Figure 1

Schematic setups for observing two-photon phase superresolution. The Michelson interferometer is composed of a 50:50 beam splitter (BS) and mirrors (M1 and M2). M1 is displaced by $z/2$ to provide the optical-path difference $z$ between the two arms of the interferometer. (a) Time-forward process. This setup is composed of a nonlinear optical crystal for SPDC, an unbalanced Michelson interferometer, and two detectors for coincidence counting. The dashed-line box describes the possible states of a photon pair. Letters S and L denote the photons from the shorter and longer arms of the interferometer, respectively. When $z$ is large enough, only photon pairs SS and LL contribute to the coincidence counts. (b) Time-reversed process. This setup is composed of an unbalanced Michelson interferometer, a nonlinear optical crystal for SFG, a bandpass filter, and a detector. When $z$ is large enough, two pulses from different arms are individually converted into sum-frequency pulses by SFG. After the bandpass filter stretches the pulse widths, these pulses interfere with each other. This interference corresponds to that of photon pairs SS and LL in the time-forward process.Reuse & Permissions

Figure 2

Time-reversed experimental setup for observing two-photon phase superresolution in an unbalanced Michelson interferometer. A coherent laser pulse from a femtosecond fiber laser enters the Michelson interferometer. To observe the white-light interference fringes, the optical-path difference $z$ of the interferometer is adjusted to about zero, and the output light is detected by photodiode PD1. To observe the sum-frequency-light interference, $z$ is adjusted to about 100 $\mu$m, and the output light is converted into sum-frequency light by a BBO crystal. A grating and a slit, both of which function as a bandpass filter, pass the sum-frequency light within a narrow band (0.039 nm) before the light is detected by photodiode PD2.Reuse & Permissions

Figure 3

Measured interference fringes: (a) white light and (b) sum-frequency light. From the fitted curves in solid lines, visibilities are estimated to be $99.1\%\pm 0.2\%$ and $97.9\%\pm 0.4\%$, respectively. Relative displacement $z^{\prime }$ of M1 depends nonlinearly on piezo voltage $V$. Assuming that $z^{\prime }$ is approximated by a quadratic function of $V$, we fitted $z^{\prime }\left(V\right)$ to the measured white-light interference fringes in (a). Calibrated function $z^{\prime }\left(V\right)$ is also applied to the sum-frequency-light interference fringes in (b). Note that $z^{\prime }$ is the relative displacement of M1 from where the optical-path difference $z$ is about zero in (a) and about 100 $\mu$m in (b).Reuse & Permissions

Figure 4

Interferogram detected by PD2 for wide range of $z$. When $z$ is about zero, the light signal is proportional to the square of the white-light interference signal. On the other hand, when $\left|z\right|$ is larger than about 100 $\mu$m, the light signal exhibits two-photon phase superresolution. The coherence length of the sum-frequency-light interference is about 22 times larger than that of the white-light interference. This difference of coherence lengths is a classical analog to the large difference between one- and two-photon coherence lengths of entangled photon pairs.Reuse & Permissions