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arXiv:1403.4194 · Physical Review Letters

Quantum non-Gaussian Depth of Single-Photon States

Ivo Straka, Ana Predojević, Tobias Huber, Lukáš Lachman, Lorenz Butschek, Martina Miková, Michal Mičuda, Glenn S. Solomon, Gregor Weihs, Miroslav Ježek, Radim Filip

Published Nov 25, 2014• 69 citations
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Abstract

We introduce and experimentally explore the concept of the non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photon state against optical losses. The directly witnessed quantum non-Gaussianity withstands significant attenuation, exhibiting a depth of 18 dB, while the nonclassicality remains unchanged. Quantum non-Gaussian depth is an experimentally approachable quantity that is much more robust than the negativity of the Wigner function. Furthermore, we use it to reveal significant differences between otherwise strongly nonclassical single-photon sources.

DOI
10.1103/physrevlett.113.223603

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We introduce and experimentally explore the concept of the non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photon state against optical losses. The directly witnessed quantum non-Gaussianity withstands significant attenuation, exhibiting a depth of 18 dB, while the nonclassicality remains unchanged. Quantum non-Gaussian depth is an experimentally approachable quantity that is much more robust than the negativity of the Wigner function. Furthermore, we use it to reveal significant differences between otherwise strongly nonclassical single-photon sources.
Highly nonclassical character of optical quantum detectors, such as single-photon detectors, is essential for preparation of quantum states of light and a vast majority of applications in quantum metrology and quantum information processing. Therefore, it is both fundamentally interesting and practically relevant to investigate the nonclassical features of optical quantum measurements. Here we propose and experimentally demonstrate a procedure for direct certification of quantum non-Gaussianity and Wigner function negativity, two crucial nonclassicality levels, of photonic quantum detectors. Remarkably, we characterize the highly nonclassical properties of the detector by probing it with only two classical thermal states and a vacuum state. We experimentally demonstrate the quantum non-Gaussianity of a single-photon avalanche diode even under the presence of background noise, and we also certify the negativity of the Wigner function of this detector. Our results open the way for direct benchmarking of photonic quantum detectors with a few measurements on classical states.
We ab initio derive quantum nonclassicality and non-Gaussianity criteria for realistic single-photon sources and discuss the robustness of these quantum features in attenuating quantum channel. The derivation based only on assumptions about the structure of the detector is performed for standard autocorrelation measurement with single-photon detectors. Using both the criteria, we compare the robustness of quantum features of realistic single-photon states with the Poissonian and thermal background noise and discuss a difference between the impact of the multimode structure of thermal noise and truly Poissonian noise.
Photon addition and subtraction are well-known to render Gaussian states non Gaussian. We evaluate how their optical nonclassicality is affected by the photon addition/subtraction process. It is known that photon addition always transforms a Gaussian state into a nonclassical state. We show that photon subtraction transforms classical states into classical ones and nonclassical ones into nonclassical ones. For a quantitative analysis of the resulting nonclassicality, we use a recently introduced nonclassicality witness, the quadrature coherence scale, and compute the relative nonclassicality gain of the single-photon added/subtracted Gaussian states with respect to the original Gaussian states. For arbitrary single-mode Gaussian states, we show that this relative gain can be substantial in both cases. Whereas the gain is always higher for photon-added states, it is only slightly so for a large parameter range and in particular in the limit of high squeezing, where both gains are identical. We also analyze the Wigner negative volume of the photon-added/subtracted states. Whereas photon-added Gaussian states are never Wigner positive, we show that single-mode photon-subtracted states are Wigner positive if and only if they are weakly nonclassical or classical. Our analysis relies on explicit and general expressions for the characteristic and Wigner functions of photon added/subtracted multi-mode Gaussian states that we obtain simply.
Quantum attributes of light have been related to non-classicality so far, i. e. to incompatibility with mixtures of coherent states. The progress in quantum optics indicates that this feature does not suffice to witness exotic behavior of light. Contrary, quantum non-Gaussianity is starting to appear as a promising and applicable property reflecting interesting states of light that are suitable for quantum protocols. We investigate a newly introduced hierarchy of criteria of quantum non-Gaussianity and predict this attribute can be observed on light emitted from many single photon emitters, even if the light undergoes realistic optical losses. It contrasts negativity of Wigner function, that is unreachable in experimental platforms with losses above fifty percent.
In this paper, a quantitative investigation of the non-classical and quantum non-Gaussian characters of the photon-subtracted displaced Fock state [Formula: see text], where k is the number of photons subtracted and n is Fock parameter, is performed by using a collection of measures like Wigner logarithmic negativity, linear entropy potential, skew information-based measure, and relative entropy of quantum non-Gaussianity. It is noticed that the number of photons subtracted ( k) changes the nonclassicality and quantum non-Gaussianity in a significant amount in the regime of small values of the displacement parameter, whereas the Fock parameter ( n) presents a notable change in the large regime of the displacement parameter. In this respect, the role of the Fock parameter is found to be stronger as compared to the photon subtraction number. Finally, the Wigner function dynamics considering the effects of photon loss channel is used to show that the Wigner negativity can only be detected by highly efficient detectors.
Photon addition and subtraction render Gaussian states non-Gaussian. We provide a quantitative analysis of the change in nonclassicality produced by these processes by analyzing the Wigner negativity and quadrature coherence scale (QCS) of the resulting states. The QCS is a recently introduced measure of nonclassicality [PRL 122, 080402 (2019), PRL 124, 090402 (2020)], that we show to undergo a relative increase under photon addition/subtraction that can be as large as 200\%. This implies that the degaussification and the concomitant increase of nonclassicality come at a cost. Indeed, the QCS is proportional to the decoherence rate of the state so that the resulting states are considerably more prone to environmental decoherence. Our results are quantitative and rely on explicit and general expressions for the characteristic and Wigner functions of photon added/subtracted single- and multi-mode Gaussian states for which we provide a simple and straightforward derivation. These expressions further allow us to certify the quantum non-Gaussianity of the photon-subtracted states with positive Wigner function.