Filippo M. Miatto

Colloquium: Understanding Quantum Weak Values: Basics and Applications

The Institute of Optics, University of Rochester, Rochester, New York 14627, USAand Department of Physics, University of Ottawa, Ottawa, Ontario, Canada(published 28 March 2014)Since its introduction 25 years ago, the quantum weak value has gradually transitioned from atheoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosionof weak value experiments, its interpretation has historically been a subject of confusion. Here apragmaticintroductiontotheweakvalueintermsofmeasurablequantitiesispresented,alongwithanexplanation for how it can be determined in the laboratory. Further, its application to three distinctexperimental techniques is reviewed. First, as a large interaction parameter it can amplify smallsignals above technical background noise. Second, as a measurable complex value it enables noveltechniques for direct quantum state and geometric phase determination. Third, as a conditionedaverage of generalized observable eigenvalues it provides a measurable window into nonclassicalfeaturesofquantummechanics.Inthisselectivereview,asingleexperimentalconfigurationtodiscussand clarify each of these applications is used.

Divergence of an orbital-angular-momentum-carrying beam upon propagation

There is recent interest in the use of light beams carrying orbital angular momentum (OAM) for creating multiple channels within free-space optical communication systems. One crucial issue is that, for a given beam size at the transmitter, the beam divergence angle increases with increasing OAM. Therefore the larger the value of OAM, the larger the aperture required at the receiving optical system if the efficiency of detection is to be maintained. Confusion exists as to whether this divergence scales linearly with, or with the square root of, the beams OAM. We clarify how both these scaling laws are valid, depending upon whether it is the radius of the waist of the beams Gaussian term or the radius of rms intensity of the beam that is kept constant while varying the OAM.

Full characterization of the quantum spiral bandwidth of entangled biphotons

Spontaneous parametric down-conversion has been shown to be a reliable source of entangled photons. Among the wide range of properties shown to be entangled, it is the orbital angular momentum that is the focus of our study. We investigate, in particular, the bi-photon state generated using a Gaussian pump beam. We derive an expression for the simultaneous correlations in the orbital angular momentum, l, and radial momentum, p, of the down-converted Laguerre-Gaussian beams. Our result allows us, for example, to calculate the spiral bandwidth with no restriction on the geometry of the beams: l, p, and the beam widths are all free parameters. Moreover, we show that, with the usual paraxial and collinear approximations, a fully analytic expression for the correlations can be derived.

Limitations to the determination of a Laguerre-Gauss spectrum via projective, phase-flattening measurement

One of the most widely used techniques for measuring the orbital angular momentum (OAM) components of a light beam is to flatten the spiral phase front of a mode, in order to couple it to a single-mode optical fiber (SMOF). This method, however, suffers from an efficiency that depends on the OAM of the initial mode and on the presence of higher-order radial modes. The reason is that once the phase has been flattened, the field retains its ringed intensity pattern and is therefore a nontrivial superposition of purely radial modes, of which only the fundamental one couples to a SMOF. In this paper, we study the efficiency of this technique both theoretically and experimentally. We find that even for low values of the OAM, a large amount of light can fall outside the fundamental mode of the fiber, and we quantify the losses as functions of the waist of the coupling beam of the OAM and radial indices. Our results can be used as a tool to remove the efficiency bias where fair-sampling loopholes are not a concern. However, we hope that our study will encourage the development of better detection methods of the OAM content of a beam of light.

Determining the dimensionality of bipartite orbital-angular-momentum entanglement using multi-sector phase masks

The Shannon dimensionality of orbital-angular-momentum (OAM) entanglement produced in spontaneous parametric down-conversion can be probed by using multi-sector phase analysers [1]. We demonstrate a spatial light modulator-based implementation of these analysers, and use it to measure a Schmidt number of about 50.

High-dimensional entanglement with orbital-angular-momentum states of light

We engineer high-dimensional orbital-angular-momentum entanglement of photon pairs that emerge from a parametric down-conversion source. By means of two angular state analysers, in essence composed of a rotatable multi-sector phase plate and a single-mode fibre, we perform selective projective measurements that maximize the Shannon dimensionality D of the measured entanglement. The multi-sector phase plates have a binary phase profile along the azimuthal coordinate, and the arc sector sizes are optimized so as to maximize D .W e fi nd that the maximum dimensionality increases linearly with the number of sectors N. The potential of our method is illustrated with an experiment for N = 4, yielding D = 16.5.

Spatial Schmidt modes generated in parametric down-conversion

This paper presents the general spatial Schmidt decomposition of two-photon fields generated in spontaneous parametric down-conversion (SPDC). It discusses in particular the separation of the radial and azimuthal degrees of freedom, the role of projection in modal analysis, and the benefits of collinear phase mismatch. The paper is written in a review style and presents a wealth of numerical results. It aims at emphasising the physics beyond the mathematics, through discussions and graphical representations of key results. The two main conclusions of the paper are the finding of a better law to describe the effective dimensionality of the spatial part of the total Hilbert space and a possible novel feature of the radial Schmidt modes.

Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems

The measurement of high-dimensional entangled states of orbital angular momentum prepared by spontaneous parametric down-conversion can be considered in two separate stages: a generation stage and a detection stage. Given a certain number of generated modes, the number of measured modes is determined by the measurement apparatus. We derive a simple relationship between the generation and detection parameters and the number of measured entangled modes.

Fair sampling perspective on an apparent violation of duality

Significance In 2012, Menzel et al. reported on the results of a fundamental experiment raising questions regarding the simultaneous observation of wave-like and particle-like properties in a given quantum system. Whereas the general applicability of the duality principle to entangled subsystems is an open question, we bring the current understanding of the duality principle a step forward by theoretically deriving the strongest relations between the visibility of an interference pattern and the which-way information in a two-way interferometer such as Young’s double slit. This formalism successfully describes tests of duality where postselection on a subset of the interference pattern is applied. Our analysis even reconciles the surprising results of Menzel et al. with the duality principle in its standard form. In the event in which a quantum mechanical particle can pass from an initial state to a final state along two possible paths, the duality principle states that “the simultaneous observation of wave and particle behavior is prohibited” [Scully MO, Englert B-G, Walther H (1991) Nature 351:111–116]. Whereas wave behavior is associated with the observation of interference fringes, particle behavior generally corresponds to the acquisition of which-path information by means of coupling the paths to a measuring device or part of their environment. In this paper, we show how the consequences of duality change when allowing for biased sampling, that is, postselected measurements on specific degrees of freedom of the environment of the two-path state. Our work gives insight into a possible mechanism for obtaining simultaneous high which-path information and high-visibility fringes in a single experiment. Further, our results introduce previously unidentified avenues for experimental tests of duality.

Recovering full coherence in a qubit by measuring half of its environment

When a quantum system interacts with its environment it may incur in decoherence. Quantum erasure makes it possible to restore coherence in a system by gaining information about its environment, but measuring the whole of it may be prohibitive: Realistically, one might be forced to address only an accessible subspace and neglect the rest. In such a case, under what conditions will quantum erasure still be effective? In this work we compute analytically the largest recoverable coherence of a random qubit plus environment state and we show that it approaches 100% with overwhelmingly high probability as long as the dimension of the accessible subspace of the environment is larger than D−−√, where D is the dimension of the whole environment. Additionally, we find a sharp transition between a linear behavior and a power-law behavior as soon as the dimension of the inaccessible environment exceeds the dimension of the accessible one. Our results imply that the typical states of a qubit plus environment system admit a measurement spanning only about D−−√ degrees of freedom, any outcome of which projects the qubit on a maximally coherent state. This suggests, for instance, that in the dynamics of open quantum systems, if the interactions are known, it would in principle be possible to gain sufficient information and restore coherence in a qubit by dealing with a fraction of the physical resources.