Full counting statistics of noncommuting variables: The case of spin counts

We discuss the full counting statistics (FCS) of noncommuting variables with the measurement of successive spin counts in noncollinear directions taken as an example. We show that owing to an irreducible detector back action, the FCS in this case may be sensitive to the dynamics of the detectors, and may differ from the predictions obtained with using a naive version of the projection postulate. We present here a general model of detector dynamics and path-integral approach to the evaluation of FCS. We concentrate further on a simple “diffusive” model of the detector dynamics where the FCS can be evaluated with transfer-matrix method. The resulting probability distribution of spin counts is characterized by anomalously large higher cumulants and substantially deviates from Gaussian statistics.

Figure 1

The proposed spin current detector. An electron with velocity $\mathbf{v}$ and spin $\mathbf{S}$ induces a voltage drop in a capacitor. The electric field $\mathbf{E}$ inside the capacitor produces an Aharonov-Casher phase shift on the electrons.

Figure 2

The setup considered, in the case of three spin detectors and one charge detector.

Figure 3

(Color online) The log of probability as a function of ${\nu }_3∕{\nu }_1$ for different values of ${\nu }_1$ for the configuration studied in the text. All the curves have been shifted by ${\nu }_1^2∕2$. The upper (full line) curves correspond to the result of the FCS approach, and the lower ones (dashed line) to the PP. The black dotted curve is the limiting scaling curve given by Eq. (47).