Perturbative Approach to Landau-Zener Transitions

Guozhu Sun, Xueda Wen, Jian Chen +3 more

Chen Sun

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.

Rongyu Hu, Chen Sun

We formulate a perturbative approach for studying a class of multilevel time-dependent quantum systems with constant off-diagonal couplings and diabatic energies being odd functions of time. Applying this approach to a general multistate Landau-Zener (MLZ) model with all diabatic levels crossing at one point (named the one-crossing MLZ model), we derive analytical formulas of all its transition probabilities up to fourth order in the couplings. For one-crossing MLZ models it is difficult to obtain such analytical results by other kinds of approximation methods; thus, these perturbative results can serve as reliable benchmarks for future studies of any one-crossing MLZ models that have not been exactly solved.

Artur Mikhailovich Ishkhanyan

Peter Hänggi A, Martijn Wubs, Sigmund Kohler +2 more

A qubit may undergo Landau‐Zener transitions due to its coupling to one or several quantum harmonic oscillators. First we show that for a qubit coupled to one oscillator, Landau‐Zener transitions can be used for single‐photon generation and for the controllable creation of qubit‐oscillator entanglement, with state‐of‐the‐art circuit QED as a promising realization. Second, for a qubit coupled to two cavities, we show that Landau‐Zener sweeps of the qubit are well suited for the robust creation of entangled cavity states, in particular symmetric Bell states, with the qubit acting as the entanglement mediator. Finally, for a qubit coupled to an environment or bath we propose to employ dissipative Landau‐Zener sweeps of the qubit for the detection of bath properties. At the heart of our proposals lies the calculation of the exact Landau‐Zener transition probability for the qubit, by summing all orders of the corresponding series in time‐dependent perturbation theory.

Duan Suqing, Libin Fu, Jie Liu +1 more