We experimentally demonstrate a fundamental relationship between the weak value of an observable and the complex zero of the response function of a system by employing weak measurement on spin Hall shift of a Gaussian light beam. Using this relationship, we show that extremely large weak values far beyond its upper bound in the conventional linear-response regime can be experimentally obtained from the position of the minima of the pointer intensity profile corresponding to the real part of the complex zero of the response function. The imaginary part of the complex zero, on the other hand, is related to the spatial gradient of the geometric phase of light, which in this particular scenario evolves due to the weak interaction and the pre- and postselections of the polarization states. These relationships between the weak value and the complex zeros of the response function may provide new insights on weak measurements in a wide class of physical systems. Extraction of large weak values outside the usual domain of its validity and quantification of small interaction parameters using a physically meaningful and experimentally accessible system property such as the response function may open up a new paradigm of weak measurement.
We experimentally demonstrate a fundamental relationship between the weak value of an observable and the complex zero of the response function of a system by employing weak measurement on spin Hall shift of a Gaussian light beam. Using this relationship, we show that extremely large weak values far beyond its upper bound in the conventional linear-response regime can be experimentally obtained from the position of the minima of the pointer intensity profile corresponding to the real part of the complex zero of the response function. The imaginary part of the complex zero, on the other hand, is related to the spatial gradient of the geometric phase of light, which in this particular scenario evolves due to the weak interaction and the pre- and postselections of the polarization states. These relationships between the weak value and the complex zeros of the response function may provide new insights on weak measurements in a wide class of physical systems. Extraction of large weak values outside the usual domain of its validity and quantification of small interaction parameters using a physically meaningful and experimentally accessible system property such as the response function may open up a new paradigm of weak measurement.
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.
Geometric Spin Hall effect of light (SHEL) has attracted considerable attention because of the universality of this variant of the spin Hall shift and its independence on the light-matter interactions and materials properties. However, the magnitude of geometric SHEL is typically very small, in the sub-wavelength domain, making it difficult for direct experimental observation. Here, we have applied weak measurement schemes to amplify and faithfully observe the geometric SHEL. In our experiment, the input beam is pre-selected in linear polarization basis (p- or slinear polarization) and post selections are done at nearly orthogonal linear polarizations (small angle ϵ away from the exact orthogonal).This results in weak value amplification (~cotϵ) of the resulting shift of the beam centroid. Moreover, this process also leads to selective conversion of spatial to angular nature of geometric SHEL, which along with the weak value amplification leads to manifold amplification of the resulting SHEL, enabling its reliable experimental detection. A key feature of our weak measurement scheme is that both the weak perturbation (tiny geometric SHEL) and the post selection are done simultaneously by a single optical element, namely, a linear polarizer. The dependence of the weak value of the geometric SHEL on the pre and the post selection of polarization states in both linear and elliptical basis were also investigated in details. The specifics of the different weak measurement schemes, various intriguing experimental manifestations of the weak value amplified geometric SHEL, their analysis / interpretation via polarization operator based treatment of weak measurements is presented here.
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.
The weak value, the average result of a weak measurement, has proven useful for probing quantum and classical systems. Examples include amplifying small signals, investigating quantum paradoxes, and elucidating fundamental quantum phenomena such as geometric phase. A key characteristic of the weak value is that it can be complex, in contrast to a standard expectation value. However, typically only either the real or imaginary component of the weak value is determined in a given experimental setup. Weak measurements can be used to, in a sense, simultaneously measure noncommuting observables. This principle was used in the direct measurement of the quantum wave function. However, the wave function's real and imaginary components, given by a weak value, are determined in different setups or on separate ensembles of systems, putting the procedure's directness in question. To address these issues, we introduce and experimentally demonstrate a general method to simultaneously read out both components of the weak value in a single experimental apparatus. In particular, we directly measure the polarization state of an ensemble of photons using weak measurement. With our method, each photon contributes to both the real and imaginary parts of the weak-value average. On a fundamental level, this suggests that the full complex weak value is a characteristic of each photon measured.
The extraordinary concept of weak value amplification and a number of other seemingly
counter-intuitive wave phenomena are manifestations of fine interference effects.
In this report, we show that weak value amplification naturally arises in the cases of
nearly destructive interference between the two orthogonal states of the observable which
mimics the corresponding near destructive interference between the eigenstates of the
measuring observable as a consequence of nearly mutually orthogonal pre and post selections
of the system states. In order to elucidate this, we first experimentally demonstrate
a weak measurement concept that uses near destructive interference of two paths
of an interferometer having slightly rotated linear polarization states of light. By generating
nearly destructive interference between the two paths with small amplitude and
phase offsets of the waves, we observe both real and imaginary (respectively) weak value
amplification of the small polarization rotation leading to large rotation of the polarization
vector orientation and dramatic changes in the circular polarization descriptor
fourth Stokes vector element of light, respectively. We go on to demonstrate that the giant
enhancement of magneto-optical Kerr rotation, by which we can claim that we have
found a non-modulation based and cost-effective magneto-optical Kerr effect (MOKE)
measurement. At the end, we show successful weak value amplification of spin Hall shift
in magneto-optical sample in both longitudinal and polar MOKE. Our ongoing work is
to find out the relation between magneto-optical Kerr rotation and spin Hall shift of light
in magnetic material.
Abstract Geometric phases and weak values (WVs) are two fundamental concepts that were originally introduced in the realm of quantum mechanics. In the course of time, it became clear that the two concepts apply in both the quantum domain and the classical domain. Moreover, the two concepts proved to be intimately connected with one another, as disclosed by Sjöqvist some years ago (Sjöqvist 2006 Phys. Lett. A 359 187). WVs were so named in reference to very weak couplings between a system’s observable, e.g. light polarization, and its measuring device, the ‘pointer’. However, the actual definition of WVs does not involve the system-pointer coupling strength. We have addressed the strong coupling regime both theoretically and experimentally, thereby obtaining geometric phases out of WVs. We report experimental results as a proof-of-concept, using classical light beams and single-shot images. Our approach applies without essential modifications also to single photons.