1. INTRODUCTION

Semiconductor quantum dots (QDs) are an excellent solid-state platform for opto-electronic, photonic, and quantum information processing devices due to their large oscillator strength and discrete density of states [1]. These attributes manifest as a highly nonlinear optical response that can be leveraged for all-optical switching and modulation with extremely low operating energy [2–8]. Within this material class, self-assembled QDs are well suited for low-power optical communications due to their strong amplitude and phase response spanning nanoseconds (exciton recombination), milliseconds (electron spins), and seconds (nuclear spins) [9].

Characterizing a QD’s optical response function across these timescales is essential for linking the intrinsic charge carrier and exciton dynamics to device performance. Nonlinear optical spectroscopy techniques are particularly effective for accessing these dynamics in single QDs and ensembles; the amplitude and phase of the nonlinear signal reveal details of optical dephasing mechanisms, recombination channels, and multi-particle interactions and correlations [10–16]. The majority of nonlinear spectroscopy techniques explore a material’s optical properties by driving inter-band transitions with a control field and measuring how that control field is converted to a signal field. In turn, this dependence provides a measurement of the material’s transfer function, which contains all of the information of the optical interaction. Looking at this in reverse, if the optical properties can be controlled through material design or the application of external fields, one can engineer the transfer function to produce an ideal optical modulator. Indeed, strong synergy exists between precision spectroscopy and the development of novel optical modulators, where new materials and techniques in the former drive innovation in the latter.

To fully demodulate the amplitude and phase of a QD’s nonlinear optical response, careful attention to the details of the detection method is required. In typical experiments, a “strong” pump field creates an excitation of a dipole-allowed excitonic transition, which is then sensed by a “weak” probe field [Fig. 1(a)]. The pump field is typically modulated and the probe signal is detected at the modulation frequency using a lock-in amplifier. This measurement is sensitive to the modulated transmission of the probe beam but does not provide any phase information, which is a detection method analogous to standard incoherent optical communications. An enhanced technique that is widely employed is to heterodyne the transmitted probe field with a strong local oscillator (LO) and detect the power in the radio-frequency beatnote between the probe and the LO. This measurement provides the time average of the in-phase ($I$) and quadrature ($Q$) components of the signal while suppressing noise from scatter of the pump field [14,17]. Parallels can be drawn here with more advanced coherent optical communication techniques, such as phase, quadrature amplitude, and single-sideband modulation [18]. Advantages of heterodyne pump–probe spectroscopy include shot-noise-limited detection of a weak probe beam, separation of co-linear, co-polarized pump and probe beams, and detection of the amplitude and phase of the transmitted probe. Despite these advantages, measurements of the power in $I$ and $Q$ do not provide any details of their dynamics, which prevents full demodulation of the nonlinear signal unless initial assumptions about the optical response are made. For example, if the system under study exhibits a non-instantaneous response to the pump field, e.g., a non-zero delay $\tau$ in Fig. 1(a), cross talk between $I$ and $Q$ of the lock-in signal prevents complete separation of the amplitude and phase.

 

Fig. 1. (a) Resonant optical excitation of a QD charged exciton transition by a strong pump field. The modulated optical response is transferred to the amplitude $M(t)$ and phase $\varphi (t)$ of a weak probe field that can be delayed by $\tau$ from the modulated pump. (b) The amplitude and phase of the probe signal can be constructed from measurements of its in-phase ($I$) and quadrature ($Q$) components. (c) The magnitude of the amplitude ($\widehat{M}$) and phase ($\widehat{\varphi }$) modulation are related through Kramers–Kronig and are sensitive to the detuning of continuous-wave pump and probe optical frequencies, $\mathrm{\Delta }f=f_{\text{probe}}-f_{\text{pump}}$.

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In this work, we demonstrate that these limitations can be circumvented by recording the dynamics of both quadrature components of a continuous-wave two-color pump–probe signal from InAs/GaAs QDs—enabling complete reconstruction of amplitude and phase modulation of the nonlinear signal. The QDs are embedded in a single-mode ridge waveguide to enhance the nonlinear light–matter interaction, providing amplitude and phase sensitivity at average optical powers approaching the single-photon level. Reconstruction of the optical response for various pump photon numbers and detunings demonstrates that appreciable and controllable amplitude modulation and nonlinear phase shift up to $\sim 0.09\times \pi \mathrm{rad}$. (16°) are achieved at few-picowatt average optical power ($\sim 1\mathrm{aJ}$ energy during the QD lifetime) without requiring an optical cavity that relies on stringent resonance-matching conditions. Demodulation of the heterodyned signal in a fully phase-coherent manner demonstrated here enables sensitive measurements of the QD optical response across multiple timescales, providing a new method for probing exciton and spin dynamics from nanoseconds to milliseconds. In turn, the ability to control the QD’s optical amplitude and phase response may find applications as a highly configurable single element quadrature modulator, which may facilitate novel optical modulation schemes, such as Hilbert transformation for faint photonic communications [19].

2. EXPERIMENTAL TECHNIQUE AND DEVICE STRUCTURE

A. Phase-Sensitive Pump–Probe Spectroscopy

To illustrate the importance of resolving the amplitude and phase of both quadrature components of the modulated optical response, let us consider a pump–probe experiment implemented using an intense pump field resonant with a QD exciton transition, which partially saturates the absorption $\alpha L$ where $L$ is the sample thickness. Modulation of the pump amplitude leads to a differential change $\mathrm{\Delta }\alpha L$ that is imparted onto the transmitted amplitude $M(t)$ of a weak quasi-resonant probe field. Through the Kramers–Kronig relations, $\mathrm{\Delta }\alpha L$ leads to a corresponding modulation of the refractive index $\mathrm{\Delta }n$, which introduces a nonlinear optical phase shift $\varphi (t)$ onto the probe field [Fig. 1(a)]. When mixed with a frequency-shifted LO, the signal is comprised of a beatnote at the carrier frequency given by the probe–LO offset and complex sidebands at the pump modulation frequency that contain all of the signal information.

Assuming an instantaneous optical response to the pump, it is straightforward to demodulate and measure the signal using a standard dual-phase lock-in amplifier. In this case, pure amplitude modulation of the probe appears in the $I$ component of the signal, whereas pure phase modulation primarily appears in $Q$ [Fig. 1(b) when setting the initial phase offset ${\varphi }_0=0$]. In the case of continuous-wave excitation, the magnitude of the amplitude ($\widehat{M}$) and phase ($\widehat{\varphi }$) modulation can be mapped out as the pump ($f_{\text{pump}}$) and probe ($f_{\text{probe}}$) frequency detuning is varied, providing insight into the spectral response of the nonlinear optical properties [Fig. 1(c)]. Measurements of $\widehat{M}$ have revealed details of exciton absorption, dephasing, and relaxation dynamics through coherent population oscillations and spectral hole burning [16]. Similar experiments using pulsed optical excitation have been particularly useful for characterizing the electronic spin dynamics in a single charged QD [20,21] and carrier heating and differential gain dynamics in QD optical amplifiers [22].

The assumption of an instantaneous optical response relies, in general, on whether the exciton and carrier dynamics are faster than the pump modulation frequency. If these timescales are comparable, then amplitude and phase modulation can be present in both components of the lock-in signal, and a simple interpretation is not possible when only measuring the power of the heterodyned signal. In fact, in the case of only pure amplitude modulation, a non-instantaneous optical response can appear as a distortion or delay ${\tau }_M$ of the amplitude response $M(t)$ with respect to the modulated pump waveform, as illustrated in Fig. 1(a). A delay ${\tau }_M$ mixes $I$ and $Q$, resulting in an apparent phase modulation arising from pure amplitude modulation. A similar argument applies for a delay ${\tau }_{\varphi }$ of the phase modulation waveform. Techniques in which the nonlinear signal is demodulated to obtain only the magnitudes $\widehat{M}$ and $\widehat{\varphi }$, such as optical detection using a lock-in amplifier or power measurements using a radio frequency spectrum analyzer, are not sensitive to these delays, limiting their ability to fully separate amplitude and phase modulation contributions. In the following subsections, we discuss how we achieve full reconstruction of $M(t)$ and $\varphi (t)$ using heterodyne-detected hole-burning spectroscopy.

B. Quantum Dot Ridge Waveguide Device

The QD waveguide device examined in this work is illustrated in the schematic diagram and scanning electron microscope image in Fig. 2(a). The 1.5 μm wide ridge waveguide consists of a single layer of InAs/GaAs QDs between two ${\mathrm{Al}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ layers grown using molecular beam epitaxy. A single optical mode is confined to the waveguide, which is patterned using optical lithography and inductively coupled plasma etching. A 50 nm thick silicon nitride layer is deposited for surface passivation and electrical insulation. Gold electrodes separated by 2 μm are patterned on both sides of the waveguide for capacitance-voltage measurements, which indicate that on average each QD is charged with a single hole from unintentional background doping introduced during the growth [15]. The sample is held in a confocal microscope setup for continuous-wave (CW) pump–probe measurements in transmission. The pump and probe are coupled into and collected from the waveguide using 0.55 numerical aperture objective lenses each mounted on a three-axis piezoelectric nanopositioner. The lenses and sample are hermetically sealed with 15 Torr of ultrahigh purity helium exchange gas in a stainless steel insert, which is fixed at 4.2 K inside a liquid helium bath magneto-cryostat. The QD photoluminescence is centered at 1045 nm with an inhomogeneous linewidth of $\sim 50\mathrm{nm}$ due to spatial variation in the QD size and shape [23].

 

Fig. 2. (a) Illustration (left) and false-color scanning electron microscope image (right) of the quantum dot ridge waveguide device. (b) Schematic diagram of the pump–probe experimental setup. The pump and probe derived from two independently tunable continuous-wave laser diodes (LD1 and LD2) are fiber coupled into the QD ridge waveguide held at 4.2 K. The beatnote at the probe/local oscillator difference frequency $f_{\mathrm{pr}}=61\mathrm{MHz}$ is detected using a fast photodiode and radio-frequency spectrum analyzer shown in (c). The pump amplitude is modulated at $f_{\mathrm{MOD}}=5\mathrm{kHz}$, leading to a differential change in the amplitude and phase of the probe that appears as upper (USB) and lower (LSB) sidebands at $+5$ and $-5\mathrm{kHz}$ relative to the 61 MHz beatnote. The probe/local oscillator signal is also demodulated into its in-phase ($I$) and quadrature ($Q$) components, which enables complete reconstruction of the amplitude and phase modulation dynamics.

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C. Experimental Setup

A schematic diagram of the pump–probe experimental setup is depicted in Fig. 2(b). The pump is derived from the first-order diffraction of a continuous-wave laser diode (LD1, $<200\mathrm{kHz}$ linewidth, $\sim 1050\mathrm{nm}$ center wavelength) through an acousto-optical modulator (AOM), which is then coupled into the magneto-cryostat using polarization-maintaining fiber. The amplitude of the pump is modulated with a sinusoidal waveform at $f_{\mathrm{MOD}}=5\mathrm{kHz}$ by controlling the driving power to the AOM. The pump is resonant with the ground state charged-exciton transition of approximately 3–5 QDs. The pump power incident on the waveguide is varied from 50 pW to 1 nW, which we later show is equivalent to approximately 0.5–10 pump photons per QD lifetime in the waveguide. The output from a second laser diode (LD2, $<200\mathrm{kHz}$ linewidth, $\sim 1050\mathrm{nm}$ center wavelength) is split into two beams: a weak probe (2 pW average power) that is fiber coupled into the cryostat, and a local oscillator that is routed around the sample and recombined with the probe for heterodyne detection. The probe frequency is shifted relative to the local oscillator by $f_{\mathrm{pr}}=61\mathrm{MHz}$ using an additional AOM. The beatnote between the transmitted probe and local oscillator is detected using a fast photodiode in a resonant LC circuit, enabling shot-noise-limited detection of the heterodyne signal, $S(t)\propto M(t)\times \cos [2\pi f_{\mathrm{pr}}t+\varphi (t)]$. Through interaction with the QDs, the modulated pump imparts a differential change in the probe amplitude and phase that is fully represented in the photocurrent, $S(t)$. A simple power measurement reveals sidebands on the 61 MHz probe/local oscillator beatnote at $f_{\mathrm{MOD}}=\pm 5\mathrm{kHz}$ in the radio-frequency spectrum shown in the left panel of Fig. 2(c). This measurement indicates how much power is transferred from the carrier to the sidebands, but information on the exact mechanism responsible for this is lost by averaging over the individual quadratures. The power in the sidebands can be measured as the pump–probe detuning is varied, providing details of the charged exciton dephasing and lifetime [23].

The upper and lower sidebands in the power spectrum arise from both amplitude and phase modulation; however, measurements of only the sideband power cannot provide the requisite phase information to uniquely distinguish these contributions, similar to optical detection using a lock-in amplifier discussed previously. Complete reconstruction of the amplitude and phase response of the signal is possible through real-time measurements of each quadrature. Shown in the right panel of Fig. 2(c), the electronic signal, $S(t)$, is equally divided and then mixed with a reference at the probe/local oscillator beatnote frequency $f_{\mathrm{pr}}$, with a 90° phase shift imparted onto one reference arm. The output from each mixer is sampled in real time (20 MHz bandwidth) to provide $I(t)$ and $Q(t)$. These components can be expressed as $I(t)=M(t)\times \cos [\varphi (t)]$ and $Q(t)=M(t)\times \sin [\varphi (t)]$, enabling complete separation and reconstruction of the amplitude and phase dynamics. Fluctuations in the relative phase between the probe and local oscillator are passively stabilized through vibration isolation and an airtight enclosure for the experiment, and phase drift is eliminated by triggering data acquisition on the rising edge of the modulated waveform driving the pump AOM.

3. RESULTS AND DISCUSSION

To demonstrate the limitations associated with power measurements, we first present in Fig. 3(a) the modulation depth of the upper ($P_{\mathrm{USB}}$, $+5\mathrm{kHz}$) and lower ($P_{\mathrm{LSB}}$, $-5\mathrm{kHz}$) sidebands as a function of pump–probe detuning. The modulation depth is determined by comparing the power in the sidebands and the signal–LO beatnote. Maximum modulation occurs at zero detuning, and the half-width at half-maximum of a Lorentzian fit function (solid lines) yields the transform-limited charged exciton homogeneous linewidth that has been previously characterized [23]. The upper and lower sideband lineshapes are symmetric, i.e., their difference $\mathrm{\Delta }{P}_{\mathrm{SB}}=P_{\mathrm{USB}}-P_{\mathrm{LSB}}$ is zero and independent of detuning [dashed red curve in Fig. 3(a)]. When applying a $\mathbf{B}=1.5\mathrm{T}$ magnetic field in the Faraday configuration, the sideband modulation depth increases by nearly a factor of 2 and the lineshapes are asymmetric, leading to a dispersive profile of $\mathrm{\Delta }{P}_{\mathrm{SB}}$ [Fig. 3(b)]; however, since both sidebands can have amplitude and phase modulation components, in principle, the origin of the asymmetry cannot be determined from measurements of the power spectrum alone owing to the lack of phase information.

 

Fig. 3. (a) Modulation depth of the lower and upper sidebands (solid and dashed blue curves, respectively) and their difference (dashed red curve). The sideband lineshapes are fit with a Lorentzian function (solid black curve). (b) Application of a $\mathbf{B}=1.5\mathrm{T}$ external magnetic field in Faraday geometry results in frequency-dependent sideband asymmetry, which is illustrated by the dispersive lineshape of the difference in the sidebands. (c) The signal amplitude modulation dynamics with (blue symbols and curve) and without (red symbols and curve) an applied magnetic field. The pump amplitude modulation waveform is depicted by the dashed curve. (d) The delay ${\tau }_M$ between the probe amplitude modulation and the waveform driving the pump AOM. The difference in the delay with and without the applied magnetic field leads to an average phase shift of $0.10\pm 0.03\mathrm{rad}$. (e) The phase modulation dynamics exhibit similar behavior with an average shift with applied magnetic field of $0.12\pm 0.04\mathrm{rad}$, shown in (f).

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This ambiguity can be circumvented through quadrature demodulation as discussed in the previous section. Real-time measurements of the amplitude and phase are shown near zero pump–probe detuning in Figs. 3(c) and 3(e), respectively. For $\mathbf{B}=0\mathrm{T}$, the waveforms (red symbols) are delayed with respect to the pump (dashed black curve) due to lag between driving the pump AOM and subsequent amplitude modulation imparted onto the pump beam. It is apparent from sinusoidal fits to $M(t)$ and $\varphi (t)$ (red solid lines) that they oscillate in phase, i.e., their relative delay is zero within estimated uncertainty ($2\pi f_{\mathrm{MOD}}[{\tau }_M-{\tau }_{\varphi }]=0.02\pm 0.05\mathrm{rad}$). We find that ${\tau }_M$ and ${\tau }_{\varphi }$ are independent of pump–probe detuning [red symbols in Figs. 3(d) and 3(f)], whereas the magnitude of the amplitude and phase (modulation depths $\widehat{M}$ and $\widehat{\varphi }$) exhibit absorptive and dispersive lineshapes consistent with the Kramers–Kronig relations for absorption and refractive index, shown in Fig. 4(a) (blue and red symbols, respectively). Not surprisingly, increasing the number of pump photons by nearly 1 order of magnitude enhances $\widehat{M}$ and $\widehat{\varphi }$, as shown in Fig. 4(b). The total phase modulation $\mathrm{\Delta }\widehat{\varphi }$ [indicated in Fig. 4(a)] is shown in Fig. 4(d) for various pump photons in the waveguide. The data are well described by a saturation model for a two-level system and are fit with $\mathrm{\Delta }\widehat{\varphi }(N)=\mathrm{\Delta }\widehat{\varphi }(\infty )/(1+N^{-1})$, where $N$ is the photon number normalized to the QD recombination lifetime [24]. The fit provides a straightforward method for estimating the number of pump photons absorbed by each QD, since $\mathrm{\Delta }\widehat{\varphi }(N=1)\equiv \mathrm{\Delta }\widehat{\varphi }(\infty )/2$. For large $N$ approaching 10 ($\sim 1\mathrm{aJ}$ pump energy), we observe an asymptotic value of $\mathrm{\Delta }\widehat{\varphi }=0.09\times \pi \mathrm{rad}$., which is a factor of $\sim 5$ larger compared to single molecules and atoms [25,26] and is the same order of magnitude as the nonlinear phase shift achieved with QD photonic and micropillar cavities [2,3,6,27].

 

Fig. 4. Phase modulation amplitude $\widehat{\varphi }$ (red squares) is shown as a function of pump–probe detuning for $\mathbf{B}=0\mathrm{T}$ and (a) 0.4 pump photons and (b) 2 pump photons, and for (c) 0.4 pump photons and $\mathbf{B}=1.5\mathrm{T}$. Normalized differential absorption lineshapes (blue circles) and Lorentzian fits are shown for reference. (d) Maximum nonlinear phase shift $\mathrm{\Delta }\widehat{\varphi }$, as indicated in (a), as a function of the number of pump photons in the waveguide for $\mathbf{B}=0\mathrm{T}$. The data are fit with a saturation model for a two-level system (solid line).

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Compared to the power measurements of the upper and lower sidebands, the advantages of full amplitude and phase demodulation are apparent when examining the change in dynamics when the external magnetic field is applied. Shown in Figs. 3(c) and 3(e), the applied field induces a lead for both $M(t)$ and $\varphi (t)$ compared to the waveforms for $\mathbf{B}=0\mathrm{T}$. The delays ${\tau }_M$ and ${\tau }_{\varphi }$ relative to the modulated pump are shown in Figs. 3(d) and 3(f) versus pump–probe detuning. Similar to $\mathbf{B}=0\mathrm{T}$, ${\tau }_M$ is independent of the detuning but clearly exhibits a shift $2\pi f_{\mathrm{MOD}}[{\tau }_M(1.5\mathrm{T})-{\tau }_M(0\mathrm{T})]=0.10\pm 0.03\mathrm{rad}$ averaged over all detunings. Similarly, for phase modulation we find that $2\pi f_{\mathrm{MOD}}[{\tau }_{\varphi }(1.5\mathrm{T})-{\tau }_{\varphi }(0\mathrm{T})]=0.12\pm 0.04\mathrm{rad}$. The fact that the ${\tau }_M$ and ${\tau }_{\varphi }$ are equal also for $\mathbf{B}=1.5\mathrm{T}$ implies that no additional modulation mechanisms are introduced by the magnetic field, i.e., the amplitude and phase are related through Kramers–Kronig. Consistent with this observation, the amplitude- and phase-modulation depths both increase by nearly a factor of 2 with applied field [Fig. 4(c)]. We emphasize here that the dependence of the delays ${\tau }_{M,\varphi }$ on the magnetic field are responsible for the sideband asymmetry shown in Fig. 3(b), which is not evident from the power measurements of the sidebands alone.

The millisecond delays ${\tau }_{M,\varphi }$ imparted by the magnetic field suggest that spin dynamics play an important role in the amplitude and phase modulation. Nuclear spins are unlikely to be responsible for such a large shift, since the timescales associated with build-up and decay of spin polarization from the $\sim {10}^5$ nuclei residing in the QD are at least 1 order of magnitude slower than the modulation frequencies used here [28,29]. We therefore speculate that the amplitude and phase modulation delays are associated with electron and hole spin relaxation, which has been studied extensively in this system, including its dependence on magnetic field [9,30]. In the remainder of our analysis, we show that the modeling employed in these previous works, and the spin relaxation rates that were experimentally determined, accurately predict the results we obtain from our quadrature pump–probe technique.

Electron and hole spin relaxation effectively couples the excited and ground states, respectively, of the positively charged exciton transition, illustrated in Fig. 5(a) by the four-level energy diagram under an applied magnetic field. The ground states consist of the spin-up ($\Uparrow$) and spin-down ($\Downarrow$) hole states with a Zeeman energy splitting $E_h={\mu }_B{g}_h\mathbf{B}$, where ${\mu }_B$ is the Bohr magneton, $g_h$ is the hole $g$-factor, and $\mathbf{B}$ is the Faraday magnetic field strength. The dipole-allowed optical transitions connect $\Uparrow \to \Uparrow \Downarrow \uparrow$ and $\Downarrow \to \Uparrow \Downarrow \downarrow$, where the charged exciton states $\Uparrow \Downarrow \uparrow$ and $\Uparrow \Downarrow \downarrow$ are Zeeman split by $E_e={\mu }_B{g}_e\mathbf{B}$, where $g_e$ is the electron $g$-factor. Coupling between the excited and ground states occurs at the electron (${\gamma }_e$) and hole (${\gamma }_h$) spin relaxation rates, respectively.

 

Fig. 5. (a) Energy level diagram for the positively charged exciton with an applied magnetic field in the Faraday geometry. (b) Solutions to the rate equation model for the energy diagram in (a). A schematic illustration of the (c) frequency response function and (d) phase angle of the output signal from the QD acting as a high-pass optical filter. The cut-off frequency ${\gamma }_r$ is determined by the rate at which carriers from $\Uparrow \Downarrow \downarrow$ repopulate the ground state $\Downarrow$ through spin pumping along the indirect relaxation channel.

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Direct spontaneous recombination is the dominant excited state relaxation channel because ${\mathrm{\Gamma }}_{\mathrm{sp}}\gg {\gamma }_e,{\gamma }_h$ (gigahertz versus kilohertz) [9]. The direct transition leads to a differential transmission signal at the pump modulation frequency because the carriers can follow the pump dynamics with $f_{\mathrm{MOD}}\ll {\mathrm{\Gamma }}_{\mathrm{sp}}$; however, because $f_{\mathrm{MOD}}\approx {\gamma }_e,{\gamma }_h$, our measurements are also sensitive to electron and hole spin relaxation dynamics. An indirect transition initiated by coupling between $\Uparrow \Downarrow \downarrow \leftrightarrow \Uparrow \Downarrow \uparrow$ via an electron spin flip adds an additional component to the nonlinear signal through optical spin pumping with a characteristic rate ${\gamma }_r$ determined by the time required to repopulate the depleted ground state. For $f_{\mathrm{MOD}}\gg {\gamma }_r$, the electron spin in the excited state is effectively “frozen” and the ground state remains depleted; thus the probe is sensitive only to the direct spontaneous recombination dynamics. For $f_{\mathrm{MOD}}\ll {\gamma }_r$, the system can respond fast enough to repopulate the ground spin state through the indirect channel, leading to subsequent absorption of the probe. In effect, the charged exciton system acts as a first-order high-pass optical filter with a cut-off frequency given by the indirect channel relaxation rate ${\gamma }_r$. A qualitative illustration of a high-pass filter frequency response function and phase angle imparted onto the probe is shown in Figs. 5(c) and 5(d). For $f_{\mathrm{MOD}}>{\gamma }_r$, the QD optical filter imparts a small positive phase shift $\vartheta ={\tan }^{-1}({\gamma }_r/f_{\mathrm{MOD}})$ on the modulation response. From our measurements of an average $\vartheta =2\pi f_{\mathrm{MOD}}{\tau }_{M,\varphi }\approx 0.11\mathrm{rad}$ discussed previously and with $f_{\mathrm{MOD}}=5\mathrm{kHz}$, we find that the filter cut-off frequency is ${\gamma }_r=0.55\mathrm{kHz}$, corresponding to a response time ${\tau }_r=1.8\mathrm{ms}$. At zero magnetic field, optical spin pumping and re-pumping of the degenerate $\Uparrow \to \Uparrow \Downarrow \uparrow$ and $\Downarrow \to \Uparrow \Downarrow \downarrow$ transitions effectively eliminates the filtering behavior of the charged QD [15,31].

The measured QD filter cut-off frequency is similar to the electron spin relaxation rate ${\gamma }_e\approx 1\mathrm{kHz}$ previously obtained using magneto-optical spectroscopy [30], suggesting that the filter response is governed by electron spin relaxation. To show this, we solve a system of coupled rate equations describing the charged exciton system in Fig. 5(a). Because the Zeeman splittings $E_h$ and $E_e$ are larger than the transition homogeneous linewidth [32], for simplicity we consider the case in which our pump is resonant only with the higher energy transition $\Downarrow \to \Uparrow \Downarrow \downarrow$, but similar results are obtained for QDs in which our pump is resonant with the $\Uparrow \to \Uparrow \Downarrow \uparrow$ transition. After optically pumping $\Downarrow \to \Uparrow \Downarrow \downarrow$, the charged exciton can relax directly via spontaneous recombination and indirectly via optical spin pumping. To isolate the latter, we solve the system of equations for the indirect channel $\Uparrow \Downarrow \downarrow \to \Uparrow \Downarrow \uparrow \to \Uparrow \to \Downarrow$ using the following expressions:

Equation (1) is solved for the population dynamics of each state, which are shown in Fig. 5(b). We use ${\gamma }_h=4\mathrm{kHz}$ taken from [33] for $\mathbf{B}=1.5\mathrm{T}$. We allow ${\gamma }_e$ to vary as a free parameter to fit ${\gamma }_r$, which is defined as the rate at which the $\Downarrow$ ground state is repopulated through optical spin pumping, to the measured QD filter cut-off frequency discussed previously. We set the rates ${\gamma }_{e/h}^{\prime }={\gamma }_{e/h}\times \exp (-E_{e/h}/k_{b}T)$, where $k_b$ is the Boltzmann constant and $T=4.2\mathrm{K}$ is the sample temperature. We use electron and hole $g$-factors of $g_e=0.3$ and $g_h=4$ for the excited and ground state splitting [33], although the exact values used here have minimal impact due to the low sample temperature used in this work, i.e., the reverse spin flip rates ${\gamma }_{e/h}^{\prime }$ do not significantly affect the system dynamics. We find that the spin pumping rate ${\gamma }_r$ is equivalent to the measured cut-off frequency for an electron spin relaxation rate ${\gamma }_e=0.57\mathrm{kHz}$ (lifetime ${\tau }_e=1.74\mathrm{ms}$). This value is consistent with previous measurements of the spin lifetime at $\mathbf{B}=1.5\mathrm{T}$ [29] and is also similar to the cut-off frequency, which supports our assertion that electron spin relaxation governs the charged exciton optical response at kilohertz frequencies.

4. CONCLUSION AND OUTLOOK

We have implemented a phase-sensitive heterodyne-detection scheme for amplitude and phase measurements of the spectral hole-burning response of self-assembled InAs QDs. This technique enables complete reconstruction of the modulation dynamics of the nonlinear optical signal, providing shot-noise-limited detection sensitivity across multiple timescales from nanoseconds to milliseconds. We observe an ultra-narrow spectral hole in the QD absorption profile and a corresponding dispersive lineshape of the phase response, which reflect the nanosecond optical coherence time of charged excitons in the QDs. Simultaneously, we measure electron spin dynamics on a millisecond timescale, as this manifests as a magnetic-field-dependent phase offset of the amplitude and phase modulation waveforms. We emphasize that while these effects appear as frequency-dependent modulation of the sideband power in the radio-frequency spectrum of the signal, power measurements cannot provide the requisite phase information to distinguish between modulation of the amplitude and phase—an inherent advantage of time-resolved heterodyne detection.

The phase-sensitive measurements discussed here also present new opportunities for quadrature modulation at faint photon levels. We have shown that the modulated amplitude and phase dynamics can be precisely controlled by tuning several independent parameters, including the pump power, optical frequency detuning, pump modulation frequency, and through the application of an external magnetic field to manipulate the electronic wavefunctions and energy levels. In principle, the ridge waveguide design enables broadband control of the phase and amplitude limited only by the QD inhomogeneous linewidth, since the pump sets the resonance frequency. Phase shifts approaching $\pi$ might be achieved by increasing the density and number of QD layers in the waveguide, but at the expense of a reduction in the efficiency [5]. Larger phase shifts at lower optical power may be possible by optimizing the optical mode overlap with the QDs and by tailoring the electronic wavefunctions using external magnetic and electric fields [23,34].