A mathematical extension of the weak value formalism to the simultaneous measurement of multiple parameters is presented in the context of an optical focused vector beam scatterometry experiment. In this example, preselection and postselection are achieved via spatially-varying polarization control, which can be tailored to optimize the sensitivity to parameter variations. Initial experiments for the two-parameter case demonstrate that this method can be used to measure physical parameters with resolutions at least 1000 times smaller than the wavelength of illumination.The concepts of weak value and weak measurement were introduced by Aharonov, Albert and Vaidman in 1988 [1][2][3] as an alternative to the standard measurement formalism of quantum mechanics. For a quantity associated with an operator B, a standard measurement is related to the expected value Φ|B|Φ / Φ|Φ , where Φ is the state vector for the quantum state being measured. Since the state is normalized, the inner product in the denominator is typically taken as unity. Clearly, for Hermitian operators, this expected value is real and limited to the range of values spanned by the eigenvalues of B. On the other hand, weak measurements are based on weak values defined as Φ post |B|Φ pre / Φ post |Φ pre , where Φ pre and Φ post are preselected and postselected states. It is easy to see that there is no bound to a weak value since the denominator can be made arbitrarily small by appropriate preselection and postselection. In fact, weak values need not even be real-valued. Weak measurements have been employed, for example, to measure very small angular deviations with great precision [4][5][6][7].
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