Electric-circuit analog of Landau-Zener tunneling using time-varying elements

TL;DR

The paper addresses whether Landau-Zener tunneling can occur in classical systems by constructing a time-varying electric-circuit analog. Using two RLC channels with linearly varying capacitors, the authors show that a generalized probability for norm-unconserved dynamics reproduces the quantum LZT dependence P ~ exp(-π Δ^2/(2 α')), with Δ the crossing gap and α' the linear sweep rate, verified through linearization near the crossing. They establish a mapping to the quantum problem via linearization and block-diagonalization, providing a general method to simulate time-dependent quantum models with circuits, and show that nonreciprocal coupling does not alter the outcome due to a similarity transform. The approach offers a robust framework for exploring more complex LZT and dynamical phenomena in classical circuits, including potential extensions to nonlinear or non-Hermitian regimes, and suggests experimental feasibility with time-varying capacitors and specialized circuit elements. This work thus bridges quantum dynamics and classical circuit physics, enabling accessible simulations of time-dependent quantum behaviors.

Abstract

Landau-Zener tunneling (LZT) is a fundamental dynamical phenomenon, ubiquitous in various quantum systems. Here, we propose a time-varying electric circuit to address the question of whether the quantum LZT can occur in classical systems. Although the underlying differential equation of motion is quite different from the Schrödinger equation and the instantaneous frequency spectrum of the proposed circuit is not linear, the probability of the LZT in circuits (circuit LZT for short), based on our new definition for norm-unconserved systems, still follows the laws of the LZT in quantum systems, co-determined by the linear sweeping rate $α'$ and the frequency gap $Δ$, i.e., approaching the analytical value $\exp(-πΔ^2/2α')$, regardless of whether the coupling is reciprocal or nonreciprocal. The deep relationship between the circuit LZT and its quantum counterpart can be established through a linearization and block-diagonalization process. Our proposal provides a general method for simulating time-dependent quantum models using time-varying electric circuits, which has been lacking in previous studies, and paves the way for studying more complicated LZT and other dynamical phenomena in circuits and other classical systems.

Figures (5)

• Figure 1: (a) The circuit diagram for circuit LZT, composed of resistors (R), inductors (L), and time-varying capacitors (C). $V_{1,2}(t)$ are the voltages of node 1 and 2. (b) The instantaneous eigenfrequencies $\omega_\pm(t)$ (solid lines) and the channel frequencies $\omega_{1,2}(t)$ (dotted lines) with parameters $(r,b)=(30,1/300)$, independent of $v$. The dashed vertical line indicates the crossing point $at_c\approx0.03$, at which the frequency gap $\Delta\approx 0.18$. The rectangles including two circles demonstrate the voltage distributions $\tilde{V}_{\pm,i}(t)$ on the two nodes for the corresponding instantaneous eigenmodes.
• Figure 2: Reciprocal case with $v=1$. (a), (b) Time evolution of the moduli of coefficients $|c_{1,2}(t)|$ (blue solid and orange dashed lines) for sweeping rate $a=10^{-4}$, calculated by Eqs. \ref{['eq:1stode']} and \ref{['eq:proj2channel']} with initial values $~\mathcal{U}^T(0)=[1,0,0,0]$ (a) and $[0,1,0,0]$ (b). The gray solid lines are the moduli of reference coefficients $|c^{(r)}_{1,2}(t)|$ (see the text for the definition). (c) Time evolution of remaining probabilities $P_{1,2}^{(c)}(t)$ of channels 1 and 2 corresponding to (a) and (b), respectively, calculated by Eq. \ref{['Pt']}. The horizontal black dotted lines denote the values of $\exp{(-\pi/2\tilde{\alpha}')}$ for comparison, with the dimensionless sweeping rate $\tilde{\alpha}'\approx0.24$ (c), $2.4$ (f), and $24$ (i). (d)-(f), (g)-(i) The same as (a)-(c), except with $a=10^{-3}$ (d-f) and $10^{-2}$ (g-i). The parameters $r$ and $b$ are set the same as those in Fig. \ref{['fig1']}. The vertical gray dashed lines in all figures indicate the time $t_c$ such that $at_c\approx0.03$.
• Figure 3: Nonreciprocal case with $v=1.25$. All other settings are the same as in Fig. \ref{['fig2']}.
• Figure 4: The schematic diagram for the linearly time-varying capacitance $C(t)=(a t + b)C_0$ (the left panel circled out by the dashed rectangle). The right panel demonstrates the INIC and the circuit multiplier. The detailed description can be found in the text.
• Figure 5: Reciprocal ciruit LZT with initial values $\mathcal{U}^T(0)=[1,0,5,0]$ and $[0,1,0,5]$ instead of $[1,0,0,0]$ and $[0,1,0,0]$ in Fig. \ref{['fig2']}, respectively. All other settings are the same as in Fig. \ref{['fig2']}.