Linear and nonlinear quantum Zeno and anti-Zeno effects in a nonlinear optical coupler

Quantum Zeno and anti-Zeno effects are studied in a symmetric nonlinear optical coupler, which is composed of two nonlinear (${\chi }^{\left(2\right)}$) waveguides that are interacting with each other via the evanescent waves. Both waveguides operate under second harmonic generation. However, to study quantum Zeno and anti-Zeno effects one of them is considered as the system and the other one is considered as the probe. Considering all the fields involved as weak, a completely quantum mechanical description is provided, and the analytic solutions of Heisenberg's equations of motion for all the field modes are obtained using a perturbative technique. Photon number statistics of the second harmonic mode of the system is shown to depend on the presence of the probe, and this dependence is considered as quantum Zeno and anti-Zeno effects. Further, it is established that as a special case of the momentum operator for ${\chi }^{\left(2\right)}-{\chi }^{\left(2\right)}$ symmetric coupler we can obtain momentum operator of ${\chi }^{\left(2\right)}-{\chi }^{\left(1\right)}$ asymmetric coupler with linear (${\chi }^{\left(1\right)}$) waveguide as the probe, and, in such a particular case, the expressions obtained for Zeno and anti-Zeno effects with nonlinear probe (which we referred to as nonlinear quantum Zeno and anti-Zeno effects) may be reduced to the corresponding expressions with linear probe (which we referred to as the linear quantum Zeno and anti-Zeno effects). Linear and nonlinear quantum Zeno and anti-Zeno effects are rigorously investigated, and it is established that, in the stimulated case, we may switch between quantum Zeno and anti-Zeno effects just by controlling the phase of the second harmonic mode of the system or probe.

Figure 1

Schematic diagrams of (a) a symmetric and (b) an asymmetric nonlinear optical coupler of interaction length $L$ in a codirectional propagation of different field modes involved. The symmetric coupler is prepared by combining two nonlinear (quadratic) waveguides operating under second harmonic generation, and in the asymmetric coupler one nonlinear waveguide of the symmetric coupler is replaced by a linear waveguide.

Figure 2

(a) Variation in linear Zeno parameter with rescaled interaction length for $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\alpha =5,\beta =3,$ and $\delta =1$ and $-1$ in the stimulated case (smooth and dashed lines) and $\delta =0$ in the spontaneous case (dot-dashed line). (b) In the spontaneous case, $\alpha =10$ with $\beta =3,6,$ and 7 for smooth, dashed, and dot-dashed lines, respectively. (c) $\alpha =10,\beta =8,$ and $\delta =-4,0,$ and 4 for smooth, dashed, and dot-dashed lines, respectively. The quantities shown in the plots are dimensionless.

Figure 3

(a) Nonlinear Zeno parameter with $\frac{{\mathrm{\Gamma }}_a}{k}=\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\alpha =5,\beta =3,$ and $\gamma =2,\delta =1$, and $-1$ in the stimulated case (smooth and dashed lines) and $\gamma =-2,\delta =1$, and $-1$ in dot-dashed and dotted lines. (b) $\alpha =10,\gamma =3,\delta =1$ with $\beta =3,6,$ and 7 for smooth, dashed, and dot-dashed lines, respectively. (c) The change in nonlinear Zeno parameter with changing the phase of $\alpha$ or $\beta$ by an amount of $\pi$ for $\alpha =\beta =6,$ and $\gamma =3,\delta =2$. It is also observed that the change in phase of $\alpha$ is equivalent to change in phase of $\beta$. All the quantities in these plots are dimensionless.

Figure 4

(a) Linear Zeno parameter in the spontaneous case for $\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\alpha =5,\beta =3$ for $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2}\left(5\times {10}^{-2}\right)$ in smooth blue (dashed red) line. (b) A similar observation in the stimulated case with $\delta =1$ and all the remaining values same as (a). In (c) and (d), the effect of phase mismatch in the spontaneous and stimulated (with $\delta =1$) cases of linear Zeno parameter is shown, respectively. The remaining parameters are $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2}$ with $\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}\left({10}^{-1}\right)$ in the smooth blue (dashed red) line. Here, all the quantities are dimensionless.

Figure 5

Dependence of nonlinear Zeno parameter on the nonlinear coupling constant is depicted in (a) and (b). In (a) the nonlinear Zeno parameter is shown with rescaled interaction length for $\frac{{\mathrm{\Gamma }}_a}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\alpha =5,\beta =3,\gamma =2,$ and $\delta =1.$ The smooth (blue) and dashed (red) lines correspond to $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2}$ and $\frac{{\mathrm{\Gamma }}_b}{k}=5\times {10}^{-2},$ respectively. Similarly, in (b) the smooth (blue) and dashed (red) lines correspond to $\frac{{\mathrm{\Gamma }}_a}{k}={10}^{-2}$ and $\frac{{\mathrm{\Gamma }}_a}{k}=5\times {10}^{-2},$ respectively, with $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2}$ and all the remaining values as in (a). In (c) the nonlinear Zeno parameter is shown in the smooth blue (dashed red) line with $\frac{{\mathrm{\Gamma }}_a}{k}=\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ for $\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3}\left(1.1\times {10}^{-1}\right).$ Similarly, in (d) the nonlinear Zeno parameter is shown in the smooth blue (dashed red) line with $\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3}$ for $\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}\left({10}^{-1}\right).$ The quantities shown in these plots are dimensionless.

Figure 6

Variation in (a) linear and (b) nonlinear Zeno parameter with linear coupling constant (in the units of ${\mathrm{m}}^{-1}$) and interaction length (in meters) are shown for $\frac{{\mathrm{\Gamma }}_a}{k}=\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\alpha =6,\beta =4,\gamma =2,$ and $\delta =1$.

Figure 7

Variation in linear Zeno parameter with phase mismatch between fundamental and second harmonic modes in system waveguide and rescaled interaction length in (a) spontaneous and (b) stimulated cases are shown for $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2}$ with $\alpha =6,\beta =4,$ and $\delta =0$ and 1 in (a) and (b), respectively. (c) The dependence of the nonlinear Zeno parameter on the phase mismatch between fundamental and second harmonic modes in system waveguide and rescaled interaction length for $\frac{{\mathrm{\Gamma }}_a}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_a}{k}=1.1\times {10}^{-3}$ and $\gamma =2,\delta =1$ with all remaining values same as (a) and (b). In (d), the effect of phase mismatch between fundamental and second harmonic modes in probe waveguide and rescaled interaction length on the nonlinear Zeno parameter are shown for $\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with all the remaining values as (c). The quantities shown in the plots are dimensionless.

Figure 8

Variation in linear and nonlinear Zeno parameters shown in three-dimensional plots in Figs. 7–7 are illustrated via equivalent contour plots. Here, all the four contour plots corresponding to Figs. 7–7 are obtained using the same parameters. In (a) and (b), the yellow regions illustrate the region for quantum anti-Zeno effect, while the blue regions correspond to quantum Zeno effect. In (c) and (d), along with the regions of Zeno and anti-Zeno effects, variation of magnitude of Zeno parameters are also shown with different colors (see the color bars in the right side of the figures). The quantities in the plots are dimensionless.

Figure 9

Variation in linear Zeno parameter in (a) spontaneous and (b) stimulated cases with photon numbers of linear modes in both the waveguides ($\alpha$ and $\beta$) are shown for $\frac{{\mathrm{\Gamma }}_b}{k}={10}^{-2},\frac{\mathrm{\Delta }{k}_b}{k}={10}^{-3}$ with $\gamma =\frac{\alpha }{2}$ and $\delta =\frac{\beta }{3}$ after rescaled interaction length $kz=1$. A similar behavior to the linear Zeno parameter is observed for nonlinear Zeno parameter in the stimulated case. The quantities shown in the plots are dimensionless.