Phase Estimation with Weak Measurement Using a White Light Source

We report results of a high precision phase estimation based on a weak measurement scheme using a commercial light-emitting diode. The method is based on a measurement of the imaginary part of the weak value of a polarization operator. The imaginary part of the weak value appeared due to the measurement interaction itself. The sensitivity of our method is equivalent to resolving light pulses of the order of a attosecond and it is robust against chromatic dispersion.

Figure 1

Weak measurement of the photon polarization. Photons emitted from the source are preselected by a polarization beam splitter (PBS) in a state $|{\psi }_{\mathrm{pre}}⟩$, undergo weak measurement interaction by passing through a birefringent plate, and are postselected at a nearly orthogonal state $|{\varphi }_{\mathrm{post}}⟩$ by a second PBS. (a) The wave packets with orthogonal polarizations are delayed after the birefringent plate, one relative to the other, by a few attoseconds, but after the second PBS, they interfere in the postselected wave packet being shifted in proportion to the real part of the weak value of the polarization operator. (b) The shift of the spectrum of the light is proportional to the imaginary part of the weak value of the polarization operator.

Figure 2

Detailed experimental setup: Photons emitted from a LED (produced by the Epitex company), with measured central wavelength ${\lambda }_0=808\mathrm{nm}$ and a spectral width of $\Delta \lambda =38.8\mathrm{nm}$ are collected by two lenses, pass through a pinhole, and enter the first PBS (a Wollaston prism), which preselects the polarization state $|{\psi }_{\mathrm{pre}}⟩$. Then, the light passes two true zero order HWPs with their optical axes perpendicular to each other and at 45° to the axis of the first prism. The plane of the second HWP is tilted by the angle $\theta$ with respect to the first HWP. A second Wollaston prism with its optical axis at an angle $(\pi /2)-(\beta /2)$ with respect to the first one postselects a polarization state which is nearly orthogonal to the preselected one. The light is then collected by a single-strand optical fiber and sent to spectrometer with sampling period of 0.02 nm and a range of 715–915 nm. A 1 mm thick ZnSe crystal is inserted to introduce a strong dispersion. FC stands for fiber collector.

Figure 3

Experimental results. The shift of the spectrum of light $\delta \lambda$ as a function of a corresponding phase shift $\alpha$. The spectral shifts are shown for three different values of postselection parameter $\beta$. The square (red), circle (blue), and triangle (green) points correspond to $\beta =0.000$, 0.004, and 0.014, respectively. The red, blue, and green lines are the theoretical predictions. The lower part shows the connection between $\alpha$ and the tilt angle $\theta$ which is controlled in the experiment as explained in the text; see Eq. (10).

Figure 4

The shift of the spectrum of light $\delta \lambda$ for light passing through a thick ZnSe crystal. The square (red) and circle (blue) points correspond to $\beta =0.000$ and 0.004, respectively. The triangles (green) also correspond to $\beta =0.004$, but for filtered light having significantly smaller spectrum width. The red, blue, and green lines are the theoretical predictions.