Fig. 1. (a) Experimental setup. The intelligent states with various uncertainties in logarithmic radial position such as (b1) λ=200.00 (Δln r=0.05), (b2) λ=50.00 (Δln r=0.1), (b3) λ=22.22 (Δln r=0.15), and (b4) λ=12.50 (Δln r=0.2) are displayed on the SLM. In addition, a set of well-elaborated holograms for analyzing the hyperbolic momentum is also displayed on the same SLM. A combination of these two holograms with fixed λ=12.50 and (b5) PH=10, (b6) PH=15, (b7) PH=20, and (b8) PH=25 is shown in the middle column of (b). (b9–b12) show the resultant light beams in the Fourier plane.

Fig. 2. Experimental observation of hyperbolic momentum spectrum for (a) λ=61.73 (Δln r=0.09), (b) λ=29.59 (Δln r=0.13), and (c) λ=17.30 (Δln r=0.17). The insets show the corresponding intelligent states displayed on the SLM.

Fig. 3. Experimental measurement of the product of the uncertainties in logarithmic radial position and hyperbolic momentum for intelligent states. The red line represents the theoretical bound of ℏ/2.

Fig. 4. Experimental measurement of the product of the uncertainties in logarithmic radial position and hyperbolic momentum for rigid slits. The red line represents the lower bound of ℏ/2 in the uncertainty relation (that can only be achieved by the intelligent states) as demonstrated in Eq. (4).