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Physical Review A

Effects of pulse shape on strongly driven two-level systems

C. W. S. Conover

Published Dec 19, 2011• 21 citations
Publisher

Abstract

We present an experimental study of the dynamics of a two-level system driven by strong nonresonant electromagnetic pulses as a function of pulse intensity and detuning. We have explored the qualitative and quantitative behavior of the transition probability as a function of pulse area for five different temporal profiles: Lorentzian, Lorentzian squared, hyperbolic secant, hyperbolic secant squared, and Gaussian. The two-level system consists of a fine-structure doublet in sodium Rydberg states coupled by Raman transitions driven through far-off-resonance intermediate states. The pulses are in the microwave regime and have high fidelity and uniform intensity. Experiments show that, despite the similarity in the pulse shapes, the behavior of the population transfer versus intensity depends dramatically on the temporal shape and that the spectral properties and area of the pulse do not adequately describe the response.

DOI
10.1103/physreva.84.063416

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We present an experimental study of the dynamics of a two-level system driven by strong nonresonant electromagnetic pulses as a function of pulse intensity and detuning. We have explored the qualitative and quantitative behavior of the transition probability as a function of pulse area for five different temporal profiles: Lorentzian, Lorentzian squared, hyperbolic secant, hyperbolic secant squared, and Gaussian. The two-level system consists of a fine-structure doublet in sodium Rydberg states coupled by Raman transitions driven through far-off-resonance intermediate states. The pulses are in the microwave regime and have high fidelity and uniform intensity. Experiments show that, despite the similarity in the pulse shapes, the behavior of the population transfer versus intensity depends dramatically on the temporal shape and that the spectral properties and area of the pulse do not adequately describe the response.
The problem of a two-level atom interacting with a radiation pulse is studied in the limit that the atom-field detuning times the pulse duration is much greater than unity. Owing to the large atom-field detuning, transitions result from nonadiabatic coupling of the states by the field. The transition probability for the atom to be excited following the pulse is studied as a function of field strength for five different pulse shapes: hyperbolic secant, Lorentzian, hyperbolic secant squared, Lorentzian squared, and Gaussian. It is shown that the behavior of the transition probability differs qualitatively for these pulses. An explanation of this qualitative difference is given in terms of the Massey parameter. Numerical solutions are compared with asymptotic solutions and several anomalies are noted. In the limit of large field strength, a universal expression for the transition probability is found. An interesting feature of the solutions is that, in the limit of very large field strengths, the transition probability for a Gaussian pulse can approach unity despite the fact that the pulse has an exponentially small Fourier amplitude at the atom-field detuning. This apparent violation of the energy-time uncertainty principle is explained in terms of the nonlinear atom-field interactions.
We present a study of the coherent interaction of a qubit with a pulse-shaped external field of a constant carrier frequency. We explore, theoretically and experimentally, the transition line profile (the dependence of the transition probability on the detuning) for five different pulse shapes: Rectangular, Gaussian, hyperbolic-secant, squared hyperbolic-secant, and exponential. The theoretical description for all cases but sech2{\mathrm{sech}}^{2}sech2 is based on the analytical solutions to the Schr\"odinger equation or accurate approximations available in the literature. For the sech2{\mathrm{sech}}^{2}sech2 pulse we derive an analytical expression for the transition probability using the Rosen-Zener conjecture, which proves very accurate. In addition, the same conjecture turns out to provide very accurate approximations for the Gaussian and exponential pulses. The experimental results are obtained with one of IBMQ's quantum processors. An excellent agreement between theory and experiment is observed, demonstrating some pulse-shape-dependent fine features of the transition probability profile. The mean absolute error (a measure of the accuracy of the fit) features an improvement by a factor of 4 to 7 for the analytic models compared to the commonly used Lorentzian fits. Moreover, the uncertainty of the qubit's resonance frequency is reduced by a factor of 4 for the analytic models compared to the Lorentzian fits. These results demonstrate both the accuracy of the analytic modeling of quantum dynamics and the excellent coherent properties of IBMQ's qubit.
An ultrashort strong nonresonant pulse induces transient phenomena in an optically thick medium consisting of an ensemble of nondegenerate two-level atoms (nonadiabatic transitions, transient light-shifts,\dots{}). A weak resonant pulse propagates in the system and we study theoretically and numerically the temporal and spectral modifications that are induced. Moreover, we show that the atomic media, modified by a strong pulse can act as a ``pulse-shaper'' device that can modify, in a controlled way, the shape of the weak pulse. It maps out the transient light shift induced by the strong pulse and reduces efficiently the long time dispersion tail of the weak pulse. We study and discuss the physical phenomena that allow such shaping effects and the necessary conditions to satisfy in order to observe them.
We observe microwave multiphoton transitions in Rydberg states of Na using intense subnanosecond half-cycle electric field pulses in a microwave transmission line. Transition probabilities from ns and nd states, ranging in principal quantum number from 18 to 40, to nearby nl (l\ensuremath{\geqslant}2) states are measured both as a function of pulse amplitude and the delay between two identical temporally spaced pulses. The transition rate depends strongly on both the pulse amplitude and the temporal delay between the two pulses. Nearly 100% of the population can be transferred for certain amplitudes of the half-cycle pulse. Transition probabilities are successfully described both with a dynamic Stark model and direct numerical integration of the time-dependent Schr\"odinger equation. Ramsey interference between two delayed pulses is observed that shows strong-field behavior. The two-pulse experiment is a technique for high-order multiphoton spectroscopy without concern for ac Stark shifts.
We study the dynamics of three-level $\ensuremath{\Lambda}$-type atoms driven by a coherent train of short, nonoverlapping laser pulses. We derive analytical nonperturbative expressions for the density matrix by approximating the pulses by a $\ensuremath{\delta}$ function. We demonstrate that, depending on the train parameters, several scenarios of system dynamics can be realized. We show the possibility of driving Raman transitions between the two ground states of the $\ensuremath{\Lambda}$ system, avoiding populating excited states by using pulses with effective area equal to $2\ensuremath{\pi}$. The number of $2\ensuremath{\pi}$ pulses needed to transfer the entire population from one ground state to another depends on the ratio between the Rabi frequencies of two allowed transitions. In the case of equal Rabi frequencies, the system can be transferred from one ground state to another with a single $2\ensuremath{\pi}$ pulse. When the total pulse area differs from $2\ensuremath{\pi}$ and the two-photon resonance condition is fulfilled, the system evolves into a ``dark'' state and becomes transparent to subsequent pulses. The third possible scenario is the quasi-steady-state regime when neither the total single-pulse area is equal to $2\ensuremath{\pi}$ nor is the two-photon resonance condition fulfilled. In this regime the radiative-decay-induced drop in the population following a given pulse is fully restored by the subsequent pulse. We derive an analytical expression for the density matrix in the quasi-steady-state regime. We analyze the dependence of the postpulse excited-state population in the quasi-steady-state regime on the train parameters. We find the optimal values for the train parameters corresponding to the maximum of the excited-state population. The maximum of the excited-state population in the steady-state regime is reached at an effective single-pulse area equal to $\ensuremath{\pi}$ and is equal to 2/32/32/3 in the limiting case when its radiative lifetime is much shorter than the pulse repetition period.