Generalized Schrödinger cat states and their classical emulation

We demonstrate that superpositions of coherent states and displaced Fock states, also referred to as generalized Schrödinger cat states, can be generated by application of a deformed Glauber displacement operator on number states, coherent states, and displaced Fock states. Based on such a deformed displacement operator we introduce a deformed version of the so-called Glauber photonic lattices. These novel lattices are endowed with alternating positive and negative coupling coefficients and give rise to classical analogs of Schrödinger cat states. Finally, we show that the analytic propagator of these new Glauber-Fock arrays explicitly contains the Wigner operator opening the possibility to emulate Wigner functions of the quantum harmonic oscillator in the classical domain.

Figure 1

Theoretical results for single-input excitation using a Glauber-Fock lattice of 60 waveguides and parameter $g=1$. Intensity evolution when the (a) zeroth and (c) third waveguides are excited. In these photonic arrays the states of the harmonic oscillator are mapped onto the lattice sites, and the lattice itself plays the role of the deformed displacement operator. As a result, the field distributions can be considered (b) the classical analogs of the Schrödinger cat states and (d) the third generalized Schrödinger cat state of the quantum harmonic oscillator.

Figure 2

Evolution of a coherent state $\left|\beta \right.〉$, with average photon number $\langle \widehat{n}\rangle =50$, in a deformed Glauber-Fock lattice with 150 waveguides and $g=1$. In this simulation we clearly see that a coherent state, which is described by a Poissonian intensity distribution at the input (bottom), splits into two Poisson distributions along evolution. From Eq. (15) we infer that each distribution is composed of two out-of-phase Poisson distributions.

Figure 3

Bottom: Initial intensity profile corresponding to the second displaced Fock state $\left|\beta ,2\right.〉$ with an average photon number of $\langle \widehat{n}\rangle =50$. Middle: Intensity evolution of such classical displaced Fock state. Top: Output intensity showing a superposition of DFS. The evolution is calculated using a deformed Glauber-Fock lattice of 4 cm with 160 waveguides and $g=1$.

Figure 4

Wigner functions for (a) the vacuum state $\left|0\right.〉$ and (b) the first $\left|1\right.〉$, (c) the second $\left|2\right.〉$, and (d) the third $\left|3\right.〉$ Fock states. For a coherent state the Wigner function is the same as for the vacuum state in (a) but is displaced to any of the quadrants of the phase space.

Figure 5

Light evolution along the first four waveguides of a deformed Glauber-Fock lattice with 160 waveguides and $g=1$. In (a) the field amplitude along the zeroth waveguide reproduces part of the Wigner function corresponding to the vacuum state. Similarly, in (b), (c), and (d) we depict sections of the Wigner functions corresponding to the first, second, and third Fock states, respectively. In all cases the initial field amplitude is unitary.