Landau–Zener formula

The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linear function of time.

The formula, giving the probability of a diabatic (not adiabatic) transition between the two energy states, was published separately by Lev Landau,[1] Clarence Zener,[2] Ernst Stueckelberg,[3] and Ettore Majorana,[4] in 1932.

If the system starts, in the infinite past, in the lower energy eigenstate, we wish to calculate the probability of finding the system in the upper energy eigenstate in the infinite future (a so-called Landau–Zener transition).

For infinitely slow variation of the energy difference (that is, a Landau–Zener velocity of zero), the adiabatic theorem tells us that no such transition will take place, as the system will always be in an instantaneous eigenstate of the Hamiltonian at that moment in time.

In order that the equations of motion for the system might be solved analytically, a set of simplifications are made, known collectively as the Landau–Zener approximation.

This requirement is quite restrictive as a linear change will not, in general, be the optimal profile to achieve the desired transition probability.

is the off-diagonal element of the two-level system's Hamiltonian coupling the bases, and as such it is half the distance between the two unperturbed eigenenergies at the avoided crossing, when

The simplest generalization of the two-state Landau–Zener model is a multistate system with a Hamiltonian of the form where A and B are Hermitian NxN matrices with time-independent elements.

There are exact formulas, called hierarchy constraints, that provide analytical expressions for special elements of the scattering matrix in any multi-state Landau–Zener model.

[11][12] Discrete symmetries often lead to constraints that reduce the number of independent elements of the scattering matrix.

[13][14] There are also integrability conditions that, when they are satisfied, lead to exact expressions for the entire scattering matrices in multistate Landau–Zener models.

Numerous such completely solvable models have been identified, including: Applications of the Landau–Zener solution to the problems of quantum state preparation and manipulation with discrete degrees of freedom stimulated the study of noise and decoherence effects on the transition probability in a driven two-state system.

Using the Schwinger–Keldysh Green's function, a rather complete and comprehensive study on the effect of quantum noise in all parameter regimes were performed by Ao and Rammer in late 1980s, from weak to strong coupling, low to high temperature, slow to fast passage, etc.

Concise analytical expressions were obtained in various limits, showing the rich behaviors of such problem.

Sketch of an avoided crossing . The graph represents the energies of the system along a parameter z (which may be vary in time). The dashed lines represent the energies of the diabatic states, which cross each other at z c , and the full lines represent the energy of the adiabatic states (eigenvalues of the Hamiltonian).