1. Introduction

Instantaneous circularly-polarized light irradiation injects spin-polarized carriers into various materials. Optical injection has been a powerful tool to study spin dynamics in paramagnetic [1–3] and ferromagnetic [4,5] materials, and also their hybrid systems [6,7]. This concept of optical injection is now at the basis of developments in optical spin manipulations in single semiconductor quantum dots (QDs) for quantum information processing [8–13]. Standard measurement techniques to observe photoinduced spin dynamics include transient Kerr and Faraday rotation measurements. In these conventional measurement techniques [1–5], a circularly-polarized pump pulse and a linearly-polarized probe pulse propagating along different directions are focused on the same spot on a sample, as shown in Fig. 1(a). Then, after spatially separating the probe pulse from the pump pulse, its polarization rotation is analyzed via a balanced detection technique, referred to as an optical bridge technique [1]. Therefore, accurate measurements can be realized without obstructs due to the pump pulse. However, this geometric configuration is not applicable to microscopic physical objects, such as single QDs [10,12], spin helixes [14–16], and single magnetic domains [6,7], which require the use of a microscope to achieve high spatial resolution. This in turn requires matching the laser beam diameter with an objective lens, which causes the pump and probe pulses to be collinearly incident on the objective lens, as shown in Fig. 1(b). In such a collinear configuration, the pump pulse generally prevents any accurate measurement of the probe pulse. As a simple way to workaround, a probe pulse with intensity comparable to that of the pump pulse was used in some previous measurements [6,7]. However, the extremely small beam spot resulting from the high resolution requirement usually gives rise to a high optical density for the probe pulse, which disturbs the physical conditions that were excited by the pump pulse. Thus, the probe pulse needs to be kept at a weak optical density, leading to a weak probe intensity, which makes rotation angle measurements more difficult. So far, many attempts have been made to try and prevent undesired obstructions from the pump. A two-color pump-probe technique effectively allowed blocking the pump pulse using an optical filter [14–16]. However, the energy difference between the pumping and probing states generally triggered additional energy relaxation processes that became part of the measured dynamics. As an example of a single-color measurement, a cascaded lock-in technique which modulates both the pump and probe pulses was developed [10,12], which needs a cavity system to enhance the Kerr signal [17] in order to improve the signal-noise ratio. Although a conventional measurement technique, homodyne detection was applied to optically amplify the probe rotation signal [18], creating additional experimental difficulties in actively stabilizing the interferometer. Therefore, a versatile method simultaneously resolving the above problems has not yet been established.

 

Fig. 1. Schematic drawings of (a) a conventional Kerr measurement and (b) a Kerr measurement in combination with micro-spectroscopy. (c) Illustration of the experimental setup for the heterodyne micro Kerr experiment: BS, beam splitter; PBS, polarization beam splitter; OL, microscope objective lens; DL, delay line; AOM, acoustic optical modulator; BPF, band pass filter; HWP, half wave plate; QWP, quarter wave plate; FLI, first lock-in amplifier; SLI, second lock-in amplifier; $\Delta$t, time delay.

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In this paper, we propose the application of a heterodyne detection scheme to transient photoinduced Kerr rotation spectroscopy in combination with micro-spectroscopy. A heterodyne detection scheme has already been applied to four-wave mixing and pump-probe experiments to measure weak signals [19–23]. However, the application to photoinduced Kerr rotation measurements has never been demonstrated. The heterodyne beat note signal generated by the interference of the probe and reference pulses optically amplifies changes in the probe pulse only, which allows an accurate measurement of small Kerr rotation angles with a weak probe intensity, even when the pump and probe pulses are collinear and in resonance with the same quantum state. Here, we apply the heterodyne Kerr rotation technique to observe spin dynamics in a single QD exciton without a cavity, as a test system, showing a rotation angle of few $\mu$rad. Heterodyne detection does not require interferometer stabilization. Therefore, the technique developed here can be easily applied and used to observe spin dynamics of other micro systems, while providing a novel optical readout method of single QD spins for quantum information processing.

2. Experimental methods and sample

The setup for our Kerr rotation experiment using a heterodyne detection scheme is shown in Fig. 1(c). The light source was a picosecond mode-locked Ti:sapphire laser with a repetition rate of 76 MHz. The temporal duration of the laser pulse was $\sim$ 2 ps and the spectral width was $\sim$ 0.5 meV. Two beam splitters (BS1 and BS2) divided the optical pulses into three beams: the pump, the probe, and the reference. The reference pulse has the role of a local oscillator in a heterodyne detection scheme. The polarization of the pump pulse at the sample position was controlled by a quarter wave plate (QWP) to be either in right- ($\sigma ^{+}$) or left- ($\sigma ^{-}$) circular polarization. The probe and reference pulses were linearly polarized along the vertical direction. The pump beam was passed through an optical delay line, and modulated by an optical chopper at a frequency of 1 kHz. The probe and reference pulses were frequency-upshifted by $\delta \omega _{1}$ = 20 MHz and $\delta \omega _{2}$ = 21 MHz, respectively, using acoustic optical modulators (AOMs). Then, the pump and probe pulses were recombined at BS3 and fed into an objective lens (numerical aperture NA = 0.5), which gives a beam spot diameter of $\sim$ 1 $\mu$m on the sample. The probe pulse after reflection at the sample surface was mixed at BS4 with the reference pulse by temporally and spatially overlapping the two pulses. In our experiment, the heterodyne interferometer consists of the optical paths of the probe and reference beams. One of the mixed beams from the interferometer was sent onto a photodetector, which measured a beat note signal generated by the interference of the probe and reference pulses. The measured electrical signal was fed to a band pass filter (BPF), which generates a reference signal at a frequency $\Delta \omega = \delta \omega _{2} - \delta \omega _{1}$ = 1 MHz in order to detect a heterodyne Kerr signal with a lock-in amplifier, which is discussed below. To measure the Kerr rotation angle, we applied a balanced detection (optical bridge) technique using a heterodyne beat note signal. We rotated the polarization direction of the other mixed beam by 45 degrees with a half wave plate (HWP), and then divided it into the horizonal ($H$) and vertical ($V$) polarization components with a polarization beam splitter (PBS). We adjusted the HWP precisely so that the intensities of the divided reference pulses were equal, and the $H$ and $V$ components of the probe pulse had the same intensities without the excitation of the pump pulse. A change in the probe polarization induced by the pump pulse varies the intensities of the divided probe pulses, giving the different electric field amplitudes to the $H$ and $V$ components of the probe pulse, which results in the difference between the beat note signals of the $H$ and $V$ components of the mixed beam. We measure the amplitude difference of the beat notes as a Kerr signal. The intensities of the $H$ and $V$ components of the mixed beam are given by

 

Fig. 2. (a) AFM image (scan size $20 \mu m \times 20 \mu m$) of the sample surface. (b) $\mu$-PL spectrum of a pyramid. (c) Enlargement of the $\mu$-PL spectrum around the pump QD, together with the pump laser spectrum.

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3. Results and discussion

Figure 3(a) and (b) show the Kerr rotation responses at the pyramid. In Fig. 3(a), the laser photon energy is in resonance with the exciton PL line. For Fig. 3(b), we tuned the photon energy to 1.5783 eV, slightly below the exciton PL line, which corresponds to the off-resonant excitation for each PL emission line. Figure 3(c) represents the result of the Kerr measurement at a place without pyramids when the laser photon energy was the same as that in Fig. 3(a). In all the measurements, the pump and probe densities were set to $\sim$ 650 nJ/cm$^{2}$/pulse and $\sim$ 65 nJ/cm$^{2}$/pulse at the sample surface, which corresponds to average powers of $\sim$ 0.4 $\mu$W and $\sim$ 0.04 $\mu$W, respectively. We set the average power of the reference pulse to 200 $\mu$W. The rotation angles in Fig. 2 are calibrated from the SLI output voltages using Fig. 4(c), which is discussed later. In each response, the Kerr rotation angles show a symmetric profile with respect to the $\sigma ^{+}$ or $\sigma ^{-}$ excitation, indicating that the spin-polarized carriers are correctly created in the sample by the circularly-polarized pump pulse. First, we discuss the result measured at the place without pyramids. As shown in Fig. 3(c), the Kerr rotation angle is much larger than for the others, and exhibits an instantaneous response with a decay time constant of $\sim$ 9 ps, as shown in Fig. 4(a). Generally, GaAs QWs with a well width less than several nm show their exciton PL line at an energy position higher than 1.60 eV [25,28], and are transparent around our excitation photon energy of $\sim$ 1.58 eV. Thus, the 3-nm-wide GaAs QW at the place without pyramids in our sample is also transparent, indicating that the measured Kerr rotation response without pyramids represents the spin dynamics in the GaAs substrate. Monitoring the substrate signal, which presents a large Kerr rotation angle and therefore has a large intensity, allowed us to clearly confirm the behavior of the optical amplification of the Kerr signal as a function of the reference intensity, by varying the latter. Figure 3(d) shows the dependence of the SLI output voltage on the square root of the average power of the reference pulse $\sqrt {I_{ref}}$ at zero time delay. The output voltage linearly depends on $\sqrt {I_{ref}}$. In our experiment, the FLI detects the Kerr rotation angle $\propto \theta _{K}$. The SLI measures the photoinduced Kerr signal $\propto \Delta \theta _{K}=\theta _{K}^{with}-\theta _{K}^{without}$, which is proportional to the FLI output. Therefore, the linear dependence in Fig. 3(d) indicates that the FLI output is also proportional to $\sqrt {I_{ref}}$, and that the Kerr signal is exactly amplified by the reference pulse, as in by Eq. (3). When the QD exciton is resonantly excited at the pyramid, the Kerr rotation response decomposes into fast and slow decay components, as shown in Fig. 4(a). Using a biexponential fitting, the fast decay time constant was estimated to be $\tau _{f} \sim$ 7 ps, and the slow component yielded $\tau _{s} \sim$ 80 ps. On the other hand, in the case of the off-resonant excitation at the same place, only the fast component with $\tau _{f} \sim$ 9 ps was observed. The measured fast decay time in the resonant and off-resonant excitations agree well with that of the substrate signal, indicating that the fast Kerr rotation responses at the pyramid arise from the substrate. The rotation angle of the substrate signal component is decreased at the pyramid. Since the pyramid is behaving like a protruding object, the sample surface is not flat. This protruding structure prevents the pump pulse from effectively exciting the substrate, and reduces the reflection light of the probe pulse from the substrate in the direction of the objective lens. As a result, the rotation angle of the substrate signal component is decreased at the pyramid. On the other hand, the slow component in Fig. 3(a) and Fig. 4(a) represents the spin relaxation process of the single QD exciton showing the PL line in Fig. 2, since the slow component was observed only when the resonant excitation was applied to the sample position at the pyramid. We calibrated the photoinduced Kerr rotation angle from the SLI output voltage. As shown in Fig. 4(b), we rearranged the experimental setup, and modulated the mixed beam for the balanced detection, instead of the pump pulse, by placing an optical chopper between BS4 and PBS. The polarization direction of the probe pulse was rotated by introducing a half wave plate before BS4, which is denoted by HWP(ES) in Fig. 4(b). We note that we confirmed the temporal and spatial overlap between the probe and reference pulses after the rearrangement. Then, we measured the dependence of the SLI output voltage on the rotation angle of HWP(ES) $\phi$ within a few degrees, which is shown in Fig. 4(c). The output voltage is proportional to $\phi$. At the measured pyramid, $\Delta \theta _{K}$ =17.4 mrad (1 degree) corresponds to the SLI output voltage of $\sim$ 0.3 V. We also calibrated the rotation angle in Fig. 3(c) by applying the same procedure to the sample place without pyramids. We estimated the Kerr rotation angle of the single exciton to be $\Delta \theta _{K} \sim$ 1 $\mu$rad for a time delay of 50 ps, where the fast rotation response has sufficiently decayed. The estimated Kerr rotation angle agrees well with reported values of Faraday rotation angle in macroscopic measurements for other III-V QD systems without a cavity [29], and the Faraday rotation angle in a single electron system in a single QD [9], supporting that the heterodyne Kerr rotation measurement successfully probes the spin dynamics of the single QD exciton.

 

Fig. 3. Kerr rotation responses at the pyramid for (a) the resonant excitation of the single QD exciton and (b) the off-resonant excitation of the excitons emitting the PL lines in Fig. 2. (c) Result of the Kerr rotation measurement at the place without pyramids. (d) Dependence of the SLI output voltage on $\sqrt {I_{ref}}$, together with the linear fitting.

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Fig. 4. (a) Semilog plots of the Kerr responses at the pyramid at QD exciton resonance and off-resonance, together with the result at a location without pyramids. The off-resonance and no-pyramid data are multiplied by 0.4 and 0.025, respectively. The biexponential fitting curve with $\tau _{f} = 7.0 \pm 0.5$ ps and $\tau _{s} = 80 \pm 5$ ps, and the single exponential fitting curve with $\tau _{f} = 9.1 \pm 0.5$ ps are also plotted. (b) Schematic drawing of the experimental setup for the estimation of the photoinduced Kerr rotation angle: HWP(ES), half wave plate for the estimation. (c) Dependence of the SLI output voltage on the rotation angle of HWP(ES) $\phi$ in (b). (d) Pump-probe response for the single QD exciton, together with the fitting curve with $\tau _{f} = 25 \pm 4$ ps and $\tau _{s} = 165 \pm 10$ ps.

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The developed technique for the Kerr rotation measurements is easily applied to a pump-probe (PP) measurement by shutting one detector for the balanced detection in Fig. 1(c). Figure 4(d) shows the PP response for the same QD exciton in Fig. 2. The PP response shows a biexponential decay curve, the same as our previous study [22]. From a biexponential fitting, the fast decay time was estimated to be $\tau _{f} \sim$ 25 ps, which represents relaxation processes of exciton excited states in other QDs present in the same pyramid [22]. The long component was $\tau _{s} \sim$ 165 ps, corresponding to the lifetime of the resonantly excited QD exciton. These decay time constants agree well with those of our previous study [22]. Finally, we briefly discuss the photoinduced spin dynamics in our QD experiment. The circularly-polarized $\sigma ^{+}$ and $\sigma ^{-}$ pump pulses create angular momentum $J_\textrm {z}$ = 1 and -1 excitons, respectively. Since our QDs are neutral QDs, the Kerr rotation signal solely originates from the population difference between the spin-polarized $J_\textrm {z}$ = 1 and -1 excitons. Thus, the spin relaxation time is limited by the exciton lifetime, which is different from spin relaxation processes in single-charged QDs showing much longer spin relaxation time [9–11,13] . In addition, our previous studies showed that the average energy spacing of the exciton states in our QDs was estimated to be $\sim$ 2 meV [22,26], which is smaller than other neutral QDs [25,30,31], indicating a weak quantum confinement of our QDs. Therefore, our QDs are easily affected by the phonon scatterings, which gives the shorter spin relaxation time than the exciton lifetime.

4. Conclusion

We developed a heterodyne Kerr rotation spectroscopy technique, which can be used simultaneously in combination with micro-spectroscopy. We applied this technique to the measurement of the photoinduced spin dynamics in the single QD exciton. The heterodyne detection distinctly extracted a change in the probe pulse even when the pump and probe pulses were collinear, and the pump pulse was in resonance with the probing state. The strong reference pulse optically amplified the Kerr signal through the interference generating the heterodyne beat note signal, which realized an accurate measurement of the small Kerr rotation angle via the weak probe pulse. We believe that the developed technique will be useful to study microscopic spin systems, and a powerful tool as an optical readout method for spin qubits in quantum information processing. In addition, by scanning the pump or probe pulse, time-resolved photoinduced spin images will be clearly obtained.