Effects of measurement backaction in the stabilization of a Bose-Einstein condensate through feedback

We apply quantum filtering and control to a particle in a harmonic trap under continuous position measurement, and show that a simple static feedback law can be used to cool the system. The final steady state is Gaussian and dependent on the feedback strength and coupling between the system and probe. In the limit of weak coupling, this final state becomes the ground state. An earlier model by Haine et al. [Phys. Rev. A 69, 13605 (2004)] without measurement backaction showed dark states: states that did not display error signals, thus remaining unaffected by the control. This paper shows that for a realistic measurement process this is not true, which indicates that a Bose-Einstein condensate may be driven toward the ground state from any arbitrary initial state.

Figure 1

(Color online) Final expected energy (in units of $\hslash \omega$) Eq. (15) as a function of the feedback parameter $k_p$ (in units of $\omega$) for $\alpha =0.09$. The horizontal line represents the ground-state energy, while the cross represents the calculated optimal control strength.

Figure 2

(Color online) Final expected energy (in units of $\hslash \omega$) Eq. (15) as a function of coupling strength $\alpha$ (in harmonic oscillator units compatible with the choice $\hslash =m=\omega =1$) for $k_p=-1.35$. The horizontal line represents the ground-state energy.

Figure 3

(Color online) $Q$ functions at different times (corresponding to the letters in Fig. 4) for the evolution of the dark state ${\pi }_0$ (a) under Eq. (21), for $k_p=-1.35$ and $\alpha =0.3$.

Figure 4

Time evolution (in units of $1∕\omega$) of the expected energy (in units of $\hslash \omega$) for the same parameters as in Fig. 3. Solid line represents a single trajectory, dashed line is the result after averaging over 48 realizations of the noise in Eq. (21), and dot-dashed line is the solution for an initial Gaussian state [see Eq. (13)]. The horizontal line corresponds to the steady-state solution given by Eq. (15) and the letters to the times at which the $Q$ functions in Fig. 3 are plotted.