Information and backaction due to phase-contrast-imaging measurements of cold atomic gases: Beyond Gaussian states

We further examine a theory of phase contrast imaging (PCI) of cold atomic gases, first introduced by us in Phys. Rev. Lett. 112, 233602 (2014). We model the PCI measurement by directly calculating the entangled state between the light and the atoms due to the ac Stark shift, which induces a conditional phase shift on the light depending upon the atomic state. By interfering the light that passes through the BEC with the original light, one can obtain information of the atomic state at a single-shot level. We derive an exact expression for a measurement operator that embodies the information obtained from PCI, as well as the backaction on the atomic state. By the use of exact expressions for the measurement process, we go beyond the continuous variables approximation such that the non-Gaussian regime can be accessed for both the measured state and the postmeasurement state. Features such as the photon probability density, signal, signal variance, Fisher information, error of the measurement, and the backaction are calculated by applying the measurement operator to an atomic two-spin-state system. For an atomic state that is initially in a spin-coherent state, we obtain analytical expression for these quantities. There is an optimal atom-light interaction time that scales inversely proportional to the number of atoms, which maximizes the information readout.

Figure 1

Schematic configuration for the off-resonant light-matter interaction for phase contrast imaging considered. The incident light is detuned from atomic resonance realizing an ac Stark shift. The atoms in the state $|\psi \rangle$ involving the ground state internal levels $f_i$ with the incident light and become entangled with the light. Every Fock state of the atoms produce a phase shift $\widehat{\varphi }$. Finally, the light which passes through the atoms $|\gamma \rangle$ and the light which does not pass through $|\chi \rangle$ are interfered, and photons are counted to obtain a signal $\mathcal{S}$.

Figure 2

The PCI photon count probability density (12) in terms of relative and average photon numbers $u, v$. The parameters for the figures are $N=1000, {\left|\gamma \right|}^2={\left|\chi \right|}^2=50, {\theta }_0=\frac{\pi }{2}, {\varphi }_{\gamma }={\varphi }_{\chi }=0$, and the atom-light interaction time $G\tau$ is as shown in each panel.

Figure 3

The (a) signal and (b) variance of the PCI measurement of a spin-coherent state with an initial state parameter $cos{\theta }_0=|{\alpha }_0{|}^2-{\left|{\beta }_0\right|}^2$. The curve for $G\tau =\frac{1}{N}$ in (b) is magnified by a factor of 10 for visibility. Parameters used are ${\left|\gamma \right|}^2={\left|\chi \right|}^2=50, N=1000$, and ${\varphi }_{\gamma }={\varphi }_{\chi }=0$. The dimensionless atom-light interaction time $G\tau$ is as marked.

Figure 4

The error of the PCI measurement for an initial spin-coherent state with Bloch sphere angle ${\theta }_0$. The interaction time is as shown in each panel. For (a), (b), (c), and (d) the solid line is (51) and the dashed line is (52). In all panels ${\varphi }_{\chi }={\varphi }_{\gamma }=0$.

Figure 5

(a) Fisher information for a PCI measurement of a spin-coherent state with Bloch sphere parameter ${\theta }_0$. (b) Error estimated from Fisher information as defined in (55). The parameters for the figures are ${\left|\gamma \right|}^2={\left|\chi \right|}^2=50, N=1000$, and ${\varphi }_{\chi }=0$. The atom-light interaction time $G\tau$ is as shown in the figure. For all curves, the solid line is the numerical curve, and the dashed line is the theoretical curve.

Figure 6

The $Q$ function of the atomic BEC state after PCI imaging for various atom-light interaction times $G\tau$, as shown in the panels. The initial condition of the state of the atomic BEC is ${\theta }_0=\pi /2, {\phi }_0=\pi /2$, and the parameters for the panels are ${\left|\gamma \right|}^2={\left|\chi \right|}^2=15, N=300$, and ${\varphi }_{\gamma }={\varphi }_{\chi }=0$.