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Physical Review A

Perturbative solution to the time-dependent two-level problem and the validity of the Rosen-Zener conjecture

R. T. Robiscoe

Published Mar 1, 1983• 17 citations
Publisher

Abstract

We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope $V(t)$. We derive an approximate solution for the system's transition amplitude $P(\ensuremath{\infty})$ which is correct to the third order of perturbation theory, and which applies to all pulses $V(t)$ with finite first and second moments which obey the following: $\mathrm{lim} {t}^{3}V(t)=0$, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Our form of solution for $P(\ensuremath{\infty})$ provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.

DOI
10.1103/physreva.27.1365

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Extension of the Rosen-Zener solution to the two-level problem

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Time of Zener tunneling

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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.

Physical Review Letters•May 22, 1989•114 citations
We consider the time-dependent two-level problem of quantum mechanics, where the levels are coupled by a radio-frequency pulse with an arbitrary time-dependent envelope V(t)V(t)V(t). We derive an approximate solution for the system's transition amplitude P(\ensuremath∞)P(\ensuremath{\infty})P(\ensuremath∞) which is correct to the third order of perturbation theory, and which applies to all pulses V(t)V(t)V(t) with finite first and second moments which obey the following: limt3V(t)=0\mathrm{lim} {t}^{3}V(t)=0limt3V(t)=0, as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞. Our form of solution for P(\ensuremath∞)P(\ensuremath{\infty})P(\ensuremath∞) provides a criterion for judging the validity of a solution previously conjectured by Rosen and Zener, and it is generally useful for providing line-shape details in many cases of practical interest.
We solve the problem of two discrete quantum levels which are coupled by a time-dependent radio-frequency pulse $W(t)=V(t) {e}^{i\ensuremath{\nu}t}$, where the envelope function is of a form suggested by Rosen and Zener: $V(t)={V}_{0}sech(\frac{\ensuremath{\pi}t}{T})$. When a level damping constant $\ensuremath{\gamma}$ is included, in the manner of Bethe-Lamb theory, the solutions show new features which are not expected on the basis of a sudden-approximation theory, where V(t)=constV(t)=\mathrm{const}V(t)=const over the pulse duration TTT. Various transient effects such as "ringing" are not present in the extended Rosen-Zener solution; these effects are related to the large impulsive forces at the step discontinuities in the sudden approximation. The final-state level amplitudes can be quite different depending on the size of the pulse rise time TTT as compared with the system Bohr period $\frac{1}{\ensuremath{\omega}}$. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden ($\ensuremath{\omega}T\ensuremath{\ll}1$) and adiabatic ($\ensuremath{\omega}T\ensuremath{\gg}1$) approximations. A model of a "quasisudden" step function is also constructed, and remarks are made on the validity of a certain conjecture by Rosen and Zener.
We consider a two-level system where a pair of discrete quantum states of arbitrary widths and separation are coupled by a general time-dependent external field with a pulselike matrix element, V(t)V(t)V(t). Using a transformation suggested by Robinson [Phys. Rev. A 24, 2239 (1981)] and applying the WKB approximation, a quasiadiabatic solution is constructed for the system's transition amplitude. This solution is useful in estimating transition amplitudes in the asymptotic region for a wide variety of smoothly varying pulses V(t)V(t)V(t).
The general semiclassical time-dependent two-state problem is considered for a specific field configuration referred to as the generalized Rosen-Zener model. This is a rich family of pulse amplitude- and phase-modulation functions describing both non-crossing and term-crossing models with one or two crossing points. The model includes the original constant-detuning non-crossing Rosen-Zener model as a particular case. We show that the governing system of equations is reduced to a confluent Heun equation. When inspecting the conditions for returning the system to the initial state at the end of the interaction with the field, we reformulate the problem as an eigenvalue problem for the peak Rabi frequency and apply the Rayleigh-Schrödinger perturbation theory. Further, we develop a generalized approach for finding the higher-order approximations, which is applicable for the whole variation region of the involved input parameters of the system. We examine the general surface in the 3D space of input parameters, which defines the position of the n-th order return-resonance, and show that the section of the general surface is accurately approximated by an ellipse. We find a highly accurate analytic description through the zeros of a Kummer confluent hypergeometric function. From the point of view of the generality, the analytical description of mentioned curve for the whole variation range of all involved parameters is the main result of the present paper.
The time-dependent theory of the three-step absorption of three different light pulses in the electronic four-level systems is considered using the time-dependent perturbation theory. The first, the second, and the third pulse are in resonance with transitions from level 0 to level 1, from level 1 to level 2, and from level 2 to level 3, respectively. The spectral and temporal behaviour of the probability that the fourth level is excited at the moment of time t depends on the energy relaxation constants of the excited electronic levels, the frequencies of the maxima, the spectral widths (and so the durations) of the light pulses, and the time delays between the pulses. To conclude, there are 12 parameters that affect the spectra in the simple model used. We calculate and analyse the case where the frequencies of the maxima of the first and the second pulse are fixed and the frequency of the maximum of the third pulse varies. On the whole, three lines may exist, one of which corresponds to the coherent contribution and the others do not.
Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.
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{)}^{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.