Measuring average of non-Hermitian operator with weak value in a Mach-Zehnder interferometer

Quantum theory allows direct measurement of the average of a non-Hermitian operator using the weak value of the positive semidefinite part of the non-Hermitian operator. Here, we experimentally demonstrate the measurement of the weak value and the average value of non-Hermitian operator by an interferometric technique. Our scheme is unique as we can directly obtain the weak value from the interference visibility and the phase shift in a Mach-Zehnder interferometer without using any weak measurement or postselection. Both the experiments discussed here were performed with laser sources, but the results would be the same with average statistics of single-photon experiments. Thus, the present experiment opens up the possibility of measuring the weak value and the average value of a non-Hermitian operator without weak interaction and postselection, which can have several technological applications.

Figure 1

Schematic for the Mach-Zehnder interferometer. Optical components $R$ are placed in one arm and $U^{†}$ in the other.

Figure 2

Experimental Mach-Zehnder setup. We have a PBS or polarizer oriented along the horizontal in one arm. To compensate for elliptic polarization components after reflection from the first beam splitter, we place a quarter-wave plate (QWP) before the half-wave plate.

Figure 3

Collinear (top) vs noncollinear configuration (bottom) of the Mach-Zehnder interferometer: Consider the above beam splitter to be the second beam splitter of the Mach-Zehnder interferometer shown in the experimental schematic in Fig. 2.

Figure 4

Obtaining visibility from spatial fringes. We first find the peaks (blue dots) and dips (red dots) of the interference profile. Then we fit the peaks and dips individually with the Gaussian profile $Aexp[-a{(x-x_0)}^2]$ to obtain the blue solid curve (top curve) and the red solid curve(bottom curve), respectively. If $A_p$ and $A_d$ are the amplitudes of the Gaussian obtained from fitting the peaks and dips, respectively, then the visibility is obtained by $V=\frac{A_p-A_d}{A_p+A_d}$.

Figure 5

The top row has profiles with lower fringe widths compared to the bottom row. The left column has higher visibility compared to the right column.

Figure 6

The expectation value of a class of non-Hermitian operators described in Eq. (21) as a function of the HWP angle placed in one arm of the MZ interferometer when the other arm has a PBS in place.

Figure 7

Visibility as a function of the HWP angle placed in one arm of the MZ interferometer.

Figure 8

Weak value of $R$ inferred from the MZ setup.

Figure 9

Experimental setup to measure the weak value of $R$ with the preselection state $\langle +|$ and the postselection state given by $\langle +|U\left(\theta \right)$, where $U\left(\theta \right)$ is the HWP angle that is used to vary postselection.

Figure 10

Absolute of weak value of $R$ with preselection $|+\rangle$ and postselection $|\varphi \rangle =U\left(\theta \right)|\psi \rangle$.

Figure 11

Absolute value of expectation value of $A$ in the state $|+\rangle$.