Direct measurement of time-frequency analogs of sub-Planck structures

Exploiting the correspondence between the Wigner distribution function and the frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate the existence of chessboardlike interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase-space representation of compass states. Using superpositions of four electric pulses as the realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for the superposition of two Schrödinger-cat-like states in a position-momentum phase space.

Figure 1

Schematic view of our experimental setup, where, with the help of a pulse shaper, a superposition of pulses is created and then measured in a SHG FROG setup. The resulting FROG spectrograms are presented in Figs. 3 and 4. Here, (a) a pulse shaper composed of a set of grating, lens, polarization maintain (PM) circulator, polarization maintain (PM) collimator, and spatial light modulator (SLM); (b) SHG FROG composed of a beam splitter (BS) and a motorized stage in one arm to control the time delay $\tau$, a nonlinear crystal (BBO) for the second-harmonic generation (SHG), and a spectrometer.

Figure 2

Typical experimental profiles for our compass states are shown as a function of (a) relative frequency ${\nu }_r=\omega /\left(2\pi \right)$ and (b) time $t$. Here, the four pulses are concurrently separated by ${\omega }_0/\pi =3.3$ THz in frequency and $2t_0=4$ ps in time.

Figure 3

Compass state in the phase space obtained by (a) a FROG map measured for $t_0=2$ ps, and (b) the numerical simulation calculated for a superposition of four identical Gaussian pulses.

Figure 4

Zooms of the central interference structure of FROG maps measured for (a) $t_0=1.25$ ps, (b) $t_0=1.75$ ps, and (c) $t_0=2.5$ ps. As a comparison, the corresponding simulations calculated for the compass states built from perfect Gaussian pulses are shown in (d)–(f), respectively.

Figure 5

(a) A central frequency $(\nu =383.36$ THz) cross section of the FROG map measured for $t_0=2.5\mathrm{ps}$. Here, blue dots depict the measured intensity values for a given time delay $\tau$ introduced between two input state copies in the arms of the FROG apparatus. (b) Average areas $\mathrm{\Delta }\tau \mathrm{\Delta }\omega$ of the rectangles appearing between the zero lines in the interference structure of the central part of the FROG maps. Values depicted by red or gray dots correspond to two independent sets of experimental data, respectively. The region below the uncertainty relation limit of $0.5$ is denoted in yellow.