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arXiv:2209.09044 · Physical Review A

Sequential unsharp measurement of photon polarization

Петер Адам, Lajos Diósi

Published Jun 20, 2023• 3 citations
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Abstract

We propose a general experimental scheme based on binary trees of partially polarizing beam splitters (PPBSs) for realizing sequential unsharp measurements of photon polarization. The sharpnesses and the bases of the particular photon polarization measurements can be chosen arbitrarily by using corresponding PPBSs and phase plates in the setup. In the limit of low sharpnesses the scheme can realize sequential weak measurements, too. We develop a general formalism for describing sequential unsharp measurements of photon polarization in which the particular unsharp measurements are characterized by appropriate measurement operators. We show that a straightforward experimental realization of this model is the proposed scheme. In this formalism the output polarization states after the sequential measurement and any correlation functions characterizing the measurement results can be easily calculated. Our model can be used for analyzing the consequences of applying postselection and reselection in the measurement. We derive the anomalous mean value for an unsharp polarization measurement with postselection and the anomalous second-order correlation function for the sequential unsharp measurement of photon polarization with reselection. We show that these anomalies can be easily measured using the proposed scheme.

DOI
10.1103/physreva.107.063706

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We propose a general experimental scheme based on binary trees of partially polarizing beam splitters (PPBSs) for realizing sequential unsharp measurements of photon polarization. The sharpnesses and the bases of the particular photon polarization measurements can be chosen arbitrarily by using corresponding PPBSs and phase plates in the setup. In the limit of low sharpnesses the scheme can realize sequential weak measurements, too. We develop a general formalism for describing sequential unsharp measurements of photon polarization in which the particular unsharp measurements are characterized by appropriate measurement operators. We show that a straightforward experimental realization of this model is the proposed scheme. In this formalism the output polarization states after the sequential measurement and any correlation functions characterizing the measurement results can be easily calculated. Our model can be used for analyzing the consequences of applying postselection and reselection in the measurement. We derive the anomalous mean value for an unsharp polarization measurement with postselection and the anomalous second-order correlation function for the sequential unsharp measurement of photon polarization with reselection. We show that these anomalies can be easily measured using the proposed scheme.
We discuss the abstract structure of sequential weak measurement (WM) of general observables. In all orders, the sequential WM correlations without postselection yield the corresponding correlations of the Wigner function, offering direct quantum tomography through the moments of the canonical variables. Correlations in spin-12\frac{1}{2}21​ sequential weak measurements coincide with those in strong measurements, they are constrained kinematically, and they are equivalent with single measurements. In sequential WMs with postselection, an anomaly occurs, different from the weak value anomaly of single WMs. In particular, the spread of polarization $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ as measured in double WMs of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}$ will diverge for certain orthogonal pre- and postselected states.
A scheme is discussed for measuring NNNth-order coherences of two orthogonally polarized light fields in a single spatial mode at very limited experimental cost. To implement the scheme, the only measurements needed are the NNNth-order intensity moments after the light beam has passed through two quarter-wave plates, one half-wave plate, and a polarizing beam splitter for specific settings of the wave plates. It is shown that this method can be applied for arbitrarily large NNN. A set of explicit values is given for the settings of the wave plates, constituting an optimal measurement of the NNNth-order coherences for any NNN. For Fock states the method introduced here corresponds to a full state tomography. Applications of the scheme to systems other than polarization optics are discussed.
A sequential measurement of two non-commuting quantum observables results in a joint probability distribution for all output combinations that can be explained in terms of an initial joint quasi-probability of the non-commuting observables, modified by the resolution errors and back-action of the initial measurement. Here, we show that the error statistics of a sequential measurement of photon polarization performed at different measurement strengths can be described consistently by an imaginary correlation between the statistics of resolution and back-action. The experimental setup was designed to realize variable strength measurements with well-controlled imaginary correlation between the statistical errors caused by the initial measurement of diagonal polarizations, followed by a precise measurement of the horizontal/vertical polarization. We perform the experimental characterization of an elliptically polarized input state and show that the same complex joint probability distribution is obtained at any measurement strength.
One of the most intriguing aspects of quantum mechanics is the impossibility of measuring at the same time observables corresponding to noncommuting operators, because of quantum uncertainty. This impossibility can be partially relaxed when considering joint or sequential weak value evaluation. Indeed, weak value measurements have been a real breakthrough in the quantum measurement framework that is of the utmost interest from both a fundamental and an applicative point of view. In this Letter, we show how we realized for the first time a sequential weak value evaluation of two incompatible observables using a genuine single-photon experiment. These (sometimes anomalous) sequential weak values revealed the single-operator weak values, as well as the local correlation between them.
Entanglement is a fundamental resource for quantum information science. However, bipartite entanglement is destroyed when one particle is observed via projective (sharp) measurements, as is typically the case in most experiments. Here, we show experimentally that, if instead of sharp measurements, one performs many sequential unsharp measurements on one particle that are suitably chosen depending on the previous outcomes, then entanglement is preserved and it is possible to reveal quantum correlations through measurements on the second particle at any step of the sequence. Specifically, we observe that pairs of photons entangled in polarization maintain their entanglement when one particle undergoes three sequential measurements and that each of these can be used to violate a Clauser-Horne-Shimony-Holt inequality. This proof-of-principle experiment demonstrates the possibility of repeatedly harnessing two crucial resources, entanglement and Bell nonlocality, that, in most quantum protocols, are destroyed after a single measurement. The protocol we use, which in principle works for an unbounded sequence of measurements, can be useful for randomness extraction.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.