## Abstract

Pump-probe quantum state tomography was applied to the transmission of a coherent state through an In(Ga)As based quantum dot optical amplifier during the interaction with an optical pump pulse. The Wigner function and the statistical moments of the field were extracted and used to determine the degree of population inversion and the signal-to-noise ratio in a sub-picosecond time window.

© 2014 Optical Society of America

Light-emitting devices based on the recombination of excitons in semiconductors have become nearly omnipresent in the 60 years since their invention. This material class offers a huge spectrum of design possibilities for optimizing optical and electronic properties. In particular, the approach of restricting the active zone in one or more spatial dimensions to create quantum wells (QWs) or quantum dots (QDs), has led to the development of extremely flexible devices, such as lasers and semiconductor optical amplifiers (SOAs) [1].

Key defining properties of optical amplifiers are the rate of gain recovery and the noise figure. Both noise and gain, and their interplay have been well studied in QD SOAs [2–8]. Recent investigations of the properties of QD SOAs have turned their attention to coherent effects. It has been shown that under typical operating conditions, signatures of coherent light-matter interaction can be observed on a light pulse transmitted through a QD-SOA [9, 10]. The relationship of relative intensity noise to second-order coherence in amplified spontaneous emission (ASE) has also been investigated [11, 12]. These studies mark a renewed interest in the quantum optical properties of semiconductor-based emitters [13]. However, the pioneering techniques for ultrafast quantum noise measurements of second-order coherence and photon-number distributions [14, 15] have not become mainstream. This is despite their potential for resolving ASE and gain recovery dynamics if they were to be combined in a pump-probe setting.

In this contribution, we present a combined pump-probe (PP) and quantum state tomography (QST) technique that delivers a full characterization of the quantum state of the probe field, whose ultrafast dynamics–induced by a pump pulse–can be measured. QST based on balanced heterodyne detection (BHD) reconstructs the Wigner function of the quantum state in a mode that is identical to a local oscillator (LO) pulse, which also functions as a spectral and temporal gate [16, 17]. We have demonstrated PP-QST on a standard QD SOA. It was possible to simultaneously extract information on the gain and the instantaneous noise figure of an optical amplifier, with a sensitivity at the quantum noise limit. Furthermore, we apply general results of optical amplifier theory to infer the degree of population inversion from the Wigner function data. Given that the degree of population inversion determines the signal-to-noise ratio, it is this which sets the ultimate quantum-noise-limit in device performance [18–20].

The device was a standard In(Ga)As based QD-SOA, described in greater detail in [21]. Spectra of amplified spontaneous emission (ASE) are shown in Fig. 1(b), which display two inhomogeneously broadened peaks originating from the QD ground state (GS) at 1280 nm, and the QD excited state (ES) at 1195 nm.

The experiment is based on balanced heterodyne detection (BHD). The probe beam that emerged from the sample (the signal beam) was interfered with an at least ten times brighter reference beam (the LO) on a symmetric beam splitter; see Fig. 1(a). Balancing was achieved using weak optical attenuation in one arm. The common-mode-rejection ratio (CMRR) of intensity noise in the LO was found to be better than 35 dB. Correct operation and adequate CMRR of the heteroyne detector (New Focus model 2117-FS) was confirmed by checking for linearity of the measured quadrature variance (for a vacuum state as the signal) against optical power [22]. Clearance of shot-noise above dark noise was 9.2 dB (variance). The frequency of the LO was shifted before the beam splitter by an acousto-optic modulator, and a slowly-varying time-delay introduced to the LO so that all quadrature angles were adequately sampled. The resulting difference photocurrent *i*_{diff} contained a beat frequency whose amplitude was directly proportional to the quadrature amplitudes of the electric field of the signal beam. This signal was mixed electronically down to zero-frequency using a lock-in amplifier (SRS-SR844), whose two output channels (cosine and sine) were recorded as a time series via a modified computer audio interface (22.05 kHz rate, 100 s duration). The phase of the signal beam relative to the LO was extracted, so as to calibrate the quadrature projection angle, while a histogram of the quadrature fluctuations at that angle was acquired. A numerical inverse Radon transformation using filtered back-projection algorithm was applied to the scaled histogram data, which then yielded the Wigner function of the quantum state of the signal beam. See Fig. 1(c) for an example histogram and Fig. 2(e) for its corresponding Wigner function. To investigate dynamic effects, a pump beam was also injected into the sample, and the measurement procedure repeated for a sequence of pump/probe delay times. All the laser beams were derived from the one master oscillator of the Toptica FemtoFiberPro laser system, which drives two independent nonlinear fiber amplifiers emitting a broad supercontinuum with a repetition rate of 75.4 MHz. The pump and the probe/LO beams were spectrally selected to have a central wavelength (and spectral width) of 1180 nm (45 nm), and 1280 nm (13 nm), respectively. The pulse durations were approximately 150 fs (pump) and 200 fs (probe). The pump and probe optical powers, measured prior to their coupling into the sample, were 500 *μ*W and 5 pW, respectively. The LO was set to 200 *μ*W. Optical losses of all the relevant optical components were characterized, as was the spatial mode matching between the signal and LO beams. The transmission of the probe through the SOA was 0.042 at the 5 mA injection current. The estimated in- and out coupling efficiency was *η*_{in} = 0.05, and *η*_{out} = 0.90 respectively. The total efficiency of the heterodyne detector was 0.12 (which includes the quantum efficiency of the photodiodes 0.88; optical components 0.61; and probe-LO mode-overlap 0.22). The input state was characterized via a bypass with total detection efficiency 0.0069 (including a factor of 0.068 intentional attenuation).

The quantum state of an optical field is fully characterized by its Wigner distribution [17], which is related to the state’s density matrix *ρ̂* via:

*X̂*

^{+}=

*â*

^{†}+

*â*and

*X̂*

^{−}= i(

*â*

^{†}−

*â*), are the amplitude and phase quadrature operators of an electric field

*Ê*(

*t*) =

*X̂*

^{+}cos(

*ωt*) +

*X̂*

^{−}sin(

*ωt*) defined in terms of the annihilation

*â*and creation

*â*

^{†}operators, with the boson commutation relation [

*a*,

*a*

^{†}] = 1. The Wigner distribution will give the marginal probability distribution at an arbitrary projection angle

*θ*using the rotated quadrature operator

*X̂*=

^{θ}*X̂*

^{+}cos

*θ*+

*X̂*

^{−}sin

*θ*such that

*W*(

*X*

^{+},

*X*

^{−}) when provided with a complete set of

*P*. For sparse sampling of

_{θ}*P*along

_{θ}*θ*from 0 to

*π*, a numerical inverse Radon transformation can be employed. The experimental acquisition of

*P*is achieved using BHD, where the signal of interest is interfered with a bright LO field with a phase

_{θ}*θ*relative to the signal, which is assumed to be in a coherent state |

*β*〉 such that

*β*= |

*β*|

*e*

^{iθ}and with

*f*

_{L}the spectral mode function. The measured difference current ${i}_{\text{diff}}\propto \left|\beta \right|\int \text{d}\omega \hspace{0.17em}\hspace{0.17em}{f}_{\text{L}}(\omega ){\widehat{X}}_{\text{sig}}^{\theta}(\omega )$ thus delivers the quadrature amplitude at projection angle

*θ*which is scaled by the mean field of the LO [16]. Note that the quantum and classical noise of the LO do not appear in the expression. The measurement can be normalized to the quadrature standard deviation obtained from a vacuum state (prepared by simply blocking the signal beam). By repetition of the measurement over many ensemble members, the marginal probability distribution

*P*can thus be acquired.

^{θ}Exemplary Wigner functions constructed from the experiment for a driving current of 50 mA are shown in Figs. 2(a)–2(f). The detector/instrument noise sets the limit on resolution in quadrature space (Fig. 2(a)). Using the vacuum state in Fig. 2(b), as a calibration reference, one can read the mean values and standard deviations of the amplitude and phase quadratures in units of vacuum states. ASE emission alone and the input state characterized via the bypass are shown in Figs. 2(c) and 2(d), respectively. The input state is nearly a coherent state, cf. Fig. 2(b). Injecting the probe field from Fig. 2(d) into the SOA generates the amplified state shown in Fig. 2(e). It is dominated by ASE emission, and displaced by the amplified probe field. Note that the Wigner functions have not been corrected for instrument noise, nor for finite detection/coupling efficiency. The influence of the former is small given that a vacuum state was easily resolved, see Fig. 2(a); while the latter markedly affects the size of the amplitude and noise, which will be addressed in the quantitative analysis that follows.

Ultrafast dynamics in the quantum state of light emitted by the QD-SOA was observed using PP QST. The measurement was exclusively sensitive to transitions from the GS of the QD. Evident in the pump delays series of Fig. 2(f), with the pump resonant to the QD-ES, we witness in the Wigner function a clear decrease in amplification of the signal, as well as a decrease in the noise of both quadratures. The qualitative influence of pumping the ES is to deplete both the gain and noise of the amplified probe which is resonant to the GS. We note that the amplitude and phase noise are affected in more-or-less equal proportion.

The quantum noise transfer function of an optical amplifier has been derived in the Heisenberg picture in [20]. The active medium is modeled as an ensemble of two-level systems, with *N*_{1} and *N*_{2} the initial populations of GS and ES, respectively, throughout the amplifier. The Hamiltonian of the system is expressed as *Ĥ*_{int} = i*ħg*∑* _{j}*((|2〉〈1|

_{j})

^{†}

*â*− |1〉〈2|

_{j}â^{†}), where the sum is over all QDs. The coupling constant

*g*is equal to half the vacuum Rabi frequency. The amplifier is divided into

*M*segments of propagation duration

*τ*. Having defined the intensity gain

*G*= [1 + (

*gτ*)

^{2}(

*N*

_{2}−

*N*

_{1})]

*, one obtains the statistical mean*

^{M}*μ*and variance

*σ*

^{2}of the amplitude and phase quadrature operators that exit the optical amplifier at the last segment:

*R*=

*N*

_{2}/(

*N*

_{1}+

*N*

_{2}) can be calculated as:

*R*= 1, and for no inversion

*R*= 0. In an experimental setting using BHD that measures ${\widehat{X}}_{\text{meas}}^{+}$, linear losses due to detection inefficiency, the in/out-coupling to the amplifier, and mode-overlap with the LO must be accounted for with the model: $\mu \left({\widehat{X}}_{\text{meas}}^{+}\right)=\sqrt{\eta}\mu \left({\widehat{X}}_{\text{out}}^{+}\right)$ and ${\sigma}^{2}\left({\widehat{X}}_{\text{meas}}^{+}\right)=\eta {\sigma}^{2}\left({\widehat{X}}_{\text{out}}^{+}\right)+(1-\eta ){\sigma}^{2}\left({\widehat{X}}_{\text{vac}}^{+}\right)$. Hence, the losses are modeled as a beamsplitter of reflectivity

*η*where the mode of interest interferes with an orthogonal mode ${\widehat{X}}_{\text{vac}}^{+}$ that is in a vacuum state |0〉. Given that $\mu \left({\widehat{X}}_{\text{vac}}^{\pm}\right)=0$ and ${\sigma}^{2}\left({\widehat{X}}_{\text{vac}}^{\pm}\right)=1$, and following characterization of

*η*, one can account for the effect of losses and hence infer

*R*experimentally. The input state was thus characterized using the bypass path. The measurement corrected for losses and inferred to a point immediately inside the SOA (i.e. accounting for

*η*

_{in}) was found to be $\mu \left({\widehat{X}}_{\text{in}}^{+}\right)=11.6$ and ${\sigma}^{2}\left({\widehat{X}}_{\text{in}}^{+}\right)=1.01$; which is essentially a coherent state.

The influence of the pump pulse is more clearly expressed when the quadrature mean and variance of the Wigner functions are plotted as functions of PP time delay (black circles in Fig. 3(a)–3(e)). We observe a transient reduction in both amplitude and variance of the signal for a weak GS probe pulse (Figs. 3(a) and 3(b), respectively). Also the ASE quadrature variance alone shows this behavior (Figs. 3(c)), which allows us to conclude that we have observed a real population effect, that is not destructed by the probe pulse. In traditional PP experiments, at these ultrafast timescales, coherent dynamics and population dynamics can merge, and are hard to quantify independently [23]. In our experiment we can separate those effects, which is a clear advantage of the method. Values obtained without a pump pulse are displayed as red crosses in Fig. 3. An apparent time dependence in these data reflects a drift of the excitation/detection system, and thus gives an estimate of the error margins of the experimental accuracy. The amplitude SNR is affected by the pump. The output SNR of 5.7 (5.1 at PP minimum) was less than the input SNR of 11.5, which agrees with the 3 dB penalty of a phase-insensitive amplifier. The SOA device performance was momentarily degraded by the influence of the pump.

Figure 3(d) shows the intensity gain derived from the mean amplitude as shown in Fig. 3(a). The GS gain is reduced by 50% (3 dB) in the interaction of the pump pulse with the ES population. The solid blue line represents a convolution of the instrument response of 270 fs and an impulse response with a recovery time of 350 fs (dashed line). As we are working close to the ES transparency at a current of 50 mA, this very transient gain depletion is expected. The inferred degree of population inversion is depicted in Fig. 3(e). In this calculation, the mean and variance data go into Eq. (5). We see a time-dependence similar to the case of the intensity gain, with a reduction in the population inversion *R* from 64% down to 60%.

In conclusion, we have combined the PP and QST techniques to observe how a QD SOA transforms the quantum state of a probe pulse that is tuned to the QD GS, while optically pumping the QD ES. From the Wigner functions thus obtained, the device gain, the amplified noise, and the excess noise due to ASE could be measured relative to the quantum noise limit and on an ultrafast time scale. This information was used to infer the absolute degree of population inversion in the gain medium, which revealed a depletion and recovery of the population inversion on a sub-picosecond time scale.

## Acknowledgments

The device was fabricated by the group of Dieter Bimberg. This research was funded by Deutsche Forschungsgemeinschaft via Sfb 787, and the GRK 1558.

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