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PHYSICS the field amplitude Φ into a product of single-photon amplitude; all of the interesting quantum optical effects discussed below are derivatives of this feature. Φ can be represented as a super- position of separable states using the Schmidt decomposition (19–21)
∞ X √ Φ(qs , qi ) = λn un (qs )vn (qi ), [5] n
where the Schmidt modes un (qs ) and vn (qi ) are the eigenvec- tors of the signal and the idler reduced density matrices, and P the eigenvalues λn satisfy the normalization n λn = 1 (20). The number of relevant modes serves as an indicator for the degree of inseparability of the amplitude, i.e., photon entangle- ment. Common measures for entanglement include the entropy P −1 P 2 Sent = − n λn log2 λn or the Schmidt number κ ≡ n λn . The latter is also known as the inverse participation ratio as it quantifies the number of important Schmidt modes or the effec- Fig. 1. Sketch of the proposed quantum imaging setup. A broadband (2) tive joint Hilbert space size of the two photons. In a maximally pump pulse with the wavevector kp propagates through a χ crystal, gen- erating an entangled photon pair denoted as signal and idler. The photons entangled wavefunction, all modes contribute equally. are distinguished either by polarization (type II) or by frequency (type I) and The spatial profile of the photons in the transverse plane are separated by a beam splitter (BS). The signal photon interacts with the (perpendicular to the propagation direction) can be expanded sample and can be further frequency dispersed and collected by a “bucket” and measured using a variety of basis functions; e.g., Laguerre– detector Ds with no spatial resolution. The idler is spatially resolved in Gauss (LG) or Hermite–Gauss (HG) has been demonstrated the transverse plane by the detector Di. The two photons are detected in experimentally (22, 23). These sets satisfy orthonormality coincidence (Eq. 12). R 2 P 0 d q un (q)vk (q)= δnk and closure relations n un (q)vn (q )= (2) 0 δ (q − q ). The deviation of λn from a uniform (flat) dis- tribution reflects the degree of entanglement. Perfect quan- Thanks to favorable scaling, weak fields can be used to study tum correlations correspond to maximal entanglement entropy fragile samples to avoid damage. and thus a flat distribution of modes. This is further clarified by the closure relations, which demonstrate the convergence Spatial Entanglement into a point-to-point mapping in the limit of perfect trans- Various sources of entangled photons are available, from quan- verse entanglement. The biphoton amplitude exhibits two lim- tum dots (14) to cold atomic gases (15) and nonlinear crystals, iting cases for the infinite inverse participation ratio which are and are reviewed in ref. 4. A general two-photon state can be demonstrated in Fig. 2. When the sinc function in Eq. 3 is written in the form
X (µs ) (µi ) † † |ψi = Φ(ks , ki ) a a |0s , 0ii , [2] ks ki ks ,µs ki ,µi ks ,ki
(ν) † where k is polarization, ak,ν ak,ν are field annihilation (cre- ation) operators, and Φ(ks , ki ) is two-photon amplitude. In the paraxial approximation the transverse momentum {qs , qi } and the longitudinal degrees of freedom are factorized. The trans- verse amplitude of the photon pair generated using a parametric down converter takes then the form (4, 16–18)