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arXiv:2005.09963 · Journal of Optics

Determination of spatial quantum states by using point diffraction interferometry

Quimey Pears Stefano, Lorena Rebón, Claudio Iemmi

Published Sep 24, 2020• 1 citation
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Abstract

Abstract We present a method to reconstruct pure spatial qudits of arbitrary dimension d , which is based on a point diffraction interferometer. In the proposed scheme, the quantum states are codified in the discretized transverse position of a photon field, once they are sent through an aperture consisting in d rectangular regions, with an extra region that provides a phase reference. To characterize these photonic quantum states, the complete phase wavefront is reconstructed through a phase-shifting technique. Combined with a multipixel detector, the acquisition can be parallelized, and only four interferograms are required to reconstruct any pure qudit, independently of the dimension d . We tested the method experimentally, for reconstructing states of dimension d = 6 randomly chosen. A mean fidelity values of 0.95 is obtained. Additionally, we develop an experimental scheme that allows to estimate phase aberrations affecting the wavefront upon propagation, and thus improve the quantum state estimation. In that regard, we present a proof-of-principle demonstration that shows the possibility to correct the influence of turbulence in a free-space communication, recovering mean fidelity values comparable to the propagation free of turbulence.

DOI
10.1088/2040-8986/abbb5c

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•Sep 1, 2018•5 citations
Abstract We present a method to reconstruct pure spatial qudits of arbitrary dimension d , which is based on a point diffraction interferometer. In the proposed scheme, the quantum states are codified in the discretized transverse position of a photon field, once they are sent through an aperture consisting in d rectangular regions, with an extra region that provides a phase reference. To characterize these photonic quantum states, the complete phase wavefront is reconstructed through a phase-shifting technique. Combined with a multipixel detector, the acquisition can be parallelized, and only four interferograms are required to reconstruct any pure qudit, independently of the dimension d . We tested the method experimentally, for reconstructing states of dimension d = 6 randomly chosen. A mean fidelity values of 0.95 is obtained. Additionally, we develop an experimental scheme that allows to estimate phase aberrations affecting the wavefront upon propagation, and thus improve the quantum state estimation. In that regard, we present a proof-of-principle demonstration that shows the possibility to correct the influence of turbulence in a free-space communication, recovering mean fidelity values comparable to the propagation free of turbulence.
We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension ddd, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only 4d4d4d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of d2{d}^{2}d2. By using this technique, we have experimentally reconstructed a large number of states ranging from d=2d=2d=2 up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with ddd slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the ddd slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d\ensuremath{-}3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.
We present a quantum state tomography method that enables the reconstruction of arbitrary d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using only d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are spatially multiplexed on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.
We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary} d−d-d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only} d+1d+1d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of d=6−d=6-d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 424242 to only 777.
Spatial qudits are D-dimensional (D ≥ 2) quantum systems carrying information encoded in the discretized transverse momentum and position of single photons. We present a proof-of-principle demonstration of a method for preparing arbitrary pure states of such systems by using a single phase-only spatial light modulator (SLM). The method relies on the encoding of the complex transmission function corresponding to a given spatial qudit state onto a preset diffraction order of a phase-only grating function addressed at the SLM. Fidelities of preparation above 94% were obtained with this method, which is simpler, less costly, and more efficient than those that require two SLMs for the same purpose.
We present a method for preparing arbitrary pure states of spatial qudits, namely, D -dimensional ( ) quantum systems carrying information in the transverse momentum and position of single photons. For this purpose, a set of D slits with complex transmission are displayed on a spatial light modulator (SLM). In a recent work we have shown a method that requires a single phase-only SLM to control independently the complex coefficients which define the quantum state of dimension D . The amplitude information was codified by introducing phase gratings inside each slit, and the phase value of the complex transmission was added to the phase gratings. After a spatial filtering process, we obtained in the image plane the desired qudit state. Although this method has proven to be a good alternative to compact the previously reported architectures, it presents some features that could be improved. In this paper we present an alternative scheme to codify the required phase values that minimizes the effects of temporal phase fluctuations associated to the SLM where the codification is carried out. In this scheme, the amplitudes are set by appropriate phase gratings addressed at the SLM, while the relative phases are obtained by a lateral displacement of these phase gratings. We show that this method improves the quality of the prepared state and provides very high fidelities of preparation for any state. An additional advantage of this scheme is that a complete modulation is obtained by shifting the grating by one period; hence the encoding is not limited by the phase modulation range achieved by the SLM. Numerical simulations, that take into account the phase fluctuations, show high fidelities for thousands of qubit states covering the whole Bloch sphere surface. Similar analyses are performed for qudits with D = 3 and D = 7.
In this Letter, we propose a simple optical architecture based on phase-only programmable spatial light modulators, in order to characterize general processes on photonic spatial quantum systems in a >2 Hilbert space. We demonstrate the full reconstruction of typical noises affecting quantum computing, such as amplitude shifts, phase shifts, and depolarizing channels in dimension =5. We have also reconstructed simulated atmospheric turbulences affecting a free-space transmission of qudits in dimension =4. In each case, quantum process tomography was performed in order to obtain the matrix that fully describes the corresponding quantum channel, ℰ. Fidelities between the states are experimentally obtained after going through the channel, and the expected ones are above 97%.