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arXiv:1312.2981 · Physical Review Letters

Amplification of Angular Rotations Using Weak Measurements

Omar S. Magaña‐Loaiza, Mohammad Mirhosseini, Brandon Rodenburg, Robert W. Boyd

Published May 21, 2014• 223 citations
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Abstract

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.

DOI
10.1103/physrevlett.112.200401

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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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Using the profound interferometric philosophy of weak-value amplification, we propose a simple, general, and robust polarization method for the amplification and quantification of small magnitudes of all possible polarization anisotropy effects in a single experimental embodiment. The approach is experimentally realized by introducing a weak coupling between the polarization degree of freedom of light and the path degree of freedom in a Mach-Zehnder interferometer in the presence of a weak anisotropy effect. Real and imaginary weak-value amplifications of different polarization anisotropy effects are manifested as characteristic changes in the relevant Stokes vector elements at the exit port of the interferometer, which follow orthogonal trajectories in the Poincar\'e sphere. The proof-of-concept experiment demonstrates that by using this scheme, one can faithfully extract and quantify an anisotropy parameter that is smaller than the typical sensitivity of measurement of a given Stokes parameter of a traditional polarimeter by a large weak-value amplification factor. This opens up the possibility of a sample measuring weak-value polarimeter for studying the rich variety of fundamental optical effects and for materials characterization and precision metrology.

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Background: Weak measurement involves weak coupling between the system and the measuring device (pointer) enables large amplification and high precision measurement of small physical parameters. The outcome of this special measurement procedure involving nearly mutually orthogonal pre- and post-selection of states in such weakly interacting systems leads to weak value that can become exceedingly large and lie outside the eigenvalue spectrum of the measured observable. This unprecedented ability of weak value amplification of small physical parameters has been successfully exploited for various metrological applications in the optical domain and beyond. Even though it is a quantum mechanical concept, it can be understood using the classical electromagnetic theory of light and thus can be realized in classical optics. Objective: Here, we briefly review the basic concepts of weak measurement and weak value amplification, provide illustrative examples of its implementation in various optical domains. The applications involve measuring ultra-sensitive beam deflections, high precision measurements of angular rotation, phase shift, temporal shift, frequency shift and so forth, and expand this extraordinary concept in the domain of nano-optics and plasmonics. Methods: In order to perform weak value amplification, we have used Gaussian beam and spectral response as the pointer subsequently. The polarization state associated with the pointer is used as pre and post-selection device. Results: We reveal the weak value amplification of sub-wavelength optical effects namely the Goos-Hänchen shift and the spin hall shift. Further, we demonstrate weak measurements using spectral line shape of resonance as a natural pointer, enabling weak value amplification beyond the conventional limit, demonstrating natural weak value amplification in plasmonic Fano resonances and so forth. The discussed concepts could have useful implications in various nano-optical systems to amplify tiny signals or effects. Conclusion: The emerging prospects of weak value amplification towards the development of novel optical weak measurement devices for metrological applications are extensively discussed.

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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
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.
Weak measurement, which can amplify a weak signal, has shown great significance in precision measurements. The amplification is usually realized through the weak value and the propagation factor. We show that the orbital angular momentum (OAM) can provide another dimension for amplification that is linearly proportional to the OAM number. We employ OAM to measure the spin Hall effect of light and demonstrate that the OAM-enabled amplification is compatible with the weak value amplification and the propagation amplification. This work is probable to promote the application of OAM in precision measurements.
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
Using the profound interferometric philosophy of weak-value amplification, we propose a simple, general, and robust polarization method for the amplification and quantification of small magnitudes of all possible polarization anisotropy effects in a single experimental embodiment. The approach is experimentally realized by introducing a weak coupling between the polarization degree of freedom of light and the path degree of freedom in a Mach-Zehnder interferometer in the presence of a weak anisotropy effect. Real and imaginary weak-value amplifications of different polarization anisotropy effects are manifested as characteristic changes in the relevant Stokes vector elements at the exit port of the interferometer, which follow orthogonal trajectories in the Poincar\'e sphere. The proof-of-concept experiment demonstrates that by using this scheme, one can faithfully extract and quantify an anisotropy parameter that is smaller than the typical sensitivity of measurement of a given Stokes parameter of a traditional polarimeter by a large weak-value amplification factor. This opens up the possibility of a sample measuring weak-value polarimeter for studying the rich variety of fundamental optical effects and for materials characterization and precision metrology.
Background: Weak measurement involves weak coupling between the system and the measuring device (pointer) enables large amplification and high precision measurement of small physical parameters. The outcome of this special measurement procedure involving nearly mutually orthogonal pre- and post-selection of states in such weakly interacting systems leads to weak value that can become exceedingly large and lie outside the eigenvalue spectrum of the measured observable. This unprecedented ability of weak value amplification of small physical parameters has been successfully exploited for various metrological applications in the optical domain and beyond. Even though it is a quantum mechanical concept, it can be understood using the classical electromagnetic theory of light and thus can be realized in classical optics. Objective: Here, we briefly review the basic concepts of weak measurement and weak value amplification, provide illustrative examples of its implementation in various optical domains. The applications involve measuring ultra-sensitive beam deflections, high precision measurements of angular rotation, phase shift, temporal shift, frequency shift and so forth, and expand this extraordinary concept in the domain of nano-optics and plasmonics. Methods: In order to perform weak value amplification, we have used Gaussian beam and spectral response as the pointer subsequently. The polarization state associated with the pointer is used as pre and post-selection device. Results: We reveal the weak value amplification of sub-wavelength optical effects namely the Goos-Hänchen shift and the spin hall shift. Further, we demonstrate weak measurements using spectral line shape of resonance as a natural pointer, enabling weak value amplification beyond the conventional limit, demonstrating natural weak value amplification in plasmonic Fano resonances and so forth. The discussed concepts could have useful implications in various nano-optical systems to amplify tiny signals or effects. Conclusion: The emerging prospects of weak value amplification towards the development of novel optical weak measurement devices for metrological applications are extensively discussed.
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.