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arXiv:cond-mat/9807216 · Physical review. B, Condensed matter

Coherent control for a two-level system coupled to phonons

H. Castella, R. Zimmermann

Published Mar 15, 1999• 19 citations
arXivHTMLPDF

Abstract

The interband polarization induced by two phase-locked pulses in a semiconductor shows strong interference effects depending on the time, ${\ensuremath{\tau}}_{1},$ separating the pulses. The four-wave mixing signal diffracted from a third pulse delayed by $\ensuremath{\tau}$ is coherently controlled by tuning ${\ensuremath{\tau}}_{1}.$ The four-wave mixing response is evaluated exactly for a two-level system coupled to a single LO phonon. In the weak coupling regime it shows oscillations with the phonon frequency which turn into sharp peaks at multiples of the phonon period for a larger coupling strength. Destructive interferences between the two phase-locked pulses produce a splitting of the phonon peaks into a doublet. For fixed $\ensuremath{\tau}$ but varying ${\ensuremath{\tau}}_{1},$ the signal shows rapid oscillations at the interband transition frequency, whose amplitude exhibits bursts at multiples of the phonon period.

DOI
10.1103/physrevb.59.r7801

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The interband polarization induced by two phase-locked pulses in a semiconductor shows strong interference effects depending on the time, \ensuremathτ1,{\ensuremath{\tau}}_{1},\ensuremathτ1​, separating the pulses. The four-wave mixing signal diffracted from a third pulse delayed by \ensuremathτ\ensuremath{\tau}\ensuremathτ is coherently controlled by tuning \ensuremathτ1.{\ensuremath{\tau}}_{1}.\ensuremathτ1​. The four-wave mixing response is evaluated exactly for a two-level system coupled to a single LO phonon. In the weak coupling regime it shows oscillations with the phonon frequency which turn into sharp peaks at multiples of the phonon period for a larger coupling strength. Destructive interferences between the two phase-locked pulses produce a splitting of the phonon peaks into a doublet. For fixed \ensuremathτ\ensuremath{\tau}\ensuremathτ but varying \ensuremathτ1,{\ensuremath{\tau}}_{1},\ensuremathτ1​, the signal shows rapid oscillations at the interband transition frequency, whose amplitude exhibits bursts at multiples of the phonon period.
Biexcitonic polarization excited in a ZnSe quantum well is coherently controlled with a pair of phase-locked laser pulses. The coherent control of this four-particle correlation is directly verified by observing a spectrally resolved four-wave-mixing signal in direction of the first order of diffraction of the pulse pair. By using polarization selection rules for the generation of biexcitons and changing the interpulse delay of the pulse pair the oscillations of the four-wave-mixing signal are studied at the energetic position of the excitonic as well as the biexcitonic resonance. The excitonic and biexcitonic signals exhibit a phase shift in dependence on the interpulse delay which clearly indicates the separate coherent control of the excitonic and biexcitonic polarization.
Summary form only given. In a coherent control four-wave mixing (FWM) experiment, a first pulse, 1, can excite a certain electron-hole pair density n/sub eh/ in a semiconductor. Because this density is connected with interband coherence, a second phase locked and time-delayed pulse, 1', adjusted for destructive interference, can lead to a coherent reemission of photons. As a result, a first light pulse has created electron-hole pairs and a second light pulse has annihilated these electron-hole pairs again. A third pulse, leads to a FWM-signal and can thus probe the remaining density. This form of coherent control has been demonstrated on excitons and on continuum transitions. In the low density regime, one can argue, however, that one merely sees the interference of two independent FWM-signals on the detector. While there is no true contradiction in theory, the latter reasoning obviously questions severely whether such experiments are really evidence for the coherent control of the carrier density.
In recent years, transient four-wave mixing (FWM) techniques have been used to investigate the dephasing times of non-equilibrium carriers in semiconductor heterostructures by monitoring the decay of the macroscopic polarization. Initially, this decay was measured by temporally-integrating the scattered FWM signal as a function of time delay between two (or more) pump pulses. More recently, the macroscopic polarization decay has been measured by time-resolving the FWM signal by cross correlating it with an ultrashort laser pulse via frequency up-conversion in a nonlinear crystal. When excitons with slightly different energies are excited, oscillations (or beats) have been observed both in the time-integrated FWM signal and in the time-resolved FWM signal. Such beats have been observed, for example, between light- and heavy-hole excitons and between excitons in wells of different widths. Both the spectral behavior1 of the time-integrated FWM signal and the temporal behavior2 of the time-resolved signal have been shown to allow the distinction between the polarization interference associated with two independent oscillators and the quantum beats associated with two coupled oscillators which share a common level. In addition, it has been demonstrated3 that the quantum beats produced by incident pulses with parallel polarizations are exactly out of phase (by π) with the beats produced by orthogonally polarized pulses. In each case1-3, the period and phase of the beats and the polarization of FWM signal were explained by using a six-level model, which excluded many-body effects, to describe the J=1/2 conduction and J=3/2 valence states. Within the last few months, however, it has been shown that a complete description of the polarization selection rules for the FWM signal requires excitation-induced dephasing4 or disorder-induced coupling of the σ + and σ – excitonic transitions5.
A theoretical investigation on the coherent control of optical transient four-wave mixing interactions in two-level systems with two intense few-cycle propagating laser pulses of central angular frequencies $\ensuremath{\omega}$ and $3\ensuremath{\omega}$ is reported. By numerically solving the full Maxwell-Bloch equations beyond the slowly varying envelope and rotating-wave approximations in the time domain, the nonlinear coupling to the optical field at frequency $5\ensuremath{\omega}$ is found to depend critically on the initial relative phase $\ensuremath{\phi}$ of the propagating pulses: the coupling is enhanced when the pulses interfere constructively in the center $(\ensuremath{\phi}=0)$, while it is nearly suppressed when they are out of phase $(\ensuremath{\phi}=\ensuremath{\pi})$.
We analyze theoretically the quantum beat and the polarization interference phenomena observed as oscillations in pulsed four-wave mixing. We show that resolution of the signal with respect to the delay between the incoming pulses as well as the detection frequency provides valuable information on the nature of the oscillations. Most importantly, it is very simple to distinguish between polarization interference from two independent optical transitions and quantum beats from a three-level system. In case of polarization interference, the phase of the oscillations shifts \ensuremath{\pi} when the detected frequency passes any of the resonances. In case of quantum beat, the phase has little dependence on the detected frequency near the resonance.
A critical dependence of the quantum interference on the optical Stark spectral shift produced when two-color laser pulses interact with a two-level medium is observed. The four-wave mixing of two ultrashort phase-locked $\ensuremath{\omega}\text{\ensuremath{-}}3\ensuremath{\omega}$ laser pulses propagating coherently in a two-level system depends on the pulses' relative phase. The phase dominating the efficiency of the coupling to the anti-Stokes Raman component is found to be determined by the sign of the total ac Stark shift induced in the system, in such a way that the phase sensitivity disappears precisely where the ac Stark effect due to both pulses is compensated. A coherent control scheme based on this phenomenon can be contemplated as the basis for nonlinear optical spectroscopy techniques.