Continuous measurement feedback control of a Bose-Einstein condensate using phase-contrast imaging

We consider the theory of feedback control of a Bose-Einstein condensate (BEC) confined in a harmonic trap under a continuous measurement constructed via nondestructive imaging. A filtering theory approach is used to derive a stochastic master equation (SME) for the system from a general Hamiltonian based upon system-bath coupling. Numerical solutions for this SME in the limit of a single atom show that the final steady-state energy is dependent upon the measurement strength, the ratio of photon kinetic energy to atomic kinetic energy, and the feedback strength. Simulations indicate that for a weak measurement strength, feedback can be used to overcome heating introduced by the scattering of light, thereby allowing the atom to be driven toward the ground state.

Figure 1

(Color online) Diagram illustrating how one could perform a nondestructive density measurement on a BEC. The BEC is illuminated with highly off-resonant laser light. The interaction between the field and the atoms is registered as a phase shift on any light scattered from the condensate. Such a phase shift is measurable by homodyne detection in the phase quadrature.

Figure 2

(Color online) Plots of the energy for simulations of 500 paths. Simulations were for a normalized Gaussian function centered at $x=2.0$. Parameters chosen were $c_1=2.0$, $w=3000.0$, and $\eta =6.0$. The measurement strength (in order from bottom to top) is $\tilde{\alpha }=2.0$ (red), 10.0 (blue), 20.0 (green), 40.0 (black), 80.0 (cobalt), 160.0 (margenta). Full lines represent the mean, while dotted lines indicate the standard error. Notice that those simulations with $\tilde{\alpha }\le 20.0$ converge to the same average steady-state energy.

Figure 3

(Color online) 500 path simulations of the energy for an initial normalized Gaussian state centered at $x=2.0$, $w=3000.0$, $\tilde{\alpha }=10.0$, and $c_1=2.0$. The Lamb-Dicke parameter (in order from bottom to top) is $\eta =4.0$ (red), 6.0 (blue), 8.0 (green), 10.0 (black). Full lines represent the mean, while dotted lines indicate the standard error. This plot shows that relatively small increases in $\eta$ result in large increases in the final average energy.

Figure 4

(Color online) 100 path simulations of the energy in the single particle approximation with feedback terms proportional to $x$ (red/highest), $x$ and $x^2$ (blue/middle), and $x,x^2,x^3$ (green/lowest). The parameters chosen were $w=3000.0$, $\tilde{\alpha }=2.0$, and $\eta =8.0$. Full lines represent the mean, while dotted lines indicate the standard error. The initial states chosen were normalized Gaussian functions centered at $x=2.0$. The constants of proportionality for each feedback term were set to 2.0.