Closed Loop Coherent Control of Electronic Transitions in Gallium

ARTICLE pubs.acs.org/JPCA

Closed Loop Coherent Control of Electronic Transitions in Gallium Arsenide Sima Singha,† Zhan Hu,*,‡ and Robert J. Gordon*,† † ‡

Department of Chemistry, University of Illinois at Chicago, Chicago, Illinois 60607, United States Institute of Atomic and Molecular Physics, Jilin University, Changchun, P. R. China 130021 ABSTRACT: A genetic algorithm was used to control the photoluminescence (PL) from GaAs(100). A spatial light modulator (SLM) used feedback from the emission to optimize the spectral phase profile of an ultrashort laser pulse. Most of the experiments were performed using a sine phase function to optimize the integrated PL spectrum over a specified wavelength range, with the amplitude and period of the phase function treated as genetic parameters. An order of magnitude increase in signal was achieved after only one generation, and an optimized waveform, consisting of three equally spaced pulses approximately 0.8 ps apart, was obtained after 15 generations. The effects of fluence, polarization, relative phase of the subpulses, and spectral range of the optimized PL were investigated. In addition, preliminary experiments were performed using the phases of individual pixels of the SLM as genetic variables. The PL spectrum is identified with recombination of electron-hole pairs in the L-valley of the Brillouin zone. Control is achieved by coherent manipulation of plasma electrons. It is proposed that hot electrons excite lattice phonons, which in turn scatter carriers into the L-valley.

I. INTRODUCTION This paper explores the possibility of using shaped ultrafast laser pulses to control electronic transitions in condensed matter. To appreciate the context of this work, it is useful to review briefly the history of coherent control. Although lasers have been used to initiate chemical reactions for over 40 years, active control of chemical processes did not emerge until the mid-eighties, when the theoretical basis was developed for using photoinduced interference to control chemical branching ratios and population transfer.1,2 Two paradigms emerged from these theories, one in the frequency domain3 and the other in the time domain.4 The former is a chemical analogue of Young’s two-slit experiment, in which quantum mechanical interference between competing pathways is controlled by varying the relative phase of two continuous (or long pulse) lasers used to initiate different reaction paths.5 In the latter, ultrashort laser pulses are used to create a wave packet that is steered toward a desired goal.6 By the late nineties, most coherent control experiments used the time domain approach, taking advantage of the versatility of pulseshaping techniques7 to control the evolution of a wave packet. Two further paradigms emerged in these experiments, one using open loops, and the other using closed loops to manipulate the reactants. In the former, a predesigned laser pulse, based on a hypothesized mechanism, is synthesized, and the response of the target is used to test the hypothesis.8 In the latter, feedback from the observables (e.g., the product distribution or energy of specific products) is used to modify the laser pulse shape, so that after a number of generations an optimal pulse is found that maximizes the yield, subject to specified constraints.9 Deducing r 2011 American Chemical Society

the mechanism from the optimal pulse poses an additional challenge, which may be addressed by post-analysis of the optimal pulse shape,10,11 auxiliary experiments to modify the optimum pulse,12 and molecular simulations.13 A hybrid approach is also useful, in which a parametrized pulse shape based on a mechanistic model is introduced, and the optimal parameters are found using an evolutionary algorithm (EA).14 A key issue in all control experiments is the effect of the environment on the laser-induced coherence. Early experiments were performed with isolated molecules, in which decoherence is limited to internal degrees of freedom.15 More recently, control of reactions and energy flow has been demonstrated in the liquid phase,16 and evidence has been found for coherent dynamics even in very complex molecules.17-20 Many experiments have also been performed in the solid state, using weak pulses to control carrier populations,21 electron currents,22 phonon oscillation,23 and photoemission.24 These studies used pulse intensities well below the damage threshold of the material, so that the dephasing time was longer than the pulse width of the laser. A number of experiments have also been performed above the damage threshold to determine the effect of pulse shape on surface ablation.25-31 Although closed loop experiments successfully optimized the yields, energies, and distribution of ablation products, no evidence Special Issue: Victoria Buch Memorial Received: November 14, 2010 Revised: January 19, 2011 Published: February 22, 2011 6093

dx.doi.org/10.1021/jp110869f | J. Phys. Chem. A 2011, 115, 6093–6101

The Journal of Physical Chemistry A

Figure 1. Photoluminescence spectra obtained with an unoptimized triple pulse (A = 0.4π) at a fluence of 13.5 J/cm2, with the pulse spacing equal to 5.5 (blue squares) and 6 (red circles) times the LO phonon period of GaAs. The black curve was obtained with a transform-limited pulse. The inset shows the enhancement ratio at 450.8 nm and a fluence of 12.8 J/cm2. Numbered arrows indicate multiples of the period of the longitudinal optical phonon.

has been reported of phase-coherent effects at intensities high enough to melt the material. Against this background we posed the question of whether it is possible to devise a pulse train that could produce a coherent response from a crystalline material at intensities high enough to disrupt its lattice. Our idea was to use a sequence of resonant pulses to induce large amplitude motion of the lattice, which could be used as the engine for controlling structural and/or electronic changes of the material. At first glance, this may seem to be an impossible task because the phonon structure collapses once the lattice begins to melt. The hope was that the material would retain its crystalline structure beneath the molten surface layer long enough to respond coherently to the pulse train. A preliminary open loop study32 (hereafter referred to as paper I) validated this hypothesis. A train of three 800 nm laser pulses was focused on a sample of GaAs(100) in air, and the photoluminescence (PL) spectrum was recorded. As shown in Figure 1, a single, transform-limited (TL) pulse produced a conventional laser-induced breakdown (LIB) spectrum, featuring two Ga emission lines riding on top of a broad continuum. The spectrum did not change when a train of three pulses separated by 5.5 times the longitudinal optical phonon period, τ, was used instead. When the pulse spacing was set equal to 6τ, however, a new emission band between 390 and 460 nm emerged. This band was identified as the emission produced by electron-hole recombination in the L-valley of the Brillouin zone.33 At a fixed wavelength the PL was found to oscillate with the pulse spacing (see the inset to Figure 1), with a periodicity approximately equal to the phonon period, suggesting that the L-valley was populated by phonon-hole scattering. Further details of the mechanism are found in paper I and in the Discussion. In the present paper we continue our study of GaAs, using a closed-loop evolutionary algorithm to enhance the PL in different regions of the spectrum. In paper I, after trying a variety of pulse shapes, we settled on a train of three equally spaced pulses of comparable amplitude. Here we wish to determine whether pulse shapes found with an evolutionary algorithm might be even more effective in inducing the L-valley emission. One strategy is

ARTICLE

Figure 2. Schematic drawing of the apparatus. Transform-limited 800 nm laser pulses are shaped into a pulse train using a spatial light modulator (SLM). The sample is mounted on a computer driven 3D translation stage, which exposes a fresh sample surface to each laser shot. The photoluminescence is dispersed with a spectrograph and detected with a CCD camera.

to use a completely random spectral phase distribution and let the material “tell” the laser what pulse shape works best. A second strategy is to continue using a sequence of regularly spaced pulses, while using feedback to optimize their number, spacing, and relative amplitudes. Other properties of the pulses that we investigated are the total fluence, the laser polarization angle, and the relative phase of adjacent pulses in the train.

II. EXPERIMENTAL METHODS The apparatus used here, depicted schematically in Figure 2, is essentially the same as the one employed in paper I and in our previous studies of double pulse ablation.34,35 Briefly, a 25 mm square GaAs(100) wafer is mounted on an automated 3D translation stage, which advances the crystal 20-40 μm after every laser shot, with larger spacing at higher pulse energy. The output of a regeneratively amplified Ti:sapphire laser, operated in single shot mode, passes through (i) a half-wave plate and Rochon polarizer used to adjust the pulse energy, (ii) a dualmask, 640 pixel liquid crystal spatial light modulator (SLM) used to shape the laser pulse, (iii) a second half-wave plate used to rotate the polarization plane, and (iv) a 0.25 NA microscope objective used to focus the beam onto the target. After the pulse-shaping and focusing optics, the pulse has an energy of 0.7-8.5 μJ, a temporal width of 62 fs, a focal diameter of 3.6 μm, and a Rayleigh range of 24 μm. The laser beam is incident on the target at a 30° angle from the normal, and, except where indicated otherwise, it is p-polarized (i.e., polarized in the plane of incidence). Under our experimental conditions, a pulse energy of 1 μJ produces a fluence of 8.32 J/cm2 and an intensity of 1.34 1014 W/cm2 at the target surface. The fluorescence from the irradiated crystal is collected with a convex lens (f = 50.8 mm) along an axis perpendicular to the laser beam and focused onto the entrance slit of a spectrograph. The entire spectrum, typically between 360 and 490 nm, is captured in a single shot with a nongated, thermoelectrically cooled CCD camera. Five shots are typically accumulated to improve the signal/noise ratio. Pulse shaping is accomplished using the phase mask of the SLM. The multiphoton intrapulse interference phase scan (MIIPS) technique36 is used to generate transform-limited pulses. Most 6094

dx.doi.org/10.1021/jp110869f |J. Phys. Chem. A 2011, 115, 6093–6101

The Journal of Physical Chemistry A of the experiments described in this paper used a train of regularly spaced pulses generated by a sine phase function,37 ψðωÞ ¼ A sin½ðω - ω0 Þt þ j ¼ A sin½2πðm - m0 Þ=T þ j ð1Þ

Figure 3. Calculated pulse shapes generated with the spatial light modulator using a sine phase function. In panel a the sine phase period, T, is fixed at 50 SLM pixels, and the intensity is normalized at each value of A, as indicated by the graded colors shown in the relative amplitude bar. Panel b shows the ratio of the amplitudes of the side peaks to that of the central peak of the pulse train. The solid curve is for the peaks closest to the central peak, and the dotted curve is for the second nearest neighbors.

ARTICLE

In this equation, ω is the laser frequency, ω0 is the central frequency, and m and m0 are corresponding pixel numbers. The amplitude, A, determines the number of peaks in the pulse train, T determines their spacing, and j is the phase offset of the nearest neighbor satellite peaks relative to the central peak, modulo 2π. In the case of a three pulse train, a nonzero value of j adds a phase j to the waves in the first pulse and subtracts the same phase from the third pulse, while leaving the central pulse unchanged (see Figure 1 of paper I). Figure 3 illustrates the properties of the pulse trains that can be generated with the sine phase function. Panel a shows the amplitudes of the subpulses as a function of A for the case of T = 50 pixels. Generally, the pulse train consists of an odd sequence of evenly spaced pulses distributed symmetrically about a central pulse. The length of the pulse train grows with A, although for some values of A one or more pulses may be very weak or even absent. Panel b shows the ratios of the amplitudes of the nearest neighbor and second nearest neighbor peaks to that of the central peak. The spacing between peaks is given by ΔtðpsÞ ¼ 15:13=T ð2Þ where T is measured in pixels. To compare the synthesized wave forms with the calculated ones, we measured the electric field using an interferometric cross-correlation technique.38 This method uses one mask of the SLM to generate the sine phase and the second mask to generate a delayed pulse that moves across the pulse train and interferes with it. The interferometric output is measured with a photodiode. The result for a three-pulse train with A = 0.4π and T = 19.00 pixels, shown in Figure 4b, compares well with the calculated field. Closed loop experiments were performed using a genetic algorithm39 (GA) to find a set of spectral phases that maximize the integrated PL over a selected range of wavelengths. In experiments using a sine phase function, the gene set (known as an individual) consists of the A and T parameters in eq 1. In principle, it is possible to optimize A and T without use of a GA by scanning through all values of these parameters. This is not

Figure 4. Simulated and measured electric field of the pulse trains generated with the SLM. Pulse trains in panels a and b were generated using a sine phase function with A = 0.4π and T = 19.00 pixels. Pulse trains in panels c and d were generated using phases optimized with an unparametrized genetic algorithm, starting from a random distribution. 6095



ARTICLE

practical, however, because the finite size of the GaAs wafer limits the number of laser shots per experiment. In other, preliminary experiments, the phases of individual pixels were optimized. In this case, the individual consists of 150 phases mapped nonlinearly onto the central 350 pixels of the SLM to cover the laser spectrum. A nonlinear projection is used so that the center of the spectrum, which has a larger spectral intensity, receives a denser set of genes. In both the sine phase and pixel optimizations, an initial collection of 46 individuals is selected randomly, and the resulting PL spectrum is recorded for each one. The fitness of each individual set is ranked by integrating the PL spectrum over a selected interval. A new group of individuals is then generated from the best half of the previous generation by crossover and mutation of the genes. Three types of crossover operations are used: (i) multiple point crossover, (ii) expanded linear crossover, in which each gene has the same expansion parameter, R, and (iii) expanded nonlinear crossover, in which each gene has a different value of R. (See the BLX-R crossover operation described in ref 40.) Each gene has a 40% probability of having the first crossover operation, 10% probability for the second, and 50% for the third. After each generation, the fitness values are again calculated and ranked, and the process is repeated until the evolution converges.

III. RESULTS As discussed in the previous section, most of the experiments were performed using a sine phase function to optimize the PL signal over a specified wavelength range. In these experiments, the parameters A and T are treated as genetic variables, while the phase, fluence, and polarization were fixed. Parts a and b of Figure 5 show the results of setting j = 0 and varying A and T. In Figure 5a the spectrum was optimized over the entire band (390-450 nm). The lowermost curve is the LIB spectrum obtained with a TL pulse. The middle curve is the first generation output, and the uppermost curve is the spectrum obtained after 25 generations. The inset shows the integrated intensity as a function of generation number, with the arrow indicating the value for a TL pulse. We observe an immediate jump of the integrated intensity by a factor of 5.3 in the first generation, when the sine phase is turned on. The optimized integrated intensity, obtained after 11 generations, is enhanced by an additional factor of 1.7. In Figure 5b, the signal is optimized between 420 and 440 nm. The integrated signal for the TL pulse is very small in this spectral region. The integrated PL jumps by a factor of 35 after one generation and by an additional factor of 2.5 after 15 generations. During the course of the evolution, the entire spectrum shifts to longer wavelengths so as to maximize the signal in the targeted wavelength range, while maintaining a memory of the Ga peak at 403.3 nm. A limited number of experiments was also performed using random phases for individual pixels. Figure 5c shows the result of optimization over the entire spectrum. A single generation shows barely any change in the PL spectrum or its intensity. Approximately 80 generations are required before the signal starts to level off with an enhancement factor of 3.2, and it is not clear whether convergence has been reached. In this case the Ga lines are very pronounced even after many generations. The finite size of the GaAs wafer restricted the number of laser shots possible with a single sample, and further work is required to determine whether the EA converged.

Figure 5. Photoluminescence spectra obtained with the GA using the sine function with φ = 0 and the random phase distribution: (a) sine phase optimized for 390-450 nm at a fluence of 13.1 J/cm2; (b) sine phase optimized for 420-440 nm at 12.8 J/cm2; (c) random phase optimized for 390-450 nm at 13.1 J/cm2. In all three panels, the blue squares and red circles are the spectra obtained in the first and last generations, respectively, while the black curves are obtained with a transform-limited pulse. The insets show the spectrally integrated intensity at different generations, and the arrows indicate the integrated intensity obtained with a transform-limited pulse.

The optimum pulse shapes achieved by the two methods differ qualitatively. For the sine function, the pulse shape is necessarily a train of regularly spaced pulses. In Figure 5a, the converged parameters are A = 0.293π and T = 24.4 pixels. The A parameter corresponds to a three-pulse train. (Note that the ratios in Figure 3b correspond to field amplitudes, not intensities, and that the intensities of the second nearest neighbors for this A value are only 1% of the central peak.) The T value corresponds to a pulse spacing of 620 fs, which is approximately 5τ. Very similar parameters were obtained in Figure 5b, namely A = 0.288π and T = 23.6 pixels. These small differences nevertheless produce very different PL spectra. For the random phase (i.e., individual pixel) optimization, the pulse shape shown in Figure 4d is very complex, with no obvious periodicity. It should be noted that the random phase pulse has appreciable amplitude out to (8 ps, with the integrated fluence outside of the interval of (2 ps contributing 19% of the total. It is interesting to see if the oscillatory structure previously observed in the open loop experiment at a fixed PL wavelength 6096



ARTICLE

Figure 6. Scan of the enhancement ratio of the integrated PL spectrum vs T for A = 0.251π (squares) and 0.4π (circles).

Figure 8. Fluence dependence of the sine function parameters and the integrated PL signal for the spectra shown in Figure 7. Panels a and b show the optimized amplitude and period of the sine phase function. Panel c shows the enhancement ratio (triangles) and integrated intensity difference (squares) of the optimized vs transform limited pulses.

Figure 7. Photoluminescence spectra at different fluences, obtained by optimizing parameters A and T of a sine phase, with φ = 0. The target function is the spectral intensity integrated from 390 to 450. The maximum of each spectrum is normalized relative to one obtained with a transform-limited (“single”) pulse at the same fluence. The single pulse spectrum shown is for 8.4 J/cm2.

(λ = 450.8 nm in Figure 1) survives integration over wavelength. In the upper trace of Figure 6, we selected the value of A = 0.251π obtained by genetically optimizing A and T over 370-490 nm and then scanned T. In the lower trace we kept the value of A = 0.4π used in the open loop experiment and again scanned T, while integrating over the entire PL spectrum. In both cases the enhancement ratio (i.e., the ratio obtained with the pulse train divided by that obtained with a TL pulse) of the integrated signal is peaked near 5τ. Additional structure is observed with irregular spacing. As pointed out in paper I, the oscillations resonant with the phonon period are very sensitive to fluence and the sine A parameter. The remainder of the experiments used only the sine phase function. Figure 7 shows the PL spectrum as a function of laser fluence, optimized between 390 and 450 nm. A blue shift in the spectral maximum with fluence is evident, with some local deviations (e.g., at 13.4 J/cm2). Between 13.4 and 70.7 J/cm2,

the spectral peak shifts by 8.9 nm to the blue, which is a sizable fraction of the 15.4 nm full-width at half-maximum of the 24.5 J/cm2 spectrum. Several additional trends emerge in Figure 8. Both A and T tend to decrease with fluence. The drop in A indicates a decreasing ratio of the intensity of the satellite peaks to that of the central peak, while the decrease in T shows that the pulse spacing increases with fluence. Especially interesting is the fluence dependence of the PL signal. In the open loop experiment (Figure 2c of paper I), the PL has a sharp threshold at 12 J/cm2, peaks at 14 J/cm2, and drops to 15% of the maximum at 24 J/cm2. Here, the threshold lies below 8 J/cm2. The signal reaches a plateau around 40 J/cm2 and shows no sign of dropping off at 70 J/cm2. In Figures 9 and 10 we explore the effects of varying the phase offset, j, for fixed values of A and T. In Figure 9, we adopted the value of A = 0.4π used in the open-loop experiment and set T = 15.60. This value of T was chosen by scanning over the pulse spacing for λ = 440 nm and selecting the value of T that gave the largest signal. Panel a shows that the PL spectrum is very sensitive to the phase. Panels b and c show slices of the 2D plot for fixed j and λ, respectively. For j = 0.0333π (spectrum a), the PL spectrum is red-shifted close to the edge of the L-valley bandgap, with a peak emission at 440 nm, whereas for j = 0.467π (spectrum b) the emission is shifted to the blue and overlaps the two Ga lines. Holding λ constant at 420 nm (slice c), the intensity is maximized for j near 0.5π, whereas for λ = 440 nm the signal is maximized near j = 0. In Figure 10, A and T were obtained using the GA to 6097



ARTICLE

Figure 9. Effect of the relative phase, φ, on the photoluminescence spectrum obtained with A = 0.4π and T = 15.596 pixels at a fluence of 12.9 J/cm2. The colored 2D plot (panel a) shows the spectral intensity as a function of φ. The intensity curve for each wavelength is normalized, as indicated by the Relative Intensity bar. Panel b shows vertical slices of the 2D plot taken at φ = 0.0333π and 0.467π (curves a and b, respectively), indicated by the corresponding arrow heads in panel a. The unlabeled black curve in panel b is the spectrum obtained with a transform-limited pulse. Panel c shows horizontal slices of the 2D plot taken at λ = 420 and 440 nm (curves c and d, respectively), indicated by corresponding arrow heads in panel a.

Figure 10. Same as Figure 9, but with A = 0.788π and T = 25.94 pixels. Curves a-c correspond to vertical slices of the 2D plot at φ = þ0.333π, -0.2π, and þ0.233π, and curves d and e correspond to horizontal slices at λ = 386 and 410 nm.

maximize the integrated signal from 390 to 450 nm. The spectral slices in this case are quite different from those in Figure 9. Slices taken at j = -0.333π, -0.2π, and þ0.233π (spectra a-c) are all peaked between 400 and 420 nm, and the Ga lines are almost completely obscured; λ-slices at 386 and 410 nm (spectra d and e) are maximized near j = -0.3π and þ0.3π, respectively. In a final experiment we explored the effect of laser polarization. In Figure 11 we optimized the spectrum over the range

390-450 nm for different fixed values of the polarization angle, θ. As in the open loop experiment (Figure 2b of paper I), the PL enhancement is greatest for p-polarized light and falls off as the polarization is rotated into the plane of the crystal. (We attribute the shifting of the peak to θ = 8° to experimental error.) The A values correspond to a three-pulse train, declining from ∼0.27π to 0.23π as θ increases from 0 to 48° from the normal. 6098



ARTICLE

Figure 11. Effect of laser polarization on the photoluminescence spectrum optimized over 390-450 nm at a fluence of 13.2 J/cm2. The inset is the ratio of the spectrally integrated signal for the optimized sine phase to that produced by a transform limited pulse.

Figure 12. Band structure of GaAs (reproduced with permission from ref 30; copyright Elsevier 2006). The arrows superimposed on the diagram depict the mechanism, showing the absorbed and emitted photons (vertical arrows) and the scattered electrons (solid circle and magenta arrow) and holes (open circle and orange arrow).

IV. DISCUSSION In paper I we identified the PL spectrum with the emission produced when electrons and holes recombine in the L-valley of the Brillouin zone. The mechanism for this process is depicted in Figure 12, which shows the electronic band structure of GaAs.33 The direct band gap of GaAs at room temperature is 1.42 eV,41 which is slightly less than the energy of a single 800 nm photon. Electrons excited to the conduction band are readily scattered

thermally into the L-valley. Normally such electrons rescatter back into the Γ-valley,42 where they recombine with the much heavier holes and emit in the infrared. Although the holes have sufficient thermal energy to cross into the L-valley, most of them lack linear momentum along the 111 reciprocal lattice direction required to make this transition. In our proposed mechanism, the laser-excited longitudinal optical (LO) phonons act as an optical switch by transferring momentum to the holes, so that some fraction of them may recombine with electrons in the L-valley. The band gap of the L-valley is 2.91 eV at 300 K and decreases with temperature.43 This property can be used to deduce the local temperature from the spectrum. For example, the 460 nm edge of spectrum a of Figure 9b corresponds to a local temperature of 625 K,41 which is consistent with our observations in paper I. The blue-shifting of the emission with increasing fluence seen in Figure 7 is consistent with increasing local temperatures. This effect was not observed in the open loop experiments of paper I, where the L-valley emission was restricted to a much narrower range of fluences. Examination of the GA data in Figure 5 suggests that there may be two mechanisms (or two components of a single mechanism) at play. The sine phase experiments (panels a and b) show a sudden jump in integrated PL after just one generation, followed in the second component by a slow growth until convergence is reached. In the first generation, the abovementioned optical switch driven by the three-pulse train opens a gate, allowing carriers to enter the L-valley. In the case of Figure 5b, the very large first-generation jump may be explained by the fact that only the low energy region of the band is optimized; a very large enhancement is possible because there is very little emission in this region before the gate is opened. The oscillatory structure in Figure 1 suggests that the mechanism for this process is carrier-phonon scattering. The more irregular pattern seen in Figure 6 suggests that, under closed loop control, nonresonant impulsive excitation may also be important. The slow growth step after the first-generation jump indicates that fine-tuning of the spacing and amplitudes of the three pulses control the rate of passage through that gate. In contrast, the random phase experiment (panel c) shows only a slow growth 6099



Figure 13. Cross correlation measurements of the intensities of various pulse trains. The uppermost trace was generated with an unoptimized sine phase function and is typical of the pulses used in the open-loop experiments in paper I. The middle trace was obtained by optimizing the amplitude and period of the sine phase function, integrating over the entire PL spectrum. The bottom trace was obtained using random phases, also integrating over the entire PL spectrum.

component. From the present data we cannot tell whether the random phase EA has converged. The envelope of the pulse train obtained after 95 generations using the random phase function (see Figure 13) extends over a period of 15 ps and contains two principle peaks that are similar to the first two peaks of the optimized sine phase pulse train. The much more rapid convergence of the sine phase algorithm is expected because it contains only two genetic variables. We cannot say at this time whether the random phase solution would eventually converge to the same three-pulse train. Increasing the number of individuals might increase the quality of the data and thereby help determine when the evolution has converged. We turn now to the effects of laser phase and polarization. These properties play an important role in the mechanism for coupling photons with the crystal lattice. We start by noting that the laser intensities needed to induce the PL emission are 2 orders of magnitude greater than the threshold for melting. It is known that at these high intensities a plasma is formed with an initial density comparable to that of the solid material.44,45 Plasmas in this overdense regime are opaque to 800 nm radiation. How can the laser penetrate this medium to excite the underlying lattice? Of various possible mechanisms discussed in paper I, one that satisfies the conditions of our experiments is Brunel or vacuum heating.46 The essential feature of this process is an acceleration of electrons in the plasma by the ponderomotive force of the laser field, which is derived from the gradient of the electric field. Ballistic electrons produced by this mechanism can penetrate the plasma and transmit sufficient energy to the lattice to excite an optical phonon. For this effect to occur, the electric field of the laser must have a component parallel to the plasma gradient.47 This condition implies that p-polarized light should be most effective in exciting the plasma by this mechanism. This effect was already observed in the open loop experiments of paper I, where it was found that the PL was strongest for θ = 0 and vanished for θ > 40°. Here we find that with the genetic algorithm we are able to obtain a substantial PL intensity with angles greater than 48°.

ARTICLE

The role of the laser phase is more speculative. Depending on the phase of the laser beam, the ponderomotive force can extract an electron from the plasma and either send it into the vacuum or return it to the plasma with keV energy.48 It is surprising, however, that the plasma retains a memory of this phase between successive pulses in the train. In the open loop experiments of paper I, we nevertheless found that the PL at a fixed wavelength was significantly enhanced or diminished by scanning over j. Here we show that under closed loop control the entire PL spectrum can be shifted to the blue or the red by manipulating the phase offset of the laser pulses. This mechanism could be tested by measuring simultaneously the photoemission current, which should have a phase dependence opposite to that of the photoluminescence. Previous studies using a GA to optimize the ablation of Si and Al showed that long pulses with complex structure (several hundred femtoseconds in one case28 and several tens of picoseconds in other cases25-27) produced much greater yields of atomic and molecular ions than did short, TL pulses. A theoretical analysis26 showed that a long pulse that continuously adapts to the changing phase composition of the superheated material is much more effective than a short pulse that only launches the phase transition. In the simplest terms, the earlier part of the optimized pulse melts the surface, and later parts of the pulse interact with the liquid phase by continuously adapting to the changing optical and thermal properties of the evolving material. To a large extent, this effect could be accomplished by simply stretching a single pulse28-30 or by using a double pulse with a variable delay.35 This type of thermodynamic control is incoherent. In the present experiment, control with the sine phase function is clearly coherent. It is difficult to imagine how shifting the fringes of the side pulses by less than 1 fs could have an effect in an incoherent mechanism. Whether the phase control is quantal or classical, however, is presently an open question.

V. CONCLUSIONS This work extends our earlier study of the coherent control of photoluminescence (PL) in GaAs by using an evolutionary algorithm to optimize the spectral phase of the laser pulse. Using a sine phase function to generate a train of equally spaced pulses, we obtained an order of magnitude increase in the wavelengthintegrated emission after a single generation, and an additional factor of 2 enhancement was achieved after convergence of the feedback loop. The sensitivity of the spectrum to the relative phases of the satellite pulses is convincing evidence of a coherent mechanism. In our prior study we identified the PL spectrum with recombination of electrons and holes in the L-valley of the Brillouin zone. We propose that coherent control arises from manipulation of electrons in the plasma, which impulsively excite lattice phonons, which in turn scatter carriers into the L-valley. The strong first-generation enhancement of the signal relative to that obtained with a single transform-limited pulse shows that the essential properties of the requisite waveform are satisfied by a simple train consisting of three equally spaced pulses. Qualitatively different behavior was obtained using an algorithm that optimizes the phases of individual pixels in the pulse shaper. It is possible, however, that the algorithm did not converge, and further work is required to ascertain whether the mechanisms induced by the sine phase and random phase functions might be different. In future experiments we will expand the gene pool of the sine phase function to include A, T, and j, as well as possibly 6100


The Journal of Physical Chemistry A the fluence, to determine how the fully optimized waveform for this function compares with the converged random phase approach.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: Z.H., [email protected]; R.J.G., [email protected].

’ ACKNOWLEDGMENT Support by the National Science Foundation under grant nos. CHE-0640306 and CHE-0848198, by the National Science Foundation of China under grant nos. 10774056 and 10974070, and by Fundamental Research Funds for the Central Universities of China under grant no. 200903371 is gratefully acknowledged. ’ REFERENCES (1) Shapiro, M.; Brumer, P. Principles of Quantum Control of Molecular Processes; Wiley: New York, 2003. (2) Rice, S. A.; Zhao, M. Optical Control of Molecular Dynamics; Wiley: New York, 2000. (3) Brumer, P.; Shapiro, M. Acc. Chem. Res. 1989, 22, 407. (4) Tannor, D. J.; Rice, S. A. Adv. Chem. Phys. 1988, 70, 441. (5) Zhu, L.; Kleiman, V.; Li, X.; Lu., S.; Gordon, R. J. Science 1995, 70, 441. (6) Scherer, N. F.; Carlson, R. J.; Matro, A.; Du., M.; Ruggiero, A. J.; Romero-Rochin, V.; Cina, J.; Fleming, G. R.; Rice, S. A. J. Chem. Phys. 1991, 95, 1487. (7) Weiner, A. M. Rev. Sci. Instrum. 2000, 71, 1929. (8) Meshulach, D.; Silberberg, Y. Nature 1998, 396, 239. (9) Judson, R, S,; Rabitz, H. Phys. Rev. Lett. 1992, 68, 1500. (10) White, J. L.; Pearson, B. J.; Bucksbaum, P. H. J. Phys. B 2004, 37, L399. (11) Lindinger, A.; Lupulescu, C.; Vetter, F.; Plewicki, M.; Weber, S. M.; Merli, A.; W€oste, L. J. Chem. Phys. 2005, 122, 024312. (12) Rey-de-Castro, R.; Rabitz, H. Phys. Rev. A 2010, 81, 063422. (13) Cardoza, D.; Baertschy, M.; Weinacht, T. J. Chem. Phys. 2005, 123, 074315. (14) Savolainen, J.; Fanciulli, R.; Dijkhuizen, N.; Moore, A. L.; Hauer, J.; Buckup, T.; Motzkus, M.; Herek, J. L. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 7641. (15) Branderhorst, M. P. A.; Londero, P.; Wasylczyk, P.; Brif, C.; Kosut, R. L.; Rabitz, H.; Walmsley, I. A. Science 2008, 320, 638. (16) Carroll, E. C.; White, J. L.; Florean, A. C.; Bucksbaum, P. H.; Sension, R. J. J. Phys. Chem. A 2008, 112, 6811. (17) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T. K.; Mancal, T.; Cheng, Y. C.; Blankenship, R. E.; Fleming, G. R. Nature 2007, 446, 782. (18) Herek, J. L.; Wohlleben, W.; Cogdell, R. J.; Zeidler, D.; Motzkus, M. Nature 2002, 417, 533. (19) Prokhorenko, V. I.; Nagy, A. M.; Waschuk, S. A.; Brown, L. S.; Birge, R. R.; Miller, R. J. D. Science 2006, 313, 1257. (20) Kuroda, D. G.; Singh, C. P; Peng, Z.; Kleiman, V. D. Science 2009, 326, 263. (21) Heberle, A. P.; Baumberg, J. J.; K€ohler, K. Phys. Rev. Lett. 1995, 75, 2598. (22) Dupont, E.; Corkum, P. B.; Liu, H. C.; Buchanan, M.; Wasilewski, Z. R. Phys. Rev. Lett. 1995, 74, 3596. (23) Cho, G. C.; K€utt, W.; Kurz, H. Phys. Rev. Lett. 1990, 65, 764. (24) Petek, H.; Heberle, A. P.; Nessler, W.; Nagano, H.; Kubota, S.; Matsunami, S.; Moriya, N.; Ogawa, S. Phys. Rev. Lett. 1997, 79, 4649. (25) Stoian, R.; Mermillod-Blondin, A.; Bulgakova, N. M.; Rosenfeld A.; Hertel, I. V.; Spyridaki, M.; Koudoumas, E.; Tzanetakis, P.; Fotakis, C. Appl. Phys. Lett. 2005, 87, 124105.

ARTICLE

(26) Colombier, J. P.; Combis, P.; Rosenfeld, A.; Hertel, I. V.; Audouard, E.; Stoian, R. Phys. Rev. B 2006, 74, 224106. (27) Guillermin, M.; Liebig, C.; Garrelie, F.; Stoian, R.; Loir, A. S.; Audouard, E. Appl. Surf. Sci. 2009, 255, 5163. (28) Dachraoui, H.; Husinsky, W. Phys. Rev. Lett. 2006, 97, 107601. (29) Gunaratne, T. C.; Zhu, X.; Lozovoy, V. V.; Dantus, M. J. Appl. Phys. 2009, 106, 123101. (30) Gunaratne, T.; Kangas, M.; Singh, S.; Gross, A.; Dantus, M. Chem. Phys. Lett. 2006, 423, 197. (31) Hergenr€oder, R.; Miclea, M.; Hommes, V. Nanotechnology 2006, 17, 4065. (32) Hu, Z.; Singha, S.; Gordon, R. J. Phys. Rev. B 2010, 82, 115205. (33) Blakemore, J. S. J. Appl. Phys. 1982, 53, R123. (34) Hu, Z.; Singha, S.; Liu, Y.; Gordon, R. J. Appl. Phys. Lett. 2007, 90, 131910. (35) Singha, S.; Hu, Z.; Gordon, R. J. J. Appl. Phys. 2008, 104, 113520. (36) Lozovoy, V. V.; Pastirk, I.; Dantus, M. Opt. Lett. 2004, 29, 775. (37) Wollenhaupt, M.; Pr€akelt, A.; Sarpe-Tudoran, C.; Liese, D.; Bayer, T.; Baumert, T. Phys Rev. A 2006, 73, 063409. (38) Galler, A.; Feurer, T. Appl. Phys. B: Laser Opt. 2008, 90, 427. (39) Mitchell, M. An Introduction to Genetic Algorithms (Complex Adaptive Systems); MIT Press: Cambridge, MA, 1998. (40) Herrera, F; Lozano, M. IEEE Trans. Evol. Comput. 2000, 4, 43. (41) Lautenschlager, P.; Garriga, M.; Logothetidis, S.; Cardona, M. Phys. Rev. B 1987, 35, 9174. (42) Nuss, M. C.; Auston, D. H.; Capasso, F. Phys. Rev. Lett. 1987, 58, 2355. (43) Varshni, Y. P. Physica (Amsterdam) 1967, 34, 149. (44) Sokolowski-Tinten, K.; Bialkowski, J.; Boing, M.; Cavalleri, A.; von der Linde, D. Phys. Rev. B 1998, 58, R11805. (45) Kim, A. M.-T.; Callan, J. P.; Roeser, C. A. D.; Mazur, E. Phys. Rev. B 2002, 66, 245203. (46) Brunel, F. Phys. Rev. Lett. 1987, 59, 52. (47) Wilks, S. C.; Kruer, W. L. IEEE J. Quantum Electron. 1997, 33, 1954. (48) Dombi, P.; Apolonski, A.; Lemell, Ch.; Paulus, G. G.; Kakehata, M.; Holzwarth, R.; Udem, Th.; Torizuka, K.; Burgd€ orfer, J.; H€ansch, T. W.; Krausz, F. New J. Phys. 2004, 6, 39.

6101


Closed Loop Coherent Control of Electronic Transitions in Gallium

Recommend Documents