Quantum state engineering by click counting

We derive an analytical description for quantum state preparation using systems of on-off detectors. Our method will apply the true click statistics of such detector systems. In particular, we consider heralded quantum state preparation using correlated light fields, photon addition, and photon subtraction processes. Using a postselection procedure to a particular number of clicks of the detector system, the output states reveal a variety of quantum features. The rigorous description allows the identification and characterization of fundamentally unavoidable attenuations within given processes. We also generalize a known scenario of noiseless amplification with click detectors for the purpose of the preparation of various types of nonclassical states of light. Our exact results are useful for a choice of experimental parameters to realize a target state.

Figure 1

A source $S$ generates quantum correlated light. One beam is measured with a click detector system $D$. The other beam will be further processed only in case of a $k$-click event of the detector.

Figure 2

The output photon distributions $p_n$ is shown for the photon-photon correlated input in Eq. (8) ($\omega =0.25$) triggered to $k$ clicks. For a small number of clicks, $k\ll N=64$, and a high quantum efficiency, $\eta =0.95$, the heralding yields a good estimate to a $k$-photon state.

Figure 3

An incident light beam is split on a beam splitter into two outgoing beams. One of them is detected with an on-off detector system. Only states that correspond to $k$ clicks of the detector will be further processed.

Figure 4

The figures show the output $P$ function of a $k$-click subtraction process for $N=16$ on-off diodes with a quantum efficiency $\eta =0.8$. The transmission is $t=0.7$; the mean thermal photon number of the input state is $\overline{n}=0.5$.

Figure 5

A nonlinear crystal (NL) is pumped by a pump beam (dashed line). An incident signal field and vacuum are mixed in this nonlinear medium. One of the generated output beams is measured with a click detector system. The other output is triggered to $k$ clicks of the detector.

Figure 6

The plots show the $P$ function of a $k$-click conditioned thermal state generated by a click counting detector with $N=16$ on-off diodes and a quantum efficiency $\eta =0.8$.The considered squeezing corresponds to $\mu =cosh\xi =1.4$. The mean thermal photon number of the input state is $\overline{n}=0.5$.

Figure 7

A combination of an initial addition followed by the subtraction of photons with click counting detectors is shown. The first detector $D{}_1$ consists of $N_1$ diodes with a quantum efficiency of ${\eta }_1$. The second detector $D{}_2$ consists of $N_2$ diodes with a quantum efficiency of ${\eta }_2$. The final outcome is conditioned to $k_1$ clicks of the first and $k_2$ clicks of the second detector.

Figure 8

Contour plot of the output $P$ function for a coherent input state with $\beta =1/\sqrt{2}$. Both detectors consist of the same number of diodes $N_1=N_2=4$ with identical quantum efficiencies of only ${\eta }_1={\eta }_2=0.5$. The parametric process is described by $\mu =3/2$ and for the beam splitter holds $t=2/3$. The number of the row is equal to the number of additive clicks, $k_1=1,2,3,4$, whereas the column counts the subtractions, $k_2=0,1,2,3$. A dark orange color represents a highly negative contribution, whereas gray depicts positive parts of the $P$ function.