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Physical Review A

Direct measurement of transverse-mode entanglement in two-photon states

H. Di Lorenzo Pires, Carlos Henrique Monken, Martin P. van Exter

Published Aug 5, 2009• 80 citations
Publisher

Abstract

We introduce and experimentally demonstrate a method to measure the Schmidt number of pure two-photon states entangled in transverse-mode structure. Our method is based on the connection between the Schmidt decomposition in quantum theory and the coherent-mode decomposition in classical coherence theory. We apply the method to two-photon states generated by spontaneous parametric down conversion and show that our results are in excellent agreement with numerical calculations based on the Schmidt decomposition.

DOI
10.1103/physreva.80.022307

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We introduce and experimentally demonstrate a method to measure the Schmidt number of pure two-photon states entangled in transverse-mode structure. Our method is based on the connection between the Schmidt decomposition in quantum theory and the coherent-mode decomposition in classical coherence theory. We apply the method to two-photon states generated by spontaneous parametric down conversion and show that our results are in excellent agreement with numerical calculations based on the Schmidt decomposition.
We characterize entanglement in two-qubit pure states encoded in transverse momenta of twin photons obtained from spontaneous parametric down-conversion. Two alternate methods are employed: (i) measurement of conditional interference patterns and (ii) measurement of the marginal probability that yields the single-photon interference pattern. Both methods are local with classical communication and rely on Schmidt decomposition of the quantum state, which is generated by letting the photons propagate through an appropriate lens system. In both cases we can obtain the concurrence either through the Schmidt coefficients in the first method or directly relating the concurrence to the visibility of the interference pattern in the second one.
We report a proof-of-principle experiment demonstrating that appropriately chosen set of Hermite-Gaussian modes constitutes a Schmidt decomposition for transverse momentum states of biphotons generated in the process of spontaneous parametric down-conversion. We experimentally realize projective measurements in the Schmidt basis and observe correlations between appropriate pairs of modes. We perform tomographical state reconstruction in the Schmidt basis, by direct measurement of single-photon density matrix eigenvalues.
The Schmidt number quantifies the number of modes and is mainly used as a measure for the quality of entanglement. We theoretically compute the photon distribution of type-I spontaneous parametric down conversion (SPDC) with an arbitrary Schmidt number. The photon distribution is used for a novel method to measure the Schmidt number. This method requires only two on–off single-photon detectors with no photon number or temporal resolution. The method works in the strong pumping regime where high photon numbers are non-negligible. We experimentally demonstrate the method for type-II SPDC. The easy and fast measurement of the Schmidt number has a broad range of applications from the calibration of strong pump SPDC and entanglement sources to multi-photon quantum interference and Gaussian boson sampling.
Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98%\%% fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.
We measure the transverse entanglement of photon pairs on their propagation from the near to the far field of spontaneous parametric down-conversion (SPDC). The Fedorov ratio, depending on the widths of conditional and unconditional intensity measurements, is shown to be only able to characterize entanglement in the near and far field zones of the source. Therefore we also follow a different approach. By evaluating the first-order coherence of a subsystem of the state we can quantify its entanglement. Unlike previous measurements, which determine the Fedorov ratio via intensity correlations, our setup is sensitive to both phase and modulus of the biphoton state and thus always grants experimental access to the full transverse entanglement of the SPDC state. It is shown theoretically that this scheme represents a direct measurement of the Schmidt number.