Fig. 1: NOON state generation scheme.
The four spheres on the left represent the initial state, with white indicating an empty site. Cyan and blue correspond to M and P particles, respectively. In time-evolved states, gradient colors are used to indicate that the state of a site is entangled with the rest of the system - superposition states for each step are shown in the framed legend. The bars connecting the spheres denote tunneling between nearest-neighbor sites, while the rectangles represent applied external fields to sites 1–3 (ν) and 2–4 (μ). In Protocol I, the system initially evolves for time tm − tμ, towards the u-NOON state. Then, a field is applied across sites 2–4 for time tμ (dashed brown time line), to encode a phase. Finally, the light cyan halo portrays a projective measurement process at site 3, denoted by \({{{{{{{\mathcal{M}}}}}}}}\). The outcomes \(\left|0\right\rangle\) and \(\left|M\right\rangle\) signify which of two possible NOON states results across sites 2–4. Similarly in Protocol II, the system first evolves for time tm − tν, then a field is applied to sites 1–3 for time tν to implement the phase π/2. Next, the system evolves for time tm − tμ, after which a field is applied to sites 2–4 to encode a phase during time tμ. This results in a NOON state across sites 2–4, without performing a measurement procedure.
