Which-way information in a nested Mach-Zehnder interferometer

Recently, A. Danan et al. [Phys. Rev. Lett. 111, 240402 (2013)] performed a much-discussed experiment in which which-way information was obtained from the light in a nested Mach-Zehnder interferometer by weak measurement. The presented analysis using the two-state vector formalism drew the conclusions that the photons followed disconnected paths. We analyze this experiment using standard quantum optical methods and arrive at analytical expressions that match the experimental results without the need for such disconnected photon paths. We also propose a simple amendment to the experiment capable of displaying new phenomena, highlighting the advantages of our description.

Figure 1

The schematic of the experiment. The symbols labeled $A, B, C, E, F$ denote the vibrating mirrors, and $D$ denotes a two-cell detector. The two dashed lines denote blind input arms of beam splitters, and the two dotted lines denote output arms, which are not measured in the experiment.

Figure 2

A tilted mirror and the original and reflected coordinate systems. The system $(x_r,y_r,z_r)$ corresponds to $(x,y,z)$ reflected with respect to the actual location of the mirror, while $(x^{\prime },y^{\prime },z^{\prime })$ is reflected with respect to ${45}^{\circ }$. The $y$ axis is not affected by the reflection ($y=y^{\prime }=y_r$) and is not plotted. Both the tilt angle $\theta$ and the beam divergence are vastly exaggerated.

Figure 3

The relative amplitudes of oscillations at frequencies ${\omega }_A$ and ${\omega }_B$ (red solid line) compared to that at ${\omega }_C$ (blue dashed line) in the power spectrum of (14) as a function of the static phase difference ${\varphi }_A-{\varphi }_C$. The value of $\zeta \left(z_D\right)$ was chosen arbitrarily to illustrate the point.