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Abstract
Quantum spectroscopy in solids directly detects nonlinear changes created exclusively by quantum fluctuations of light. So far, it has been realized only by projecting a large set of measurements with a coherent-state laser to a specific quantum-light response. We present two complementary experimental approaches to realize intense and ultrafast thermal-state sources. We investigate the effects of continuous excitation from a superluminescent diode (SLD) as well as an ensemble-averaging technique using phase-modulated pulses. By measuring excitonic nonlinearities in gallium arsenide, we demonstrate that the experimentally realized thermal-state source produces significantly reduced many-body nonlinearities compared to a coherent-state excitation. We also review experimental approaches toward future realization of quantum spectroscopy with thermal states.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical and electronic properties of solids are determined by quasiparticles that are electron–hole clusters created by many-body interactions. Therefore, such complex systems should optimally be studied and designed based on accessing and controlling individual quasiparticles to assign and realize specific quantum effects. Quantum (optical) spectroscopy for semiconductors was introduced in Ref. [1] outlining how quantum-optical photon correlations of an ultrafast laser pulse directly excite equivalent electron–hole clusters/correlations in a semiconductor. This idea was functionalized in Ref. [2] where it was demonstrated that a laser pulse in a Schrödinger-cat state directly accessed the biexciton, a molecule of two electrons and holes, with much higher precision than classical light sources could. This capability was utilized in Ref. [3] to discover a new quasiparticle, a dropleton, consisting of more than four electron–hole pairs confined inside a small droplet.
In extended solids, quantum-spectroscopic sensitivity arises only when optical pulses create substantial nonlinearities, much beyond what few photons can induce. Unfortunately, flexible quantum-light sources do not yet exist in this regime such that semiconductor quantum-spectroscopy successes have so far been realized using a cluster-expansion transformation approach [2–4], where a quantum-light response was constructed from a large set of coherent-state light measurements.
To predict how the quantum statistics of a light source would affect the excited nonlinearities in a semiconductor, we explore the first steps toward realizing semiconductor quantum spectroscopy with actual light sources, not relying on projection. Since it is not known how a thermal light source would excite nonlinearities, we follow the most straightforward comparison of using ultrafast pulses with either a coherent- or thermal-state quantum statistics. Despite both belonging to the class of semiclassical sources, computations [1] predict that thermal excitations yield a strong reduction of scattering-induced line broadening, which is particularly visible at elevated excitation levels. As major benefits of our approach, ultrafast thermal-state pulses are much simpler to create than equivalent-intensity true quantum-light sources, and using them allows experimental exploration of simple quantum-statistics induced changes in solids.
We consider two independent ways of realizing strong-intensity, thermal state sources with which to demonstrate sensitivity of a system response to quantum statistics of the exciting light source. In one approach, we detect differential absorption of a GaAs heavy-hole exciton using continuous wave excitation from both a source having coherent statistics and having thermal statistics. Additionally, we measure the transient absorption for an ensemble of phase-modulated pulses which have the statistical properties of thermal light.
2. Quantum aspects of light in spectroscopy
Light quantization and its consequences to light–matter interactions have been discussed in multiple books, and we only briefly summarize the key aspects based on Ref. [5]. Any transversal field $\hat {E}(\textbf {r},t)$, at position $\textbf {r}$ and time $t$ can be expressed using a mode expansion
For a single-mode light source, only one $\textbf {q}$ mode is needed in the analysis. Often field properties are relevant at length scales where $\textbf {u}_{\textbf {q}}(\textbf {r})$ is constant while a specific light-polarization direction is selected for detection. With these simplifications, we only need to analyze a scalar field $\hat { E}(0,t)$ at a reference point $\textbf {r}=0$, yielding
where ${\mathcal E} \equiv {\mathcal E}_\textbf {q} \textbf {u}_\textbf {q} \cdot \textbf {e}_\textrm {det}$ combines the vacuum-field amplitude, studied single-mode function, and its scalar component along the direction of detection set by an unit vector $\textbf {e}_\textrm {det}$. In this form, the boson operators are expressed in the Heisenberg and are dynamic. Formally speaking, the selected single mode can be either spatial or temporal modes. For the latter case, also ${\mathcal E}(t)$ can be conceived as dynamics as discussed, e.g., in Ref. [5].2.1 First- and second-order coherence
Optical coherences of quantized light were originally quantified by Glauber [6]. In this formulation, the first-order correlation function is
The $\hat {E}^{(\pm )}(t)$ represent scalar projections of a multi-mode field in direction $\textbf {e}_\textrm {det}$ whereas the second form applies for single-mode fields. This function is normalized to produce $g^{(1)}(\tau =0) = 1$ for vanishing delay $\tau$ between the measured fields. All factorizable fields, i.e. $\langle \hat {E}^{(-)}(t) \, \hat {E}^{(+)}(t-\tau ) \rangle = \langle \hat {E}^{(-)}(t) \rangle \, \langle \hat {E}^{(+)}(t-\tau ) \rangle$ yield $g^{(1)}(\tau ) = 1$, indicating first-order optical coherence. In this context, quantum-mechanical expectation values are defined as $\langle \hat {O} \rangle \equiv \textrm {Tr}\left [ \hat {O} \hat {\rho } \right ]$ for any operator $\hat {O}$ and quantum state of light, defined by a density matrix $\hat {\rho }$. For fields that do not possess first-order optical coherence, $g^{(1)}(\tau )$ decays with increasing $|\tau |$, and its decay constant defines the coherence time $\tau _c$.
The second-order correlation function,
The second-order correlation function can be measured by splitting a beam into two paths, measuring coincidence counts between the two beams on two separate detectors, and delaying the optical path length of one beam with respect to the other. An interferometer based on the Hanbury Brown-Twiss design [7] is commonly used to measure single photon sources (characterized as having a $g^{(2)}(0) = 0$) correlates the measurement time of single photons on avalanche photodiode detectors (or photomultiplier tubes) in each arm [8,9]. Thermal light emitted from narrow resonances of a gas discharge tube has also been measured with photomultiplier tubes [10] because the resonances have very long coherence times.
The thermal sources and sample resonances we study here have coherence times that are less than 1 ps, requiring significantly higher timing resolution than can be offered by a traditional intensity interferometer. We therefore use balanced homodyne detection, which has been described in detail in Refs. [11,12]. Balanced homodyne detection utilizes an ultrafast local oscillator (LO) pulse to measure the temporal correlation of noise with a timing resolution given by the duration of the LO.
Correlation functions (4) and(5) can also be defined with respect to space as well by replacing time arguments by spatial positions. While the coherence time is related to the light source’s spectral bandwidth, the spatial coherence is related to the light source’s momentum distribution. Since the momentum distribution is limited by the momentum of a photon, the emission from two points separated by less than half the light source’s wavelength cannot be separated. A single mode light source having a very narrow momentum distribution then has a large spatial resolution given by $\Delta x = 1/2\Delta k$.
2.2 Relevant single- and multi-mode quantum-light states
An ideal laser source outputs a single-mode light whose quantum properties are defined by a coherent state $| \beta \rangle$. Mathematically, a coherent state is as an eigenstate of the annihilation operator [6]:
where the eigenvalue $\beta$ is a complex number, an amplitude of the field. A coherent state exhibits coherence, i.e. $g^{(1)} = g^{(2)}= \cdots = 1$, to all orders. The quantized light mode has a Hamiltonian, $\hat {H}_\textbf {q} \equiv \hbar \omega \left ( B^\dagger _\textbf {q} B_\textbf {q} + \frac {1}{2} \right )$, and its eigenstates,Thermally emitted light is distributed in terms of both light modes and Fock states. It can be assigned by a thermodynamic density matrix $\hat {\rho }_\textrm {thermal} \equiv \frac {1}{Z} \textrm {exp}\left (- \frac {1}{k_B T} \sum _\textbf {q} \hat {H}_\textbf {q} \right )$, where $T$ assigns the temperature and the partition function $Z$ normalizes the distribution. In a thermal state, all modes can be separated, such that a single-mode thermal state becomes [13]:
A thermal state can also be expressed in terms of coherent states using a relation
2.3 Coherent vs. thermal states
Many aspects of quantized fields can be understood by analyzing their photon number distribution $P_n \equiv \textrm {Tr}[\hat {\rho }\, \left | n \right \rangle \left \langle n \right |]$ which produces expression (9) for a thermal state and a Poisson distribution
for a coherent state with $\bar {n} =\langle B^\dagger B \rangle$ photons on average. Thermal and coherent-state photon distributions are compared in Fig. 1 for the same average number of photons $\bar {n}$. While the coherent photon distribution peaks near $\bar {n}$, the thermal distribution is highest at $n=0$, and much broader. This correctly implies thermal light has much larger photon-number fluctuation than a coherent state with a matching mean photon number.
Fig. 1. Comparison of photon number probability distributions for coherent and thermal light sources, both with $\bar {n} = 10$.
This comparison becomes particularly clear when analyzing normally ordered expectation values to $J$th order, $\langle \left [ B^\dagger \right ]^J \, B^J \rangle$. A coherent state produces
By defining $\hat {n} \equiv B^\dagger B$ as the photon number operators, one can determine the photon number fluctuations
3. Quantum spectroscopy with continuous thermal excitations
Coulomb bound electron–hole pairs produce strong absorption resonances below the bandgap of a direct-gap semiconductors [14]. These resonances change greatly as function of optical excitation level, which is controlled by the varying the intensity of an optical pump pulse. As a typical trend, a sharp exciton resonance becomes gradually broader for elevated densities [15] due to excitation-induced dephasing [16–18]. Eventually, the exciton resonance saturates completely [19–22] because Pauli blocking prevents excitonic binding at too high electron–hole densities, yielding excitonic Mott transition. In typical GaAs quantum wells (QWs), the Mott density [23,24] is roughly $10^{11}$ excitons/cm$^2$ for resonant optical excitations.
These excitation-induced effects have been predicted to be substantially weaker for thermal-state than coherent-state excitations [25], which we will use as a basis to develop and verify thermal-state sources for quantum spectroscopy for semiconductors. We use a 10-QW GaAs sample such that the thermal-state source should have a photon density of at least $10^{12}$ photons/cm$^2$ to reach the Mott density. To produce sufficient light absorption, this source should match the exciton absorption linewidth of approximately 1 meV at a photon energy of 1.55 eV (800 nm), corresponding to the $1s$ exciton energy, $E_{1s}$. In practice, the photon density must be about two orders of magnitude higher – approximately $10^{14}$ photons/cm$^2$ because of reflection losses at dielectric interfaces. In an effort to achieve a source that satisfies these parameters, we explore several experimental approaches to create incoherent photon statistics for quantum spectroscopy.
An incandescent light bulb is a simple device that produces thermal, multi-mode light. The bulb, like the sun, emits blackbody radiation which has frequency dependence of the spectral radiance of a black body object at temperature $T$ according to Planck’s law [26]:
Even though there are $\sim 10^{11}$ photons/cm$^2$ arriving in a second from a blackbody source, the photon density within a time window of the lifetime of semiconductor excitation (about $1$ ns) is much smaller than the density required for creating nonlinearities in the QW system. The maximum photon density in a 1 meV bandwidth emitted from a hot blackbody such as a halogen bulb (100 W, 5000 K) within 1 ns is less than 100 photons/cm$^2$, which is more than 10 orders of magnitude below the required photon density for generating sufficient excitonic nonlineatites. To achieve photon densities of $10^{12}$ photons/cm$^2$ in a 1 meV bandwidth about $E_{1s}=1.55$ eV and within a 1 ns time window would require temperatures of over $10^{14}$ K, which is unreasonable for practical applications. While thermal events generated from electron-positron or hadron colliders (100 GeV of thermal energy corresponds to $10^{15}$ K) may produce the intense thermal light necessary for generating many-body configurations in GaAs QWs. However, since imaging and filtering light from a collider appears experimentally unfeasible, we consider other potential sources of bright thermal light needed for directly observing the effects of a high-intensity thermal source in a solid-state system.
3.1 Superluminescent diode
Superluminescent diodes (SLD) are $p$–$n$ diodes that emit light through spontaneous electron–hole recombination and generate amplified spontaneous emission over a wide range of wavelengths [27]. The front facet of our SLDs are anti-reflection coated and the back facet is cut at an angle to prevent any feedback to the gain medium. Both aspects eliminate onset to lasing by suppressing stimulated emission. As a main advantage, such a SLD can emit at a narrower bandwidth than a blackbody source, leading to brighter emission within a small frequency window.
Specifically, we use a superluminescent diode (SLD-381-HP2 produced by Superlum), which can emit $\sim$20 mW of amplified spontaneous emission with 40 meV bandwidth out of a single-mode fiber. The spectrum emitted from the diode is a smooth Gaussian with no spectral spikes that would indicate partial lasing. The single-mode fiber-coupled output ensures that the emission has a single spatial mode. Since the diode emits an excess of optical power, we are able to filter the spectrum with a grating based spectral filter, which is very stable and tunable. As a coherent source, we use an external-cavity diode laser. To compare thermal vs. coherent-state sources, we couple them into a single-mode fiber so that exchanging light sources is straight forward, which also ensures that the single-spatial modes of the two sources are identical.
We study a sample containing ten 10 nm GaAs QWs separated by 10-nm thick Al$0.3$Ga$0.7$As barriers. Cooled to 4 K in a cryostat, the QW heavy-hole 1s exciton resonance at $E_{1s}= 1547$ meV is well separated from all other absorption resonances. Figure 2(a) presents thermal (red area), coherent (blue area), and excitonic probe transmission spectrum (gray area). We use the transmitted probe spectrum to compare the response of the GaAs QWs to different light statistics. The probe pulse is spectrally filtered above 1549 meV to not excite the light-hole exciton or heavy-hole continuum states. The absorption dip in the spectrum results from a heavy-hole exciton at 1547 meV. The coherent and thermal light sources are tuned to the heavy-hole 1s exciton resonance wavelength at low excitation density, and the thermal source is spectrally filtered to match the exciton lineshape.
Fig. 2. (a) Probe transmission spectrum through the GaAs sample (gray shaded region) along with the thermal excitation (red region) and the coherent excitation source (blue region). The absorption dip at 1547 meV corresponds to the 1s exciton of GaAs. (b) Nonlinear signal is generated by amplitude modulated pump beam and frequency shifted probe beam. Amplitude modulation can be performed with a mechanical chopper and frequency shifting is performed with acousto-optic modulators (AOMs). A local oscillator (LO) beam, which is frequency shifted by a different frequency than the probe beam, interferes with the signal on a detector. Since the modulation on the detector corresponding to the interference between the differential absorption signal of interest and the LO is unique, we can isolate the signal with a lock-in detector tuned to that modulation frequency.
Both the coherent- and thermal-state sources are continuous while the coherent source has a much narrower linewidth than the exciton, and its measured linewidth here is actually spectrometer resolution limited. The thermal-state source has a 1 meV linewidth and a corresponding $\tau _c = 4$ ps coherence time in a $g^{(2)}$ measurement. In other words, the $g^{(2)}$ of the thermal light state decays from thermal $g^{(2)}=2$ to coherent $g^{(2)}=1$ much faster than the duration of absorption, which is determined by the excitation lifetime. Even though this coherent–thermal pair is not temporally identical, it is useful to check whether few-ps-long thermal features are sufficient to generate quantum-statistical differences in a response when the sources are made strong enough.
3.2 Heterodyne detection of pump nonlinearities
We compare the excitations induced by coherent and thermal light sources by measuring the spectrally-resolved differential absorption of the probe spectrum due to a pump excitation. The pump is either a continuous coherent-state or a continuous thermal-state source focused to a diffraction limited 700 nm spot to maximize the intensity at the sample of the light sources. Using diffraction-limited beams forces us to use a collinear pump–probe beam geometry.
Since all the beams are spectrally and spatially overlapping, we need some way of distinguishing the nonlinear differential absorption signal from all the other sources of light that impinge the detector. In order to measure the nonlinear optical response described by the field $E_{\textrm {NL}}$, we heterodyne detect the field with an external local oscillator (LO) pulse that we route around the sample, shown in Fig. 2(b). Detection using a LO serves two purposes. 1) By setting the timing of the experiment, the LO guarantees that the measured signal results from exciton pumping by the continuous pump beam and probing by the coherent pulse. If, alternatively, the pump and probe were intensity modulated, one would also be sensitive to a change in the pump absorption after the probe pulse excites the sample due to the probe effectively pumping the system. 2) The integrated intensity of the continuous pump is high, and because the pump is spectrally resonant with the transition of interest, spectrally-resolved intensity measurements are noisy. By using a very bright local oscillator to extract the signal field in the time domain, we are able to optically amplify the probe field without amplifying the shot noise due to the pump.
To isolate the resonant signal, we shift the frequencies of a probe beam and the LO, which is coherent with the probe, so that the interference between them is amplitude modulated at a frequency that is unique to that interference term. The frequency shifts applied to the probe and LO with acousto-optic modulators are $\omega _\textrm {probe}$ and $\omega _\textrm {LO}$, respectively. The pump beams, neither of which are coherent with the probe and LO, are amplitude modulated by a square chop with frequency $\omega _{\textrm {pump}}$. The fields of the various beams are:
We measure the interference between the nonlinear signal and the LO as a function of the delay between the probe and LO pulses. This time-domain measurement and the optical spectrum of the interference are related by a Fourier transform, but it is necessary to measure the signal phase in order to go between these domains. To measure the phase of the signal, we generate the probe and LO pulses in an interferometer and use the synchronous detection techniques detailed in [28–30]. We acquire spectra of the nonlinear signal by Fourier transforming the measured time-domain signal.
3.3 Continuous thermal vs. coherent-source spectroscopy
Figure 3 compares differential absorption spectra excited with 80 µW pump beams having coherent and thermal statistics, each with roughly $6 \times 10^{12}$ photons/cm$^2$ within 100 ps. We have also measured nonlinear spectra for three times higher photon densities, which nearly saturates the exciton, indicating that the measured nonlinearities are significant. We observe that these coherent vs. thermal state light sources yield very similar nonlinear responses for measurements at both 7 K and 65 K. In this specific case, the relative quantum-statistical insensitivity is not completely unexpected because the thermal state is temporally different, exhibiting thermal character only for 4 ps, as discussed in Sec. 3.1.
Fig. 3. Real part of spectrally resolved nonlinear signal. Here we compare nonlinear signals resulting from thermal (red) and coherent (blue) excitation. We show minimal difference in the induced signal for both low (left) and high (right) temperatures.
To assess the expected single-mode thermal state response when its temporal ($g^{(2)}(t) = 2$ for all times) and spectral characteristics are identical to a coherent state, we use the method of projection [2,3] from a set of coherent state responses, $R_{\textrm {coh}}$, onto that of a thermal state
Figure 4 presents the time-integrated nonlinear signals resulting from thermal- and coherent-state excitations as well as the projected thermal-state response. These results indeed show that the used temporally multi-mode thermal source produces greater nonlinearities than the projected single-mode thermal source. We assign this difference to thermal character that exists in the used multi-mode source much shorter than its duration.
Fig. 4. Time-integrated nonlinear response as a function of pump power for a thermal excitation source (red and maroon represent two separate measurements) and a coherent excitation source (blue). These are also compared to a projected thermal response (black line) calculated using the measured coherent response. These measurements are shown for low temperature (left) and high temperature (right).
4. Ensemble averaging for phase-modulated pulses
Since careful experimentation with a continuous (and narrow-band) thermal source does not excite a measurably distinct many-body configuration from a coherent source, we seek to excite the sample with a pulsed thermal source. If we could compare excitation using a broadband pulsed thermal source and a pulsed coherent source, we could identically reproduce the conditions required to see pure quantum statistics effects on quasiparticles. Since creating a pulsed thermal source is experimentally challenging, in this section we develop theory and experimental means of shaping an ensemble of coherent pulses to be quasi-thermal.
4.1 Density-Matrix Ensemble (DME) Averaging
Born’s rule defines the functional principles how measurements must be perceived according to quantum mechanics. Assuming that a quantity $q$ is measured, a single elementary detection event produces only one outcome that follows the probability distribution $P(q)$ of $q$. If quantity $q$ is measured under identical conditions for $N$ times, the quantum-mechanical expectation value is directly connected to the measurement via
As a possibility to perform a new class of measurements, we lift the requirement that the repeated measurements have to be performed under identical initial conditions. We basically want to allow for measurements where the initial conditions, i.e. the light excitations, are changed in a controlled manner from pulse to pulse. The intention is to introduce a well-defined ensemble of classical excitation pulses such that we obtain new schemes to measure optical response to a well-defined, single-mode thermal-state pump. For this purpose, we generalize Eq. (19) by defining an ensemble-averaged quantity
Each of the $\langle q \rangle ^\textrm {ens}_r$ is defined by their own quantum distribution $P^{\textrm {ens}}_r(q)$ and connection $\langle q \rangle ^\textrm {ens}_r = \int dq \; q \; P^\textrm {ens}_r(q)$, based on Eq. (19). Once used in Eq. (20), we find that the quantum theory predicts the ensemble average via
The introduced ensemble averaging procedures can equivalently be presented via ensemble-averaged quantum statistics. In particular, Eqs. (20)–(22) have a one-to-one correspondence to the ensemble-averaged density matrix
4.2 DME-averaging algorithm for thermal state
The proposed DME-averaging procedure can rather easily be implemented in current experiments if the measurement is already performed via repetition. This is the typical scenario in most of the optical experiments since one needs to apply repeated measurements anyway to determine the desired quantity via the elementary detection processes. Instead of using identical excitation pulses in the repetition sequence, one now alters the each pulse in a controlled manner in order to generate the desired $\hat {\rho }_\textrm {eff}$ via DME averaging.
To realize $\hat {\rho }_\textrm {eff}$ of a thermal state, we chose a combination of coherent-state laser pulses that are suitably phase-randomized. To accomplish phase randomization, we prepare a collection of phase masks to be applied to a spatial light modulator (SLM) in a pulse-shaper device [31]. The phase function $\Delta \phi (\omega )$ for each mask to be applied for each realization is divided into discrete spectral bins of size $\Delta \omega _\textrm {bin}$. The phase of the field within each bin is chosen randomly (using a pseudo-random number generator) in the interval $0<\Delta \phi (\omega )<2 \pi$. This procedure randomizes the spectral phase of successive mode-locked pulses from a Ti:sapphire laser. The concept of phase randomization is illustrated in Fig. 5(a). The light from a pulse of a mode-locked laser travels through a pulse-shaper device that is able to manipulate the spectral phase of the pulse.
Fig. 5. (a) A spectrum from a mode-locked ti:sapphire laser with a random spectral phase function with spectral bin sizes of $\Delta \omega _\textrm {bin} =200 \,\mu$eV. There are $N_\textrm {bins}=30$ spectral bins across this spectrum. (b) Cross-correlation traces for six different scrambled realizations. All traces are normalized so that the area is unity. For comparison, the normalized transform-limited pulse (flat spectral phase mask) is shown as a dashed red line.
Classically, the unscrambled light pulse is described by a spectral amplitude $E^0(\omega )$ that is binned into $N^\textrm {bin}$ spectral regions $E^0_j(\omega )$, labeled by $j$. When a random phase $\Delta \phi _j(r)$ is assigned to to each spectral bin of the $r$-th realization, the total field becomes
for the $r$-th realization. This field is still a single-mode coherent-state pulse because the the pulse shaper does not substantively alter the quantum aspects of light. Therefore, this phase-randomization can be expressed quantum mechanically via a single photon operator for the light mode matching the $r$-th realization where the individual Boson operators $\hat {B}_j$ define the quantum properties within the bin $j$. Since each bin is a coherent state $|\beta _j \rangle$, they produce defined uniquely by a coherent amplitude $\beta _j$.As a test of the quantum statistics of the ensemble average, we first compute the ensemble-average Eq. (20) for the proposed randomization procedure, yielding
To verify thermal state behavior for $\hat {\rho }^\textrm {eff}$, we need to verify property 14. Using a straight forward derivation, following the same argumentation as in Eqs. (29)–30, we end up with
To test the opposite limit where the number of spectral bins $N_\textrm {bins}=0$, we expect that we recover the statistics for the coherent state since there would be no phase modulation across the spectrum. For this situation, the ensemble sums each have just one term, and so $\left \langle \left \langle [\hat {B}^\dagger ]^J \hat {B}^K \right \rangle \right \rangle = \left (\beta ^\star \right )^J \beta ^K$, as a coherent state should according to Eq. (13).
5. Characterization of average statistics
We present the procedure for characterizing the statistics of scrambled pulses using DME averaging. We refer to the phase-randomized pulses introduced in Sec. 4.2 as ‘scrambled’ pulses. To perform experiments with the scrambled realizations, we prepare a group of randomized phase masks. A set of $N_\textrm {real}$ phasemasks is created with random phases for each frequency bin $\Delta \omega _\textrm {bin}$ using a pseudo-random number generator. We characterize the statistics of the set of pulse realizations through time-domain measurements. Typical cross-correlation traces for the scrambled realizations compared with the transform-limited (flat mask) case are shown in Fig. 5(b). We observe that the pulses are stretched out in time, and that the intensity at any given delay $\tau$ fluctuates between realizations.
We then quantify the second-order correlation function $g^{(2)}(0)$ of Eq. (5) and its $k$-th order generalizations [32,33]:
To measure $g^{(k)}$ of the DME-averaged ensemble, we histogram $P_m$ as follows: The temporal intensity profile of a single pulse realization can be approximated by a cross-correlation measurement with the short-duration (120 fs) probe pulse. We measure the cross-correlations for $N_\textrm {ens}=500$ different pulse realizations. We then histogram the intensity distribution at a fixed delay, and assign $P^\textrm {eff}_m$ of the ensemble from the normalized histogram. The $g^{(k)}$ can then be deduced using formula (32).
Figure 6(a) presents the measured $P_m$ and Fig. 6(b) the $g^{(k)}$ as function of $k$ for three representative bin numbers $N_\textrm {bin}$. The single-bin case produces $g^{(k)}$ close to one (slightly below) as expected for a coherent state. As the bin number is increased, so does $g^{(k)}$. For $N_\textrm {bin}=20$ (red line), we find $g^{(2)}$ close to 2, as expected for a thermal state. The higher order $g^{(k)}$ are also much closer to $k!$ than for a coherent state, which verifies that the DME averaging indeed produces outcomes close to an ideal thermal state.
Fig. 6. (a) Probability distribution function for the normalized cross-correlation values, where the single bin $P_\textrm {n}$(10 photons) = 1. (b) $k^\textrm {th}$-order coherence values calculated by Eq. (32).
The deviations from the perfect $g^{(k)}=k!$ limit follow from the finite $N_\textrm {ens}$ and $N_\textrm {bin}$. Having a finite number of realizations means that the histogram intervals must be chosen to be large enough to count multiple values in each interval, but small enough so that the photon number distribution can be observed with some resolution. An interval size spacing of 10% of the total intensity combined with the normalization procedure described earlier satisfies these two criteria.
Since realizing $g^{(2)}=2$ is the most important criteria in realizing a quasi-thermal-state source, we have systematically measured $g^{(2)}$ as function of $N_\textrm {bin}$, as shown in Fig. 7(a). The measured $g^{(2)}$ (squares) indeed approaches two, as predicted by theory (red line). For perfectly working phase masks, the scrambled pump spectra $S^\textrm {scram}_\textrm {avg}(\omega ) = \frac {1}{N_\textrm {ens}} \sum _r^{N_\textrm {ens}} S^\textrm {scram}_r(\omega )$ should be identical to the unscrambled, flat-mask spectrum $S^\textrm {flat}(\omega )$ (with a flat spectral phase mask). Figure 7(b) shows the convergence of $S^\textrm {scram}_\textrm {avg}(\omega )$ for three representative $N_\textrm {ens}$. Only the single-element ensemble (green line) deviates from the flat mask spectrum (red line). In particular, the standard error of the normalized spectra $\sigma ^2 = \sum _i \left [ \left ( S^\textrm {scram}_\textrm {avg}(\omega _i)- S^\textrm {flat}(\omega _i) \right ) /S^\textrm {flat}(\omega _i)\right ]^2$ is below $10^{-6}$ for $N_\textrm {ens}>50$ realizations.
Fig. 7. (a) Measured second-order coherence function values for several different spectral bin widths. Measured values correspond to the trend predicted in Eq. (31). (b) The average scrambled pump spectrum (shown for $N_\textrm {ens}=1,14,50$) converges to the flat mask spectrum as the number of realizations $N_\textrm {ens}$ is increased.
6. Excitonic nonlineatities with a DME averaged source
We can now proceed in using the DME-averaged thermal source to study whether it changes the optical nonlinearity of a semiconductor as predicted by quantum spectrocopy. We therefore perform spectrally resolved transient absorption measurements of the QW system after excitation by each scrambled realization, following the methods described in Refs. [2,15]. Between each measurement, we change the phase mask on the spatial light modulator in the pulse shaper to modulate the spectral phase of the pump light source, and use the DME algorithm 25 to compute the thermal-state response, based on the characterization in Sec. 5..
Since the pulse-shaper has some realization-dependent losses, we monitor the pump input power $P_\textrm {IN}^\textrm {pump}$ continuously, adjusting the power of each phase mask realization so that the total power $P_\textrm {IN}^\textrm {pump}$ is the same for all realizations. Additionally, the reflected and transmitted pump powers are measured continuously. These three calibrated measurements allow calculation of the pump power absorbed by the QW system $P_\textrm {QW}^\textrm {pump}$ as described in [15]. We then spectrally resolve the absorption $\alpha (\omega ,r)$ of a probe pulse that arrives at the sample a delay $\tau$ after the pump pulse. Technically, we record a phase-mask-$r$ dependent absorption as the spectrum $S_r(\omega ) \equiv \alpha (\omega ,r)$ to be averaged via Eq. (25).
From each probe-spectrum measured at one of the powers and phase mask realizations, we extract the peak height and the center resonance position, as shown in Fig. 8 for flat-phase mask (blue) and scrambled (green) with $N_\textrm {bin}=20$ bins (across a 6 meV bandwidth) as function of average pump power. Based on the calibration in Sec. 5., the flat-phase case represents a coherent state and scrambled case represents a thermal-state source. We observe that the scrambled ensemble shows a spread of peak values whereas the flat phase is fixed, which is expected for a thermally fluctuating source. Even more so, the scrambled source produces significantly lower nonlinearities because the resonance peak decreases less and center position increases less than the flat-phase source does for elevated pump power. This measured result is consistent with the theory prediction [1] that thermal state creates weaker many-body nonlinearities than an equivalent coherent-state excitation.
Fig. 8. Nonlinear observables as a function of power. (a) Peak heights of $\beta _\textrm {QW}$ and (b) center resonance positions are shown for varying $P_\textrm {QW}^\textrm {pump}$. Fits (solid lines) through each set of points clarify the trends for the observables.
To distinguish the changes in the nonlinearities even clearer, Fig. 9(a) presents the DME-averaged spectra, according to Eq. (25), for scrambled (green line), flat-phase (blue line), and the linear response (red line) without pump. The difference in the nonlinear response is strikingly different; the probe absorption spectra are less saturated for the scrambled realizations compared with the flat-phase pulse. To gauge this difference with the expected thermal vs. coherent-state response, we have performed an independent analysis by varying the coherent-state pulse intensity continuously. We have then projected the expected thermal state response from the coherent-state measurements using Eq. (25). Figure 9(b) compares the probe absorption for a coherent state (blue line) and the projected thermal-state (thermal state) with matching intensity. The quantum-statistical difference of this strongly resembles that of obtained with the DME averaging in Fig. 9(a). This comparison independently verifies that the used DME-averaged source matches well with a thermal state, and that a thermal state indeed produces a significant reductions in optical nonlinearities.
Fig. 9. (a) The average spectral response from the scrambled realizations reveals the HH $1s$-resonance as less saturated than from coherent excitation with the same $P_\textrm {IN}^\textrm {pump}$. (b) Similarly, the projected thermal response determined from applying a set of coherent data using Eq. (18) also reveals less saturation compared with coherent excitation for the same pump excitation density ($4.1\times 10^9$ electron-hole pairs / cm$^2$/layer).
7. Conclusions and Outlook
We have explored two approaches toward direct optical excitation of exciton populations in a GaAs QW system using thermal statistics: continuous excitation from a superluminescent diode (SLD), and an ensemble-averaging technique using phase-modulated pulses.
We have demonstrated a nonlinear response induced by thermal light emission collected from a SLD; however, we measured no significant difference in response between this source and a coherent source. We have found clear reasons a nonlinear response to a continuous thermal light did not match the expected thermal-state response. Quantum spectroscopy ideally changes only the quantum statistics, and our SLD and a coherent-state source neither have a matching spectrum nor a temporal aspect, which makes quantum-statistical comparisons challenging.
One could potentially develop different ideas to make SLD and a coherent-state source temporally and spectrally more identical. It would be interesting to repeat this experiment with a coherent-state source having a short coherence time. The full bandwidth of a SLD is 40 meV, corresponding to a coherence time of 50 fs. The challenge with this approach is that a single short coherent pulse looks very different to a sample resonance than the long SLD temporal profile. A better way to match spectral and temporal aspects (to only change quantum statistics) would be to compare the broadband thermal response to that of a very high repetition rate coherent laser with a spectrum that matches the SLD spectrum. Microresonator-based frequency combs have been demonstrated with repetition rates between 10 and 1000 GHz [34]. Therefore, sources with such high repetition rate could create a coherent source that is comparable in time and spectrum to a broadband thermal source.
We did not follow this extension of the work in this paper, but instead introduced density matrix ensemble (DME) averaging as the means to minimize spectral and temporal changes between the coherent and thermal sources. Technically, we realized DME averaging with phase-modulated pulses accomplished through a pulse-shaping technique. We have demonstrated that phase-randomization across a finite number of spectral bins results in an ensemble of pulses having thermal statistics. We have used DME pulses to excite our excitonic system as a pump pulse, and demonstrate that the nonlinearities are significantly reduced compared to those created by a matching coherent-state. Using projection, we have confirmed that these results are fully consistent with differences between coherent and thermal-state responses. However, we also have identified that while the spectrum of the coherent and DME ensemble match, their temporal duration does not. Therefore, our results are only indicative that changes in nonlinearities can be attributed exclusively to changes in quantum statistics.
Nevertheless, the DME concept also introduces a perfect way to realize temporally and spectrally matching coherent vs thermal state. One could start from a perfect single-mode laser that outputs a coherent state $|\beta \rangle$ where $\beta$ also assigns the amplitude of the field. If one modulated only $\beta$ randomly on a shot-by-shot basis, and $\beta$ were randomly distributed according to a thermal-state distribution $P(\beta ) = \frac {1}{\pi \,\bar {n}}\,e^{-\frac {|\beta |^2}{\bar {n}}}$, the ensemble would define a single-mode thermal state 10 while fully conserving the spectrum and temporal aspects. Technically, this could be realized by letting the phase of each laser pulse vary randomly, which follows naturally by not implementing phase stabilization between pulses, and by using a random amplitude modulator, e.g. a single-mask SLM that changes only the amplitude, on a shot-by-shot level. This latter part is slow and cumbersome to realize as long as the amplitude modulators are relatively slow.
However, this can be done more easily following the DME approach we have demonstrated. We have shown that instead of modulating pulses on a shot-by-shot level, we can change the order of data acquisition and repeat the measurement multiple times at amplitude $|\beta \rangle$, and then change the amplitude according to a thermal distribution between the ensembles. If one wishes, one does not even have to randomize the intensity $|\beta |^2$ between collecting coherent-state responses $R(\beta )$, but collect them with a monotonically increasing intensity sweep. In practice, one would discretize $\beta$ in $\Delta \beta$-sized intervals and have each $\beta _j = (j-\frac {1}{2})\,\Delta \beta$ ($j=1,2,\ldots$) value appear $N_\textrm {ens} P(\beta _j) |\beta _j|\Delta \beta$ times in the DME ensemble with $N_\textrm {ens}$ elements. It is straightforward to show that this approach produces the projection 11 at the limit of small $\Delta \beta$.
This completes a full circle in terms of different levels of thermal-state source strategies, from the spectrally shaped SLD thermal-state to the DME-based approach and the projection 11 implementation. The specific choice of source strategies depends on the application. The SLD-approach is a "natural" source because it does not require active external control. Implementing DME averaging follows from a simple external control sequence while the projection requires post-selection-type data analysis. All approaches are appropriate to demonstrate some aspects of a thermal-state.
In our investigations, only the DME-averaged pulses create similar results as the projection approach, i.e. a pure thermal state, does. However, a SLD approach could also produce a thermal-state response in systems with a much faster coherence time, or equivalently spectrally broader nonlinearities, than our GaAs excitons have. In such situations, any pump within the spectral width of the resonance will produce a similar response, reducing requirements of having identical spectra to see quantum-statistical differences. One test would be to measure integrated second harmonic generation of the two distinct pump sources because it has typically spectral broad nonlinearities. A second test could be use a different sample such as monolayer molybdenum diselenide because it has a shorter response time compared to GaAs [35]. The nonlinear response of such a sample is also very strong because of the enhanced Coulomb interactions resulting from the decreased Coulomb screening by the monolayer. Since one could excite a substantial electron–hole densities with thermal light in a short window of time, this semiconductor system is possibly a good candidate for exploring differences in the nonlinear response between coherent and thermal light.
In summary, we have presented a comprehensive study of how to realize high-intensity thermal-state and thermal-light sources for quantum spectroscopic purposes. We have introduced a new scheme of density matrix ensemble (DME) averaging, and realized it with phase-modulated pulses. The results show that DME averaging produces significantly reduced excitonic broadening, as expected for excitations using a thermal state compared with a coherent state. We also have discussed possible shortcomings and extensions of the presented methodology.
Funding
National Science Foundation (0868895, 1222301, 1253236); National Key Research and Development Program of China (2014AA014402); California State University East Bay, Division of Academic Affairs (2019-20 Faculty Support Grant).
Acknowledgments
R. P. Smith acknowledges support for this project by a 2019-20 Faculty Support Grant from the California State University, East Bay Division of Academic Affairs.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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