Measurement backaction on the quantum spin-mixing dynamics of a spin-1 Bose-Einstein condensate

We consider a small $F=1$ spinor condensate inside an optical cavity driven by an optical probe field, and subject the output of the probe to a homodyne detection, with the goal of investigating the effect of measurement backaction on the spin dynamics of the condensate. Using the stochastic master equation approach, we show that the effect of backaction is sensitive to not only the measurement strength but also the quantum fluctuation of the spinor condensate. The same method is also used to estimate the atom numbers below which the effect of backaction becomes so prominent that extracting spin dynamics from this cavity-based detection scheme is no longer practical.

Figure 1

Schematic diagram of homodyne detection.

Figure 2

Evolution of the fractional population in spin-0: $N_0/N$. (a) and (b) show the conditional and deterministic evolution, respectively, with the initial state $|1\rangle$ and $q^{\prime }=0$. The red dotted line is for nonmeasurement case $(\xi =0)$, while the blue dashed line and gray solid line are for the weak measurement $(\xi =0.5)$ and strong measurement $(\xi =2)$ cases, respectively. In (c), the initial state is $|b\rangle$ (see text) and $\xi =0.1$. The blue dashed line shows the conditional evolution, while the gray solid line shows the deterministic one. The red dotted line is for the nonmeasurement case.

Figure 3

Left column: Evolution of normalized particle number in spin-$0$ component under a single run of measurements with different strength (a) $\xi =0.001$, (b) $0.01$, and (c) $0.1$. The initial state is the antiferromagnetic ground state (see the text) with total atom number $N=100$. The corresponding Fourier spectra are shown in the right column.

Figure 4

Same as Fig. 3, except that for a ferromagnetic condensate. Insets show details of the oscillations.

Figure 5

(a) Time evolutions of $\langle {\mathrm{N̂}}_0\rangle /N$ for the $q^{\prime }=10$ (upper curves) and $q^{\prime }=0$ (lower curves) case with the same initial state $|0,N,0\rangle$. The blue solid curves correspond to the evolution without measurement. The gray dotted curves display a variety of evolutions conditioned on measurement outcomes with measurement strength $\xi =0.01$, and the red dashed curves are given by averaging over 10 conditional evolutions. (b) The number fluctuation of spin-0 component $\text{Var}({\mathrm{N̂}}_0)=\sqrt{\langle {\mathrm{N̂}}_0^2\rangle -\langle {\mathrm{N̂}}_0\rangle {}^2}$, with solid and dashed lines corresponding to the $q^{\prime }=10$ and $q^{\prime }=0$ case, respectively.

Figure 6

Comparison between the evolution without measurement (a) and the deterministic evolution (b) for the $q^{\prime }=0$ case in a long time scale.

Figure 7

Measurement outcomes given by averaging over ${10}^2$ (outmost gray), ${10}^3$ (inner blue), and ${10}^4$ (innermost red) photodetector currents. The total atom number $N$ is $10$ in (a) and $100$ in (b), and other parameters are the same as in Fig. 5. The insets show the deterministic evolution of $\langle {\mathrm{N̂}}_0\rangle$.