Figure 1

Nondeterministic (a) preparation and (b) measurement of NOON-states for phase super-resolution, as described in the text. (i) Photon counting after a 50% beam splitter to measure $⟨N{|}_1$ & $⟨0{|}_2{\kappa }_N=1/2^N$. (ii) Coherent-state detection, $⟨\alpha {|}_1$ & $⟨\alpha {|}_2$, via a 50% beam splitter and two homodyne detectors to measure amplitude and phase ${\kappa }_N^{ii}={\kappa }_N^i/\sqrt{2\pi N}$. (iii) and (iv) are the corresponding time-reversed processes.Reuse & Permissions

Figure 2

Ideal multiports, $U_{\mathrm{multi}}$: (a) symmetric $3\times 3$ and (b) asymmetric $4\times 4$ ($6\times 6$), constructed from $2\times 2$ beam splitters with reflectivities as shown [polarizing beamsplitter (PBS)]. Laser is input to modes 1, 2, with no light to modes 3–6. In (a) the internal phase, $\beta$, ensures each input transforms to an equal superposition of the three outputs; in (b) polarization rotations $U_1$ to $U_3$ set the phases of the singles fringes. (c),(d) Experimental realizations of (a),(b). In (c) the indicated HWP’s are set to 22.5° to form $1/2$ beam splitters with the beam displacers, the third is set to 17.6° so that $2/3$ of the light intensity takes the upper path; the angle of the tilted HWP sets $\beta$, its optic axis is at 0°. In (d), the beam splitter for modes 3, 4 is a pellicle; for modes 5, 6 it is a microscope slide set at a small angle of incidence, to avoid polarization effects. Reflectivities are not ideal, rates are equalized with lossy coupling. $U_1$ to $U_3$, are realized using wave plates: the orientation and tilt of each wave plate was adjusted so that one detector reaches an interference minimum every 22.5° for $N=4$; every 15° for $N=6$.Reuse & Permissions

Figure 3

(a)–(c) $N$-fold coincidence rates as a function of phase, $\varphi$, respectively, exhibiting 3, 4, and 6 distinct oscillations within a single phase cycle. The main source of uncertainty is Poissonian statistics: error bars represent the square root of the count rate. The solid line is a fit to a product of $N$ sinusoidal fringes, as explained in the text. (d)–(f) Corresponding singles rates, each exhibiting only one oscillation per phase cycle. Error bars are contained within the data points; solid lines are the individual sinusoidal fringes obtained from (a)–(c). In (d) D1 to D3 are, respectively, indicated by black, blue, and red; in (e)–(f) D1 to D6 are indicated by black, gray, blue, cyan, red, and pink. Ideally, the phase differences between adjacent fringes is $2\pi /N$, we find: $N=3$, {122°,119°}; $N=4$, {92°,90°,90°}; and $N=6$, {55°,66°,56°,62°,56°}.Reuse & Permissions