Weak Values in an Electronic Mach-Zehnder Interferometer

Aharonov et al. introduced the concept of weak values [1] as statistical averages of weak measurements on a quantum system whose states both before and after the measurement are known. As opposed to the standard value, the weak value can lie outside the spectrum of eigenvalues of the measured operator, can be negative for positive defined quantities, and can even be complex.
This Demonstration shows the real and imaginary parts of the weak value of charge in one arm of an electronic Mach–Zehnder interferometer [2].



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An electronic Mach-Zehnder interferometer [2] is sketched in the following figure.
Panel a represents an electronic Mach–Zehnder interferometer realized in a Hall bar. The electrons' states are edge states (full lines). Inter-edge tunneling (dashed lines) takes place at the quantum point contacts. Panel b represents a double electronic Mach–Zehnder interferometer, where the second Mach–Zehnder interferometer is used as a detector of charge in arm 2 of the first interferometer.
In this system the preselection is determined by the injection from source D1 and by the transmission and reflection coefficients of QPC A. Similarly, the postselection corresponds to the detection of electrons in the drain D2 and can be controlled by the transmission and reflection coefficient of QPC B. Physically one can control the voltage bias at S1, the Aharonov-Bohm flux (and hence the phase difference between electrons traveling through arms 1 and 2), the temperature, and finally the duration of the measurement. The result of the measurement cannot be interpreted in terms of single electrons, but many-body effects have to be considered.
In the Demonstration it is possible to choose whether to plot the charge's weak value, (in units of the electron charge), as a function of the voltage bias (measured in ), the phase difference, or the temperature (measured in mK). (The phase difference includes the Aharonov–Bohm flux and the phases of the transmission coefficients of both the quantum point contacts.) In all cases it is possible to control the other parameters, including the transparencies of both the quantum point contacts and the duration of the measurement (measured in nsec). In the upper panel of the Demonstration we also plot the upper bound for standard quantum average of the charge in arm 2 (the lower bound is obviously 0): the weak value is clearly beyond the marked bounds.
[1] Y. Aharonov, D. Z. Albert, and L. Vaidman, "How the Result of a Measurement of a Component of the Spin of a Spin-1/2 Particle Can Turn Out to Be 100," Phys. Rev. Lett. 60, 1988 p.1351.
[2] V. Shpitalnik, Y. Gefen, A. Romito, "Tomography of Many-Body Weak Values: Mach-Zehnder Interferometry," arXiv:0805.2737.
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