Quantum chaos in PT symmetric quantum systems

TL;DR

This work investigates how $\mathcal{PT}$-symmetry interacts with quantum chaos in a non-Hermitian driven system by studying the $\mathcal{PT}$-symmetric quantum kicked rotor. The authors extend chaos diagnostics to complex spectra using the complex level spacing ratio (CLSR) and a non-Hermitian-aware OTOC, identifying three phases: $\mathcal{PT}$-symmetric integrable, $\mathcal{PT}$-symmetric chaotic, and $\mathcal{PT}$-broken chaotic, with transitions $T_1$, $T_2$, and $T_3$. They show that CLSR robustly distinguishes these phases, matching Poisson, GOE, and GinOE benchmarks across Hermitian and non-Hermitian regimes, and that a normalized OTOC reveals Lyapunov-like chaotic growth whose exponent peaks inside the PT-broken region. The study highlights a nuanced link between PT-symmetry breaking and chaos in a driven non-Hermitian quantum system, offering a framework for chaos quantification via OTOCs in non-Hermitian settings and guiding future explorations in related models and experiments.

Abstract

In this study, we explore the interplay between $\mathcal{PT}$-symmetry and quantum chaos in a non-Hermitian dynamical system. We consider an extension of the standard diagnostics of quantum chaos, namely the complex level spacing ratio and out-of-time-ordered correlators (OTOCs), to study the $\mathcal{PT}$-symmetric quantum kicked rotor model. The kicked rotor has long been regarded as a paradigmatic dynamic system to study classical and quantum chaos. By introducing non-Hermiticity in the quantum kicked rotor, we uncover new phases and transitions that are absent in the Hermitian system. From the study of the complex level spacing ratio, we locate three regimes -- one which is integrable and $\mathcal{PT}$-symmetry, another which is chaotic with $\mathcal{PT}$-symmetry and a third which is chaotic but with broken $\mathcal{PT}$-symmetry. We find that the complex level spacing ratio can distinguish between all three phases. Since calculations of the OTOC can be related to those of the classical Lyapunov exponent in the semi-classical limit, we investigate its nature in these regimes and at the phase boundaries. In the phases with $\mathcal{PT}$-symmetry, the OTOC exhibits behaviour akin to what is observed in the Hermitian system in both the integrable and chaotic regimes. Moreover, in the $\mathcal{PT}$-symmetry broken phase, the OTOC demonstrates additional exponential growth stemming from the complex nature of the eigenvalue spectrum at later times. We derive the analytical form of the late-time behaviour of the OTOC. By defining a normalized OTOC to mitigate the effects caused by $\mathcal{PT}$-symmetry breaking, we show that the OTOC exhibits singular behaviour at the transition from the $\mathcal{PT}$-symmetric chaotic phase to the $\mathcal{PT}$-symmetry broken, chaotic phase.

Figures (10)

• Figure 1: Left The CLSR and RLSR as functions of the kicking strength $K$, calculated for $N = 10095$ with $\hbar_{\text{eff}} = 0.2$ and $\lambda = 0$ (Hermitian). Horizontal lines in orange (blue) correspond to values of the RLSR (CLSR) for Poisson and GOE statistics. The figure shows that the CLSR is as good an indicator of the transition from integrability to chaos as the more commonly employed RLSR. Right The CLSR as a function of the non-Hermiticity parameter $\lambda$, calculated for $N = 10095$ with $\hbar_{\text{eff}} = 0.2$ and $K = 0.15$. Horizontal lines represent the values of CLSR for the Poisson and GOE distributions. The figure resembles the transition observed for the CLSR on the left obtained by varying $K$ with $\lambda = 0$. However, it is obtained by tuning only $\lambda$ instead of $K$. This suggests that the system can transition from a $\mathcal{PT}$-symmetric integrable phase to a $\mathcal{PT}$-symmetric chaotic phase with increasing non-Hermiticity while the value of $K$ is kept constant.
• Figure 2: (a) The CLSR for varying values of the kicking strength $K$ and the non-Hermiticity $\lambda$, computed for $N = 6005$, $\hbar_{\text{eff}} = 0.2$. The dashed lines $T_2$ and $T_1$ represent the transitions across the $\mathcal{PT}$-symmetric integrable phase to $\mathcal{PT}$-symmetry broken chaotic phase and the $\mathcal{PT}$-symmetric chaotic phase to $\mathcal{PT}$-symmetry broken chaotic phase, respectively. Line $T^*$ is the Hermitian transition from the integrable to chaotic phase. $T_3$ marks a transition similar to $T^*$ in which $K$ is held constant and $\lambda$ varied. (b): The CLSR across the transition Main panel: $T_2$Inset$T_1$, calculated for $N = 6005$. The horizontal lines show the standard values of the CLSR for the GOE and Poisson distributions. (c): The maximum imaginary part of an energy eigenvalue, $\alpha$, across the transition Main panel: $T_2$Inset$T_1$, calculated for $N = 6005$. It can be seen that while $\alpha$ shows an abrupt change along $T_2$, it seems to increase smoothly along $T_1$. The CLSR on the other hand, shows an abrupt transition along both $T_2$ and $T_1$.
• Figure 3: The normalized OTOC $\Tilde{C}(t)$ vs $t$ across the transition along Top$T_1$ and Bottom$T_2$ shown in Fig. \ref{['fig:CLSR_MAT']}, calculated for system size $N = 2^{14}$. For $T_2$ (bottom), the values $(\lambda,K) = \{(0.001,0.1),(0.002,0.456),(0.003,1.135)\}$ correspond to the $\mathcal{PT}$-symmetry unbroken integrable phase, while the values $(\lambda, K) = \{ (0.006,5.734),(0.01,15.0) \}$ correspond to the $\mathcal{PT}$-symmetry broken chaotic phase. For $T_1$ (top), the values $(\lambda,K) = \{(10^{-5},30),(10^{-4},30),(10^{-3},30)\}$ correspond to the $\mathcal{PT}$-symmetry unbroken chaotic phase, while the values $(\lambda, K) = \{ (0.005,30),(0.001,30.0) \}$ correspond to the $\mathcal{PT}$-symmetry broken chaotic phase.
• Figure 4: The above figure provides a comparison between the observed $\mathcal{PT}$-symmetric chaotic $\to$ PT-symmetry broken chaotic regime transition probed by the CLSR and the accompanying LE values for the same set of parameters for $K$ and $\lambda$. These have been plotted for various values of $\hbar_{\text{eff}}$ that span three orders of magnitude. We observe that this transition observed by the CLSR is accompanied by a peak in the LE. Note that different values of $K$ have only been used to ensure the transition is properly captured, since, as previously stated, the $\mathcal{PT}$-symmetric chaotic regime shrinks as $\hbar_{\text{eff}}\to 0$.
• Figure 5: The figure above shows the transition from the PT-symmetric integrable regime to the PT-symmetric chaotic regime for different system sizes that increase by a factor of $2$. $\hbar_{\text{eff}} = 0.2$ and $K=30$ for all the plots. It can be seen that the transition, as obtained from the CLSR gets sharper as the system size ($N$) increases. There is a slight shift in the value of $\lambda$ at which the transition occurs and this shift appears to decrease with successively increasing values of $N$. The peak in the LE also appears to remain at a nearly fixed value of $\lambda$ as the system size is varied.