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Optical coherent control and phase shift of excitonic and biexcitonic four-wave-mixing polarization
Optics Communications 218 (2003) 415–421 www.elsevier.com/locate/optcom
Optical coherent control and phase shift of excitonic and biexcitonic four-wave-mixing polarization T. Voss *, H.G. Breunig, I. R€ uckmann, J. Gutowski Institut f€ur Festk€orperphysik, Universit€at Bremen, P.O. Box 330440, D-28334 Bremen, Germany Received 22 January 2003; received in revised form 19 February 2003; accepted 24 February 2003
Abstract The optical coherent control of the exciton–biexciton system is studied in four-wave-mixing (FWM) experiments on a ZnSe single-quantum well. Coherent control is achieved by applying a pair of phase-locked laser pulses which produce an FWM signal in combination with an additional third pulse. Oscillations as a function of the interpulse delay time are observed in the FWM signal at the energetic position of the exciton and biexciton resonance, respectively. A phase shift between these oscillations is found for certain orders of arrival of the excitation pulses at the sample. This clearly demonstrates that the coherent control acts in different ways: if the phase shift is observed the FWM polarization is coherently controlled. Otherwise, the coherent control just involves the polarization induced by the two phase-locked pulses. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 78.47.+p; 71.35.)y; 73.21.Fg Keywords: Coherent control; Four-wave mixing; Excitons; Biexcitons; ZnSe
1. Introduction By use of the coherent-control technique it has become possible in the last few years to study chemical reactions [1–3] as well as fundamental properties and dynamics of quantum systems in physics by selectively manipulating the wavefunctions of these systems on ultrashort time scales [4–11]. As an experimental method to achieve such *
Corresponding author. Tel.: +41-421-218-4431; fax: +41421-218-7318. E-mail address: [email protected] (T. Voss).
a control a pair of phase-locked laser pulses can be used to excite the sample [5–7,12–14]. In this case the second pulse will change the excitation created by the first pulse depending on their relative phase. In this way, coherent control offers the outstanding possibility to select the output of chemical reactions as well as to enhance or suppress the relative contributions of certain quantum mechanical excitations by carefully adjusting the relative phase of the two pulses. The coherent control has also been extensively used in the field of semiconductor physics where it has proven to be a fruitful experimental technique
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01273-2
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T. Voss et al. / Optics Communications 218 (2003) 415–421
to study the dynamics of excitons [5,9,15] and exciton–phonon interactions [16] in nanostructures such as quantum wells, microcavities, [17,18] and quantum dots[19]. Because of the short dephasing times of excitations in semiconductors of the order of several picoseconds only, femtosecond laser pulses must be used to study their dynamics. For coherent-control experiments it is even necessary to adjust and stabilize the pulses with a time resolution of a fraction of the optical periods. Apart from the coherent control of one-photon excitations usually used for the creation of, e.g., excitonic polarization also two-photon transitions which do not couple directly to the light field have been shown to be controllable in the coherent regime [20,21]. In a four-wave-mixing (FWM) experiment, where a nonlinear signal generated by the interplay of different polarizations in the sample is observed in a background-free direction, we have recently demonstrated that biexcitonic polarization which is generated by the absorption of two photons can be coherently controlled by applying a phase-locked pulse pair [22]. This result is of particular importance because biexcitonic effects strongly influence the nonlinear optical response of a semiconductor as has been shown in several FWM experiments [23–25] which exploit the polarization selection rules for the formation of a bound-biexciton polarization: since biexcitons are composed of two excitons with opposite spins their creation may be prevented by applying laser pulses with exclusively the same circular polarization state (co-circular polarization) [26] (see Fig. 1). As it is possible to coherently control excitonic and biexcitonic polarization in a semiconductor quantum well simultaneously but separately, this control promises even more direct insight into the fundamental dynamics contributing to the final optical response of the material. For the investigation of the coherent control of the exciton–biexciton system II–VI semiconductors provide an especially suitable model system because of their large biexciton binding energies. Nevertheless, the results obtained with this model system can be easily transferred to other samples which possess excitonic and biexcitonic resonances. By spectrally resolving the FWM signal the different spectral contributions of the exciton (XT) and the exciton–
Fig. 1. Energy levels and transitions of the exciton–biexciton system. rþ and r correspond to the two different circular polarization states of the photons involved in the transition, Exx denotes the biexciton binding energy.
biexciton transition (EBT) of the ZnSe quantum well can be easily separated (see Fig. 1). In this paper we present a detailed experimental study of the coherent control of the exciton–biexciton system in a ZnSe quantum well for crosslinearly polarized excitation pulses. The coherent control is analyzed by recording the time-integrated but spectrally resolved degenerate FWM signal. In extension to our previous work [22] it is studied in both directions of the first order of diffraction which are not equivalent in this case where a pulse pair is applied from direction k1 and an additional single pulse is incident from a different direction k2 . Furthermore, we analyze three different sequences of our excitation pulses: (i) if the phase-locked pulse pair hits the sample first the coherent control affects the k1 polarization only; (ii) if the single k2 pulse hits the sample first our experiments show that the FWM polarization is coherently controlled by the phase-locked pulse pair; (iii) if the single pulse hits the sample at a time between the two k1 pulses coherent control of the FWM polarization is observed, too. In the FWM signal these distinctions manifest themselves by the fact that a phase shift between the signals at the XT and the EBT related to their energetic separation is observed, but only for certain sequences of the excitation pulses in real time. This clearly demonstrates the different microscopic mechanisms which are responsible for the coherent control depending on the order of arrival of the laser pulses at the sample. These different mecha-
T. Voss et al. / Optics Communications 218 (2003) 415–421
nisms are also not restricted to the ZnSe system which is used for the experiments, but they apply to all kinds of coherent-control experiments where the FWM technique is used to monitor the coherent control.
2. Experimental setup The time-integrated but spectrally resolved degenerate transient FWM experiments are carried out on a 4.8 nm ZnSe single-quantum well which is embedded into ZnMgSSe cladding layers and was pseudomorphically grown by molecular beam epitaxy on a (0 0 1) GaAs substrate. For the experiments which are performed in transmission geometry at a temperature T ¼ 4 K the substrate was carefully removed by chemical etching. The setup of the coherent-control experiment is shown in Fig. 2. A phase-locked pulse pair which is generated by use of an actively stabilized Michelson interferometer with a temporal resolution of 40 attoseconds is incident on the sample from a direction k1 . The temporal separation of the pulses will be denoted as tint . An additional single pulse incident from a different direction k2 is delayed with a mechanical delay line by a time tdel with respect to one pulse of the pulse pair. The interplay of the polarizations induced by the k1 and k2 pulses leads to diffracted signals. These FWM signals are recorded in both directions 2k2 k1 and 2k1 k2 . The first direction corresponds to the first order of
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diffraction of the single pulse, the second to the first order of diffraction of the pulse pair. For both directions tdel will be positive if the diffracted pulse arrives at the sample last, i.e., for the direction 2k1 k2 tdel will be positive if the single k2 pulse arrives first whereas for 2k2 k1 tdel is positive if the single pulse arrives last. By use of Pockels cells in each laser beam the polarization states of the k1 and k2 pulses are set to a cross-linear (xy) configuration. This configuration is favorable for our experiment since the signal at the XT is strongly reduced and its strength is now comparable to the EBT [27,28]. All laser pulses have the same intensity and are generated by a frequency-doubled, self-modelocked Ti:sapphire laser producing nearly transform-limited pulses with a temporal width of 120 fs (full width at half maximum). The spectral position of the pulses is chosen such that the absorptions at the exciton and biexciton resonance are of comparable strengths. In the pulse-transmission spectra no signature of resonant twophoton absorption towards the bound-biexciton state can be observed. This shows that in the experiments discussed here the biexcitonic polarization is excited in two-step processes with two photons of different energies. The FWM signal is spectrally resolved by a grating spectrometer with a resolution of 0.03 nm and finally recorded with a liquid-nitrogen cooled CCD camera.
3. Phase shift between excitons and biexcitons
Fig. 2. Experimental setup for the coherent control experiments. The FWM signal is recorded in both directions 2k1 k2 (first order of diffraction of the pulse pair) and 2k2 k1 (first order of diffraction of the single pulse) which are not equivalent.
Fig. 3 shows contour plots of conventional twopulse (one k1 and one k2 pulse) FWM signals for both diffraction directions. For these measurements one arm of the interferometer has been closed and the signals are recorded spectrally resolved as a function of the delay tdel between the k1 and the k2 pulse. Both FWM contour plots show comparable dephasing rates at the XT and the EBT (the signal at the EBT decays faster by a factor of 2) and a clear beating for positive delay times at the XT. This beating has been previously explained with the inhomogeneous broadening of the resonances in the sample [24] and is present in the FWM signal even in the low density regime
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Fig. 3. Contour plots of the two-pulse FWM signal measured for (xy) polarized incident pulses as a function of tdel in direction 2k1 k2 (left) and 2k2 k1 (right). The signals are shown on a logarithmic gray scale over three orders of magnitude.
which is used for all of our experiments. The FWM traces evidently reflect the symmetry of the experiment as expected for only two applied pulses. In all following experiments (i) tdel is set to different, but fixed values, (ii) the second pulse from the interferometer is added and (iii) the FWM signal is recorded as a function of the interpulse delay tint between the two k1 pulses. In Fig. 4 the coherent-control FWM signal observed in the direction 2k2 k1 is shown for a range of tint ¼ 0–6 fs (left panel) and tint ¼ 351:5– 357:5 fs (right panel) recorded for two fixed values of (a) tdel ¼ þ1:5 ps and (b) tdel ¼ 0:3 ps. As the interpulse delay tint is changed the FWM signals show oscillations corresponding to the successive change between constructive and destructive interference of the induced polarizations. Around
Fig. 4. Contour plots of the coherent-control FWM signals for two different ranges of tint and (a) tdel ¼ þ1:5 ps and (b) tdel ¼ 0:3 ps observed in direction 2k2 k1 . The signals are shown on a logarithmic gray scale over two orders of magnitude.
tint ¼ 0 fs the spectral components of the two k1 pulses are still in phase which means that constructive and destructive interference occurs as a function of tint simultaneously for all measured energies. Consequently, a simultaneous switching of the FWM signal at the XT and EBT can be clearly seen in the left panel of Fig. 4. The asymmetric shape of the peaks at the XT is due to the influence of higher Coulomb correlations and has been analyzed in detail elsewhere [15]. The situation changes completely if the coherent-control FWM signal is recorded in a range around tint ¼ 350 fs. Now, the polarizations at the XT and EBT which are induced by the first pulse of the pulse pair have accumulated a phase difference of D/ ¼ Exx =h tint p when the second k1 pulse hits the sample (Exx denotes the biexciton binding energy which is 6.1 meV for the quantum well used here). Intuitively, one would expect that a phase shift between the signal at the XT and the EBT as a function of tint should be observed, i.e., that a maximum of the emission at the XT (EBT) and simultaneously a minimum at the EBT (XT) should occur at this inter-pulse delay, if for the coherent-control experiment the exciton–biexciton system could be treated as a system of two independent resonances. This is indeed the case for the configuration with a negative delay time tdel ¼ 0:3 ps (b). However, the signal recorded for a positive delay time tdel ¼ þ1:5 ps (a) does not show the behavior of two independent resonances. No phase shift between the signals at the XT and EBT is observed in this case. With the same experimental arrangement the FWM signal emitted in the other FWM direction 2k1 k2 is observed, i.e., the first order of diffraction of the pulse pair. The results are shown in Fig. 5 for the same ranges of tint as in Fig. 4, again for a negative tdel ¼ 0:6 ps (a) and a positive tdel ¼ þ0:6 ps (b). The results clearly show the same remarkable dependence of the phase shift between the signals at the XT and the EBT on the delay time. However, in this direction the phase shift is observed for the positive delay time whereas no phase shift can be observed for the negative delay time. The phase shift between the signals at the XT and EBT has been interpreted as a clear signature of the coherent control of
T. Voss et al. / Optics Communications 218 (2003) 415–421
Fig. 5. Contour plots of the coherent-control FWM signals for two different ranges of tint and (a) tdel ¼ 0:6 ps and (b) tdel ¼ þ0:6 ps observed in direction 2k1 k2 . The signals are shown on a logarithmic gray scale over two orders of magnitude.
biexcitonic polarization in the FWM direction elsewhere [22]. The phase shift becomes even more clearly visible in Fig. 6 where cuts through the signals at the XT (solid line) and EBT (dotted line) of the right panel of Fig. 5 are shown. In (a) the two signals are in phase whereas in (b) the signal at the XT is shifted by half a period relative to the signal at the EBT which remains unchanged when compared to (a). The fact that in the direction 2k2 k1 the phase shift is observed for a negative tdel whereas in the direction 2k1 k2 for a positive tdel suggests that
Fig. 6. Coherent-control FWM signals in direction 2k1 k2 spectrally recorded at the XT (solid line) and at the EBT (dotted line) shown for (a) tdel ¼ 0:6 ps and (b) tdel ¼ þ0:6 ps. In (b) the phase shift between the emission at the XT and the EBT is clearly visible.
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the occurrence of the phase shift does not depend on the fixed delay time tdel at which the coherentcontrol FWM signal is recorded but on the order of arrival of the three laser pulses at the sample in real time. The same results have been found also with another similar sample (ZnSe 10 nm singlequantum well, results not shown) which proves that no specific properties of the used sample cause the observed effect. To verify this conclusion the experiment is also performed with the single k2 pulse arriving at a time between the two phase-locked k1 pulses. Fig. 7 shows the observed phase shift between the signals in direction 2k2 k1 at the XT and EBT for three different values of tdel in a range of tint ¼ 0–1700 fs. It can be clearly seen that within the time resolution of the experimental setup making use of three pulses with a temporal width of 120 fs each the phase between the FWM signals at the XT and the EBT starts to shift immediately at tint 0 fs for the negative delay time. In the other two cases a phase shift is observed only if the second k1 pulse from the interferometer arrives after the k2 pulse, i.e., only if tint > tdel . In all three configurations the phase shift evolves as a function of tint according to the energetic separation between the XT and the EBT. This is shown in Fig. 7 where linear functions with a fixed slope calculated from the measured energetic separation between the two resonances (solid lines) are fitted to the experi-
Fig. 7. Phase shift between the FWM signal at the XT and EBT in dependence on tint for three different values of tdel . The slope of the solid lines has been calculated from the energy separation between the XT and the EBT.
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mental data. By this an excellent agreement between the measured and the calculated phase shifts is obtained.
4. Discussion From the results shown above it is clear that a phase shift between the FWM signal at the XT and the EBT only occurs if the single k2 pulse hits the sample in real time before at least one of the two phase-locked k1 pulses from the interferometer. This can be understood by considering the three different sequences of the laser pulses that have been experimentally analyzed. In the configurations where the k2 pulse arrives at the sample first a polarization with a wave vector k2 is excited inside the sample. When the first k1 pulse hits the sample it will create an FWM polarization propagating in both directions 2k2 k1 and 2k1 k2 . Upon its arrival at the sample the second k1 pulse will also create an FWM polarization with the remaining k2 polarization. These different FWM polarizations then interfere according to their relative phase which results in their coherent control. Since in our experiment the FWM signal consists of different spectral components due to the simultaneous excitation of the XT and the EBT obviously a phase shift between these components occurs if tint is increased because they have acquired different phases before the arrival of the second k1 pulse. Note that this configuration differs from the usual one [12–15] and has not been analyzed in detail so far since in most experiments in which coherent control is monitored by detecting FWM signals the single pulse arrives at the sample last and as a weak probe pulse. In the usual case the two phase-locked pulses excite an effective excitonic and biexcitonic polarization with a wave vector k1 which depends on their interpulse delay, i.e., their relative phase. The additional k2 pulse finally ÔprobesÕ this effective polarization which results in FWM signals at the XT and the EBT being, consequently, always in phase. Our findings demonstrate that the exciton–biexciton system must not be treated as two uncoupled resonances since in this case a phase shift would be expected
also in this configuration. Furthermore, it shows again that the biexcitonic polarization is excited in two-step processes: if resonant two-photon absorption was involved in the creation of the biexcitonic polarization the phase shift would be expected for all sequences of the excitation pulses as in the case of two uncoupled resonances. This argument also explains the third case in which the single k2 pulse arrives at the sample between the two k1 pulses. Upon its arrival the k2 pulse creates an FWM polarization inside the sample together with the existing k1 polarization excited by the first of the phase-locked pulses. The second k1 pulse will also create an FWM polarization and again the two FWM polarizations will interfere according to their relative phase. However, this relative phase between these FWM polarizations is now determined by the time between the arrival of the k2 pulse and the second k1 pulse at the sample. This exactly corresponds to the experimental results shown in Fig. 7 that show the starting of the phase shift only for tint > tdel for the direction 2k2 k1 .
5. Conclusions We have presented a detailed experimental analysis of the simultaneous coherent control of the exciton–biexciton system in a ZnSe singlequantum well. The coherent control was detected in FWM experiments using pulses of equal intensity. Our results show that if the single k2 pulse arrives first the FWM polarization is coherently controlled by the phase-locked pulse pair. In this case a phase shift between the FWM signal at the XT and the EBT was observed corresponding to the relative phase which the different spectral components of the FWM signal have acquired at the arrival of the second k1 pulse. If the single k2 pulse arrives at the sample last it ÔprobesÕ the effective polarization induced by the phase-locked pulse pair. In this case in which the k1 polarization is coherently controlled no phase shift between the FWM signal at the XT and EBT could be observed which shows the important impact of higher Coulomb correlations, in particular boundbiexciton polarization on the whole FWM signal.
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Additionally, the case where the single k2 pulse arrives at a time between the two k1 pulses was analyzed. In this case also a phase shift between the XT and the EBT was found, however, the relative phase of the two spectral components was now determined by the time between the arrival of the k2 and the second k1 pulse. This again unambiguously verifies that the observed phase shift is due to a coherent control of the FWM polarization inside the sample. These results offer a new and direct way to analyze the influence of biexcitons on the FWM signal by selectively enhancing or suppressing certain components of the signal by use of the coherent control technique. Further investigations and a detailed theoretical analysis are currently in progress.
Acknowledgements The authors thank J. N€ urnberger, W€ urzburg University, for providing the sample. This work has been supported by the Deutsche Forschungsgemeinschaft (Grants No. Gu 252/12 and Ne 525/8).
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Optical coherent control and phase shift of excitonic and biexcitonic four-wave-mixing polarization T. Voss *, H.G. Breunig, I. R€ uckmann, J. Gutowski Institut f€ur Festk€orperphysik, Universit€at Bremen, P.O. Box 330440, D-28334 Bremen, Germany Received 22 January 2003; received in revised form 19 February 2003; accepted 24 February 2003
Abstract The optical coherent control of the exciton–biexciton system is studied in four-wave-mixing (FWM) experiments on a ZnSe single-quantum well. Coherent control is achieved by applying a pair of phase-locked laser pulses which produce an FWM signal in combination with an additional third pulse. Oscillations as a function of the interpulse delay time are observed in the FWM signal at the energetic position of the exciton and biexciton resonance, respectively. A phase shift between these oscillations is found for certain orders of arrival of the excitation pulses at the sample. This clearly demonstrates that the coherent control acts in different ways: if the phase shift is observed the FWM polarization is coherently controlled. Otherwise, the coherent control just involves the polarization induced by the two phase-locked pulses. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 78.47.+p; 71.35.)y; 73.21.Fg Keywords: Coherent control; Four-wave mixing; Excitons; Biexcitons; ZnSe
1. Introduction By use of the coherent-control technique it has become possible in the last few years to study chemical reactions [1–3] as well as fundamental properties and dynamics of quantum systems in physics by selectively manipulating the wavefunctions of these systems on ultrashort time scales [4–11]. As an experimental method to achieve such *
Corresponding author. Tel.: +41-421-218-4431; fax: +41421-218-7318. E-mail address: [email protected] (T. Voss).
a control a pair of phase-locked laser pulses can be used to excite the sample [5–7,12–14]. In this case the second pulse will change the excitation created by the first pulse depending on their relative phase. In this way, coherent control offers the outstanding possibility to select the output of chemical reactions as well as to enhance or suppress the relative contributions of certain quantum mechanical excitations by carefully adjusting the relative phase of the two pulses. The coherent control has also been extensively used in the field of semiconductor physics where it has proven to be a fruitful experimental technique
0030-4018/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0030-4018(03)01273-2
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T. Voss et al. / Optics Communications 218 (2003) 415–421
to study the dynamics of excitons [5,9,15] and exciton–phonon interactions [16] in nanostructures such as quantum wells, microcavities, [17,18] and quantum dots[19]. Because of the short dephasing times of excitations in semiconductors of the order of several picoseconds only, femtosecond laser pulses must be used to study their dynamics. For coherent-control experiments it is even necessary to adjust and stabilize the pulses with a time resolution of a fraction of the optical periods. Apart from the coherent control of one-photon excitations usually used for the creation of, e.g., excitonic polarization also two-photon transitions which do not couple directly to the light field have been shown to be controllable in the coherent regime [20,21]. In a four-wave-mixing (FWM) experiment, where a nonlinear signal generated by the interplay of different polarizations in the sample is observed in a background-free direction, we have recently demonstrated that biexcitonic polarization which is generated by the absorption of two photons can be coherently controlled by applying a phase-locked pulse pair [22]. This result is of particular importance because biexcitonic effects strongly influence the nonlinear optical response of a semiconductor as has been shown in several FWM experiments [23–25] which exploit the polarization selection rules for the formation of a bound-biexciton polarization: since biexcitons are composed of two excitons with opposite spins their creation may be prevented by applying laser pulses with exclusively the same circular polarization state (co-circular polarization) [26] (see Fig. 1). As it is possible to coherently control excitonic and biexcitonic polarization in a semiconductor quantum well simultaneously but separately, this control promises even more direct insight into the fundamental dynamics contributing to the final optical response of the material. For the investigation of the coherent control of the exciton–biexciton system II–VI semiconductors provide an especially suitable model system because of their large biexciton binding energies. Nevertheless, the results obtained with this model system can be easily transferred to other samples which possess excitonic and biexcitonic resonances. By spectrally resolving the FWM signal the different spectral contributions of the exciton (XT) and the exciton–
Fig. 1. Energy levels and transitions of the exciton–biexciton system. rþ and r correspond to the two different circular polarization states of the photons involved in the transition, Exx denotes the biexciton binding energy.
biexciton transition (EBT) of the ZnSe quantum well can be easily separated (see Fig. 1). In this paper we present a detailed experimental study of the coherent control of the exciton–biexciton system in a ZnSe quantum well for crosslinearly polarized excitation pulses. The coherent control is analyzed by recording the time-integrated but spectrally resolved degenerate FWM signal. In extension to our previous work [22] it is studied in both directions of the first order of diffraction which are not equivalent in this case where a pulse pair is applied from direction k1 and an additional single pulse is incident from a different direction k2 . Furthermore, we analyze three different sequences of our excitation pulses: (i) if the phase-locked pulse pair hits the sample first the coherent control affects the k1 polarization only; (ii) if the single k2 pulse hits the sample first our experiments show that the FWM polarization is coherently controlled by the phase-locked pulse pair; (iii) if the single pulse hits the sample at a time between the two k1 pulses coherent control of the FWM polarization is observed, too. In the FWM signal these distinctions manifest themselves by the fact that a phase shift between the signals at the XT and the EBT related to their energetic separation is observed, but only for certain sequences of the excitation pulses in real time. This clearly demonstrates the different microscopic mechanisms which are responsible for the coherent control depending on the order of arrival of the laser pulses at the sample. These different mecha-
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nisms are also not restricted to the ZnSe system which is used for the experiments, but they apply to all kinds of coherent-control experiments where the FWM technique is used to monitor the coherent control.
2. Experimental setup The time-integrated but spectrally resolved degenerate transient FWM experiments are carried out on a 4.8 nm ZnSe single-quantum well which is embedded into ZnMgSSe cladding layers and was pseudomorphically grown by molecular beam epitaxy on a (0 0 1) GaAs substrate. For the experiments which are performed in transmission geometry at a temperature T ¼ 4 K the substrate was carefully removed by chemical etching. The setup of the coherent-control experiment is shown in Fig. 2. A phase-locked pulse pair which is generated by use of an actively stabilized Michelson interferometer with a temporal resolution of 40 attoseconds is incident on the sample from a direction k1 . The temporal separation of the pulses will be denoted as tint . An additional single pulse incident from a different direction k2 is delayed with a mechanical delay line by a time tdel with respect to one pulse of the pulse pair. The interplay of the polarizations induced by the k1 and k2 pulses leads to diffracted signals. These FWM signals are recorded in both directions 2k2 k1 and 2k1 k2 . The first direction corresponds to the first order of
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diffraction of the single pulse, the second to the first order of diffraction of the pulse pair. For both directions tdel will be positive if the diffracted pulse arrives at the sample last, i.e., for the direction 2k1 k2 tdel will be positive if the single k2 pulse arrives first whereas for 2k2 k1 tdel is positive if the single pulse arrives last. By use of Pockels cells in each laser beam the polarization states of the k1 and k2 pulses are set to a cross-linear (xy) configuration. This configuration is favorable for our experiment since the signal at the XT is strongly reduced and its strength is now comparable to the EBT [27,28]. All laser pulses have the same intensity and are generated by a frequency-doubled, self-modelocked Ti:sapphire laser producing nearly transform-limited pulses with a temporal width of 120 fs (full width at half maximum). The spectral position of the pulses is chosen such that the absorptions at the exciton and biexciton resonance are of comparable strengths. In the pulse-transmission spectra no signature of resonant twophoton absorption towards the bound-biexciton state can be observed. This shows that in the experiments discussed here the biexcitonic polarization is excited in two-step processes with two photons of different energies. The FWM signal is spectrally resolved by a grating spectrometer with a resolution of 0.03 nm and finally recorded with a liquid-nitrogen cooled CCD camera.
3. Phase shift between excitons and biexcitons
Fig. 2. Experimental setup for the coherent control experiments. The FWM signal is recorded in both directions 2k1 k2 (first order of diffraction of the pulse pair) and 2k2 k1 (first order of diffraction of the single pulse) which are not equivalent.
Fig. 3 shows contour plots of conventional twopulse (one k1 and one k2 pulse) FWM signals for both diffraction directions. For these measurements one arm of the interferometer has been closed and the signals are recorded spectrally resolved as a function of the delay tdel between the k1 and the k2 pulse. Both FWM contour plots show comparable dephasing rates at the XT and the EBT (the signal at the EBT decays faster by a factor of 2) and a clear beating for positive delay times at the XT. This beating has been previously explained with the inhomogeneous broadening of the resonances in the sample [24] and is present in the FWM signal even in the low density regime
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Fig. 3. Contour plots of the two-pulse FWM signal measured for (xy) polarized incident pulses as a function of tdel in direction 2k1 k2 (left) and 2k2 k1 (right). The signals are shown on a logarithmic gray scale over three orders of magnitude.
which is used for all of our experiments. The FWM traces evidently reflect the symmetry of the experiment as expected for only two applied pulses. In all following experiments (i) tdel is set to different, but fixed values, (ii) the second pulse from the interferometer is added and (iii) the FWM signal is recorded as a function of the interpulse delay tint between the two k1 pulses. In Fig. 4 the coherent-control FWM signal observed in the direction 2k2 k1 is shown for a range of tint ¼ 0–6 fs (left panel) and tint ¼ 351:5– 357:5 fs (right panel) recorded for two fixed values of (a) tdel ¼ þ1:5 ps and (b) tdel ¼ 0:3 ps. As the interpulse delay tint is changed the FWM signals show oscillations corresponding to the successive change between constructive and destructive interference of the induced polarizations. Around
Fig. 4. Contour plots of the coherent-control FWM signals for two different ranges of tint and (a) tdel ¼ þ1:5 ps and (b) tdel ¼ 0:3 ps observed in direction 2k2 k1 . The signals are shown on a logarithmic gray scale over two orders of magnitude.
tint ¼ 0 fs the spectral components of the two k1 pulses are still in phase which means that constructive and destructive interference occurs as a function of tint simultaneously for all measured energies. Consequently, a simultaneous switching of the FWM signal at the XT and EBT can be clearly seen in the left panel of Fig. 4. The asymmetric shape of the peaks at the XT is due to the influence of higher Coulomb correlations and has been analyzed in detail elsewhere [15]. The situation changes completely if the coherent-control FWM signal is recorded in a range around tint ¼ 350 fs. Now, the polarizations at the XT and EBT which are induced by the first pulse of the pulse pair have accumulated a phase difference of D/ ¼ Exx =h tint p when the second k1 pulse hits the sample (Exx denotes the biexciton binding energy which is 6.1 meV for the quantum well used here). Intuitively, one would expect that a phase shift between the signal at the XT and the EBT as a function of tint should be observed, i.e., that a maximum of the emission at the XT (EBT) and simultaneously a minimum at the EBT (XT) should occur at this inter-pulse delay, if for the coherent-control experiment the exciton–biexciton system could be treated as a system of two independent resonances. This is indeed the case for the configuration with a negative delay time tdel ¼ 0:3 ps (b). However, the signal recorded for a positive delay time tdel ¼ þ1:5 ps (a) does not show the behavior of two independent resonances. No phase shift between the signals at the XT and EBT is observed in this case. With the same experimental arrangement the FWM signal emitted in the other FWM direction 2k1 k2 is observed, i.e., the first order of diffraction of the pulse pair. The results are shown in Fig. 5 for the same ranges of tint as in Fig. 4, again for a negative tdel ¼ 0:6 ps (a) and a positive tdel ¼ þ0:6 ps (b). The results clearly show the same remarkable dependence of the phase shift between the signals at the XT and the EBT on the delay time. However, in this direction the phase shift is observed for the positive delay time whereas no phase shift can be observed for the negative delay time. The phase shift between the signals at the XT and EBT has been interpreted as a clear signature of the coherent control of
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Fig. 5. Contour plots of the coherent-control FWM signals for two different ranges of tint and (a) tdel ¼ 0:6 ps and (b) tdel ¼ þ0:6 ps observed in direction 2k1 k2 . The signals are shown on a logarithmic gray scale over two orders of magnitude.
biexcitonic polarization in the FWM direction elsewhere [22]. The phase shift becomes even more clearly visible in Fig. 6 where cuts through the signals at the XT (solid line) and EBT (dotted line) of the right panel of Fig. 5 are shown. In (a) the two signals are in phase whereas in (b) the signal at the XT is shifted by half a period relative to the signal at the EBT which remains unchanged when compared to (a). The fact that in the direction 2k2 k1 the phase shift is observed for a negative tdel whereas in the direction 2k1 k2 for a positive tdel suggests that
Fig. 6. Coherent-control FWM signals in direction 2k1 k2 spectrally recorded at the XT (solid line) and at the EBT (dotted line) shown for (a) tdel ¼ 0:6 ps and (b) tdel ¼ þ0:6 ps. In (b) the phase shift between the emission at the XT and the EBT is clearly visible.
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the occurrence of the phase shift does not depend on the fixed delay time tdel at which the coherentcontrol FWM signal is recorded but on the order of arrival of the three laser pulses at the sample in real time. The same results have been found also with another similar sample (ZnSe 10 nm singlequantum well, results not shown) which proves that no specific properties of the used sample cause the observed effect. To verify this conclusion the experiment is also performed with the single k2 pulse arriving at a time between the two phase-locked k1 pulses. Fig. 7 shows the observed phase shift between the signals in direction 2k2 k1 at the XT and EBT for three different values of tdel in a range of tint ¼ 0–1700 fs. It can be clearly seen that within the time resolution of the experimental setup making use of three pulses with a temporal width of 120 fs each the phase between the FWM signals at the XT and the EBT starts to shift immediately at tint 0 fs for the negative delay time. In the other two cases a phase shift is observed only if the second k1 pulse from the interferometer arrives after the k2 pulse, i.e., only if tint > tdel . In all three configurations the phase shift evolves as a function of tint according to the energetic separation between the XT and the EBT. This is shown in Fig. 7 where linear functions with a fixed slope calculated from the measured energetic separation between the two resonances (solid lines) are fitted to the experi-
Fig. 7. Phase shift between the FWM signal at the XT and EBT in dependence on tint for three different values of tdel . The slope of the solid lines has been calculated from the energy separation between the XT and the EBT.
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mental data. By this an excellent agreement between the measured and the calculated phase shifts is obtained.
4. Discussion From the results shown above it is clear that a phase shift between the FWM signal at the XT and the EBT only occurs if the single k2 pulse hits the sample in real time before at least one of the two phase-locked k1 pulses from the interferometer. This can be understood by considering the three different sequences of the laser pulses that have been experimentally analyzed. In the configurations where the k2 pulse arrives at the sample first a polarization with a wave vector k2 is excited inside the sample. When the first k1 pulse hits the sample it will create an FWM polarization propagating in both directions 2k2 k1 and 2k1 k2 . Upon its arrival at the sample the second k1 pulse will also create an FWM polarization with the remaining k2 polarization. These different FWM polarizations then interfere according to their relative phase which results in their coherent control. Since in our experiment the FWM signal consists of different spectral components due to the simultaneous excitation of the XT and the EBT obviously a phase shift between these components occurs if tint is increased because they have acquired different phases before the arrival of the second k1 pulse. Note that this configuration differs from the usual one [12–15] and has not been analyzed in detail so far since in most experiments in which coherent control is monitored by detecting FWM signals the single pulse arrives at the sample last and as a weak probe pulse. In the usual case the two phase-locked pulses excite an effective excitonic and biexcitonic polarization with a wave vector k1 which depends on their interpulse delay, i.e., their relative phase. The additional k2 pulse finally ÔprobesÕ this effective polarization which results in FWM signals at the XT and the EBT being, consequently, always in phase. Our findings demonstrate that the exciton–biexciton system must not be treated as two uncoupled resonances since in this case a phase shift would be expected
also in this configuration. Furthermore, it shows again that the biexcitonic polarization is excited in two-step processes: if resonant two-photon absorption was involved in the creation of the biexcitonic polarization the phase shift would be expected for all sequences of the excitation pulses as in the case of two uncoupled resonances. This argument also explains the third case in which the single k2 pulse arrives at the sample between the two k1 pulses. Upon its arrival the k2 pulse creates an FWM polarization inside the sample together with the existing k1 polarization excited by the first of the phase-locked pulses. The second k1 pulse will also create an FWM polarization and again the two FWM polarizations will interfere according to their relative phase. However, this relative phase between these FWM polarizations is now determined by the time between the arrival of the k2 pulse and the second k1 pulse at the sample. This exactly corresponds to the experimental results shown in Fig. 7 that show the starting of the phase shift only for tint > tdel for the direction 2k2 k1 .
5. Conclusions We have presented a detailed experimental analysis of the simultaneous coherent control of the exciton–biexciton system in a ZnSe singlequantum well. The coherent control was detected in FWM experiments using pulses of equal intensity. Our results show that if the single k2 pulse arrives first the FWM polarization is coherently controlled by the phase-locked pulse pair. In this case a phase shift between the FWM signal at the XT and the EBT was observed corresponding to the relative phase which the different spectral components of the FWM signal have acquired at the arrival of the second k1 pulse. If the single k2 pulse arrives at the sample last it ÔprobesÕ the effective polarization induced by the phase-locked pulse pair. In this case in which the k1 polarization is coherently controlled no phase shift between the FWM signal at the XT and EBT could be observed which shows the important impact of higher Coulomb correlations, in particular boundbiexciton polarization on the whole FWM signal.
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Additionally, the case where the single k2 pulse arrives at a time between the two k1 pulses was analyzed. In this case also a phase shift between the XT and the EBT was found, however, the relative phase of the two spectral components was now determined by the time between the arrival of the k2 and the second k1 pulse. This again unambiguously verifies that the observed phase shift is due to a coherent control of the FWM polarization inside the sample. These results offer a new and direct way to analyze the influence of biexcitons on the FWM signal by selectively enhancing or suppressing certain components of the signal by use of the coherent control technique. Further investigations and a detailed theoretical analysis are currently in progress.
Acknowledgements The authors thank J. N€ urnberger, W€ urzburg University, for providing the sample. This work has been supported by the Deutsche Forschungsgemeinschaft (Grants No. Gu 252/12 and Ne 525/8).
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