Faraday-imaging-induced squeezing of a double-well Bose-Einstein condensate

We examine how nondestructive measurements generate spin squeezing in an atomic Bose-Einstein condensate confined in a double-well trap. The condensate in each well is monitored using coherent light beams in a Mach-Zehnder configuration that interacts with the atoms through a quantum nondemolition Hamiltonian. We solve the dynamics of the light-atom system using an exact wave-function approach, in the presence of dephasing noise, which allows us to examine arbitrary interaction times and a general initial state. We find that monitoring the condensate at zero detection current and with identical coherent light beams minimizes the backaction of the measurement on the atoms. In the weak atom-light interaction regime, we find the mean spin direction is relatively unaffected, while the variance of the spins is squeezed along the axis coupled to the light. Additionally, squeezing persists in the presence of tunneling and dephasing noise. For long atom-light interaction time, we equally show our methods can be used to prepare non-Gaussian correlated spin states.

Figure 1

A double-well trap containing atoms that are allowed to tunnel between the wells. The atoms in each well are imaged using identical laser light in the Mach-Zehnder interferometer configuration. The solid black line is the double-well trap. The symmetric wave function is the ground-state $|g\rangle$ wave function and the antisymmetric line is first excited state $|e\rangle$ belonging to the double-well trap, respectively.

Figure 2

The conditional probability of finding the atoms in the state $|k\rangle$. (a) The solid line is calculated numerically from (20) while the dashed line is given by (28). (b) The numerically computed probability density from (20) at large values of $gt$, $gt\sim 1/\sqrt{N}$, and the dashed line is given by (28). The parameters for the plots are $N=200$, ${\alpha }_l={\alpha }_r=\sqrt{20}$, $\alpha =0$, and $\beta =1$.

Figure 3

The conditional Husimi Q function. The parameters for the plots are $N=200$, $|{\alpha }_l|=|{\alpha }_r|=\sqrt{20}$, $\alpha =0$, and $\beta =1$. The parameter $gt$ is as shown in the figure. For ease of visualization, the positive $z$ axis is flipped and points downwards.

Figure 4

The conditional Husimi Q function. The parameters for the plots are $N=30$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, $\mathrm{\Omega }=\pi /4$. and ${\alpha }_l={\alpha }_r=2$. The parameters for the strength of the coupling $g$ and time $\mathrm{\Omega }t$ are as shown in the figure. For ease of visualization, the positive $z$ axis is flipped and points downwards.

Figure 5

The normalized conditional mean of $J_x$, $J_y$, and $J_z$ with dephasing parameter $\gamma =0$. The atom-light interaction strengths are as shown in the figure. The parameters for the figure are $N=30$, $\mathrm{\Omega }=\pi /4$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, ${\alpha }_r={\alpha }_l=2$.

Figure 6

The normalized conditional variance of $J_x$, $J_y$, and $J_z$ at dephasing parameter $\gamma =0$. The atom-light interaction strengths are as shown in the figure. The dotted line at normalized conditional variance of $J_x$ equal to zero is a guide for the eye. The parameters for the figure are $N=30$, $\mathrm{\Omega }=\pi /4$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, ${\alpha }_r={\alpha }_l=2$.

Figure 7

The conditional Husimi Q function. The parameters are $N=30$, $\mathrm{\Omega }=\pi /4$, $g=0.1\mathrm{\Omega }/N$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, and ${\alpha }_l={\alpha }_r=2$. The dephasing rate $\gamma$ (row) and the parameter proportional to time $\mathrm{\Omega }t$ (columns) are as indicated. For ease of visualization, the positive $z$ axis is flipped and points downwards.

Figure 8

The normalized conditional mean of $J_x$, $J_y$, and $J_z$ at different values of dephasing parameter $\gamma$. The measurement strength $g=0.1\mathrm{\Omega }/N$ for all plots. The solid line is for $\gamma =0$, the dashed line is for $\gamma =0.1g$, the dash-dotted line is for $\gamma =g$, and the dotted line is for $\gamma =3g$. The parameters for the figure are $N=30$, $\mathrm{\Omega }=\pi /4$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, ${\alpha }_r={\alpha }_l=2$.

Figure 9

The normalized conditional variance of $J_x$, $J_y$, and $J_z$ at different values of dephasing parameter $\gamma$. (a) Presented in this column are the normalized variances of $J_x$, $J_y$, and $J_z$. The dotted line at $4{\left(\mathrm{\Delta }{J}_x\right)}^2/N=0$ is a guide for the eye. (b) An enlarged plot of the normalized variance of $J_x$ showing its behavior around unity for different values of the dephasing parameter. The solid line at $4{\left(\mathrm{\Delta }{J}_x\right)}^2/N=1$ is a guide for the eye. The parameters for the figures are $N=30$, $\mathrm{\Omega }=\pi /4$, $g=0.1\mathrm{\Omega }/N$, $\alpha =\sqrt{0.001}$, $\beta =\sqrt{0.999}$, and ${\alpha }_r={\alpha }_l=2$.