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Physical Review Letters

Time of Zener tunneling

Kieran J. Mullen, Eshel Ben‐Jacob, Yuval Gefen, Z. Schuss

Published May 22, 1989• 114 citations
Publisher

Abstract

The time of Zener tunneling, ${\ensuremath{\tau}}_{z}$, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is \ensuremath{\Delta} and the time rate of change of the energy is \ensuremath{\alpha}, that in the adiabatic limit ${\ensuremath{\tau}}_{z}$ scales as \ensuremath{\Delta}/\ensuremath{\alpha} and in the sudden limit ${\ensuremath{\tau}}_{z}$ scales as (\ensuremath{\Elzxh}/\ensuremath{\alpha}${)}^{1/2}$. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.

DOI
10.1103/physrevlett.62.2543

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The time of Zener tunneling, \ensuremathτz{\ensuremath{\tau}}_{z}\ensuremathτz​, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is \ensuremath{\Delta} and the time rate of change of the energy is \ensuremath{\alpha}, that in the adiabatic limit \ensuremathτz{\ensuremath{\tau}}_{z}\ensuremathτz​ scales as \ensuremath{\Delta}/\ensuremath{\alpha} and in the sudden limit \ensuremathτz{\ensuremath{\tau}}_{z}\ensuremathτz​ scales as (\ensuremath{\Elzxh}/\ensuremath{\alpha})1/2{)}^{1/2})1/2. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.
We compare the standard Zener tunneling problem with one in which the unperturbed states do not intersect. We find that even when their adiabatic energy spectra are identical, the two cases have dramatically different asymptotic tunneling probabilities. The time of Zener tunneling, a measure of the duration of the transition, is also different for the two. The latter is determined by calculating the transition probability as a function of time, for both the sudden and adiabatic limits. We also develop analytic asymptotic expansions for the tunneling probability for these limits. The results are discussed in terms suitable for comparison with experiment.
The time-dependent behavior of a two-level system interacting with a quantum oscillator system is analyzed in the case of a coupling larger than both the energy separation between the two levels and the energy of the quantum oscillator ($\ensuremath{\Omega}<\ensuremath{\omega}<\ensuremath{\lambda}$, where $\ensuremath{\Omega}$ is the frequency of the transition between the two levels, $\ensuremath{\omega}$ is the frequency of the oscillator, and $\ensuremath{\lambda}$ is the coupling between the two-level system and the oscillator). Our calculations show that the amplitude of the expectation value of the oscillator coordinate decreases as the two-level system undergoes the transition from one level to the other, while the transfer probability between the levels is staircaselike. This behavior is explained by the interplay between the adiabatic and nonadiabatic regimes encountered during the dynamics with the system acting as a quantum counterpart of the Landau-Zener model. The transition between the two levels occurs as long as the expectation value of the oscillator coordinate is driven close to zero. On the contrary, if the initial conditions are set such that the expectation values of the oscillator coordinate are far from zero, the system will remain locked on one level.
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.
The authors consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H( in t). In the adiabatic limit, in <<1, the transition probability P(+,-) from one energy level (labelled by -) at time t=- infinity to the other (labelled by +) at time t=+0OI is known to behave as P(+,-)=exp(- alpha -1 in -1 ) exp( alpha 0 )(1+O( in )). Using a simple iterative procedure generating Hamiltonians H 0 =H,H 1 ,. . .H N+1 , they compute the full asymptotic expansion of the transition probability ln P(+,-)=- alpha -1 in -1 + alpha 0 + j=0 N alpha j in j +O ( in N+1 ) For all N>or=0.
This paper presents analytic formulas for various transition times in the Landau-Zener model. Considerable differences are found between the transition times in the diabatic and adiabatic bases, and between the jump time (the time for which the transition probability rises to the region of its asymptotic value) and the relaxation time (the characteristic damping time of the oscillations which appear in the transition probability after the crossing). These transition times have been calculated by using the exact values of the transition probabilities and their derivatives at the crossing point and approximations to the time evolutions of the transition probabilities in the diabatic basis, derived earlier [N. V. Vitanov and B. M. Garraway, Phys. Rev. A 53, 4288 (1996)], and similar results in the adiabatic basis, derived in the present paper.