Super-resolving single-photon number-path-entangled state and its generation

In this paper, we propose two protocols for generating super-resolving single-photon (SRSP) number-path-entangled states from general maximally number-path-entangled NOON states. It is our purpose to show that, contrary to popular belief, a field in a single-photon state can carry multifold phase information. We also show that both protocols generate the desired state with different probabilities depending on the type of detectors being used. Such SRSP number-path-entangled states preserve high resolving power but lack the requirement of a multiphoton absorbing resist, which may serve as a proof-of-principle prototype for quantum lithography in the future.

Figure 1

Two beam splitters with the same reflectance $\rho =r^2$ are employed to reduce the number of the photons being transmitted. The existence of detectors $c$ and $d$ reveals the which-way information of the reflected photons, and the incoming NOON states in modes $a$ and $b$ collapse into a separable state in mode $a^{\prime }$ or $b^{\prime }$. Therefore the relative phase is lost, and the phase-resolving power is destroyed.

Figure 2

A single “unit”: Fig. 1 plus an extra 50:50 beam splitter which erases the which-way information; the relative phase is preserved at the output. The dashed lines connecting detectors $c$ and $d$ represent the fact that coincidence detection is only needed under certain circumstances which will be specified in later sections. On the other hand, the 50:50 beam splitter is always necessary.

Figure 3

A stacking of units to reduce $|N_{\mathrm{o}}{::0\rangle }^{N_{\mathrm{o}}\varphi }$, with $N_{\mathrm{o}}=2l+1$ being an odd number. Each rectangle represents the unit described in Fig. 2. Notice we count from the output for the ease of calculation. Unit $i$ with a reflectance of $r_i$ is the $i$th last unit before the output. It has a NOON state $|N_i{::0\rangle }^{N_{\mathrm{o}}\varphi }$ as the input state. Note that although the number of photons decreases from left to right, the relative phase is always $N_{\mathrm{o}}\varphi$, which is exactly what we need to generate a SRSP number-path-entangled state.

Figure 4

The maximized probabilities of transmitting a ${|1::0\rangle }^{N\varphi }$ state following different protocols [see Eqs. (10), (16), and (24)] are plotted as a function of the input photon number. The squares represent the probabilities for a general input state such as in Eq. (3) without photon-number-resolving detections, while the circles represent the probabilities with photon-number-resolving detections. It is clear that with increasing the input photon number, number-resolving detection maintains around 40$\%$ efficiency, while coincidence detection drops to zero rapidly.