Numerical simulations of optical centroid measurements with nonclassical fields

Optical imaging methods are typically restricted to a resolution of the order of the probing light wavelength ${\lambda }_p$ by the Rayleigh diffraction limit. This limit can be circumvented by making use of multiphoton detection of correlated $N$-photon states, having an effective wavelength ${\lambda }_p/N$. However, the required $N$-photon detection usually renders these schemes impractical. To overcome this limitation, recently, so-called optical centroid measurements have been proposed which replace the multiphoton detectors with an array of single-photon detectors. Complementary to the existing approximate analytical results, we explore the approach using numerical experiments by sampling and analyzing detection events from the initial-state wave function. This allows us to quantitatively study the approach also beyond the constraints set by the approximate analytical treatment, to compare different detection strategies, and to analyze other classes of input states.

Figure 1

Momentum (a)–(c) and the corresponding position (d)–(f) space probability distributions of a two-photon NOON state (top row), joint Gaussian state with $N=2$ and $B=\beta ={\lambda }^{-1}$ (middle row), and correlated cat state with $\alpha =i$ (bottom row).

Figure 2

Model for the detection setup. Panel (a) shows an array of detectors of size $d_0$. The red arrows indicate the central positions of the individual detectors. Panel (b) shows the corresponding array with double detector size $2d_0$. Compared to (a), only half of the possible detection positions occur. Panel (c) shows the array of (b) suitably shifted that now the possible detection events cover those of (a) which are missing in (b).

Figure 3

Results of the centroid method for NOON states with different $N$, shown against the centroid position in units of the single-photon transverse wavelength $\lambda$. Continuous blue, dashed red, and dotted green curves correspond, respectively, to $N=2$, 3, and 4. The increase in the number of fringes clearly shows the resolution enhancement with $N$.

Figure 4

Root-mean-square deviation of the recovered two-photon centroid distribution from the original wave function position space distribution as a function of the detector size. Note the logarithmic scale.

Figure 5

Root-mean-square deviation of the recovered two-photon centroid distribution from the original wave function position space distribution as a function of the detector shift. The different curves show results for the detector sizes (i) down-pointing triangles $\lambda$, (ii) up-pointing triangles $\lambda /2$, (iii) blue dots $0.3\lambda$, (iv) green squares $\lambda /4$, and (v) magenta diamonds $\lambda /10$. Note the logarithmic scale.

Figure 6

Centroid analysis for detector size $\lambda /4$. The continuous red curve shows the probability distribution for a two-photon NOON state. The blue dots indicate the results of a centroid measurement with particular detector settings. In (a), the detection array is not shifted with respect to the origin of the probability distribution, such that one of the centroid data points coincides with the center position $X=0$. (b) The detection array is shifted to $s=-13\lambda /200$, which corresponds to a minimum in the rms deviation in Fig. 5. The blue dots show the centroid data with the same scaling as in (a). The green asterisks show this data fitted to the original distribution.

Figure 7

Centroid analysis for detector size $\lambda$. The figure is analogous to Fig. 6, except for the detector size.

Figure 8

Effect of statistical fluctuations on the centroid methods for small detector sizes. The rms deviation was evaluated (i) black dots, once for ${10}^6$ events, (ii) red squares, twice for two subsets with $5\times {10}^5$ events, (iii) blue diamonds, five times for subsets of $2\times {10}^5$ events, and (iv) green triangles, ten times for subsets of ${10}^5$ events. In all cases, the same ${10}^6$ events were analyzed, and the respective subsets were chosen disjunct. The figure shows the results averaged over the respective subsets. Note the logarithmic scale.

Figure 9

(a) Root-mean-square deviation as a function of detector size. Results are shown for a two-photon NOON state. The blue squares are obtained using method I discussed in the text, whereas the red dots are obtained using method II. The green diamonds and the black triangles are obtained using method II with one-half and one-third of the total number of events used in (a), respectively. The inset shows a magnification at small detector sizes.

Figure 10

Root-mean-square deviation versus detector size for NOON states using method I. Results are compared for number of photons $N=2$ (lower blue curve), $N=3$ (middle red curve), and $N=4$ (upper green curve). The magenta dotted, gray dashed and black dot-dashed lines indicate linear fits to the curves over the indicated detector range for $N\in \{2,3,4\}$, respectively.

Figure 11

Root-mean-square deviation versus detector size for a jointly Gaussian state using method I. The parameters are the same as in Fig. 1.

Figure 12

$\langle k_n^2\rangle ={\lambda }^{-1}$. (a)–(c) Total multiphoton absorption rate and (d)–(f) peak multiphoton absorption rate versus the spot size reduction factor for jointly Gaussian state. The top, middle, and bottom rows correspond to $N\in \{2,3,4\}$, respectively. The data points obtained via numerical calculation are plotted as blue diamonds while the red solid line shows the theoretical predictions. We multiplied the $R_{\mathrm{tot}}$ values with $17.8,156,4100$ for $N\in \{2,3,4\}$, respectively, to normalize the data such that at $r=1$ the corresponding value on the $y$ axis becomes unity too.

Figure 13

Root-mean-square deviation plotted versus the detector size for a fixed physical size of the quantum mechanical jointly Gaussian centroid probability distribution for various $N$ using method I. The blue dashed, red dotted, and green dot dashed curves correspond, respectively, to $N=2$, 3, and 4. The parameters are explained in the text.

Figure 14

Root-mean-square deviation against detector size for a correlated coherent cat state with $\alpha =i$. Results have been obtained using method I.

Figure 15

Root-mean-square deviation versus detector size for a correlated coherent cat state with $\alpha =i$. In contrast to Fig. 14, results for a single fixed detector position are shown.

Figure 16

(a) Probability distribution for a correlated coherent cat state with $\alpha =i\sqrt{2}$ in position space. (b) Root-mean-square deviation against detector size using method I.

Figure 17

Root-mean-square deviation for correlated cat states as a function of $\left|\alpha \right|$. In all cases, two-photon detection events are considered. $N_0={10}^5$ events were considered with detector size $\lambda /100$.

Figure 18

Position space probability density along the centroid coordinate for $\left|\alpha \right|=1$ and (i) $\varphi =0$, (ii) $\varphi =\pi /8$, (iii) $\varphi =3\pi /8$, (iv) $\varphi =\pi /2$.