content_image/419393.jpg content_image/419394.jpg content_image/419391.jpg content_image/419395.jpg content_image/419405.jpg content_image/419406.jpg content_image/419400.jpg content_image/419399.jpg content_image/419401.jpg
where κ ≡ | S$_{11}$ ( z ) | $^{2}$. Similar expressions are obtained for the marginal photon distributions P$_{2}$ ( n$_{2}$, z ) and P$_{3}$ ( n$_{3}$, z ) for waveguides W$_{2}$ and S after setting κ = | S$_{12}$ ( z ) | 2 and κ = | S$_{13}$ ( z ) | $^{2}$, respectively. Let us now suppose that the input beam is a coherent state with mean photon number 〈 n 〉 = n$_{0}$ and Poissonian distribution 

From Eq.(33) it follows that the marginal photon distri- bution P$_{1}$ ( n$_{1}$ ; z ) remains Poissonian with mean photon number 〈 n$_{1}$ 〉 = n$_{0}$ | S$_{11}$ ( z ) | 2 that decays along the prop- agation distance z according to the classical decay law given by Eq.(12). Conversely, if the input beam is a pho- ton number state, i.e. a$_{n}$ = δ$_{n,n}$$_{0}$ , from Eq.(33) it follows that the marginal photon distribution P$_{1}$ ( n$_{1}$ ; z ) is given by the binomial distribution 

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| S$_{11}$ ( z ) | $^{2}$] n$_{0}$ . The binomial photon distribution highlights the particle-like behavior of photons undergoing the de- cay from waveguide W$_{1}$ and it is analogous to that cre- ated by a beam splitter when excited by a photon num- ber state in one port, and the vacuum state in the other one (see, for instance, [19]). The photons in the initially excited waveguide behave like independent classical par- ticles and tunnel into the other waveguides is ruled by a Bernoulli trial (a coin toss) with a cumulative tunneling probability given by 1 − | S$_{11}$ ( z ) | $^{2}$. When the two waveguides W$_{1}$ and W$_{2}$ are simultaneously excited by two nonclassical independent beams, quan- tum signatures of light propagation resulting from quan- tum interference can be detected by photon coincidence measurements. To describe propagation of independent beams, let us notice that for a given set of normalized and orthogonal solutions g$_{1}$ ( ρ, z ), g$_{2}$ ( ρ, z ), ... to the clas- sic wave equation (23), one can construct the n -photon number state 

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which describes excitation of the optical system with a set of independent beams with spatial profiles g$_{1}$ , g$_{2}$ ,... carrying n$_{1}$ , n$_{2}$ ,... photons ( n = n$_{1}$ + n$_{2}$ + ... ). In par- ticular, let us assume that at the input plane the waveg- uides W$_{1}$ and W$_{2}$ are excited in their fundamental modes by single photon number states, i.e. g$_{1}$ ( ρ, 0) = u$_{1}$ ( ρ ), g$_{2}$ ( ρ, 0) = u$_{2}$ ( ρ ) and 

where the operators ˆ a † 1 and ˆ a † 2 are defined by Eqs.(27) and (28). According to the classical analysis of Sec.II.B, the waves g$_{1}$ and g$_{2}$ evolve according to 

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with S$_{23}$ = S$_{13}$ for symmetry reasons [32], S$_{12}$ = S$_{21}$ , S$_{11}$ = S$_{22}$ [see Eqs.(12-14)], and | S$_{11}$ | $^{2}$+ | S$_{12}$ | $^{2}$+ | S$_{13}$ | 2 = 1 for power conservation. From Eqs.(36), (38) and (39) it follows that the state vector of the system at the generic 

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propagation distance z is given by 

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3 that the expectation value of the field intensity for the two-photon state (40) is given by the incoherent super- position 

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the absence of interference being due to the lack of a defi- nite phase relationship between the two photons (see, for instance, [33]). The two independent beams which excite the two waveguides W$_{1}$ and W$_{2}$ behave, therefore, as two incoherent classical fields, and therefore according to the analysis of Sec.II.B half of the light power tunnels, on average, into the slab waveguide S for propagation dis- tances z ≫ l$_{d}$ . As the operators ˆ a † $_{1}$, ˆ a † 2 and ˆ a † 3 create photons in waveguides W$_{1}$, W$_{2}$ and W$_{3}$, respectively [see Eqs.(27-29)], from Eq.(40) the joint photon distribution P ( n$_{1}$, n$_{2}$, n$_{3}$ ; z ) can be readily calculated, and its nonva- nishing terms are explicitly given by 

