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arXiv:1308.4523 · Physical Review Letters

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

A. K. Pan, Prasanta Kumar Panigrahi

Published Jul 10, 2013• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.111.028901

Related papers

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

Weak Measurements with Orbital-Angular-Momentum Pointer States

Graciana Puentes, Nathaniel Hermosa, Juan P. Torres

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.

arXiv:1204.3544•Physical Review Letters•Jul 24, 2012•63 citations

Weak measurement takes a simple form for cumulants

Graeme Mitchison

A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wave function for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wave function takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by Jozsa [R. Jozsa, Phys. Rev. A 76, 044103 (2007)]. This is even more striking in the case of sequential weak measurements [G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007)]. These are carried out by coupling several pointers at different stages of the evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wave function. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.

arXiv:0710.5113•Physical Review A•May 2, 2008•12 citations

Visualizing Quantum State Rotations through Weak Measurements of Orbital Angular Momentum

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Using the technique of direct measurement, we observe the action of the angular momentum operator as the generator of rotations. Our method employs weak measurements of orbital angular momentum and strong measurements of angular position.

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Amplification of Angular Rotations Using Weak Measurements

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We present a weak measurement protocol that permits a sensitive estimation of angular rotations based on the concept of weak-value amplification. The shift in the state of a pointer, in both angular position and the conjugate orbital angular momentum bases, is used to estimate angular rotations. This is done by an amplification of both the real and imaginary parts of the weak-value of a polarization operator that has been coupled to the pointer, which is a spatial mode, via a spin-orbit coupling. Our experiment demonstrates the first realization of weak-value amplification in the azimuthal degree of freedom. We have achieved effective amplification factors as large as 100, providing a sensitivity that is on par with more complicated methods that employ quantum states of light or extremely large values of orbital angular momentum.

arXiv:1312.2981•Physical Review Letters•May 21, 2014•223 citations

Weak measurements with non-integer orbital angular momentum states

Jörg B. Götte

Weak measurements typically require two non-orthogonal states, which are not eigenstates of an observable. This is why weak values of the orbital angular momentum operator seem counterintuitive or involve additional auxiliary observables, such as polarization. We show how the use of states emerging from spiral phase plates or holograms with fractional step height can be used to construct weak values of the orbital angular momentum operator.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 25, 2014•1 citation
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.
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
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.
A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wave function for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wave function takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by Jozsa [R. Jozsa, Phys. Rev. A 76, 044103 (2007)]. This is even more striking in the case of sequential weak measurements [G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, 062105 (2007)]. These are carried out by coupling several pointers at different stages of the evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wave function. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.
We present a weak measurement protocol that permits a sensitive estimation of angular rotations based on the concept of weak-value amplification. The shift in the state of a pointer, in both angular position and the conjugate orbital angular momentum bases, is used to estimate angular rotations. This is done by an amplification of both the real and imaginary parts of the weak-value of a polarization operator that has been coupled to the pointer, which is a spatial mode, via a spin-orbit coupling. Our experiment demonstrates the first realization of weak-value amplification in the azimuthal degree of freedom. We have achieved effective amplification factors as large as 100, providing a sensitivity that is on par with more complicated methods that employ quantum states of light or extremely large values of orbital angular momentum.