Kerr nonlinearity hinders symmetry-breaking states of coupled quantum oscillators

We study the effect of Kerr anharmonicity on the symmetry-breaking phenomena of coupled quantum oscillators. Two types of symmetry-breaking processes are studied, namely, the inhomogeneous steady state (or quantum oscillation death state) and quantum chimera state. Remarkably, it is found that Kerr nonlinearity hinders the process of symmetry breaking in both the cases. We establish our results using direct simulation of the quantum master equation and analysis of the stochastic semiclassical model. Interestingly, in the case of quantum oscillation death, an increase in the strength of Kerr nonlinearity tends to favor the symmetry and at the same time decreases the degree of quantum mechanical entanglement. This paper presents a useful mean to control and engineer symmetry-breaking states for quantum technology.

Figure 1

Numerical results of the quantum model: (a) Quantum bifurcation diagram showing the variation of $y$ projection of the distance between the local maximum values of Wigner distribution in phase space with increasing $\varepsilon /k_1$ at Kerr parameter $K=0$. Insets show the Wigner function of the quantum limit cycle at $\varepsilon /k_1=0.1$, and QAD at $\varepsilon /k_1=1.4$. (b) The Wigner distribution function in phase space for a representative point ($\varepsilon /k_1=1.99$) of (a). (c) Quantum bifurcation diagram for Kerr parameter $K=0.5$. Note that the QOD state does not appear in the same scale of Fig. 1; instead, the boundary to the QOD region is moved towards larger $\epsilon /k_1$ (see Fig. 2). The inset shows the quantum limit cycle in phase space ($\varepsilon /k_1=0.25$). (d) The Wigner function distribution in phase space for a representative point ($\varepsilon /k_1=1.99$) of (c). Other parameters are $\omega =2$ and $(k_1,k_2)=(1,0.2)$.

Figure 2

(a) Two parameter phase diagram in the $\varepsilon /k_1-K$ space delineating the zone of the quantum limit cycle (Osc), QAD, and QOD states. (b) Bifurcation diagram with $K$ along the vertical dashed line of (a), i.e., $\varepsilon /k_1=2.5$. Insets show the Wigner distribution function of QOD ($K=0.02$) and QAD ($K=0.85$). Other parameters are $\omega =2$ and $(k_1,k_2)=(1,0.2)$.

Figure 3

Analytical results of the noisy classical model. (a)–(c) System without anharmonicity, $K=0$. (a) Phase space plot and (b) histogram of the $y_1$ variable at $\varepsilon /k_1=1.99$, showing bimodal shape (noisy classical OD). (c) The bifurcation diagram showing the variation of $\mathrm{\Delta }y$ (the $y$ projection of distance between two local maxima). (d)–(f) System with anharmonicity ($K=0.5$). At the same value of coupling strength ($\varepsilon /k_1=1.99$) we have the (d) phase space plot and (e) histogram of the $y_1$ variable showing unimodal shape (noisy classical AD). (f) Bifurcation diagram of $\mathrm{\Delta }y$ showing AD. Other parameters are $\omega =2$ and $(k_1,k_2)=(1,0.2)$.

Figure 4

Effect of Kerr nonlinearity on quantum entanglement. (a) Variation of negativity ($\mathcal{N}$) in $\varepsilon /k_1$ vs $K$ space. (b) Plot of negativity ($\mathcal{N}$) with parameter $\varepsilon /k_1$ corresponding to the horizontal line in (a) at $K=0$. (c) Plot of negativity with the Kerr parameter ($K$) corresponding to the vertical line in (a) at $\varepsilon /k_1=2.5$. Other parameters are $\omega =2$ and $(k_1,k_2)=(1,0.2)$.

Figure 5

Effect of Kerr nonlinearity on the quantum chimera state. (a) No Kerr ($K=0$): After a short time evolution $\mathrm{\Delta }t=1$, the initial coherent states evolve to squeezed states—shown using the Husimi $Q$ function in the phase space. The directions of squeezing are the same for $i=1–21$ oscillators forming the coherent group (indicated within [..]) and random for $i=22–49$ oscillators forming the incoherent group: the quantum signature of the chimera. (b) With Kerr ($K=1$): The signature of the quantum chimera is destroyed as all the oscillators show the ring shape symmetric Husimi $Q$ function in the phase space. The distinction of the coherent-incoherent group no longer exists. Other parameters are $k_1=1, k_2=0.2, d=10$, and $V=1.2$. For all the plots the $x$ axis represents $\text{Re}\left({\alpha }_i\right)$ ranging from $-4$ to 4 and the $y$ axis represents $\text{Im}\left({\alpha }_i\right)$ also ranging from $-4$ to 4.