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Physical Review A

Coherent excitation of a two-state system by a train of short pulses

Nikolay V. Vitanov, P. L. Knight

Published Sep 1, 1995• 61 citations
Publisher

Abstract

A theoretical nonperturbative study of the coherent excitation of a two-state system by N consecutive equally spaced identical pulses is presented. General relations between the evolution matrix elements in the cases of one and N pulses are obtained in a closed form. For pulse envelopes allowing analytical solutions, these relations enable analytical treatment of pulse-train excitation; for pulses that can only be treated numerically, they shorten the computations by a factor of N. The relations show that the multiple-pulse excitation of a two-state system can be considered as a quantum analog of the diffraction grating. The interaction dynamics substantially depends on the way the train is produced because the phase shift that is accumulated by the probability amplitudes due to the free evolution of the system during and between the pulses differs. In the limit of weak excitation, the results recover earlier conclusions based on perturbative treatments. Simple formulas are derived for the conditions for complete population inversion (CPI) and complete population return (CPR), which differ from the single-pulse ones. These general results are applied to four particular cases that allow analytical solutions: resonant, rectangular, Rosen-Zener, and Allen-Eberly pulses. A common feature in all these cases is that the number and the amplitude of oscillations in the state populations increase with the number of pulses, while their width decreases. For rectangular pulses, CPI is possible below a given value of the ratio between the detuning and the pulse area; this value increases nearly linearly with the number of pulses.For the Rosen-Zener model, CPI is found to be possible for two and more pulses, while it is impossible for a single non-resonant pulse. It is shown that for a given number of pulses there is an upper limit on the detuning for which CPI can be observed; this limit increases logarithmically with the number of pulses. For the Allen-Eberly model, CPR is established to be possible for more than one pulse, while it is impossible for a single pulse. The cases of even and odd number of pulses are shown to lead to substantially different results. The limitations imposed on the detuning by the conditions for CPR and CPI are derived and discussed.

DOI
10.1103/physreva.52.2245

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A theoretical nonperturbative study of the coherent excitation of a two-state system by N consecutive equally spaced identical pulses is presented. General relations between the evolution matrix elements in the cases of one and N pulses are obtained in a closed form. For pulse envelopes allowing analytical solutions, these relations enable analytical treatment of pulse-train excitation; for pulses that can only be treated numerically, they shorten the computations by a factor of N. The relations show that the multiple-pulse excitation of a two-state system can be considered as a quantum analog of the diffraction grating. The interaction dynamics substantially depends on the way the train is produced because the phase shift that is accumulated by the probability amplitudes due to the free evolution of the system during and between the pulses differs. In the limit of weak excitation, the results recover earlier conclusions based on perturbative treatments. Simple formulas are derived for the conditions for complete population inversion (CPI) and complete population return (CPR), which differ from the single-pulse ones. These general results are applied to four particular cases that allow analytical solutions: resonant, rectangular, Rosen-Zener, and Allen-Eberly pulses. A common feature in all these cases is that the number and the amplitude of oscillations in the state populations increase with the number of pulses, while their width decreases. For rectangular pulses, CPI is possible below a given value of the ratio between the detuning and the pulse area; this value increases nearly linearly with the number of pulses.For the Rosen-Zener model, CPI is found to be possible for two and more pulses, while it is impossible for a single non-resonant pulse. It is shown that for a given number of pulses there is an upper limit on the detuning for which CPI can be observed; this limit increases logarithmically with the number of pulses. For the Allen-Eberly model, CPR is established to be possible for more than one pulse, while it is impossible for a single pulse. The cases of even and odd number of pulses are shown to lead to substantially different results. The limitations imposed on the detuning by the conditions for CPR and CPI are derived and discussed.
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.
We consider the population inversion in a two-level system generated by a sub-one-cycle pulse excitation. Specifically, we explore the effect that the time derivative of the pulse envelope has on the Rabi dynamics. Our analysis is based on a combination of analytical, perturbative, and nonperturbative treatments and is complemented by numerical simulations. We find a shortening of the Rabi inversion period and show that complete inversion is unobtainable under resonant, ultrashort pulse conditions. The impact of nonresonant and carrier-envelope phase-dependent effects on the dynamics of two-level and multilevel systems is studied numerically, and conditions for complete population inversion are derived.
An exact analytic solution to the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant pulsed external field. The pulse has a hyperbolic-secant shape, with a sign jump (i.e. a phase step of π) at its maximum, which nullifies the overall pulse area. It is shown that off resonance, for detunings larger than the pulse bandwidth, the transition probability approaches unity as the field intensity increases. The population inversion is robust against small-to-moderate variations in the detuning and the intensity, a feature reminiscent of adiabatic passage. The population inversion, however, is not of adiabatic nature; it is induced by δ-function-shaped nonadiabatic coupling with an area of nearly π. An estimate of the required experimental resources shows that implementation with shaped femtosecond pulses is feasible.
We characterize the coherent dynamics of a two-level quantum emitter driven by a pair of symmetrically detuned phase-locked pulses. The promise of dichromatic excitation is to spectrally isolate the excitation laser from the quantum emission, enabling background-free photon extraction from the emitter. While excitation is not possible without spectral overlap between the exciting pulse and the quantum emitter transition for ideal two-level systems due to cancellation of the accumulated pulse area, we find that any additional interactions that interfere with cancellation of the accumulated pulse area may lead to a finite stationary population inversion. Our spectroscopic results of a solid-state two-level system show that, while coupling to lattice vibrations helps to improve the inversion efficiency up to 50% under symmetric driving, coherent population control and a larger amount of inversion are possible using asymmetric dichromatic excitation, which we achieve by adjusting the ratio of the intensities between the red- and blue-detuned pulses. Our measured results, supported by simulations using a real-time path-integral method, offer a new perspective toward realizing efficient, background-free photon generation and extraction.
We present an analytically exactly soluble two-state model, in which a hyperbolic-secant-shaped pulsed interaction has a phase jump of $\ensuremath{\phi}$ at the time of its maximum. The detuning has a constant part and a hyperbolic-tangent chirp term. For $\ensuremath{\phi}=0$, this model reduces to the Demkov-Kunike model, which in turn contains as particular cases three other well-known models: the Rosen-Zener, Allen-Eberly, and Bambini-Berman models. A nonzero $\ensuremath{\phi}$ induces dramatic changes in the transition probability, ranging from complete population inversion to complete population return. The analytic results are particularly instructive in the adiabatic limit and demonstrate that complete population inversion can always occur for a suitable choice of $\ensuremath{\phi}$. The jump phase $\ensuremath{\phi}$ can therefore be used as a control parameter for the two-state transition probability.
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.