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Ultracold and high-density 1s orthoexcitons in semiconductor cuprous oxide are prepared via resonant two-photon absorption of a phase-tailored femtosecond pulse, by utilizing an acousto-optic programmable dispersive filter. The stability of the quantum degenerate exciton gas is studied using excitonic Lyman spectroscopy. A density of 1016 cm−3 is realized without creating hot carriers, and the Lyman spectrum remains unchanged at this density. This result assures the stability of a spontaneous Bose–Einstein condensate of excitons at sub-Kelvin temperatures.
© 2014 Optical Society of America
1. Introduction
The ability to control the amplitude and phase of femtosecond laser pulses plays a vital role in modern ultrafast spectroscopy. Especially, a pulse shaper in a 4f configuration with paired grating and a crystal spatial light modulator (SLM) in either transmittive or reflective geometry has opened up the possibility of coherent control of material excitations [1]. Meanwhile, acousto-optic programmable dispersive filters (AOPDF) [2, 3] have found widespread applications, owing to the simplicity and higher damage threshold than SLMs. In addition to tailoring [3–6] and characterizing femtosecond pulses [7–9], AOPDFs have been applied to study various physical systems such as applying spectral shift in high-harmonic generation [10], coherent anti-Stokes Raman scattering microscopy [11], carrier-envelope phase control of femtosecond pulses [12,13], generation of ellipsoidal electron beams [14], and quantum control of a single molecule [15]. However, to our knowledge, the application of AOPDF to solid-state systems remains unexploited.
In this paper, we demonstrate that AOPDFs can be employed to manipulate intrinsic electronic excitations in solids and to use the prepared systems for fundamental studies on quantum manybody physics. We use a system that exhibits atom-like sharp resonances and thus provides full controllability via the optical phase: excitons in bulk semiconductors. At low density and low temperature, an exciton (i.e., a bound state of an electron and a hole) is a well-defined bosonic entity in a semiconductor. Among various semiconductors, Cu2O is a prime candidate for realizing Bose–Einstein condensation (BEC) of bulk excitons [16–22]. The lowest-energy 1s yellow-series excitons have large binding energy (150 meV) and are widely considered to constitute a prototypical exciton system on account of the stability of the exciton gas. In general, however, excitons become unstable at high density and eventually becomes fermionic electron-hole plasma (exciton Mott transition), because of the strong Coulomb interaction between the constituent carriers.
There is little theoretical justification that precludes the possibility of such excitonic instability at much lower densities than the conventional Mott criteria in bulk semiconductors. Indeed, recent experiments revealed that two-body inelastic collision processes shorten the exciton lifetime, significantly reducing exciton densities even at surprizingly low densities [23, 24]. It is therefore crucial to study experimentally the electronic stability in a high-density and low-temperature condition. One promising method to prepare such a highly quantum degenerate exciton state directly is resonant two-photon excitation of 1s orthoexciton states using near-infrared pulses. We demonstrated previously that resonant two-photon excitation using π-phase-step pulses [25, 26] can create ultracold orthoexcitons up to 1015 cm−3 while removing unwanted hot excitons that emerge via interband three-photon absorption processes [27]. However, preparing a higher density has still been difficult, because of the deviation from the optimal waveform that results in the three-photon processes and of the low damage threshold of a spatial light modulator. A density of 1015 cm−3 is one order of magnitude lower than the critical density for BEC at 800 mK, which is relevant with recent sub-Kelvin experiments for spontaneous formation of BEC under non-resonant excitation of 1s paraexcitons [21, 22].
Here we have resolved the difficulty and have studied the stability of the ultracold 1s orthoexcitons up to a density of 2×1016 cm−3. The stability can be confirmed by examining whether the excitons keep well-defined energy levels of their internal electronic structures. This can be done by directly measuring the absorption spectrum of internal transitions [28, 29]. The 1s–2p induced absorption shows no spectral indication of instability in such a highly quantum degenerate exciton system. The unwanted hot exciton component was suppressed by applying a carefully designed phase modulation via an AOPDF.
2. 1s orthoexcitons in Cu2O and optical phase manipulation
In a bulk semiconductor Cu2O, the 1s yellow-exciton states split into paraexciton and orthoexciton states. The lower-lying 1s paraexciton is a singlet state, whose coupling to an electromagnetic field is spin forbidden to all orders. The higher-lying 1s orthoexciton states are triply degenerate and can be created by two-photon excitation. The dipole-forbidden nature of 1s excitons in Cu2O prohibits radiative recombination, allowing them to cool to the lattice temperature during their lifetimes. At high density, two-body inelastic collisions may not allow excitons to survive for the ∼1 ns necessary to arrive at equilibrium with the lattice [30]. Since the two-body collisional decay constant has been experimentally determined to be on the order of 10−16 cm3/ns [23, 24], equilibrium with the lattice can be achieved at an exciton density of 1016 cm−3 or below. The critical temperature of BEC at 1016 cm−3 is below 1 K, and a recent sub-Kelvin experiment has revealed that paraexcitons indeed show a threshold-like explosive behavior at the BEC transition point for ideal bosons [21]. We note that the present experimental method can only be applied to orthoexcitons that can be created by two-photon excitation. However, we believe that the stability of the orthoexcitons also assures the stability of the paraexciton state because it is the lowest, nondegenerate state and thus has a simpler electronic structure.
Although the creation of 1s excitons by one-photon absorption from the ground state is dipole forbidden, the internal 1s–np transitions are dipole allowed. In addition, one can evaluate the density of 1s excitons from the differential absorption coefficient integrated over the 1s–2p transition line by the relationship [24],
where η is the background index of refraction, μ1s−2p is the transition dipole moment, E1s−2p is the energy offset, and Δα(E) is the differential absorption spectrum. The temperature of 1s excitons can also be estimated by using the fact that the 1s and 2p yellow excitons of Cu2O have different effective masses [31–33], which leads to higher transition energies for excitons with larger center-of-mass momenta.A high-density exciton gas in Cu2O can be produced by creating free electron–hole pairs by dipole-allowed transitions to a high-energy band followed by subsequent relaxation. However, due to the heat generated in inelastic collisions, the orthoexciton gas thus prepared remains at a high temperature and never reaches the quantum degenerate regime [34]. Moreover, since some residual carriers may not form excitons, it is difficult to estimate the density of excitons from the number of absorbed photons. Resonant two-photon excitation of 1s orthoexcitons offers a powerful alternative that allows one to obtain a pure exciton gas in a quantum degenerate regime. In resonant two-photon excitation with a short light pulse, every frequency pair that satisfies energy conservation contributes to the excitation. Since the two-photon resonant frequency is much less than the electronic transition frequency, the dispersion of the host crystal is small and the exciton gas thus created has an ultranarrow momentum distribution with an effective temperature well below that of the lattice [30]. Since the initial momentum distribution is fixed by the momentum selection of the excitation process, the phase-space density around the peak of the distribution would well exceed unity under strong two-photon excitation. This method allows us to prepare a quantum degenerate exciton gas, which is an alternative of the exciton BEC spontaneously formed in thermal equilibrium conditions.
At low density excitation, this method can produce a pure, ultracold exciton gas because contributions of higher-order nonlinear optical processes are negligible. However, at high density excitation, it has been experimentally demonstrated that the generation of free carriers by band-to-continuum three-photon transitions dominates. The free carriers immediately become hot excitons that do not allow one to observe the absorption spectrum of ultracold excitons [27]. To reduce this heating of the exciton ensemble, it was proposed that the spectral phase (π step) of the excitation pulse be modulated to generate of an ultracold orthoexciton gas at 1015 cm−3. As found in [27], the residual hot excitons prevent a further increase in density. In the current paper, we present an improved spectral phase-modulation technique that enables us to reach an ultracold orthoexciton density of 1016 cm−3. At this density, we examine the stability of the dense exciton ensemble by measuring 1s–2p absorption spectrum (see Fig. 1).
Fig. 1 Excitation and probe configurations of present experiment. Generation of ultracold 1s orthoexcitons in Cu2O is realized by resonant two-photon excitation using a broadband femtosecond pulse. The density and the momentum distribution of the generated orthoexcitons are detected by exciton Lyman spectroscopy. The energy of the 1s–np transition of orthoexcitons is approximately 116 meV, 129 meV, and 133 meV, respectively.
The principle by which the heating is reduced is as follows: For resonant two-photon excitation of a two-level system by a pulse, the transition probability (the transition strength) P2ph in the absence of near-resonant intermediate states is written as
where A(ω) and ϕ(ω) are the amplitude and phase of the excitation pulse, respectively. The transition amplitude is the sum of the contributions of all frequency pairs that satisfy energy conservation, in which each pair is attached with a phase equal to the sum of the spectral phases at the corresponding frequencies. The maximum amplitude is obtained when all the spectral components have the same phase (e.g., a transform-limited pulse). On the one hand, the maximum amplitude may be achieved if the spectral phase distribution is asymmetric with respect to the resonance frequency. On the other hand, for a three-photon transition into the continuum, only the temporal intensity profile matters, and the phase coherence plays no role because of the broad energy distribution of the final states. In the absence of near-resonant intermediate states the three-photon transition probability is proportional to the third order of the pulse intensity integrated over the entire pulse duration. Therefore, to reduce the three-photon transition probability, we should modulate the phase of the pulse in a way that decreases the peak intensity in comparison with that of a transform-limited pulse. At the same time, an antisymmetric spectral phase distribution with respect to the central frequency allows us to achieve the maximum two-photon transition probability. We note that there also exists a stepwise excitation path via two-photon resonant orthoexciton states to the continuum. However, if we assume that this process dominates, the simulation results cannot explain the experimental data shown in Fig. 4(b) of Ref. [27]. Therefore, we conclude that the contribution of this process is negligible.To optimize the spectral phase, one should take into account the characteristics of the phase-modulating device. In this study, we used an AOPDF, which exploits the diffraction of light by applying an acoustic pulsed wave to modulate the phase and amplitude of the incident optical pulse. The requirement that the acoustic wave fit inside the acousto-optic crystal turns out to be equivalent to requiring that the group delay (i.e., the derivative of the spectral phase) be within a certain range. This determines the upper limit of the pulse duration, which for our device is several picoseconds. Moreover, the diffraction efficiency of an AOPDF strongly depends on the temporal profile of the acoustic wave, which in turn is determined by the designed spectral phase. This feature becomes especially important for applications that require high throughput, as is in the case of the present experiment. Since the relationship between the designed spectral phase and diffraction efficiency is not trivial, we empirically searched for a spectral phase that has a relatively high diffraction efficiency. The spectral phase we employed has the form of a modulated sine function:
The quantity A corresponds to the period of the sine function, and B determines the additional chirped modulation. In the absence of the modulation (B = 0), a sine spectral phase represents a pulse comprising several subpulses regularly spaced in the time domain. The introduction of the modulation of the pitch leads to a deformation of the sub-pulses, which further reduces the three-photon transition probability. The maximum A is determined by the abovementioned condition on the group delay, and B is tuned to minimize the three-photon transition probability. The designed spectral phase is shown in Fig. 2. This phase modulation allows us to reduce the three-photon transition probability to 2%, which is five times smaller than that for the phase-modullated pulse of Ref. [27]. The overall transmittance of the phase-modulating device (including the grating pairs as mentioned below) was 30% for a transform limited pulse and 13% for the designed pulse.Fig. 2 Applied spectral phase (red curve). Note that the phase is antisymmetric about the two-photon resonance wavelength (1220 nm). The dashed curve designates a modeled spectrum of the excitation pulse.
3. Experimental setup
The experimental setup shown in Fig. 3 is essentially the same as that described in our previous paper Ref. [27]. In the present experiment, we used an AOPDF instead of a liquid crystal spatial light modulator (SLM) as the phase-modulating device. The AOPDF has a much higher damage threshold and thus allows phase modulation of more intense pulses. The output of a 1 kHz, 100 fs Ti:sapphire regenerative amplifier is split into two beams to pump two OPA systems. One beam is converted to 1220 nm with a full width at half maximum (FWHM) of 20 nm and used as a pump pulse for the resonant two-photon excitation of 1s orthoexcitons. The other beam is converted into the spectral region of around 10 μm and is used as a probe pulse to probe the internal 1s–np transitions. The duration of the probe pulse is about 100 fs. The pump pulse is sent through the phase-modulating device, which consists of an AOPDF and a grating pair. The grating pair is used twice: first to stretch the pulse to reduce nonlinear optical effects inside the acousto-optic crystal and then to recompress the pulse. A positive chirp is added in the AOPDF that exactly compensates for the negative chirp introduced by the grating pair. After phase modulation, the pump pulse irradiates the (100) surface of a naturally grown 220-μm-thick single crystal [27] of Cu2O mounted in a liquid-He cryostat and kept at 4 K. The polarization angle of the pump pulse is chosen so that the two-photon absoption strength becomes maximum. The probe pulse is transmitted through the sample with a variable delay with respect to the pump pulse and then spectrally resolved and detected by a MCT detector. Unless otherwise noted, the pump–probe delay was 5 ps. The probe signals with and without the pump-pulse irradiation are collected to calculate the differential absorption spectrum. The spot sizes of the gaussian pump and probe beams are both 250 μm at FWHM, and a 200-μm aperture that is placed on the incident surface of the sample truncates both beams.
Fig. 3 Experimental setup (see text). A grating pair was used to avoid nonlinear optical effects that are induced in the AOPDF.
4. Results and discussions
Before proceeding to high-density excitation, we first tested whether the control of the two-photon transition probability by phase modulation worked as expected. To do this, we applied to the AOPDF spectral phases that are of the same form as that shown in Fig. 2 but that are shifted in frequency. The density of the 1s excitons created by the two-photon transition was then estimated from the peak height of the 1s–2p transition line (at 116 meV) as a function of frequency shift. At a low-density excitation, the density of the excitons generated by the three-photon transition is negligible, and the signal is proportional to the two-photon transition probability. Figure 4 shows the experimental data (open circles) and the simulation results (solid curve). The two-photon transition probability becomes large when the spectral phase is nearly antisymmetric (sine like), and becomes small when the spectral phase is nearly symmetric (cosine like). The two results agree well, which shows that the phase modulation and coherent control indeed work well.
Fig. 4 Coherent control of ultracold 1s orthoexcitons at relatively low density. The relative orthoexciton density (vertical axis) is recorded as a function of the central wavelength of the added optical phase given in Eq. (3). The open circles are experimental data. The solid curve is calculated by Eq. (2).
Next, we increased the excitation pulse intensity to achieve the high-density regime. In Fig. 5, we compare the differential absorption spectra for excitations excited by a transform-limited pulse and by a phase-modulated pulse at (a) low-density excitation (1.3 mJ/cm2) and (b) high-density excitation (4.5 mJ/cm2) density excitation. At low-density excitation, the contribution of the three-photon transition is negligible, and both spectra show the presence of the cold 1s orthoexcitons directly created by two-photon excitation. The 1s–np transition lines with n = 2 to 4 are observed in order of increasing probe photon energy. At high-density excitation, the high-energy tail of the 1s–2p transition line builds up for the case of the transform-limited pulse, which signals the generation of hot excitons via the three-photon transition. However, for the phase-modulated pulse, the density of hot excitons is greatly reduced. The density of cold excitons estimated from the 1s–2p transition line is 2 × 1016 cm−3. The spectral position and lineshape of the 1s–2p transition line are nearly unchanged from the low-density limit, which indicates that the picture of an exciton remains valid at this density. Although during the following delay of several hundred picoseconds the conversion of orthoexcitons into paraexcitons was observed [35], no sign appeared of drastic effects such as the complete collapse of the exciton ensemble. However, some deviations from the theoretical calculation are also observed. First, the ratio of the strength of the 1s–3p absorption line (129 meV) to that of the 1s–2p line is found to decrease as the exciton density increases. The 1s–4p line (133 meV) even seems to vanish in Fig. 5(b). These features may be caused by many-body effects or by interactions between excitons and the residual free carriers that cannot be completely eliminated. To clarify the mechanism, further improvement in the spectral phase manipulation is required. Second, we observed that the three-photon transition probability was two times lower rather than 5 times lower. We also found that the growth from the two-photon absorption is sub-quadratic for both the cold and hot exciton densities at high excitation intensity as shown in Fig. 6. These most probably originate from self-phase modulation of the excitation pulse in the Cu2O crystal. Using the reported value of the nonlinear index of refraction [36], the estimated nonlinear phase change that occurs in passing through the sample is of the order of 1 rad, so self-phase modulation becomes important. Indeed, a simplified numerical simulation assuming plane wave propagation indicates that the two-photon transition probability should start to decrease at an excitation intensity of the same order as that actually observed, although self-phase modulation cannot explain the reduction in the three-photon transition probability. A refined simulation that includes the Kerr effect in the cross section of the beam and characterization of the excitation pulse after transmission through the sample may provide the information required to settle this point.
Fig. 5 Induced absorption spectra 5 ps after irradiation by the pump pulse. The spectral resolution is 0.1 meV. Red and blue curves correspond to transform-limited (TL) and sine-like phase modulation [Eq. (3)]), respectively. (a) Low-density excitation (1.3 mJ/cm2). (b) High-density excitation (4.5 mJ/cm2). The lattice temperature is 4.2 K. An oscillatory structure due to Fabry–Perot interference in the sample crystal (220 μm thick) is removed by numerical filtering.
Fig. 6 Pump energy density dependence of the two- and three-photon absorption. The amount of two- and three-photon absorption as a function of pump energy density is estimated at 116 meV (cold orthoexcitons) and 125 meV (hot orthoexcitons), respectively. We estimate the statistical uncertainty is less than 5% in the horizontal axis, and less than 2% in the vertical axis.
5. Conclusion
In conclusion, we generated an ultracold 1s orthoexciton gas in Cu2O by resonant two-photon excitation with a phase-modulated pulse. By measuring the absorption spectrum of the internal transitions, we demonstrated that excitons at a density of 2 × 1016 cm−3 are still well-defined quasiparticles. This density is of special importance in the context of the observation of exciton BEC in Cu2O. Although some deviations from theoretical predictions have been observed, they are most likely caused by self-phase modulation of the excitation pulse. Compensation of the self-phase modulation may allow us to improve the excitation efficiency, thereby allowing us to reach the exciton Mott transition density. Using this highly coherent excitonic matter wave for nonlinear optical applications would also be interesting [37].
Acknowledgments
We thank A. Mysyrowicz and Yu. P. Svirko for fruitful discussions. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Area ”Optical science of dynamically correlated electrons (DYCE)” 20104002, by Special Coordination Funds for Promoting Science and Technology of MEXT, Japan, by the Photon Frontier Network Program of MEXT, Japan, and by JSPS through its FIRST Program.
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