10 [21] A. Politi, M.J. Cryan, J.G. Rarity, S. Yu, and J.L. O’Brien, Science 320 , 646 (2008). [22] Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silber- berg, arXiv:0807.3938v1 [quant-ph] (2008). [23] S. Longhi, Phys. Rev. Lett. 101 , 193902 (2008). [24] T.A.B. Kennedy and E.M. Wright, Phys. Rev. A 38 , 212 (1988). [25] U. Leonhardt, Rep. Prog. Phys. 66 , 1207 (2003); J. Skaar, J.C.G. Escartin, and H. Landro, Am. J. Phys. 72 , 1385 (2004). [26] Y. Lai and H.A. Haus, Phys. Rev. A 40 , 844 (1989); E.M. Wright, Phys. Rev. A 43 , 3836 (1991); P.L. Hagelstein, Phys. Rev. A 54 , 2426 (1996) [27] A.B. Matsko and V.V. Kozlov, Phys. Rev. A 62 , 033811 (2000). [28] B. Yurke and M. J. Potasek, J. Opt. Soc. Am. B 6 , 1227 (1989); C. Sibilia, V. Schiavone, M. Bertolotti, R. Ho- rak, and J. Perina, J. Opt. Soc. Am. B 11 , 2175 (1994); B Crosignani, P Di Porto and A Treppiedi, Quantum Semiclass. Opt. 7 , 73 (1995); J.M. Fini, P.L. Hagelstein, and H. A. Haus, Phys. Rev. A 60 , 2442 (1999); D. Rand, K. Steiglitz, and P.R. Prucnal, Phys. Rev. A 71 , 053805 (2005). [29] H.I. Yuen, and J.H. Shapiro, IEEE Trans. Inf. Theory IT-24 , 657 (1978); M.I. Kolobov, Rev. Mod. Phys. 71 , 1539 (1999); M. Bache, E. Brambilla, A. Gatti, and L.A. Lugiato, Phys. Rev. A 70 , 023823 (2004); A. Aiello and J.P. Woerdman, Phys. Rev. A 72 , 060101(R) (2005); M. Tsang and D. Psaltis, Phys. Rev. A 73 , 013822 (2006). [30] In the Heisenberg picture, the state vector of the system would be constant whereas the operator ˆ φ evolve accord- [34] K. Wang, J. Phys. B 39 , R293 (2006); W.P. Grice and I.A. Walmsley, Phys. Rev. A 56 , 1627 (1997). [33] R. Ghosh and L. Mandel, Phys. Rev. Lett. 59 , 1903 (1987); L. Mandel, Rev. Mod. Phys. 71 , S274 (1999). [32] As the waveguides W$_{1}$ and W$_{2}$ are symmetrically placed at opposite sides from the slab waveguide S, if ( c$_{1}$ ( z ) , c$_{2}$ ( z ) , c$_{k}$ ( z )) is a solution to Eqs.(5-7) with ini- tial condition c$_{1}$ (0) = 1, c$_{2}$ (0) = 0 and c$_{k}$ (0) = 0, then ( c$_{2}$ ( z ) , c$_{1}$ ( z ) , c$_{k}$ ( z )) is the solution to Eqs.(5-7) with reversed initial condition c$_{1}$ (0) = 0, c$_{2}$ (0) = 1 and c$_{k}$ (0) = 0. Light tunnelled into the slab waveguide S is thus given by the same superposition of continuous modes ( ∼ ∫ dkc$_{k}$ ( z ) u$_{k}$ ( ρ )) for the two different initial conditions. This means that the last two terms on the right hand sides of Eqs.(38) and (39) are equal each other. [31] More precisely, the monochromatic limit is attained by assuming in Eqs.(21) and (22) f ( n $^{)}$( q , Ω , z ) = g ( n $^{)}$( q , z ) h (Ω$_{1}$) h (Ω$_{2}$) ...h (Ω n ), where h (Ω) is a spectrally narrow function at around Ω = 0 normalized such that ∫ d Ω | h (Ω) | 2 = 1, and g ( n $^{)}$( q , z ) satisfies Eq.(22) with Ω$_{l}$ = 0. The operator ˆ φ ( ρ ) is then formally defined as ˆ φ ( ρ ) = ∫ d Ω h (Ω) ˆ φ ( ρ, Ω). The n -photon number state | f ( n $^{)}$〉 , given by Eq.(21), takes then the simplified form f ( n = (1 / $_{(}$√$_{n}$$_{!)}$ ∫ d q g ( n $^{)}$( q , z ) ˆ φ $^{†}$( ρ$_{1}$ ) ˆ φ $^{†}$( ρ$_{2}$ ) ... ˆ φ $^{†}$( ρ n ) | 0 〉 . ing to i ℏ ( d ˆ φ/dz ) = [ ˆ φ, ˆ H ]. One can easily check that, with the Hamiltonian operator given by Eq.(18), the evolution equation of the operator ˆ φ is precisely Eq.(3), where the c-field φ is replaced by the operator ˆ φ .