Landau-Zener transition

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

If S Sx the avoided crossing is traversed diabatically and if S Sx it is traversed adiabatically. Between the two extremes of purely adiabatic and purely diabatic traversals, the probability of making the diabatic transition is given by the Landau-Zener transition probability, as has been demonstrated by Rubbmark et... 

Fig. 10.5 Energy level diagram showing the mechanism by which an n = 20 atom is ionized by a microwave field of 700 V/cm amplitude. An atom initially excited to the 20d state is brought to the point at which the n = 20 and 21 Stark manifolds intersect. At this point the atom makes a Landau-Zener transition, as shown by the inset, to the n = 21 Stark manifold and on subsequent microwave cycles makes further upward transitions as shown by the bold arrow. The process terminates when the atom reaches a sufficiently high n state that the microwave field itself ionizes the atom. This occurs at point B in this example (from...

Fig. 10.11 Schematic picture of a multicycle Landau-Zener transition. In combined static and microwave fields the oscillating field brings the atom to the avoided crossing on successive cycles, and the transition amplitudes due to successive cycles add, leading to interference, or resonances (from ref. 12).

Finally, Mahon etal.5 have observed that even at frequencies as low as 670 MHz that ionization of Na occurs at fields near 1/3n5. At such low frequencies it is impossible to explain the ionization on the basis of incoherent single cycle Landau-Zener transitions. Rather the coherent effect of many cycles of the field is required. 

The problem of two crossing points was considered by Shimshoni and Gefen [1991]. In this case the resulting value of the transition probability is dominated by an interference effect, which becomes important when the separation between crossing points is comparable with the characteristic length of interaction region for a single Landau-Zener transition. 

It predicts the adiabatic plateaus of Fig. 17, which can be interpreted as a topological quantization of the number of exchanged photons. The dips are due to nonadiabatic Landau-Zener transitions when the pulse overlap is in the neighborhood of the intersections. With a configuration of counterpropagating laser fields, perpendicular to an atomic beam, this translates into the possibility of deflection of the beam by the quantized transfer of a momentum kh(ffli +... 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

In any problem involving avoided crossings, there are two modes of variation. A physical system will evolve adiabatically, i.e. it will remain on one or other branch if the intensity A is varied slowly. If. however, the variation is rapid then diabatic or Landau-Zener transitions may occur [508]. The probability of a transition from one branch to the other at the avoided crossing is given by... 

Fig. 9.11. (a) Schematic map of the quasienergies, plotted modulo laser photon energy, as a function of the laser intensity, showing avoided crossings, (b) The shape of the laser pulse as a function of time. Under appropriate conditions (as described in the text) a generalised form of Landau-Zener transition occurs which is what one commonly calls a multiphoton transition (after J.-P. Connerade et al. [482]). 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

Figure 6 Landau-Zener transition probability in the linear potential model of the LZ type with a2 = 1.0. a2 and b2 are defined by Eq. (19). Landau-Zener means the result of Eq. (10).

The Landau-Zener transition probability pZN is as simple as the famous Landau-Zener formula of Eq. (10) but much better. Note that p is expressed in terms of a2 and b2 as... 

Figure 12 The Landau-Zener transition probability p in the LZ case. Dashed line exact, solid line Eqs. (170) and (183). (From Ref. 23.)...

The Landau-Zener transition probability yZN can be explicitly given as... 

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