Quantum noise effects with Kerr-nonlinearity enhancement in coupled gain-loss waveguides

It is generally difficult to study the dynamical properties of a quantum system with both inherent quantum noises and nonperturbative nonlinearity. Due to the possibly drastic intensity increase of an input coherent light in gain-loss waveguide couplers with parity-time $\left(\mathcal{PT}\right)$ symmetry, the Kerr effect from a nonlinearity added into the system can be greatly enhanced and is expected to create macroscopic entangled states of the output light fields with huge photon numbers. Meanwhile, quantum noises also coexist with the amplification and dissipation of the light fields. Under the interplay between the quantum noises and nonlinearity, the quantum dynamical behaviors of the systems become rather complicated. However, the important quantum noise effects have been mostly neglected in previous studies about nonlinear $\mathcal{PT}$-symmetric systems. Here we present a solution to this nonperturbative quantum nonlinear problem, showing the real-time evolution of the system observables. The enhanced Kerr nonlinearity is found to give rise to a previously unknown decoherence effect that is irrelevant to the quantum noises and imposes a limit on the emergence of macroscopic nonclassicality. In contrast to what happens in linear systems, the quantum noises exert significant impact on the system dynamics and can create nonclassical light field states in conjunction with the enhanced Kerr nonlinearity. This study on the noise involved in quantum nonlinear dynamics of coupled gain-loss waveguides can help to better understand the quantum noise effects in many nonlinear systems.

Figure 1

Setup. The light is amplified at the rate $\kappa$ in channel $A$, but is damped at the same rate in channel $B$. The two waveguide modes couple at the rate $J$. The loss channel also carries a Kerr nonlinearity with a small coefficient $\chi$.

Figure 2

Kerr-nonlinearity-induced change of a mean quadrature in the $\mathcal{PT}$-symmetric regime. The unmarked vertical axis of the plot is the dimensionless ratio $\mathrm{\Delta }|\langle {\widehat{X}}_b(\pi /2)\rangle |/|\langle {\widehat{X}}_b^0(\pi /2)\rangle |$ defined in the text, and its distribution is over time and in the parameter space. Here we use the quadratic mean of $\langle {\widehat{X}}_b^0(\pi /2)\rangle$ over an oscillation period to avoid the singularities from its vanishing values, and the corrections are calculated to the first order of the Kerr coefficient $\chi$ in Eq. (10). The system parameters are $\chi ={10}^{-9}\kappa$ and ${\alpha }_0={10}^3$.

Figure 3

Change of a mean quadrature under the enhanced Kerr nonlinearity in the symmetry-broken regime. The unmarked vertical axis of the plots is the dimensionless ratio $\mathrm{\Delta }|\langle {\widehat{X}}_b(\pi /2)\rangle |/|\langle {\widehat{X}}_b^0(\pi /2)\rangle |$ defined in the text. The upper panels, one as the front view along the time axis and the other as the corresponding back view, are about the realistic situation under the quantum noise effects. The lower ones describe the situation under the “noiseless” non-Hermitian Hamiltonian (15) together with the Kerr nonlinear action. The platforms of unit ratio indicate where the mean quadrature disappears due to decoherence. The quantum noises cause considerable delay of the Kerr nonlinearity enhancement and its consequent decoherence, as compared with the evolution under the non-Hermitian Hamiltonian without noise. Here we take the Kerr coefficient and input coherent-state amplitude in Fig. 2.

Figure 4

Time evolution of the mean quadratures. Here we use the notations ${\widehat{X}}_c\left(0\right)={\widehat{X}}_C$ and ${\widehat{X}}_c(\pi /2)={\widehat{P}}_C$, for $C=A$ (the gain channel) or $B$ (the loss channel). The solid curves represent the average quadrature values given the Kerr nonlinearity of $\chi ={10}^{-9}\kappa$, and the dashed curves stand for those of the linear coupler with $\chi =0$. The evolutions take place at $J=0.1\kappa$ for coherent light of ${\alpha }_0={10}^3$. In (d) a refined view of the exponentially accelerating oscillation is shown in the inserted plot. Due to our choice of initially sending the light into the gain channel, the dashed curves coincide with the horizontal axis in (b) and (c).

Figure 5

Measure of the influence from quantum noise correction on the expectation value of an evolved waveguide mode. The unmarked vertical axis of the plots is the dimensionless quantity $|[E{B}_1\left(t\right)/E{B}_0\left(t\right)]\langle {\beta }_1\left(t\right),{\beta }_2\left(t\right)|{\alpha }_0,0\rangle |$, which combines the relative intensity of $E{B}_1\left(t\right)$ in Eq. (13) and the decoherence effect from the exponentially accelerating oscillation. The upper frame is obtained with the Kerr coefficient $\chi ={10}^{-9}\kappa$ and the lower one with $\chi ={10}^{-5}\kappa$. A higher Kerr coefficient does not substantially enhance the effects from the additional term $E{B}_1\left(t\right)$. The amplitude of the input coherent state is taken as ${\alpha }_0={10}^3$.

Figure 6

Quantum-noise-induced deviation in a mean quadrature evolution. The thicker (purple) curves stand for the mean quadrature ${\widehat{X}}_b(\pi /2)={\widehat{P}}_B$ under the noise effects, and the thinner (blue) ones are those evolving according to the non-Hermitian Hamiltonian in (15) and Kerr nonlinear action. The plots in (a) are found with $J=0.3\kappa$. Those in (b) are obtained with $J=0.9\kappa$. The inset plots in both (a) and (b) show a longer-time realistic evolution under the noise effects. The Kerr coefficient and input coherent-state amplitude are the same as in the previous figures.

Figure 7

Emergence of the entanglement and nonclassical states. The negative values of $D_3$ indicate the entanglement of the output fields and their general nonclassicality. (a) and (b) show the quantity's evolution under the full dynamics including the quantum noise effects. The system parameters are taken as $J=0.1\kappa$ in (a) and $J=0.9\kappa$ in (b). (c) illustrates the corresponding evolution to (a), but due to the non-Hermitian Hamiltonian (15) plus the nonlinear Hamiltonian $H_{NL}$ without quantum noise. (d) is the corresponding situation to (b) also without the quantum noise effects. The plot in (b) shows that the nonclassical regime can be realized with an amplification rate $e^{\lambda t}\sim 10$. All other parameters are the same as in Fig. 4.