Figure 1

The grid shows particular section in the field density matrix ${\widehat{\rho }}_f^{\text{retr}}$ with “circles” representing the elements of the density matrix. The arrows give the directions of couplings between the $\left(n,n^{\prime }\right)$ element and its immediate neighbors if a measurement is made for different combinations of prepared and measured atomic states for atoms: (a) prepared in $\mid e⟩$ and measured in $\mid g⟩$, (b) prepared in $\mid g⟩$ and measured in $\mid e⟩$, (c) prepared in $\mid e⟩$ and measured in $\mid \pm ⟩$ (also for atoms prepared in $\mid \pm ⟩$ and measured in $\mid g⟩$), (d) prepared in $\mid g⟩$ and measured in $\mid \pm ⟩$ (also for atoms prepared in $\mid \pm ⟩$ and measured in $\mid e⟩$), and (e) both prepared and measured in $\mid \pm ⟩$. Here, we do not show the trigonometric self-modulations on the $\left(n,n^{\prime }\right)$ element itself.Reuse & Permissions

Figure 2

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, (c) three, (d) four, (e) five, and (f) six atoms, prepared in the excited state, are detected in the coherent superposition state $\mid -⟩$. The interaction strength is fixed at $\lambda {\tau }^{\prime }=\pi$ with the dimension of the density matrix chosen to be $15\times 15$. The gray scale is tuned accordingly for ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$, from the minimum (brightest) up to the maximum (darkest), with the area outside of the blocks considered being set to value zero. Note: all density matrix elements in (f) are positive.Reuse & Permissions

Figure 3

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ after (a) one, (b) two, (c) three, (d) four, (e) five, and (f) six atoms, prepared in the excited state, are detected in the coherent superposition state $\mid +⟩$. The other conditions are the same as in Fig. 2. Note: white squares in (f) correspond to negative elements.Reuse & Permissions

Figure 4

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, prepared in the ground state, are detected in the coherent superposition state $\mid +⟩$. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 5

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, prepared in the ground state, are detected in the coherent superposition state $\mid -⟩$. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 6

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, prepared in the coherent superposition state $\mid +⟩$, are detected in the excited state. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 7

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, prepared in the coherent superposition state $\mid +⟩$, are detected in the ground state. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 8

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, both prepared and measured in the coherent superposition state$\mid +⟩$. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 9

Images of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if (a) one, (b) two, and (c) six atoms, prepared in the coherent superposition state $\mid +⟩$, are measured in the coherent superposition state $\mid -⟩$. The other conditions are the same as in Fig. 2.Reuse & Permissions

Figure 10

Normalized field density matrix ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if 50 initially inverted atoms are measured in the (a) $\mid -⟩$ and (b) $\mid +⟩$ superposition states with ${\mid \xi \mid }^2=1∕2$. The interaction strength is set at $\lambda {\tau }^{\prime }=\pi ∕\sqrt{26}$.Reuse & Permissions

Figure 11

Probability density function $P\left(\theta \right)$ for the retrodictive field states if 1 (dashed curve), 3 (dashed-dotted curve), 5 (dashed-dotted-dotted curve), and 50 (solid curve) atoms, prepared in the excited state, are measured in the $\mid -⟩$ superposition state. Also shown is the case for 50 atoms (dotted curve) measured in the $\mid +⟩$ superposition state. Phase window of $\left[-\pi ,\pi \right]$ has been used and the other conditions are the same as in Fig. 10.Reuse & Permissions

Figure 12

Normalized values of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if 50 atoms, initially prepared in the $\mid +⟩$ state, are measured in the $\mid -⟩$ superposition state with (a) ${\mid \alpha \mid }^2={\mid \xi \mid }^2=0.25$ and (b) ${\mid \alpha \mid }^2={\mid \xi \mid }^2=0.75$. The interaction strength is set at $\lambda {\tau }^{\prime }=\pi ∕\sqrt{26}$.Reuse & Permissions

Figure 13

$\Delta \theta$ (dashed curve), $\Delta \theta \Delta n$ (solid curve) and$0.5\mid 1-2\pi P\left({\theta }_0\right)\mid$ (dotted curve) for retrodictive states over the whole range of ${\mid \alpha \mid }^2\left(={\mid \xi \mid }^2\right)$. All atoms are prepared in the $\mid +⟩$ state and measured in the $\mid -⟩$ state. The other conditions are the same as in Fig. 12.Reuse & Permissions

Figure 14

The modulus (normalized values) of ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ if 100 atoms, both prepared and measured in the $\mid +⟩$ superposition state with $\mid \alpha {\mid }^2=\mid \xi {\mid }^2=0.25$. The interaction strength is set at $\lambda {\tau }^{\prime }=\pi ∕\sqrt{26}$.Reuse & Permissions

Figure 15

The retrodictive field density matrix ${\widehat{\rho }}_f^{\text{retr}}\left(t_p\right)$ for (a) 50 atoms, prepared in $\mid +⟩$ and measured in $\mid -⟩$ superposition states with ${\mid \alpha \mid }^2={\mid \xi \mid }^2=0.6$. The interaction strength is set at $\lambda {\tau }^{\prime }=2\pi ∕\sqrt{24}$ with photon number $\mid 5⟩$ the $\pi$ trapping state; (b) 25 atoms, prepared in $\mid +⟩$ and measured in the ground state with $\mid \alpha {\mid }^2=0.6$. The interaction strength is set at $\lambda {\tau }^{\prime }=2\pi ∕\sqrt{23}$ with no immediate trapping states.Reuse & Permissions