Theory of anomalous Landau-Zener tunneling induced by nonlinear coupling

Abstract

We develop a general theory of Landau-Zener (LZ) tunneling in a two-level system with amplitude-dependent, sign-reversible nonlinear coupling, distinguishing it fundamentally from conventional on-site nonlinearity. Through a combination of analytical and phase-space analysis, we show that beyond a critical interaction strength, the nonlinear coupling fundamentally reshapes the adiabatic energy landscape, introducing a topological twisted and knotted structure. This structure leads to a complete breakdown of the standard exponential LZ formula, even in the adiabatic limit. Central to this anomalous behavior is the emergence of a black-hole-like fixed point, which acts as a universal attractor: upon traversing the critical region, all quantum trajectories converge to this fixed point, irreversibly erasing any memory of the initial state. From this fixed-point picture, we derive an exact analytical expression for the adiabatic tunneling probability, revealing a characteristic power-law dependence on both linear and nonlinear coupling strength. Our work establishes a paradigmatic framework for nonlinear-coupling-induced anomalous adiabaticity breaking and offers a universal mechanism for state control in driven quantum and wave systems.

Figures (6)

• Figure 1: (color online) Adiabatic energy levels $\varepsilon$ as a function of bias $\gamma$ for different regimes of the nonlinear coupling strength $\beta$. The nonlinear coupling qualitatively alters the level structure and induces four distinct topological types: (a) Type-I, a conventional avoided crossing for $\beta\gtrsim-0.9575$; (b) Type-II, a swallowtail structure for $-1<\beta\lesssim-0.9575$; (c) Type-III, a twisted-knotted structure A for $-2<\beta<-1$; and (d) Type-IV, a twisted-knotted structure B for $\beta<-2$.
• Figure 2: (color online) Phase diagram in the ($\beta,~\gamma$) parameter space, showing regions with two real adiabatic energy levels (orange) and four real adiabatic energy levels (shaded blue). The analytical boundaries $f(\beta)$, $g_1(\beta)$, and $g_2(\beta)$, derived from the fixed-point analysis described in the text (Eqs. \ref{['eq:boundary1']} and \ref{['eq:boundary2']}), are indicated and show excellent agreement with the numerically determined shaded regions. The labels I, II, III, and IV correspond to the four distinct topological types illustrated in Fig. \ref{['fig:energylevels']}.
• Figure 3: (color online) Landau-Zener tunneling probability $p$ induced by nonlinear coupling $\beta$, plotted as a function of the sweep rate $v$ for (a) the conventional avoided-crossing level structure regime (Type-I), (b) the swallowtail level structure regime (Type-II), and (c) the twisted-knotted level structure regime (Type-III and Type-IV). For comparison, panel (a) includes the standard LZ result for the linear case ($\beta = 0$). Panel (d) shows the adiabatic tunneling probability as a function of nonlinear coupling $\beta$ with $v = 0.001$; numerical data are represented by solid symbols, while the solid line corresponds to the analytical expression Eq. \ref{['eq:pad']} obtained from fixed-point analysis.
• Figure 4: (color online) Nonlinear coupling induced adiabatic following versus breakdown in quantum-state evolution. Comparison between the dynamical energy levels $\varepsilon_{\mathrm{dyn}}$ (symbols) and the adiabatic levels (solid lines) in (a) Type-I regime, (b) Type-II regime, (c) Type-III regime, and (d) Type-IV regime. In (a)-(d), the dynamical levels (circles) are obtained by evolving from the lower adiabatic branch. In (d), the dynamical levels (triangles) are evolved from the upper adiabatic branch to illustrate the distinct behavior in this regime. All dynamical evolutions are computed in the adiabatic limit with a sweep rate $\alpha = 0.001$.
• Figure 5: Fixed points $s_{\rm f}$ as a function of bias $\gamma$ for different values of the nonlinear coupling strength $\beta$. The nonlinear coupling qualitatively modifies the fixed-point topology and count. Green symbols mark the black-hole-like fixed points.