CEWQO, Central European Workshop on Quantum Information Palermo, 1 – 5 June 2007

1. Dipartimento di Fisica e Matematica, Università dell’Insubria, Como C.N.R.-I.N.F.M., U.d.R.Como Quantum and classical correlations in tripartite Gaussian states of light Alessia Allevi Alessandra Andreoni,Maria Bondani, Matteo G.A. Paris CEWQO, Central European Workshop on Quantum Information Palermo, 1 – 5 June 2007

2. Outline • Three-mode (CV) entanglement • - What is it useful for? How to produce it? • Our interaction scheme • Quantum analysis • - Theoretical description of the state • - Entanglement properties • - Photon number correlations • Experimental results • - Realization of the interaction scheme • - Characterization • Future perspectives

3. Three-mode entanglement Applications • cloning at distance (telecloning) • Murao et al., Phys.Rev. A 59, 156 (1999) • van Loock et al., Phys. Rev. Lett. 87, 247901 (2001) • communication network • van Loock et al., Phys. Rev. Lett. 84, 3482 (2000) • improvement in the discriminationof quantum operations • D’Ariano et al., J. Opt. B 4, S273 (2002) • check of different kind of entanglement • (fully inseparable states vs biseparable ones) • Olsen et al., Phys. Rev. A 74, 063809 (2006)

4. Three-mode entanglement Several schemes • one party of a TWB mixed with a coherent state Furusawa et al. (1998) • one party of a TWB mixed with the vacuum Jing et al. (2003) • three independent squeezed vacua mixed in a network of beam splitters • Aoki et al. (2003) • cascaded processes in second-order nonlinear crystals • Bradley et al. (2005); Olsen et al. (2006) • our scheme: two interlinked nonlinear interaction in a single c(2) crystal • Ferraro, Allevi et al., J. Opt. Soc. Am. B 21, 1241 (2004) • Allevi et al., Laser Phys. 16, 1451 (2006) Advantages of our scheme • use of a single crystal  compactness • non-collinear interaction geometry  remarkable experimental flexibility • type I nonlinear crystal (ooe)  no additional unwanted fields • frequency non-degeneracy  polycromatic entanglement

5. Five-fields interactions x k3 0 k2 Optical axis y k5 k4 z k1 Optical scheme for the two interlinked interactions - 2 energy-matching conditions w1+w3=w4 w3+w5=w2 - 6 phase-matching conditions with the hp. Bondani, Allevi et al., Opt.Express 14, 9838 (2006)

6. If we take the vacuum as the initial state, the evolved state takes the form: Quantum analysis The Hamiltonian corresponding to the scheme proposed is with constant of motion It is a gaussian and fully inseparable three-mode state

7. Quantum analysis Photon-number statistics Photon-number correlation coefficient between N1 and the sum of N2 +N3 The correlation is identically 1 for any value of N1, N2 and N3 !

8. Quantum analysis • Partial photon-number correlations • between N1 andN2 • between N2 andN3 • between N1 andN3

9. Quantum vs Classical Correlations The measurement of e is in principle not sufficient to discriminate between quantum and classical correlations. See for example the state generated by by sending a thermal state on two subsequent beam-splitters whose second port is left unexcited. BS The state exiting two cascated beam-splitters is never entangled, but it shows a high degree of classical correlations BS A relevant marker of non-classicality is represented by the distribution of the difference photon-number If the variance s2(d) is smaller than the shot-noise level we can assert the quantum nature of our state! Bondani, Allevi et al. (2006), Phys. Rev. A, in press

10. SGI PRE+AMP PRE+AMP Experimental setup in the mesoscopic regime He-Ne PC f5 + P-i-n 3 P-i-n 2 f3 f5 Laser Nd:YLF f2 NF f4 BBO f1 P-i-n 1 Pump fields l4=349 nm l5=1047 nm External angles Generated fields l1= 632.8 nm l2= 446.4 nm l3= 778.2 nm

11. Spatial coherence In the mesoscopic regime, it is possible to see the twin coherence areas. To correctly select the triplet, it is necessary to collect a single coherence area on each party of the triplet. To this aim, we used three pin-holes with the following diameters and distances from the BBO • FIELD a1: f1 = 30 mm, d1= 60 cm • FIELD a2: f2 = 50 mm, d2= 141.5 cm • FIELD a3: f3 = 30 mm, d3= 49 cm

12. Temporal coherence As the pulse duration of the pumps is longer than the coherence time of the interactions, the distributions of the photons are temporally multimode. mequally occupied thermal modes Experimental results: multithermal statistics The distributions of the detected photons are also multithermal with approximately the same number of temporal modes 21 modes for a2+a3 19 modes for a2 14 modes for a3 19 modes for a1

13. Experimental results: correlations As real detectors have quantum efficiency h <1, the correlation coefficient e1,2+3 for the three-mode state must be written as follows In the mesoscopic regime (Nj>>1) e1,2+3 approaches unity for any value of h. This result is also valid for the partial photon-number correlations. Correlation coefficients for the detected photons In correspondence with e1,2+3 exp we obtained a quantum noise reduction R not too larger than 1 The mean values of the detected photons (if h1 = h2 = h3) almost satisfy the photon-number conservation law:

14. Future perspectives • Demonstration of the non-classical nature • - Improvements in the sub-shot noise correlations • Demonstration of the entangled nature • - Telecloning of coherent states • Other applications • - Production of conditional states • - Imaging: • ghost-imaging • image-transfer