The electron–phonon coupling in self-assembled InGaAs quantum dots is relatively weak at low light intensities, which means that the zero-phonon line in emission is strong compared to the phonon sideband. However, the coupling to acoustic phonons can be dynamically enhanced in the presence of an intense optical pulse tuned within the phonon sideband. Recent experiments have shown that this dynamic vibronic coupling can enable population inversion to be achieved when pumping with a blueshifted laser and for rapid de-excitation of an inverted state with red detuning. In this paper we confirm the incoherent nature of the phonon-assisted pumping process and explore the temperature dependence of the mechanism. We also show that a combination of blueshifted and redshifted pulses can create and destroy an exciton within a timescale of as determined by the pulse duration and ultimately limited by the phonon thermalization time.
© 2016 Optical Society of America
Vibronic sidebands are observed in the optical spectra of many solid-state materials , with Ti:sapphire  and nitrogen-vacancy (NV) centers in diamond [3–5] being good examples. The key point about a vibronic transition is that it involves the simultaneous absorption or emission of a photon and a phonon (or phonons) as the electron jumps between two electronic states. The sideband spectra are continuous bands, although substructure can frequently be identified due to the involvement of specific phonon modes, especially at low temperatures. For example, in the case of NV centers in diamond, clear structure can be identified that corresponds to the coupling of the electronic state to the mode (), giving rise to resolved side peaks in both absorption and emission at integer multiples of the phonon energy [3,4]. The coupling to the vibrational modes is so strong that only a few percent of the emission occurs in the zero-phonon line, with most of the photons emitted from the sidebands . The relatively weak intensity of the zero-phonon line has serious consequences for practical applications of NV centers in optical quantum information processing.
The reason that the phonon sidebands in materials like diamond NV centers and Ti:sapphire are so strong is that both the electronic and vibrational modes are strongly localized on length scales similar to the unit cell size. This means that the overlap between the electronic wave functions and the phonon modes is large, and, hence, the vibronic coupling is strong. By contrast, InGaAs quantum dots (QDs) have envelope wave functions localized on much larger length scales that are determined by the size of the dot, i.e., . This means that the coupling to phonons is relatively weak, with the dominant interaction being to longitudinal-acoustic (LA) phonons via deformational potential scattering. The weak vibronic coupling in these dots gives rise to very strong emission in the zero-phonon line, with only in the sideband at cryogenic temperature [6,7]. This makes InGaAs QDs excellent single-photon sources . It also ensures that the light–matter coupling is strong, with optical dipole moments typically in the range of Debye () . Moreover, the relatively weak electron–phonon coupling leads to long coherence times that are ultimately limited only by the radiative lifetime , which facilitates their application in coherent control experiments [11–14]. This contrasts with bulk and quantum well samples, where the coherence time is only a few picoseconds at best [15–17].
The vibronic sideband in InGaAs QDs manifests itself in a number of important experimental situations. One example is the emission of photons from a nanocavity when the dot and cavity are out of resonance, with the emission involving either the absorption or emission of a phonon depending on the sign of the detuning [8,18]. Another case is the observation of QD emission when pumping in the phonon sideband in a resonance fluorescence geometry . The difference in the frequency of the pump laser and the QD exciton line facilitates spectral selection of the QD photons  and can be exploited for stabilizing the frequency of the zero-phonon line . The sideband also appears in four-wave mixing experiments .
A key point about the experiments described above is that the optical pumping is relatively weak, so that the QD–laser system is in the weak-coupling limit. In this paper we explore the other limit where the QD–laser coupling is strong, such that a dressed-state picture is appropriate. This limit was addressed in a theoretical paper in 2013 by Glässl et al., where it was predicted that strong driving in the phonon sideband could lead to exciton populations approaching full inversion . The predictions of the theory were confirmed independently by three experimental groups in 2014–2015, both for pumping of the neutral exciton and the biexciton [14,24,25]. Most recently, it has also been demonstrated that the process works in reverse, so that an inverted system can be rapidly depopulated by pumping with a laser with red detuning relative to the exciton . In these experiments the phonon coupling becomes strong when an intense laser field is present. The pulse durations in the experiments are short, and so the vibronic coupling is turned from weak to strong and back to weak again on picosecond timescales. We therefore call this process dynamic vibronic coupling (DVC). The DVC is fundamentally different from conventional excitation schemes, such as coherent Rabi oscillation [27–29] and adiabatic rapid passage protocols [30,31] where exciton–phonon coupling is usually an obstacle.
In this paper we first review the process underlying the DVC and summarize the results from our previous experiments [24,26]. After discussing the experimental methods in Section 3, we then present Ramsey interference data to demonstrate that the mechanism of DVC is incoherent, i.e., that the exciton created by phonon-sideband pumping is incoherent with the pumping laser (Section 4). We next consider the temperature dependence of the DVC in Section 5, comparing the population generated at to that at base temperature (4.2 K). Finally we show in Section 6 how we can combine pumping with a blue-detuned and a red-detuned pulse to create and destroy an exciton within . Section 7 gives the conclusions and outlook.
2. MECHANISM OF DYNAMIC VIBRONIC COUPLING
The starting point for DVC is the coupling of excitons to the acoustic phonon bath by the deformation potential [32,33]. In the absence of a laser field this leads to nonexponential pure dephasing of the excitonic dipole . The behavior of the coupled system becomes more interesting, however, when a strong laser field is applied. We consider a laser pulse with energy detuning and area :
Figure 1 illustrates the dressed state energies (green lines), admixture angle (black line), and occupancy of (blue line) against time during the passage of a pulse with . The pulse has a temporal FWHM of 16.8 ps and is centered on . The figure illustrates that the dressed state splitting rises and falls in line with the envelope of the laser pulse , and that the admixing angle follows it. The transfer of population into occurs only while the states are admixed, tuning the system from weakly to strongly vibronic and back again in approximately 20 ps.
The transfer between the dressed states shown in Fig. 1 can be exploited to achieve ultrafast incoherent excitation and de-excitation of the exciton by detuned laser pulses. Beginning with excitation, in the case of the higher energy dressed state is dominated by the crystal ground state, while the lower energy state is primarily excitonic in character, as illustrated to the right in Fig. 2(a). During the passage of the laser pulse, admixing of the states allows relaxation to occur from to by emission of an LA phonon (—blue arrows).
For a sufficiently strong pulse, most of the population relaxes into and the exciton-dominated nature of means that the population of the exciton state after the passage of the pulse is high. Figure 2(b) shows a phenomenological level diagram of the phonon-assisted excitation process, while Fig. 2(c) illustrates the competing process whereby a phonon is absorbed (—red arrows) and a photon emitted (purple arrow). The latter process is weak at low temperatures.
In the case of , the main difference is that the characteristics of the dressed states are now exchanged. During the passage of the laser pulse, relaxation by phonon emission again occurs into the state. However, as this state is now dominated by the crystal ground state, the effect of the relaxation is instead to de-excite the exciton with most of the population left in the ground state after the passage of the laser pulse. Figure 2(d) shows an illustrative schematic of this process with the emission of a photon (purple arrow) accounting for the rest of the energy difference. This emission is stimulated by the laser and, hence, the process is termed LA-phonon stimulated emission (LAPSE). Figure 2(e) illustrates the competing phonon absorption process.
The physical factors that determine the efficiency of phonon-assisted excitation and de-excitation may be considered by analyzing the parameters that enter Eq. (4). The degree of admixing increases by increasing the driving strength () or decreasing the detuning () (noting that should exceed the laser linewidth to exclude resonant coherent driving), leading to more efficient relaxation. However, both of these dependencies are modified by the properties of the phonon bath. The exciton–phonon interaction strength is characterized by the function , which increases with at first due to the rising phonon density of states, and then rolls off rapidly beyond a cutoff frequency, which is typically around 1–2 meV . The physical origin of this cutoff is the point at which the phonon wavelength is comparable to the spatial FWHM of the carrier wavefunction. As a result, the cutoff frequency depends strongly on the height of the QD, and phonon sideband measurements may be used to probe the confinement potential. A plot of derived from fitting data measured in Ref.  is shown in Fig. 3. The form of modifies the detuning dependence of the DVC and also weakens the exciton–phonon interaction for very strong driving where the effective Rabi frequency exceeds the cutoff frequency (although this regime has not been reached in the present experiments).
The processes of interest for DVC both rely on phonon emission rather than phonon absorption and are, therefore, highly sensitive to temperature. Increasing the bath temperature increases the phonon occupation, leading to a higher probability of phonon emission and thus faster relaxation. However, the probability of phonon absorption generally increases more than that of emission. As illustrated by the red arrows in Fig. 2(a), phonon absorption () causes the opposite population transfer between the dressed states to phonon emission (—blue arrows) and results in a lower final occupation of . As such, DVC processes are generally optimized by low bath temperatures to maximize the difference between phonon emission and absorption rates. The influence of temperature on phonon-assisted excitation is studied both experimentally and theoretically in Section 5.
It is only by combining all of these influences that the full spectral and power dependence of the DVC emerges. Analytical approximations do not accurately predict these dependencies; hence, numerical methods such as path integral calculations [23,24] or the master equation formalism  are employed. The result of such calculations is a broad (few meV) spectral sideband feature that appears at high driving strengths and persists until the Rabi splitting exceeds the cutoff energy for phonon coupling.
3. EXPERIMENTAL METHODS
Figure 4 gives a schematic diagram of the apparatus used in the experiments. A mode-locked Ti:sapphire laser with 76 MHz pulse repetition rate is passed through two pulse shapers to obtain independently tunable pulses from within the bandwidth of the pulses . The spectral FWHM is selected to be either 0.2 or 0.42 meV corresponding to a Fourier-transform-limited pulse duration of 16.8 or 8 ps. Both pulses pass through wave plates to permit independent control of their polarizations, and one of them follows a variable path length controlled by a delay stage to enable pump–probe experiments with precise relative time control.
The pulses are incident on the sample in a variable temperature liquid He cryostat with piezo actuators for precise positioning of the sample relative to the focused beams. The sample consists of a layer of InGaAs QDs embedded in a Schottky diode. The exciton created by the laser pulses can be measured by detecting the photocurrent (PC) from the QD when a reverse bias is applied to the diode . Further details of the experimental methods may be found in Ref. .
4. COHERENCE AND SPIN PERSISTENCE OF DYNAMIC VIBRONIC COUPLING
In Ref. , Quilter et al. demonstrated the DVC by creating a QD exciton using a slightly blueshifted laser pulse via emitting an LA phonon. Since it involves emission of a phonon, in theory the DVC should be incoherent, namely, the phase of the exciton is random relative to that of the blueshifted laser pulse. The exciton coherence time in this case is limited by the phonon relaxation time (a few picoseconds ). To demonstrate the incoherent nature of the DVC, we performed Ramsey-like interference measurements [38–40] using an unstabilized interferometer.
The QD is pumped with a blue-detuned pulse with at . The pulse area () is chosen to give an exciton population of . (Note that this is slightly higher than in Ref.  due to different laser detunings.) Once the QD has been excited, the state is then probed by a pulse resonant with the exciton at . The relative phase between the pump and probe is proportional to . If the system is coherent, the probe pulse drives the QD to either or 1 depending on the phase of the exciton with respect to the probe. The degree of coherence can then be determined by measuring the visibility of the Ramsey fringes as a function of . In our experiment, the relative phase between the pump and probe is unstabilized, and so the final state fluctuates randomly within the visibility envelope.
Figure 5 shows the time-integrated PC signal versus . The red squares show the results for phonon-sideband pumping with . A flat line is observed, with no Ramsey-like fringes, indicating that the phase of the exciton is random relative to the probe pulse. The slow increase of the PC signal in time is most probably related to the decay of the exciton population created by the pump via electron/hole tunneling before the arrival of the probe. By contrast, the black dots show the results measured when the pump pulse is tuned to resonance () and the pulse area is set to . In this fully resonant situation, Ramsey-like fringes are observed provided is shorter than or comparable to the exciton coherence time. The exciton coherence time of can be estimated from the dashed line envelope and is limited by the electron tunneling rate . The incoherent nature of the DVC is clearly demonstrated by the absence of the fluctuations in the PC signal in the red data compared with that in the black data. Our result is consistent with that reported by Weiler et al. , showing that the coherence properties of the emitted photon created via DVC is much worse than resonant CW excitation. However, Bounouar et al.  demonstrate that the photons generated by quasi-resonantly pumping the biexciton state using pulsed excitation show similar coherence properties as measured in the resonant two-photon scheme. The relation of the coherence properties of the emitted photon and exciton prepared via DVC is still unclear and require further investigation.
It should be noted that the fact that the DVC process is incoherent does not imply that the exciton spin is random. The results published in Ref.  clearly show in the phonon-assisted excitation process that the spin of the exciton is the same as that of the photon in the blueshifted laser pulse. In theory, the spin of the emitted photon in the LAPSE process should also be the same as that of the laser photon, since the phonon-assisted excitation and de-excitation are two opposite processes in the dressed state picture [see Fig. 2(a)].
Directly demonstrating the spin-preserving nature of the LAPSE process is beyond the scope of this paper, but it has been observed that the LAPSE process occurs only when the spin of the exciton and the redshifted laser pulse are the same. This spin selectivity of the LAPSE process can be demonstrated by preparing a spin-up exciton using a circularly polarized pulse and then comparing the effects of co- or cross-polarized redshifted pulses. In the text below, we call this redshifted pulse the “LAPSE pulse.” Figure 6 shows the results of such a comparison as a function of the detuning of the LAPSE pulse. The first pulse creates a reference PC level corresponding to an exciton population of 1. This reference level is calibrated by at measured with a weak () LAPSE pulse [see the black line in Fig. 6(a)]. In this spectrum, no phonon-assisted de-excitation occurs since the exciton–phonon coupling in the weak driving regime is negligible. However, in the case that a strong () co-polarized LAPSE pulse is applied, efficient de-excitation occurs, as shown by the negative sideband relative to the reference level at [see the red line in Fig. 6(a)]. By contrast, no negative sideband occurs in the cross-polarized case [see the red line in Fig. 6(b)]. (The peak and weak positive sideband in Fig. 6(b) at around can be attributed to the resonant and phonon-assisted excitation of the biexciton.) The absence of the negative phonon sideband in the cross-polarized differential spectrum unambiguously proves the spin selectivity of the LAPSE process. Based on this observation and the fact that the spin is preserved during the emission of the LA phonon as demonstrated in Ref. , we conclude that DVC is incoherent, but spin-preserving.
5. TEMPERATURE DEPENDENCE OF DYNAMIC VIBRONIC COUPLING
In previous studies, DVC processes have mainly been studied at [14,24,25], the temperature of liquid He. It is interesting, however, to consider the influence of temperature on the mechanism. Increasing the temperature increases the strength of the exciton–phonon interaction, observed, for instance, by stronger damping and frequency renormalization of excitonic Rabi rotations . As discussed in Section 2, this increases the rate of phonon relaxation between the dressed states, and, for relatively weak excitation, increases the resulting exciton population as more population relaxes during the pump pulse. This is shown by the higher population observed at positive detuning for (red triangles) compared to (black circles) in Fig. 7(a) and also by the steeper initial gradient for higher temperatures observed in the calculations of Fig. 7(b).
However, this does not illustrate the complete picture. At higher bath temperatures, the increased number of phonons increases the probability of the competing process at positive detuning, namely, the annihilation of an exciton by absorption of a phonon [see Fig. 2(c)]. This is shown in our measurements by the reduced detuning asymmetry of the phonon sidebands at in Fig. 7(a). The origin of the negative detuning signal is the creation of exciton population by phonon absorption, illustrating the increased probability of the competing absorption processes discussed in Section 2. As such, the reduced asymmetry in the phonon sidebands can be considered equivalent to the reduced difference between the probability of phonon emission and absorption.
The consequence of this competition is that, for driving that is sufficiently strong to allow complete relaxation [i.e., in the plateau region of Fig. 7(b)], the final occupancy of the exciton state is in fact lower at higher temperatures owing to the thermalization of the dressed states. This is illustrated by the reduced maximum exciton population attained at higher bath temperatures in Fig. 7(b). In summary, a higher bath temperature will lead to a faster rise with pulse area, but a lower maximum population transfer by DVC. As such, increasing the bath temperature may provide a means to enhance DVC processes when the driving is relatively weak, as in Fig. 7(a).
6. PICOSECOND TIMESCALE EXCITATION AND DE-EXCITATION
The three first experimental observations of DVC in QDs [14,24,25] focused on the LA-phonon-assisted excitation process [see Figs. 2(a) and 2(b)]. Recently Liu et al. have demonstrated the reverse process—LA-phonon-assisted de-excitation [see Fig. 2]—by observing the reduction of the exciton population of an excited QD by a redshifted laser pulse . A combination of the LA-phonon-assisted excitation and de-excitation processes allows us to create and destroy an exciton within picosecond time scales, enabling ultrafast incoherent optical switching with a single QD much faster than the exciton radiative lifetime (a few nanoseconds [24,41]).
To demonstrate this, a two-color pump–probe experiment is performed with a blueshifted (excitation) and redshifted (de-excitation) pulse with the same pulse area. The pulses are equally detuned () from the exciton. At negative delay times, the excitation pulse creates a certain exciton population corresponding to the background PC level. The de-excitation pulse arrives before excitons are created, and therefore does nothing. When the two pulses overlap, the PC signal decreases to a minimum within 20 ps, as shown in Fig. 8. This occurs due to erasure of the exciton created by the excitation pulse by the de-excitation pulse. The PC signal then slowly recovers due to the electron tunneling out from the QD before the de-excitation pulse arrives . The increase of the PC signal at is most probably related to the phonon-assisted excitation of the biexciton by a very small portion of imperfectly circularly polarized light in the excitation/de-excitation pulses. Figure 8(b) shows the calculated exciton population with parameters similar to those of Fig. 8(a). The phase relation between the two pulses has been assumed to be random, and we show the average of multiple repetitions of the simulation. The numerically complete path-integral method takes into account the electron tunneling in the PC experiment by incorporating it as a Lindblad-type relaxation term . Before the arrival of the pulses, the QD is assumed to be in the ground state and the phonon modes follow a thermal distribution. This model well reproduces the de-excitation of the QD and the subsequent recovery of the PC signal. The oscillation of the exciton population at is not clearly visible in the measured data in Fig. 8(a) due to the limited time resolution of the experiment.
The excitation/de-excitation time in our experiment is determined by the laser pulse duration. The ultimate limit is set by the phonon thermalization time (a few picoseconds ). We note that, in contrast to the optical switching scheme employing resonant coherent excitation , our scheme is robust against the fluctuation of laser power and detuning . Furthermore, the incoherent nature of the phonon-assisted relaxation process determines that no phase locking is needed, in contrast to fully coherent excitation schemes.
7. CONCLUSIONS AND OUTLOOK
In summary, we have investigated the dynamic vibronic coupling in a single InGaAs QD. The DVC enables the population inversion and ultrafast depopulation of a QD by incoherent quasi-resonant excitation with the assistance of phonons [24,26]. We prove that the DVC is an incoherent process by performing Ramsey-like interference measurements. The temperature dependence study shows that, at high temperature, the phonon-assisted absorption induced by both the blueshifted and redshifted laser pulses are enhanced, and the strong asymmetry of the phonon sidebands observed at low temperatures disappears. Furthermore, we demonstrate that it is possible to create and destroy an exciton within a pulse-width limited picosecond time scale by combining a blue-detuned and a red-detuned pulse, opening the door to ultrafast incoherent optical switching with a single QD [15,42,43]. The DVC may also be used to reduce the timing jitter of single/entangled photon sources [8,44] or make tunable single QD lasers .
Compared with resonant coherent excitation, one factor that limits the potential applications of DVC is the high laser power required to achieve an efficient phonon-assisted excitation/de-excitation. This problem may be solved by incorporating the QD into a nanocavity with a small mode volume where the light–matter interaction is strongly enhanced [8,46]. When cavities are involved, the knowledge of phonon interactions in the strong QD–phonon coupling regime becomes an essential requirement for a full understanding of the behavior of optically driven QDs. Various recent studies have shown new physical phenomena that arise from this coupling [47–51].
Engineering and Physical Sciences Research Council (EPSRC) (EP/J007544/1); Deutsche Forschungsgemeinschaft (DFG) (AX 17/7-1).
The authors thank H.Y. Liu and M. Hopkinson for sample growth.
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