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Physical Review A

Pulsed excitation of a transition to a decaying level

Nikolay V. Vitanov, Stig Stenholm

Published Apr 1, 1997• 51 citations
Publisher

Abstract

The paper presents an analytic study of a two-state system which interacts with an external pulsed field. The excited state is supposed to decay out of the system with a decay rate \ensuremath{\Gamma}. We present two exact analytic solutions, viz., the finite Landau-Zener model and the Demkov-Kunike model. We also derive an approximate formula for arbitrary pulse shapes and detunings in the limit of a large decay rate. It gives the exact result for the original Landau-Zener model and the correct asymptotics for large \ensuremath{\Gamma} in the general case. We conclude that the independence of the final ground-state population on the decay rate, recently established for the original Landau-Zener model [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)], is only valid for the latter and not for any other model. The results show that the population loss ultimately decreases at large decay rates which can be viewed as a quantum analog of the overdamped oscillator.

DOI
10.1103/physreva.55.2982

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The paper presents an analytic study of a two-state system which interacts with an external pulsed field. The excited state is supposed to decay out of the system with a decay rate \ensuremath{\Gamma}. We present two exact analytic solutions, viz., the finite Landau-Zener model and the Demkov-Kunike model. We also derive an approximate formula for arbitrary pulse shapes and detunings in the limit of a large decay rate. It gives the exact result for the original Landau-Zener model and the correct asymptotics for large \ensuremath{\Gamma} in the general case. We conclude that the independence of the final ground-state population on the decay rate, recently established for the original Landau-Zener model [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)], is only valid for the latter and not for any other model. The results show that the population loss ultimately decreases at large decay rates which can be viewed as a quantum analog of the overdamped oscillator.
Exact analytical solutions to the dissipative time-dependent Schrödinger equation are obtained for a decaying two-state system with decay rates Γ1 and Γ2 for the levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only t0 = 0 or t0 = − ∞ as for non-decaying models and may be useful for the experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses, while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.
This paper presents an analytic study of the properties of a three-state system coupled to two external laser fields on two-photon resonance. The intermediate state is generally off single-photon resonance and can decay irreversibly out of the system. The two fields are assumed to have the same pulse-shaped time dependence but may have different strengths. Various general properties for arbitrary pulse shapes, detunings and decay rates are derived and illustrated with an exactly solvable analytic model. They include the conditions for complete population transfer and depletion, complete population loss or no loss, various limitations on the values of the populations, the effect of overdamping at large decay rates and others.
For a Landau-Zener transition in a two-level system, the probability for a particle, initially in the first level, to survive the transition and to remain in the first level, does not depend on whether or not the second level is broadened [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)]. In other words, the seminal Landau-Zener result applies regardless of the broadening of the second level. The same question for the multistate Landau-Zener transition is addressed in the present paper. While for integrable multistate models, where the transition does not involve interference of the virtual paths, it can be argued that the independence of the broadening persists, we focus on nonintegrable models involving interference. For a simplest four-state model, which allows an analytical treatment, we demonstrate that the decay of the excited states affects the survival probability provided that the widths of the final states are different.
Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.
We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.
We show that the solution of a multistate system composed of NNN degenerate lower (ground) states and one upper (excited) state can be reduced by using the Morris-Shore transformation to the solution of a two-state system involving only the excited state and a (bright) superposition of ground states. In addition, there are $N\ensuremath{-}1$ dark states composed of ground states. We use this decomposition to derive analytical solutions for degenerate extensions of the most popular exactly soluble models: the resonance solution, the Rabi, Landau-Zener, Rosen-Zener, Allen-Eberly, and Demkov-Kunike models. We suggest various applications of the multistate solutions, for example, as tools for creating multistate coherent superpositions by generalized resonant $\ensuremath{\pi}$ pulses. We show that such generalized $\ensuremath{\pi}$ pulses can occur even when the upper state is far off resonance, at specific detunings, which makes it possible to operate in the degenerate ground-state manifold without populating the (possibly lossy) upper state, even transiently.