Elementary asymptotic approach to the Landau-Zener problem

Abstract

We present an asymptotic approach towards the standard Landau-Zener problem based on two linearly independent elementary waves of constant amplitude but time-dependent phase. The two contributions to this phase are quadratic and logarithmic in time and result from the linear chirp of the energies and the lowest order correction in the coupling between the two levels in the long-time limit. Indeed, our solutions subjected to initial conditions at a large but finite time in the past, are valid for large negative and large positive times. Due to their asymptotic nature they are not valid in the neighborhood of the moment when the levels cross. However, as the starting point of the dynamics moves further into the past, the time interval of the break-down of our asymptotic solutions shrinks and vanishes in the limit of the infinite past which corresponds to the standard Landau-Zener situation. Our approach explains not only every feature of the exact solution but yields deeper insights into the origin of the effects. In particular, it (i) brings to light the subtleties involved in the asymptotic limit leading to the standard expressions for the Landau-Zener transition amplitudes, (ii) identifies the logarithmic phase as the origin of the exponential transition probability amplitude, and (iii) reveals the structure of the lowest order corrections to the Landau-Zener result when the starting point is not in the infinite past.

Figures (4)

• Figure 1: Quantum dynamics in the Landau-Zener level-crossing problem. At $\tau = -\tau_{0}$ the two energy levels denoted by $\ket{a}$ and $\ket{b}$ (a) start to approach each other linearly in time and cross at $\tau=0$. We address the question of the values of the corresponding probability amplitudes $a$ and $b$ at $\tau = \tau_{0}$ when $\tau_{0}\rightarrow\infty$. In panel (b), we present the time evolution of the absolute values $\abs{a}$ and $\abs{b}$ and in (c) the phases $\varphi_a$ and $\varphi_b$ of the two probability amplitudes $a = a(\tau)$ and $b = b(\tau)$. We emphasize the sudden transitions of $\abs{a}$ and $\abs{b}$ in the vicinity of $\tau=0$ from $\abs{a(-\tau_{0})} = 1$ and $\abs{b(-\tau_{0})}=0$ to $\lim_{\tau_{0} \to \infty} \abs{a(\tau_{0})} = \exp(-\pi/(2\epsilon))$ and $\lim_{\tau_{0} \to \infty} \abs{b(\tau_{0})} = \sqrt{1-\exp(-\pi/\epsilon)}$ in the limit of $\tau_{0}\rightarrow\infty$. For positive times, $\abs{a}$ and $\abs{b}$ display the decaying Stueckelberg oscillations. The phases $\varphi_a$ and $\varphi_b$ are both negative and are either symmetric for $\varphi_a$ or antisymmetric for $\varphi_b$ with respect to $\tau = 0$. Here, we have chosen $\tau_{0}=100.0$ and $\epsilon=3.0$.
• Figure 2: Time evolution of the two probability amplitudes $a$ and $b$ in the complex plane. For large negative times, $a$ (blue line) performs a circular motion on the unit circle. Simultaneously, $b$ (red line) starts from the origin of the complex plane and rotates around it while its amplitude increases (a). In the vicinity of $\tau=0$, amplified in (b), $a$ and $b$ quickly reach their asymptotic values for large positive $\tau$, while still rotating. Whereas $a$ changes the direction of rotation close to $\tau = 0$, the amplitude $b$ maintains its direction. Here, we have chosen $\tau_{0}=100.0$ and $\epsilon=3.0$.
• Figure 3: Comparison between the exact numerical solution of eqs. (\ref{['eq:group:a:b']}) for the two probability amplitudes $a$ and $b$ in their absolute values $\abs{a}$ and $\abs{b}$ and their elementary asymptotic approximations given by eqs. (\ref{['eq:group:approx:sol:a:b:large:neg:times']}) and (\ref{['eq:group:approx:a:neg:times:with:bla']}) valid for large negative times, and \ref{['eq:group:asymptotic:exp:a:b:large:times']} for large positive times. In order to cover the complete time domain from $-\tau_{0}$ to $\tau_{0}$ and amplify the behavior at $\tau=0$, we have chosen a logarithmic scale for $\tau$. For large negative times there is an excellent agreement between the approximation and the exact result. The oscillations amplified in the two insets are a consequence of the fact that $\tau_{0}$ is large but finite, and $a$ and $b$ are given by superpositions. In the vicinity of $\tau=0$, we recognize in the approximations for both probability amplitudes $1/\tau$-singularities---they appear for negative and positive times and originate from the logarithmic phase. For large positive times, the Stueckelberg oscillations result from the interference of the two contributions in \ref{['eq:asymptotic:exp:a:large:times', 'eq:asymptotic:exp:b:large:times']} and we find again an excellent agreement. Since the $1/\tau$-decay appears in one of the interfering contributions the Stueckelberg oscillations decay with the same power law. Here, we have chosen $\tau_{0}=100.0$ and $\epsilon=3.0$.
• Figure 4: Logarithmic phase singularity in the asymptotic expression, eqs. (\ref{['eq:group:approx:a:neg:times:with:bla']}), for $a(-\abs{\tau})$. In (a), we compare and contrast the approximate phase $\Delta\phi(-\abs{\tau})$ (dashed blue) and the exact phase $\varphi_a$ (solid gray) of the probability amplitude $a$ and in (b) the corresponding phase velocities $\Delta\dot{\phi}(-\abs{\tau})$ and $\dot{\varphi}_a$. Although, \ref{['eq:group:approx:a:neg:times:with:bla']} is valid only for large negative times, we show $\Delta\phi(-\abs{\tau})$ and $\Delta\dot{\phi}(-\abs{\tau})$ in the complete domain $-\tau_{0}\leq\tau<0$. In the neighborhood of $-\tau_{0}$ the phase $\Delta\phi(-\abs{\tau})$ is quadratic in $\tau$ giving rise to a phase velocity $\Delta\dot{\phi}(-\abs{\tau})$ which is linear in $\tau$. This behavior is a consequence of the dominance of the quadratic phase compared to the logarithm. In this time domain, $\Delta\phi(-\abs{\tau})$ is an excellent approximation of $\varphi_a$. However, close to the origin that is $\tau\lesssim 0$, the logarithm dominates giving rise to a logarithmic phase singularity and a $1/\tau$-singularity in the phase velocity. Here, the two functions strongly deviate since the exact phase remains finite. In this figure, we have chosen $\tau_{0}=100.0$ and $\epsilon=3.0$.