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arXiv:1209.3110 · Physical Review A

Extracting joint weak values from two-dimensional spatial displacements

H. Kobayashi, Graciana Puentes, Yutaka Shikano

Published Nov 5, 2012• 30 citations
arXivHTMLPDF

Abstract

The joint weak value is a counterfactual quantity related to quantum correlations and quantum dynamics, which can be retrieved via weak measurements, as initiated by Aharonov and colleagues. In this paper, we provide a full analytical extension of the method described by Puentes et al. [Phys. Rev. Lett. 109, 040401 (2012)], to extract the joint weak values of single-particle operators from two-dimensional spatial displacements of Laguerre-Gauss probe states, for the case of the azimuthal index $|l|>1$. This method has a statistical advantage over previous ones since information about the conjugate observable, i.e., the momentum displacement of the probe, is not required. Moreover, we demonstrate that, under certain conditions, the joint weak value can be extracted directly from spatial displacements without any additional data processing.

DOI
10.1103/physreva.86.053805

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Comment on “Weak Measurements with Orbital-Angular-Momentum Pointer States”

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In a recent Letter (Phys. Rev. Lett. 109, 040401 (2012)), G. Puentes, N. Hermosa and J. P. Torres report a scheme for extracting higher-order weak values by using orbital-angular momentum states as pointer states. They claim that such weak values are inaccessible with a Gaussian pointer state only. In this Comment, we show that the Gaussian pointer state by itself can provide access to the higher-order weak value, if suitable pointer displacement is observed.

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Puentes, Hermosa, and Torres Reply:

Graciana Puentes, Nathaniel Hermosa, Juan P. Torres

In a preceding Comment [1] to our Letter [2] concerning weak measurements with orbital-angular-momentum (OAM) pointer states [2], Pan and Panigrahi show that the real and imaginary parts of higher-order weak values can be made accessible using Gaussian pointer states.In [2], our goal is to show that by using pointer states embedded with OAM it is possible to extract higher-order weak values where pointer states with a Gaussian shape cannot.As a consequence, our work put forward a new tool for weak amplification schemes, i.e., the use of pointer states with more general spatial forms that could be advantageous in some scenarios.We did not intend to claim that a pointer state with orbital angular momentum was needed to extract higherorder weak values in general.We clarify our point with this Reply.In our Letter we consider a specific Hamiltonian of interaction of the form H ¼ g A AP x þ g B BP y , where A, B are operators, (P x , P y ) are the pointer momentum operators, conjugate to the pointer position operators (X, Y), and g A;B are coupling constants.Moreover, we calculate the two-dimensional pointer displacement hXYi, corresponding to a specific measurement.We emphasize that we consider a particular interaction and a specific measurement, which are generally dictated by the physical system under investigation.The result, Eq. ( 4) of our Letter, shows that pointer states with OAM (l ¼ Æ1) can be used to retrieve the imaginary part of the higher-order weak values hA 2 i w and hB 2 i w , whereas this is not possible with pointer states with no OAM (l ¼ 0).In addition, Eq. ( 9) of our Letter shows, by means of a specific example with B ¼ 0, that hXYi ¼ 0 for Gaussian pointer states.On the contrary, a pointer state with l ¼ Æ1 allows us to extract the imaginary part of the weak value hA 2 i w .Pan and Panigrahi show that by measuring hX 2 i one can obtain the real part of the weak value hA 2 i w using a Gaussian pointer [see Eq. ( 4) of [1]].However, in order to access its imaginary part, they are forced to use a different interaction Hamiltonian, and a measurement observable involving noncommuting operators.The work of Pan and Panigrahi proves the strength of our proposal: One

Physical Review Letters•Jul 10, 2013•1 citation

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arXiv:2412.07440•Dec 10, 2024•0 citations
The joint weak value is a counterfactual quantity related to quantum correlations and quantum dynamics, which can be retrieved via weak measurements, as initiated by Aharonov and colleagues. In this paper, we provide a full analytical extension of the method described by Puentes et al. [Phys. Rev. Lett. 109, 040401 (2012)], to extract the joint weak values of single-particle operators from two-dimensional spatial displacements of Laguerre-Gauss probe states, for the case of the azimuthal index ∣l∣>1|l|>1∣l∣>1. This method has a statistical advantage over previous ones since information about the conjugate observable, i.e., the momentum displacement of the probe, is not required. Moreover, we demonstrate that, under certain conditions, the joint weak value can be extracted directly from spatial displacements without any additional data processing.
Here, joint spatial weak measurements with higher-order Laguerre-Gaussian (LG) point states are investigated experimentally. From the intensity patterns of the final LG point states, two dimensional position operators 〈X〉 and 〈Y〉 as well as high-order position operators 〈XY〉, 〈X2 − Y2〉, 〈X3〉, and 〈Y3〉 are extracted simultaneously, from which both the complex weak values and joint weak values of two non-commuting observables can be obtained. The enhancement of joint weak values by the postselection state are analyzed. The simple relationship between the expectation values of position operators and the azimuthal and radial indexes of LG modes allows us to identify the mode indexes directly. A simple and robust scheme based on an optical window is demonstrated to monitor the LG mode indexes. These findings deepen the understanding of the weak measurement and provide an alternate and effective method for LG mode index monitoring.
Weak measurements are a unique tool for accessing information about weakly interacting quantum systems with minimal back action. Joint weak measurements of single-particle operators with pointer states characterized by a two-dimensional Gaussian distribution can provide, in turn, key information about quantum correlations that can be relevant for quantum information applications. Here we demonstrate that by employing two-dimensional pointer states endowed with orbital angular momentum (OAM), it is possible to extract weak values of the higher order moments of single-particle operators, an inaccessible quantity with Gaussian pointer states only. We provide a specific example that illustrates the advantages of our method both in terms of signal enhancement and information retrieval.
Abstract We propose an efficient method to accurately identify two mode indices ( p and l ) of Laguerre–Gaussian (LG) beams simultaneously with weak value measurement (WVM). Concise relations between the spatial displacements from the WVM and two mode indices of LG beams have been obtained. Compared with the traditional measurements, the advantage of using our method is that LG beams with large p and l can be identified precisely. A joint spatial WVM with a looser approximate condition has been presented, which realizes better identification than the individual spatial WVM with correction terms or strict approximation. Furthermore, the signal-to-noise ratio (SNR) of the spatial WVM is also discussed.
In a preceding Comment [1] to our Letter [2] concerning weak measurements with orbital-angular-momentum (OAM) pointer states [2], Pan and Panigrahi show that the real and imaginary parts of higher-order weak values can be made accessible using Gaussian pointer states.In [2], our goal is to show that by using pointer states embedded with OAM it is possible to extract higher-order weak values where pointer states with a Gaussian shape cannot.As a consequence, our work put forward a new tool for weak amplification schemes, i.e., the use of pointer states with more general spatial forms that could be advantageous in some scenarios.We did not intend to claim that a pointer state with orbital angular momentum was needed to extract higherorder weak values in general.We clarify our point with this Reply.In our Letter we consider a specific Hamiltonian of interaction of the form H ¼ g A AP x þ g B BP y , where A, B are operators, (P x , P y ) are the pointer momentum operators, conjugate to the pointer position operators (X, Y), and g A;B are coupling constants.Moreover, we calculate the two-dimensional pointer displacement hXYi, corresponding to a specific measurement.We emphasize that we consider a particular interaction and a specific measurement, which are generally dictated by the physical system under investigation.The result, Eq. ( 4) of our Letter, shows that pointer states with OAM (l ¼ Æ1) can be used to retrieve the imaginary part of the higher-order weak values hA 2 i w and hB 2 i w , whereas this is not possible with pointer states with no OAM (l ¼ 0).In addition, Eq. ( 9) of our Letter shows, by means of a specific example with B ¼ 0, that hXYi ¼ 0 for Gaussian pointer states.On the contrary, a pointer state with l ¼ Æ1 allows us to extract the imaginary part of the weak value hA 2 i w .Pan and Panigrahi show that by measuring hX 2 i one can obtain the real part of the weak value hA 2 i w using a Gaussian pointer [see Eq. ( 4) of [1]].However, in order to access its imaginary part, they are forced to use a different interaction Hamiltonian, and a measurement observable involving noncommuting operators.The work of Pan and Panigrahi proves the strength of our proposal: One
Quantum weak measurements became extremely popular in classical optics to amplify small optical signals for fundamental interests and potential applications. Later, a more general extension, joint weak measurement has been proposed to extract weak value from a joint quantum measurement. However, the detection of joint weak value in the realm of classical optics remains less explored. Here, using the polarization-dependent longitudinal and transverse optical beam shift as a platform, we experimentally realize the quantum joint weak measurement in a classical optical setting. Polarization states are cleverly pre and post-selected, and different single and joint canonical position-momentum observables of the beam are experimentally extracted and subsequently analyzed for successful detection of complex joint weak value. We envision that this work will find usefulness for gaining fundamental insights on quantum measurements and to tackle analogous problems in optics.