minimum of the potential (Fig. 1a). At time t=0 we apply a small sudden shift to the lattice. The atom will gain extra potential energy and start to oscillate (Fig.1b). At t=T/2 (half of oscillation period) the atom will be at its classical turning point (Fig. 1c). Now if we apply another sudden shift to the lattice the atom will lose the added potential energy.
Figure1
To achieve a real-time feedback let's extend this double shift concept into real time continuous reaction shift. In order to do that we need to know where, on average, the atoms are at any moment. The optical lattice provides us with such a tool to continuously and non-destructively measure the position of the atoms. As described previously, we can measure the electric dipole force averaged over ensemble of atoms. In our case we can approximate the potential wells as harmonic, and we know that the average displacement of atoms form ht equilibrium <Dx(t)> is small compared to the lattice well spacing. Then measurement of the power difference between the beams of the lattice leads directly to the measurement of <Dx(t)>.
Now let's place a phase modulator in each arm of the lattice, and constructed a feedback lop shown in Fig. 2.
Figure 2.
The feedback circuit has variable gain to allow us to control the size of the response shift. We also can invert the circuit's output. That gives us the capability to apply response shift to decrease or increase <Dx(t)> (negative or positive gain respectively). The experimental results for the varying gains of the feedback circuit are represented in Fig. 3.
Figure 3.
The first peak is omitted here for clarity. Note that for the case of "gain -1" the oscillations are damped efficiently, and are almost gone after 1 period (thus the designation of "gain -1" in analogy with classical control theory, the other gain settings are calibrated in reference to it). Application of bigger negative feedback (gains <-2) alter the motion of the atoms completely, and we observe long-lasting oscillation at half the original period. Such behavior is consistent with the results of theoretical simulations and is due to the time delay in application of the response shift. For the case of positive feedback we observe amplification of the osculation, and coherence is preserved longer than in case of no feedback. The experimental data is in very good agreement with our theoretical simulations.
The negative feedback can be used to suppress the common-mode oscillation, such as those that are caused be mirror vibrations or air currents. The positive feedback can be used to extend coherence times of the wave packet motion.
Next we will try to compensate for the common-mode motion exited by natural causes such as mirror vibrations and air currents. The signal will still be proportional to the number of atoms and be relatively easy to feedback on. The ultimate application of this technique would be to stochastic cooling of neutral atoms. The stochastic power exchange signal is proportional to N1/2, and thus we will need significant experimental improvements. We also plan to do further study of the extended coherence that was observed for gains of apprx. +1.
For detailed description of the experiment and theoretical simulations, see our paper: "Feedback Control of Atomic Motion in an Optical Lattice", N.V. Morrow, S. K. Dutta, G. Raithel, Phys. Rev. Lett. 88, 093003 (2002).