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Physical Review A

Projective measurements in quantum and classical optical systems

Filippus Stefanus Roux, Yingwen Zhang

Published Sep 22, 2014• 15 citations
Publisher

Abstract

Experimental setups for the optical implementation of projective measurements in the Laguerre-Gaussian basis are discussed. Special attention is given to the case where the measurements are made with the aid of single-mode fibers that are used to extract the Gaussian component from an optical distribution. Although this setup can, under some conditions, implement the inner product operation accurately, the overlap with the Gaussian mode in the single-mode fiber often produces cross-talk among Laguerre-Gaussian modes with different radial indices. The extent of the cross-talk is analyzed quantitatively.

DOI
10.1103/physreva.90.033835

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Experimental setups for the optical implementation of projective measurements in the Laguerre-Gaussian basis are discussed. Special attention is given to the case where the measurements are made with the aid of single-mode fibers that are used to extract the Gaussian component from an optical distribution. Although this setup can, under some conditions, implement the inner product operation accurately, the overlap with the Gaussian mode in the single-mode fiber often produces cross-talk among Laguerre-Gaussian modes with different radial indices. The extent of the cross-talk is analyzed quantitatively.
We propose and demonstrate a simple and easy-to-implement projective-measurement protocol to determine the radial index p of a Laguerre-Gaussian (LGpl) mode. Our method entails converting any specified high-order LGp0 mode into a near-Gaussian distribution that matches the fundamental mode of a single-mode fiber (SMF) through the use of two phase screens (unitary transforms) obtained by applying a phase-retrieval algorithm. The unitary transforms preserve the orthogonality of modes before the SMF and guarantee that our protocol can, in principle, be free of crosstalk. We measure the coupling efficiency of the transformed radial modes to the SMF for different pairs of phase screens. Because of the universality of phase-retrieval methods, we believe that our protocol provides an efficient way of fully characterizing the radial spatial profile of an optical field.
As a special experimental technique, weak measurements extract very little information from the measured system and does not cause the measured state to collapse. When coupling the Laguerre-Gaussian (LG) state with a well-defined pre- and post-selected system of a weak measurement process, there will be an indirect interconnection between the expected value of coordinate operators of the final state and the mode indices of the measured LG state. The mode of the light is impacted very slightly after the weak measurement. Based on this we propose an experiment scheme and have managed to experimentally measure the mode indices of LG beams spanning from l = -6 to l = +6, p = 0 to p = +8 accurately with the final intensity distributions approximatly at their origin.
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
We experimentally demonstrate a 10 Mbit/s free-space quantum communication link using data encoding on orthogonal Laguerre–Gaussian (LG) modes with the same azimuthal index but different radial indices. Data encoding on two LGℓp modes (i.e., for ℓ=0, we encode [“0”, “1”] as [p=0, p=1], and for ℓ=1, we encode [“0”, “1”] as [p=0, p=1]) is demonstrated by employing directly modulated laser diodes and helical phase holograms. The quantum symbol error rate (QSER) of <5% is achieved at an encoding rate of 10 Mbit/s. Moreover, the influence of the circle radius (R) of the receiver phase pattern on registered photon rates and QSERs is investigated. Our results show that a receiver phase pattern whose R does not match the beam size of the LG modes would induce higher cross talk between the two encoded quantum branches.
The possibility of implementing a given photonic projective measurement with linear optics and photon detectors is discussed. This problem can be viewed as a single-shot discrimination of orthogonal pure quantum states. It is particularly shown that any two orthogonal states can be perfectly discriminated using only linear optics, photon counting, coherent ancillary states, and feedforward. This means that one can construct any binary projective measurement with these means, but without any nonclassical ancillary state. The statement holds in the asymptotic limit of a large number of these physical resources. To extend this result, we also address the problem of discriminating a simple set of three orthogonal states.