Wigner function evolution of quantum states in the presence of self-Kerr interaction

A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr medium (${\chi }^{\left(3\right)}$ nonlinearity) is presented. We numerically solved this equation taking a coherent state as an initial condition. The dissipation effects are discussed. We provide examples of quantum interference, sub-Planck phase space structures, and Gaussian vs non-Gaussian dynamical evolution of the state. The results also apply to the description of a nanomechanical resonator with an intrinsic Duffing nonlinearity.

Figure 1

(Color online) The Wigner function for $\alpha =5$ and $\tau =0$ is a Gaussian function for the coherent state (left). Due to the fact that the evolution in an ideal ${\chi }^{\left(3\right)}$ medium is periodic, we achieve the same shape of the Wigner function for $\tau =2\pi$. The distribution is an ellipse and the state becomes squeezed beginning from $\tau =0.01$ (middle). The negative values of the function start to appear for $\tau =0.04$ (right).

Figure 2

(Color online) Further steps of Wigner function evolution obtained also for $\alpha =5$ and $\tau =0.08$ (left), $\tau =0.16$ (middle), and $\tau =0.3$ (right). An interference pattern arises in the form of a “tail” of interference fringes.

Figure 3

(Color online) For the evolution parameter values close to $\tau =2\pi R$, where $R<1$ is a rational number, in some areas a destructive and in the other a constructive interference starts to dominate—left $\left(\tau =0.6\right)$ and middle figures $\left(\tau =1\right)$, thus leading to a Wigner function of a coherent superposition state—right figure $\left(\tau =\pi /3\right)$. Figures evaluated for $\alpha =5$.

Figure 4

(Color online) The superposition states obtained for $\alpha =5$ and $\tau =2\pi /5$ (left), $\tau =\pi /2$ (the compass state) (left middle), $\tau =2\pi /3$ (right middle), and $\tau =\pi$ (the Schrödinger cat state) (right).

Figure 5

(Color online) The effect of the decoherence on the Wigner function evaluated for $\alpha =2$, $\tau =0.2\pi$, the thermal noise $N=3.8\times {10}^{-19}$, and the different values of the damping constant: $\xi =0$ (left), $\xi =0.1$ (left middle), $\xi =1$ (right middle), $\xi =2$ (right). Note that the thermal noise gives no input if $\xi =0$.

Figure 6

(Color online) The coherent superposition states are extremely fragile to the decoherence process. The Wigner function of four coherent state superposition $\left(\tau =\pi /2\right)$ of the amplitude $\alpha =2$ evaluated including the thermal noise $N=3.8\times {10}^{-19}$ and the damping constant equal to $\xi =0$ (left), $\xi =0.1$ (left middle), $\xi =1$ (right middle), $\xi =2$ (right).

Figure 7

(Color online) The decoherence looks similarly for the two coherent state superposition Wigner function $\left(\tau =\pi \right)$, also of amplitude $\alpha =2$, the same thermal noise $N=3.8\times {10}^{-19}$ and the damping constants: $\xi =0$ (left), $\xi =0.1$ (left middle), $\xi =1$ (right middle), $\xi =2$ (right).

Figure 8

(Color online) The Wigner function evolution is a noisy ${\chi }^{\left(3\right)}$ medium is not periodic any more. The distribution evaluated for $\alpha =2$, $\tau =2\pi$, the thermal noise $N=3.8\times {10}^{-19}$ and the damping constant equal to $\xi =0$ (left), $\xi =0.1$ (middle), $\xi =1$ (right). The state pursues vacuum.

Figure 9

(Color online) A sub-Planck structure of the Wigner function for $\tau =\pi /2$ and $\xi =0$ (left), $\tau =\pi /2$ and $\xi =0.1$ (left middle), $\tau =\pi$ and $\xi =0$ (right middle), $\tau =\pi$ and $\xi =0.1$ (right).

Figure 10

(Color online) A sub-Planck structure of the Wigner function evaluated for $\tau =0.16$, $\tau =0.3$, $\tau =0.6$, $\tau =1$.

Figure 11

The sparse band matrix necessary to invert to solve the set of linear equations giving the Wigner function values for all points of the discretized phase space.