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arXiv:2504.02576 · Journal of Physics A Mathematical and Theoretical

Derivation of the Landau–Zener formula via functional equations

Chen Sun

Published Sep 2, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract The Landau–Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau–Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and does not involve any sophisticated mathematics. Our work provides insights into the origin of the exponential form of the Landau–Zener transition probability, and shows that the Landau–Zener formula can be viewed as a consequence of integrability, though the two-level Landau–Zener Hamiltonian itself does not satisfy the integrability conditions.

DOI
10.1088/1751-8121/ae0219

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Abstract The Landau–Zener formula describes the diabatic transition probability of a two-level system under linear driving. Its rigorous derivation typically relies on sophisticated mathematical tools, such as special functions, Laplace transforms, or contour integrals. In this work, we present a derivation of the Landau–Zener transition probability using a fundamentally different approach via functional equations. By leveraging integrability, we prove that this transition probability satisfies a functional equation, whose solutions establish the exponential form of the formula. The coefficient in the exponent is then determined through a lowest-order perturbation calculation. This derivation is rigorous and does not involve any sophisticated mathematics. Our work provides insights into the origin of the exponential form of the Landau–Zener transition probability, and shows that the Landau–Zener formula can be viewed as a consequence of integrability, though the two-level Landau–Zener Hamiltonian itself does not satisfy the integrability conditions.
The control of nonadiabatic quantum transitions is a requirement for the development of modern quantum technologies. The paradigm to understand these dynamics is the Landau-Zener model. However, for nearly a century, the exact transition probability of this model has been derived using special functions or semiclassical approximations. These conventional methods obscure the underlying physical mechanisms and transition dynamics near the avoided crossing. This cumulative thesis, resulting from the author's research activities at the Intitute of Quantum Physics at Ulm University under the supervision of Prof. Dr. Wolfgang P. Schleich, establishes an analytical framework to study the underlying structure of two-level crossing dynamics. In contrast to the standard techniques, the framework developed in this thesis is based on elementary concepts combined with complex analysis. First, we map the Landau-Zener equations onto an integro-differential equation to decouple them and formulate the Markov approximation which reduces the complicated integrals to the evaluation of Fresnel integrals. This technique reproduces in the asymptotic limit the exact probability amplitude of the Landau-Zener model. Next, to uncover the origin behind the success of the Markov approximation, we analyze the Landau-Zener equations in terms of their underlying nonlinear dynamics. First, we introduce an amplitude-and-phase approach, which explains in a clear way the emergence of the Stueckelberg oscillations through the dynamics of the phase velocity. Second, we map the two-level dynamics onto a nonlinear Riccati differential equation and show that the Markov approximation is equivalent to a linearization of the derived Riccati equation. Moreover, our approach helps to identify the nonlinear term as the reason for the breakdown of the Markov approximation close to the avoided crossing. Finally, we derive exact asymptotic expressions for the transition amplitudes using elementary functions for both negative and positive times. Connecting these two regimes via the analytical continuation of the logarithmic phase reveals that the Landau-Zener transition is mainly governed by a logarithmic phase singularity. Based on this logarithmic phase singularity, we introduce a new time scale to map the system dynamics as a path via a logarithmic phase ansatz in the complex plane. Evaluating this path via the Cauchy’s integral formula yields the exact transition probability which demonstrates that the exact solution emerges from the underlying phase singularity rather than properties of special functions.
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.
We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.
We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ on the adiabatic parameter $\ensuremath{\delta}$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\ensuremath{\eta}$ in the Landau-Zener-St\"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\ensuremath{\delta}\ensuremath{\rightarrow}0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\ensuremath{\eta}\ensuremath{\rightarrow}1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.
By the adiabatic theorem, the probability of non-adiabatic transitions in a time-dependent quantum system vanishes in the adiabatic limit. The Landau-Zener (LZ) formula gives the leading functional behavior of the probability close to this limit. On the other hand, in counterdiabatic dynamics, one achieves effectively adiabatic evolution at finite driving speed by adding an extra field which suppresses non-adiabatic transitions: the adiabatic gauge potential (AGP). We investigate the mechanism by which the AGP suppresses the transition probability, changing it from the LZ formula to exactly zero. Quantitatively, we find that adding the AGP to the Hamiltonian modifies the LZ formula by a universal prefactor, independent of the adiabatic parameter, which vanishes in the counterdiabatic regime. Qualitatively, this prefactor can be understood as arising from the Aharonov-Bohm phases generated by the AGP between different paths in the complex time plane. Finally, we show that these results extend to a class of integrable time-dependent quantum Hamiltonians by proving that the AGP preserves their integrability condition.