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The Journal of Chemical Physics

Periodic motion around the crossing point in a two-level system

J.-M. Lopez-Castillo, Abdelali Filali-Mouhim, Jean‐Paul Jay‐Gerin

Published Aug 1, 1992• 15 citations
Publisher

Abstract

We study a two-level system described by a Hamiltonian which exhibits a time-periodic variation of its diagonal matrix elements in the small coupling regime. We investigate the transition probability, the Bloch–Siegert shift, and the Rabi frequency of the multiquantum transition as functions of the amplitude and frequency of the oscillations. These quantities all exhibit a very distinct behavior depending upon whether or not the amplitude of the oscillations allows for the crossing of the levels. This behavior is monotonic for small amplitudes while it presents oscillations as a function of the frequency for larger amplitudes. These oscillations are unambiguously attributed to the Landau–Zener–Stueckelberg interference effects between the crossing point and the turning point. A discussion of the relevance of these oscillations to possible experimental observations, notably in electron-transfer reactions, is presented.

DOI
10.1063/1.463126

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We study a two-level system described by a Hamiltonian which exhibits a time-periodic variation of its diagonal matrix elements in the small coupling regime. We investigate the transition probability, the Bloch–Siegert shift, and the Rabi frequency of the multiquantum transition as functions of the amplitude and frequency of the oscillations. These quantities all exhibit a very distinct behavior depending upon whether or not the amplitude of the oscillations allows for the crossing of the levels. This behavior is monotonic for small amplitudes while it presents oscillations as a function of the frequency for larger amplitudes. These oscillations are unambiguously attributed to the Landau–Zener–Stueckelberg interference effects between the crossing point and the turning point. A discussion of the relevance of these oscillations to possible experimental observations, notably in electron-transfer reactions, is presented.
We investigate quantum interference effects in a superconducting Cooper-pair box by taking into account the possibility of tunneling processes involving one and two Cooper pairs. The quantum dynamics is analyzed in a framework of three-level model. We compute Landau–Zener probabilities for a linear sweep of the gate charge and investigate Rabi oscillations in a periodically driven three-level system under in- and off-resonance conditions. It was shown that the Landau–Zener probabilities reveal two different patterns: “step”- and “beats”-like behaviors associated with the quantum interference effects. Control on these two regimes is provided by the change of the ratio between two characteristic time scales of the problem. We demonstrate through the analysis of a periodically driven three-level system, that if a direct transition between certain pairs of levels is allowed and fine-tuned to a resonance, the problem is mapped to the two-level Rabi model. If the transition between a pair of levels is forbidden, the off-resonance Rabi oscillations involving second order in tunneling processes are predicted. This effect can be observed by measuring a population difference slowly varying in time between the states of the Cooper-pair box characterized by the same parity.
We clarify some aspects of population dynamics of a level-crossing system, in which a periodic modulation of the relative energy is superimposed on the energy change with a constant velocity. For moderate values of the off-diagonal coupling, the temporal evolution of the system is well described by a transfer-matrix formalism. It is shown that the existence probability on one level shows a series of steplike changes both in the high-frequecy limit and the low-frequency limit of the periodic modulation, reflecting the quasiquantization of the oscillating field in the former case, while reflecting the multiple real crossings in the latter case.
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.
The effect of an additional high frequency alternating field on the Bloch-Zener oscillation of a quantum particle is investigated theoretically. The effective magnitude of the band gap is reduced by the ac field, and even collapsed to zero by appropriately tuning the amplitude and the frequency of the oscillation. This results in the enhancement of the Zener tunneling with dramatic changes of the Bloch-Zener oscillation. The temporal behavior of the wave packet is clarified as due to the interplay of the Landau-Zener interference in the repeated level crossings with two different time scales, namely, the period of the Bloch oscillation, and the period of the high frequency ac field.
We investigate quantum interference effects in a superconducting Cooper-pair box by taking into account the \npossibility of tunneling processes involving one and two Cooper pairs. The quantum dynamics is analyzed in a \nframework of three-level model. We compute Landau–Zener probabilities for a linear sweep of the gate charge \nand investigate Rabi oscillations in a periodically driven three-level system under in- and off-resonance conditions. It was shown that the Landau–Zener probabilities reveal two different patterns: “step”- and “beats”-like \nbehaviors associated with the quantum interference effects. Control on these two regimes is provided by change \nof the ratio between two characteristic time scales of the problem. We demonstrate through the analysis of a periodically driven three-level system, that if a direct transition between certain pairs of levels is allowed and finetuned to a resonance, the problem is mapped to the two-level Rabi model. If the transition between pair of levels \nis forbidden, the off-resonance Rabi oscillations involving second order in tunneling processes are predicted. \nThis effect can be observed by measuring a population difference slowly varying in time between the states of \nthe Cooper-pair box characterized by the same parity
We have explored a genuinely time-dependent problem consisting of periodic crossings of energy levels. The model can be realized in several physical systems, but we refer it explicitly to a two-level ion that swings in a harmonic trap and concurrently interacts with a traveling light wave. We have solved the Schr\"odinger equation for this model by numerical integration as well as by the use of matrix continued fractions for the steady state. We investigate both the time and the frequency behavior of the model. We have also compared the exact behavior with an intuitive model based on a Landau-Zener description of each crossing followed by a simple relaxation behavior. In addition, the coherent interaction with the field is found to lead to a resonancelike behavior of the population transfer, which we can attribute to the accumulated phase of the Bloch vector. The various time scales of the problem are identified and their physical significance is elucidated.