## Physical Background

### Fock States and the Size of the Computational Basis

After the quantization of the Electro-Magnetic fields, which is roughly mode expansion of these fields, it’s possible to describe the state of the system by Fock (number) states.** These states describe the number of photons in each mode** (which is nonnegative integer). Example for $M=5$ modes and $N=3$ photons state is $|\psi\rangle=|00102\rangle$, which has 1 photon in 3rd mode and 2 photons in 5th mode. **At our case each mode represents a specific waveguide in the array.** Considering the bosonic nature of photons, the counting of states of (**exactly**) $N$ indistinguishable photons in $M$ distinguishable modes produce – $$ \tilde{N_b} = {M+N-1 \choose N}$$ where $\tilde{N_b}$ denotes the number of equiphoton states. For example - for $M=20$, $N=2$ exist $\tilde{N_b}=210$ photon states. For a general superposition of Fock states, which has at most $N_{max}$ photons in a single state, **the computational basis for calculations is the union of all states which have at most $N_{max}$ photons**. Its size is - $$ N_b = \sum_{n=0}^{N_{max}}{M+n-1 \choose n}$$ Here $N_b$ denotes the size of the computational basis. For $M=20$, $N=2$ exist $N_b={20+0-1 \choose 0}+{20+1-1 \choose 1}+{20+2-1 \choose 2}=1+20+210=231$ photon states. Notice that in general case the number of states, $N_b$, is much greater than the number of distinguished modes in waveguide array – $M$.

### Hamiltonian of the Waveguide Array and Unitary Propagation

Consider a waveguide array with nearest neighbour coupling. The propagation and the coupling between the waveguides in such an array can be modeled by the following Hamiltonian - $$ H = \beta \sum_{n=1}^M a_n^\dagger a_n +\sum_{n=1}^Mc_{n,n+1}a^\dagger_{n+1}a_n +c_{n,n-1} a^\dagger_{n-1}a_n$$ Here $\beta$ is the propagation constant of the waveguides (taken as identical) and the $c_{i,j}$ coefficients are the coupling coefficients between adjacent waveguides. Use of Heisenberg equation of motion for $a_n^\dagger$ leads to the following system of ODE's - $$i\frac{\partial a_n ^\dagger}{\partial z}=\beta a_n^\dagger+c_{n,n+1} a_{n+1}^\dagger+c_{n,n-1} a_{n-1}^\dagger$$ The solution for $a_n ^\dagger(z)$, as shown in [1], is given by - $$a_n^\dagger(z) =\sum_{l=1}^M U_{n,l}(z) a_l^\dagger(z=0) \\ U_{n,l}(z) = (e^{iHz})_{n,l}$$

**Notice that this solution aplies to any general Hamiltonian, and not only to that of nearest neighbour coupling, which I used as an example.**

### Intensity and Correlation Calculation

Using the Heisenberg representation, $a_n^\dagger(z)$ are all we need for intensity and correlation measurements. The intensity measured at the exit of waveguide $n$ is proportional to the expectation value of the photon number at that waveguide - $$\boxed{I_n(z) \propto APN(n,z) =\langle \psi_0 | a_n^\dagger(z) a_n(z) |\psi_0\rangle}$$

$APN(n,z)$ denotes the average photon number at waveguide $n$ at specified $z$ value. **Normalized average photon number can be interpreted as the probability to measure (at least) a single photon at each spesific waveguide.**

The second order correlation between waveguides $q$ and $r$ defined as [2] - $$\boxed{\Gamma_{q,r} = \langle \psi_0 | a_q^\dagger(z) a_r^\dagger(z) a_r(z) a_q(z) |\psi_0\rangle}$$

**Normalized correlation can be interpreted as the probability to measure simultaniously (at least) a single photon at waveguide $q$ and (at least) a single photon at waveguide $r$. **To stress the obvious - as we cannot measure non-zero intensity with 0 photons (state) - we cannot measure non-zero correlations with only single photon states.

#### References:

[1] Yaron Bromberg, Yoav Lahini, Roberto Morandotti, and Yaron Silberberg, "Quantum and Classical Correlations in Waveguide
Lattices", Phys. Rev. Lett. **102**, 253904
(2009)

[2] Roy J. Glauber, "The Quantum Theory of Optical Coherence", Phys. Rev. **130**, 2529 (1963)