Coherent coupling of individual quantum dots measured with phase-referenced two-dimensional spectroscopy: photon echo versus double quantum coherence

TL;DR

This paper demonstrates a phase-referenced two-dimensional spectroscopy method to investigate and distinguish coherent couplings within individual quantum dots and molecules, enhancing the understanding of quantum coherence dynamics.

Contribution

It introduces a phase referencing technique in 2D spectroscopy that simplifies phase stabilization and enables detailed analysis of coherent couplings in quantum dots.

Findings

Successful differentiation between rephasing and non-rephasing FWM configurations.

Effective monitoring of two-photon coherence dynamics.

Validation of the method through experimental detection of quantum couplings.

Abstract

We employ two-dimensional (2D) coherent, nonlinear spectroscopy to investigate couplings within individual InAs quantum dots (QD) and QD molecules. Swapping pulse ordering in a two-beam sequence permits to distinguish between rephasing and non-rephasing four-wave mixing (FWM) configurations. We emphasize the non-rephasing case, allowing to monitor two-photon coherence dynamics. Respective Fourier transform yields a double quantum 2D FWM map, which is corroborated with its single quantum counterpart, originating from the rephasing sequence. We introduce referencing of the FWM phase with the one carried by the driving pulses, overcoming the necessity of its active-stabilization, as required in 2D spectroscopy. Combining single and double quantum 2D FWM, provides a pertinent tool in detecting and ascertaining coherent coupling mechanisms between individual quantum systems, as exemplified experimentally.

Tables

Table 1. Table 1: Model parameters used to calculate the optical response of the QD molecule composed of two coupled InAs QDs.

	QD 1	QD2
resonance energy	$E_1$ = 1359.7 meV	$E_2$ = 1358.95 meV
fine-structure splitting	${\delta }_1$ = 60$\mu$eV	${\delta }_2$ = 140$\mu$eV
biexciton binding energy	${\Delta }_1$ = -3.3 meV	${\Delta }_2$ = -3.6 meV
interdot coupling	${\Delta }_{12}$=+0.09 meV

Equations

$$\mathbf{E}(\mathbf{r},t)=\sum _{j=1}^3\sum _{u_j=\pm 1}{\mathcal{E}}_j^{u_j}(\mathbf{r},t-{\tilde{\tau }}_j){\text{e}}^{i{u}_j({\phi }_j-{\tilde{\omega }}_j(t-{\tilde{\tau }}_j))}.$$

$$\begin{array}{rl}{P}_{\alpha }^{(3)}(\mathbf{r},t)=& {\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\text{d}{t}_3\text{d}{t}_2\text{d}{t}_1\sum _{\beta ,\gamma ,\delta =1}^3{R}_{\alpha \beta \gamma \delta }^{(3)}(t_3,t_2,t_1)\\ & \times E_{\beta }(\mathbf{r},t-t_3)E_{\gamma }(\mathbf{r},t-t_3-t_2)E_{\delta }(\mathbf{r},t-t_3-t_2-t_1).\end{array}$$

$$S_{{\Omega }_{\text{s}}}^{(3)}({\tau }_{3\text{s}},{\tau }_{23},{\tau }_{12})={\int }_{-\infty }^{+\infty }\text{d}t{\mathbf{P}}_{{\Omega }_{\text{s}}}(t)\cdot {\mathcal{E}}_s^{\ast }(t-{\tilde{\tau }}_s){\text{e}}^{i{\tilde{\omega }}_s(t-{\tilde{\tau }}_{\text{s}})},$$

$$\begin{array}{rl}{S}_{\text{ESE}}^{(3)}({\omega }_3,{\tau }_2,{\omega }_1)=\frac{(2\pi {)}^4}{{\hslash }^3}\sum _{e,e^{\prime }}& \left({\mathbf{d}}_{e^{\prime }g}^{\ast }\cdot {\mathcal{E}}_{\text{s}}^{\ast }({\omega }_{e^{\prime }g}-{\tilde{\omega }}_s)\right)\left({\mathbf{d}}_{e^{\prime }g}\cdot {\mathcal{E}}_2({\omega }_{e^{\prime }g}-{\tilde{\omega }}_2)\right){\text{e}}^{-i{\xi }_{e^{\prime }e}{t}_2}\\ \times & \frac{{\mathbf{d}}_{eg}\cdot {\mathcal{E}}_3({\omega }_{eg}-{\tilde{\omega }}_3)}{{\omega }_3-{\xi }_{e^{\prime }g}}\frac{{\mathbf{d}}_{eg}^{\ast }\cdot {\mathcal{E}}_1^{\ast }({\omega }_{eg}-{\tilde{\omega }}_1)}{{\omega }_1-{\xi }_{ge}},\end{array}$$

$$\begin{array}{rl}{S}_{\text{GSB}}^{(3)}({\omega }_3,{\tau }_2,{\omega }_1)=\frac{(2\pi {)}^4}{{\hslash }^3}\sum _{e,e^{\prime }}& \left({\mathbf{d}}_{e^{\prime }g}^{\ast }\cdot {\mathcal{E}}_{\text{s}}^{\ast }({\omega }_{e^{\prime }g}-{\tilde{\omega }}_s)\right)\left({\mathbf{d}}_{eg}\cdot {\mathcal{E}}_2({\omega }_{eg}-{\tilde{\omega }}_2)\right)\\ \times & \frac{{\mathbf{d}}_{e^{\prime }g}\cdot {\mathcal{E}}_3({\omega }_{e^{\prime }g}-{\tilde{\omega }}_3)}{{\omega }_3-{\xi }_{e^{\prime }g}}\frac{{\mathbf{d}}_{eg}^{\ast }\cdot {\mathcal{E}}_1^{\ast }({\omega }_{eg}-{\tilde{\omega }}_1)}{{\omega }_1-{\xi }_{ge}},\end{array}$$

$$\begin{array}{rl}{S}_{\text{ESA}}^{(3)}({\omega }_3,{\tau }_2,{\omega }_1)=-\frac{(2\pi {)}^4}{{\hslash }^3}\sum _{e,e^{\prime },f}& \left({\mathbf{d}}_{fe}^{\ast }\cdot {\mathcal{E}}_{\text{s}}^{\ast }({\omega }_{fe}-{\tilde{\omega }}_s)\right)\left({\mathbf{d}}_{e^{\prime }g}\cdot {\mathcal{E}}_2({\omega }_{e^{\prime }g}-{\tilde{\omega }}_2)\right){\text{e}}^{-i{\xi }_{e^{\prime }e}{t}_2}\\ \times & \frac{{\mathbf{d}}_{f{e}^{\prime }}\cdot {\mathcal{E}}_3({\omega }_{f{e}^{\prime }}-{\tilde{\omega }}_3)}{{\omega }_3-{\xi }_{fe}}\frac{{\mathbf{d}}_{eg}^{\ast }\cdot {\mathcal{E}}_1^{\ast }({\omega }_{eg}-{\tilde{\omega }}_1)}{{\omega }_1-{\xi }_{ge}},\end{array}$$

$$S_{\text{PE}}^{(3)}({\omega }_3,t_2,{\omega }_1)=S_{\text{ESE}}^{(3)}({\omega }_3,t_2,{\omega }_1)+S_{\text{GSB}}^{(3)}({\omega }_3,t_2,{\omega }_1)+S_{\text{ESA}}^{(3)}({\omega }_3,t_2,{\omega }_1).$$

$$S_{\text{DQC}}^{(3)}({\omega }_3,{\omega }_2,t_1)=S_{{\text{ESA}}_1}^{(3)}({\omega }_3,{\omega }_2,t_1)+S_{{\text{ESA}}_2}^{(3)}({\omega }_3,{\omega }_2,t_1)$$

$$\begin{array}{rl}{S}_{{\text{ESA}}_1}^{(3)}({\omega }_3,{\omega }_2,t_1)=-\frac{(2\pi {)}^4}{{\hslash }^3}\sum _{e,e^{\prime },f}& \left({\mathbf{d}}_{f{e}^{\prime }}^{\ast }\cdot {\mathcal{E}}_{\text{s}}^{\ast }({\omega }_{f{e}^{\prime }}-{\tilde{\omega }}_{\text{s}})\right)\left({\mathbf{d}}_{eg}\cdot {\mathcal{E}}_1({\omega }_{eg}-{\tilde{\omega }}_1)\right){\text{e}}^{-i{\xi }_{eg}{t}_1}\\ \times & \frac{{\mathbf{d}}_{e^{\prime }g}^{\ast }\cdot {\mathcal{E}}_3^{\ast }({\omega }_{e^{\prime }g}-{\tilde{\omega }}_3)}{{\omega }_3-{\xi }_{f{e}^{\prime }}}\frac{{\mathbf{d}}_{fe}\cdot {\mathcal{E}}_2({\omega }_{fe}-{\tilde{\omega }}_2)}{{\omega }_2-{\xi }_{fg}},\end{array}$$

$$\begin{array}{rl}{S}_{{\text{ESA}}_2}^{(3)}({\omega }_3,{\omega }_2,t_1)=\frac{(2\pi {)}^4}{{\hslash }^3}\sum _{e,e^{\prime },f}& \left({\mathbf{d}}_{e^{\prime }g}^{\ast }\cdot {\mathcal{E}}_{\text{s}}^{\ast }({\omega }_{e^{\prime }g}-{\tilde{\omega }}_{\text{s}})\right)\left({\mathbf{d}}_{eg}\cdot {\mathcal{E}}_1({\omega }_{eg}-{\tilde{\omega }}_1)\right){\text{e}}^{-i{\xi }_{eg}{t}_1}\\ \times & \frac{{\mathbf{d}}_{f{e}^{\prime }}^{\ast }\cdot {\mathcal{E}}_3^{\ast }({\omega }_{f{e}^{\prime }}-{\tilde{\omega }}_3)}{{\omega }_3-{\xi }_{e^{\prime }g}}\frac{{\mathbf{d}}_{fe}\cdot {\mathcal{E}}_2({\omega }_{fe}-{\tilde{\omega }}_2)}{{\omega }_2-{\xi }_{fg}}.\end{array}$$

Full text

Nuclear magnetic resonance (NMR) spectroscopy conceived phase-locked, multi-pulse techniques, yielding multi-dimensional spectra by Fourier transforming temporal sequences into respective frequency coordinates WuthrichNobelLecture ; VandersypenRMP05 . The possibility to spread the response of biological or chemical molecules of high structural complexity, especially proteins, across many axes enabled to assess their spatial form and to understand inter-atomic interactions and couplings. The idea to selectively address and evolve subsets of transitions from congested spectra via multi-pulse toolbox, and then projecting the results onto two-dimensional (2D) or three-dimensional diagrams, is a far-reaching legacy of NMR. As concern optical spectroscopy, achieving phase-stabilized sequences of laser pulses is more challenging, owing to substantially shorter optical cycles with respect to radio-frequency domain used in NMR. Thus, it has been first accomplished in mid-NIR regime HammJPCB98 ; MukamelARPC00 ; RehaultOE14 and recently conveyed into NIR-VIS range BrixnerOL04 ; BrixnerJCP04 ; BrixnerN05 ; ColliniN10 ; BaranczykAP14 , while being continuously improved employing active pulse-shaping StoneScience09 ; TurnerN10 ; Tollerud16 and phase-feedback techniques BristowOE08 ; HelbingJOSAB11 .

At a juncture of coherent spectroscopy and condensed matter physics, 2D spectroscopy provided insight into dynamics and couplings of many-body optical excitations in solids, in particular of excitons in semiconductor quantum wells StoneScience09 ; TurnerN10 ; MoodyPRL14 and novel 2D layered materials HaoNanoLett16 , as well as in ensembles of quantum dots MoodyPRB13 ; MoodyPRB13a (QDs) or nanocrystals CassetteSmall16 . A principal tool in these investigations is k-resolved four-wave mixing (FWM) spectroscopy and its extensions probing multi-wave mixing processes TurnerN10 . FWM microscopy of single QD excitons LangbeinPRL05 was accomplished by phase-sensitive optical heterodyning combined with interferometric detection, efficiently subtracting resonant background and permitting co-linear geometry of the excitation pulses. Recently, detection sensitivity of intrinsically weak single QD FWM has been enhanced substantially by using photonic nanostructures, improving the QD coupling with external laser beams FrasNatPhot16 ; MermillodPRL16 ; JakubczykACSPhot16 .

Here, we perform FWM spectroscopy of individual InAs QDs embedded in a low-Q semiconductor microcavity MaierOE14 ; FrasNatPhot16 ; MermillodOptica16 . We point out two major advancements: Firstly, we demonstrate 2D FWM constructed from two-photon coherences — known as double quantum 2D FWM KimACR09 ; KaraiskajPRL10 ; TurnerPRB11 — driven on individual transitions, specifically QD exciton-biexciton systems (GXB) FingerhutAP13 . Secondly, we introduce referencing of the FWM phase, offering convienient alternative for its active-stabilization, which is widely believed to be required in 2D spectroscopy. Using the one-quantum and two-quantum spectroscopy, we have measured single QDs and a QD molecule. A comparison of the spectra signatures to theory allowed us to identify the nature of the internal coupling mechanism in the QD molecule system. Our work shows that the combined single and double quantum 2D spectroscopy is a powerful tool to reveal and understand coherent coupling and excitation transfer mechanisms - an interdisciplinary issue spanning from biology and photo-chemistry, to quantum engineering. The results are especially pertinent for the latter area, as we open new avenues of research in quantum control of optically active nanoscopic two-level and few-level systems in solids.

To acquire the FWM spectra FrasNatPhot16 , we use a pair of 100 fs laser pulses: ${\cal E}_{1}$ and ${\cal E}_{2}$, with a variable delay $\tau_{12}$, positive for ${\cal E}_{1}$ leading. They are frequency shifted by $\Omega_{1}$ and $\Omega_{2}$, respectively, using acousto-optic deflectors. FWM heterodyne beat with a reference field ${\cal E}_{\rm R}$ is retrieved at $2\Omega_{2}-\Omega_{1}$ frequency, carrying the lowest order response ${\cal E}_{1}^{\star}{\cal E}_{2}{\cal E}_{2}$ (where $\star$ denotes complex conjugate) and also higher orders with the same phase evolution. The signal is spectrally dispersed using a spectrometer, detected with a CCD camera and retrieved in amplitude and phase by applying spectral interferometry.

As shown in Fig. 1 b, in two-beam FWM, the first pulse ${\cal E}_{1}$ induces coherence, which evolves during $\tau_{12}$, to be then converted into FWM by the second pulse ${\cal E}_{2}$, arriving a few picoseconds after ${\cal E}_{\rm R}$. The lowest electronic excitations of a neutral QD can be cast into three categories of states: a ground state ($G$), single excitons ($X$) and two-exciton states, known as biexcitons ($B$). $GX$ transitions are addressed by one-photon coherence driven by ${\cal E}_{1}$, which is converted to FWM of $GX$ and $XB$ by a density grating ${\cal E}_{1}^{\star}{\cal E}_{2}$ on $G$ and $X$ KasprzakNJP13 ; MermillodOptica16 . Inverting temporal ordering of the two light pulses, $GB$ transition can be inspected by a two-photon coherence induced by ${\cal E}_{2}$, transformed into FWM of both transitions at the arrival of ${\cal E}_{1}$ KasprzakNJP13 ; MermillodOptica16 . The simple three-level system of Fig. 1 b illustrates the case of a neutral QD driven along one of its polarization axes. For a single two-level system, like a QD trion, FWM can be only created for $\tau_{12}>0$ from one-photon coherence induced by ${\cal E}_{1}$, since the trion system cannot be doubly excited within the employed spectral bandwidth. In fact, two transitions in Fig. 1 a show strictly no signal for $\tau_{12}<0$ and are attributed to trion transitions. Therein, we also recognize pairs of exciton-biexcitons, labeled as: $GX_{1}-X_{1}B_{1}$, $GX_{2}-X_{2}B_{2}$, $GX_{3}-X_{3}B_{3}$ occurring in three distinct QDs. FWM exhibits a pronounced beating as a function of $\tau_{12}>0$, with a period corresponding to $B$ binding energy, which is induced beyond $\chi^{(3)}$ regime by high order contributions propagating at the FWM frequency LangbeinJOSAB01 ; MermillodOptica16 . Instead, for $\tau_{12}<0$ FWM is equally created on $GX$ and $XB$ transitions, with no beating.

Time-resolved FWM transient created upon the two pulse configurations displays different characteristics. For $\tau_{12}>0$, there is a phase-conjugation between ${\cal E}_{1}$ and FWM. Owing to the rephasing, FWM of an inhomogeneously broadened system has a Gaussian form, with a maximum at $t=\tau_{12}$ and temporal width inversely proportional to the probed spectral inhomogeneous broadening $\sigma$. Importantly, time-integrated amplitude of such photon echo is not sensitive on $\sigma$, instead the homogeneous broadening is probed through the $\tau_{12}$-dependence. At a level of individual transitions, $\sigma$ is accumulated due to a residual spectral wandering in time-averaged measurement PattonPRB06 ; KasprzakNJP13 ; MermillodPRL16 ; JakubczykACSPhot16 . For $\sigma$ in $\mu$eV range, which is a case even for high quality QD systems, the echo width becomes comparable or larger than the temporal sensitivity, given by the spectrometer resolution (here about 120 ps). To demonstrate formation of such a broad echo JakubczykACSPhot16 , we scan the delay $\tau_{R2}$, between ${\cal E}_{\rm R}$ and ${\cal E}_{2}$, for three different $\tau_{12}$, as shown in Fig. 2. The echo develops fully only for $\tau_{12}=200\,$ps, from its width (FWHM) $t_{\sigma}=\hbar/\sigma=(214\,\pm\,33)$ ps we retrieve spectral inhomogeneous broadening $8\ln{(2)}\sigma=8\ln{(2)}\hbar/t_{\sigma}=(17\,\pm\,3)\,\mu$eV (FWHM). Adjusting $\tau_{R2}$ permits to retrieve FWM significantly beyond the temporal resolution of the setup, as shown in the inset for $\tau_{12}=1\,$ns.

For $\tau_{12}<0$ there is no strict phase conjugation between two-photon coherence and FWM, and therefore the photon echo is absent. In Fig. 3 c we show FWM$(t,\,\tau_{12}$) maps measured on $GX_{1}$ and $X_{1}B_{1}$ transitions. As $\tau_{12}$ is increased towards more negative values, FWM decay becomes more pronounced, owing to a non-rephasing character of the signal. The two-photon coherence dynamics of $GX_{1}$ and $X_{1}B_{1}$, i.e. respective time-integrated FWM versus $\tau_{12}$, are presented in Fig. 3 a and b. From the exponential decay we retrieve two-photon dephasing $T_{TP}(GX_{1},X_{1}B_{1})=(75\,\pm\,3,\,65\,\pm\,3)\,$ps. Similar values of $T_{TP}$ are retrieved by analyzing two other $GX$-$XB$ pairs.

To illustrate couplings in the probed system of a few QDs, we Fourier-transform FWM$(\omega_{3},\,\tau_{12})$ sequences with respect to the delay $\tau_{12}$. The experimental setup is encapsulated, providing a passive stabilization of the phase during the acquisition. However, the phase relationship between FWM measured for subsequent $\tau_{12}$ is inevitably lost and can only be achieved via active-stabilization BristowOE08 ; HelbingJOSAB11 , which is not implemented here. The knowledge of the FWM phase for subsequent delays $\tau_{12}$ is a precondition to execute the Fourier transform yielding 2D FWM. In our previous works KasprzakNPho11 ; MermillodOptica16 , we have circumvented this issue by imposing a phase relationship onto the data by choosing a separated transition in the spectral domain, acting as a local oscillator, and setting its phase to zero for all delays. We then applied this phase factor globally to the full spectrum, adjusting all other frequencies versus $\tau_{12}$, accordingly. Such transformation remains justified, as long as the guiding transition to correct for, in particular exhibiting no coherent coupling, is available in the spectrum. This generally might not the case. To overcome this experimental limitation, we have conceived a post-treatment protocol permitting to reference the FWM phase, using auxiliary spectral interferences of ${\cal E}_{\rm R}$ with the driving pulses, as explained in Supplementary Material.

In Fig. 4 we present 2D FWM obtained from the set of QDs highlighted in Fig. 1. For $\tau_{12}>0$, FWM generated by all resonances driven by ${\cal E}_{1}$, forms a diagonal in the resulting 2D spectrum. This includes, single trions and neutral excitons, but also biexcitons - the latter can directly be driven by ${\cal E}_{1}$ beyond the $\chi^{(3)}$ limit MermillodOptica16 , as applied here (an example in the $\chi^{(3)}$ limit is provided in the Supplementary Fig. S6). The biexcitons are off-diagonally shifted by their respective binding energies of a few meV, and form square-like features in 2D FWM under strong excitation, i.e. close to $(\pi/2,\,\pi)$ area of $({\cal E}_{1},\,{\cal E}_{2})$ pulses. 2D FWM resulting from $\tau_{12}<0$ is shown in Fig. 4 b. FWM originates from a corresponding two-photon resonance driven by ${\cal E}_{2}$. Here, the two-photon energy corresponds to the sum GX and XB transition energies. In such non-rephasing 2D FWM, we retrieve the response of GXB systems, whereas exciton complexes without doubly excited states within the excitation bandwidth, such as singly charged QDs, do not contribute.

Fig. 5 a and c show the measured rephasing and non-rephasing 2D spectra recorded at another position at the sample. In the following, we focus on the two QDs that show up as transitions $GX_{1}$ and $GX_{2}$ on the diagonal of the rephasing spectrum with resonance energies $E_{1}$=1359.7 meV and $E_{2}$=1358.95 meV - via hyperspectral imaging KasprzakNPho11 these are found to be within 0.5$\,\mu$m vicinity (see Supplementary Fig. S7). The peak pattern highlighted by the dashed lines differs from the signatures observed in Fig. 4 in two major respects: First, spin-orbit coupling of the two circularly polarized excitons within each QD leads to linearly polarized exciton eigenstates, where each QD is described by a four-level system MermillodOptica16 . This causes a splitting of each exciton resonance on the diagonal of the rephasing spectrum into clusters of four peaks. Second, besides the $X_{1}B_{1}$ and $X_{2}B_{2}$ peaks that are redshifted along the FWM axis by the intradot biexciton binding energies $\Delta_{1}$=-3.3 meV and $\Delta_{2}$=-3.6 meV, respectively, we observe two off-diagonal cross peaks labeled $X_{2}X_{1}$ and $X_{1}X_{2}$ at the spectral positions $(\omega_{3}=E_{2};\omega_{1}=E_{1})$ (upper cross peak) and $(\omega_{3}=E_{1};\omega_{1}=E_{2})$ (lower cross peak). The appearance of these cross peaks clearly indicates a coherent interdot coupling between the two QDs: The electrostatic Coulomb coupling leads to an energy renormalization of the interdot biexciton $B_{12}$ consisting of one exciton in each QD. The biexciton shift lifts the symmetry of the lower $GX_{1}$ ($GX_{2}$) and higher $X_{1}B_{12}$ ($X_{2}B_{12}$) transitions such that the quantum pathways involving these transitions do not destructively interfere anymore and cross peaks show up KasprzakNPho11 . A level scheme of such a QD molecule including all coupling-induced energy shifts is shown in Supplementary Fig. 1. The electrostatic interaction $\Delta_{12}$ between two excitons located in two different QDs is small compared to the intradot biexciton binding energies. In fact, spectrally-resolved FWM amplitude, shown in the Supplementary Fig. S7, reveals that it is only of the order of $\Delta_{12}=90\,\mu$eV and it shifts the interdot biexciton towards higher energies, showing up as blueshifted high-energy shoulders of the exciton resonance peaks. This interpretation is supported by calculations AbramaviciusChemRev09 (see Supplementary Material) of the rephasing and non-rephasing 2D signals depicted in Fig. 5 b and d.

In the non-rephasing two-quantum spectrum, the coupling of the two QDs manifests itself in a peak pair labeled $X_{1}B_{12}$ and $X_{2}B_{12}$ at the interaction-shifted two-exciton transition $GB_{12}$ (energy $\omega_{2}=E_{1}+E_{2}+\Delta_{12}=$2718.74 meV) with FWM frequencies $\omega_{3}=E_{1}=$1359.7 meV and $\omega_{3}=E_{2}=$1358.95 meV, respectively. Theoretical calculations also suggest that exciton transfer processes between the two QDs such as dipole-induced (Förster) interaction and Dexter-type coupling via wave-function overlap are negligible SpechtPRB15 : First, these coupling types are expected to be in the $\mu$eV range KasprzakNPho11 and therefore difficult to detect at our spectrometer resolution of 25$\,\mu$eV. Second, they would lead to additional peaks for an intradot biexciton in one QD after the first pulse has created a single-exciton in the other QD. These peaks are not observed in the spectra indicating that exciton transfer elements are negligible, as elaborated in the Supplementary Material (Fig. S3).

An interesting feature about the observed QD molecule is that, in contrast to the other isolated exciton-biexciton systems, the two coupled QDs show a pronounced fine-structure splitting (FSS) of the order of 60$\,\mu$eV and 140$\,\mu$eV, respectively. This is around 5 times higher than the FSS typically present in these QDs MermillodOptica16 . Moreover, the FSS of the other isolated exciton-biexciton systems in our sample (see also Fig. 4) is not visible since the direction of the linear excitation/reference polarization was chosen to be parallel to the anisotropy axis. The observation of such a pronounced FSS only for the resonances associated with the QD molecule therefore suggests that the spatial proximity of the two coupled QDs altered the local symmetry of the confinement, changing the magnitude of the FSS and the polarization of the excitonic transitions.

In summary, we have implemented phase-referenced double quantum 2D FWM spectroscopy of individual quantum systems. By merging it with the single quantum counterpart, we have ascertained coherent couplings between excitons, optical selection rules, the structure of (bi-)exciton states, and coupling energies in single InAs QDs and in a quantum dot molecule. This methodology is appealing to infer electronic couplings and charge transfer in deterministically defined QD molecules StinaffScience06 ; ArdeltPRL16 and propagative coherence in photonic molecules KapfingerNC15 . By merging it with recently developed multi-wave mixing toolbox FrasNatPhot16 , it could be also used to visualize and control polaritonic couplings in solid state cavity-quantum electrodynamics AlbertNatComm13 .

acknowledgements

We gratefully acknowledge the financial support by the European Research Council (ERC) Starting Grant PICSEN (grant no. 306387). J.S. and M.R. acknowledge financial support by SFB 787 and GRK 1558.

I Theory Notes

The model system used for the quantum dot (QD) molecule is discussed and the third-order response function is derived. The Liouville space pathways entering the rephasing photon echo and non-rephasing double-quantum coherence signal are illustrated using double-sided Feynman diagrams AbramaviciusChemRev09 . Finally, the simulation parameters are given and the calculated spectra are discussed with respect to the coupling type of the QD molecule measured in Fig. 5 of the main manuscript.

I.1 Model system of a quantum dot molecule

We will show that the signatures observed in the measured spectrum of Fig. 5 of the main text clearly indicate an intradot and interdot coupling of excitons in a quantum dot (QD) molecule consisting of two coupled QDs with resonance energies $E_{1}$ and $E_{2}$, each with a different spin-orbit coupling $\delta_{1}$ and $\delta_{2}$, respectively. The level scheme of such a QD molecule is schematically shown in Fig. S1. There, the electrostatic Coulomb coupling shifts the energy of the doubly excited states with respect to the single exciton energies of its constituents. $\Delta_{1}$ ($\Delta_{2}$) denotes the intradot biexciton shift within QD 1 (QD 2) and $\Delta_{12}$ represents the electrostatic interdot coupling between an exciton in QD 1 and an exciton in QD 2, forming a bound interdot biexciton $B_{12}$. The optical selection rules are governed by the fine-structure splitting (FSS): they originate from an exchange interaction between the two circularly polarized excitons in one QD, resulting in a new linearly polarized basis of excitonic eigenstates MermillodOptica16 . Vertically polarized optical excitations are marked with red arrows, horizontally polarized with blue arrows in Fig. S1. Since the resonances of the two exciton levels of each QD separated by the FSS are (more or less) equally pronounced in the measured spectrum, we assume an angle of $45^{\circ}$ between the direction of the linear excitation/reference polarization and the fine-structure axis of the two dots in our theoretical calculation. This way, all interaction pathways shown in Fig. S1 contribute to the measured signal and a FSS will show up in the optical spectra.

I.2 Calculation of the rephasing photon echo and

non-rephasing double-quantum coherence

In the following, the rephasing one-quantum and non-rephasing two-quantum signals are calculated following Ref. [AbramaviciusChemRev09, ]. In a four-wave mixing (FWM) experiment in the $\chi^{(3)}$ regime, the applied optical field is composed of a sequence of three pulses with envelopes $\mathcal{E}_{j}^{u_{j}}$ (where $\mathcal{E}_{j}^{-1}=[\mathcal{E}_{j}^{+1}]^{*}$), frequencies $\tilde{\omega}_{j}$, and phases $\varphi_{j}$. Each pulse $j$ is centered at time $\tilde{\tau}_{j}$:

[TABLE]

The third-order response function links the polarization induced in the system to the applied electric field:

[TABLE]

The heterodyned signal is a function of the delay times $\tau_{ij}\equiv\tilde{\tau_{j}}-\tilde{\tau_{i}}$ between the pulses:

[TABLE]

where $\mathcal{E}_{\text{s}}(t-\tilde{\tau}_{\text{s}})$ denotes the local oscillator field envelope and $\Omega_{\text{s}}=u_{1}\Omega_{1}+u_{2}\Omega_{2}+u_{3}\Omega_{3}$ describes the specific phase combination of the detected signal.

The rephasing photon-echo (PE) frequency combination is $\Omega_{\text{s}}=\Omega_{\text{I}}=-\Omega_{1}+\Omega_{2}+\Omega_{3}$. There are three types of excitation quantum pathways contributing to the rephasing signal, denoted excited-state emission (ESE), ground-state bleaching (GSB), and excited-state absorption (ESA) and illustrated as double-sided Feynman diagrams in Fig. S2(a). The ESE and GSB pathway involve only the singly excited states $e$, whereas the ESA pathway includes also the doubly excited states $f$. Note that in the absence of many-body interactions, the two Liouville space pathways involving singly excited states (lower transitions) are completely canceled out by the pathway including the doubly excited states (upper transitions) for off-diagonal resonances and therefore, all off-diagonal peaks vanish DaiPRL12 ; KasprzakNPho11 .

The non-rephasing double-quantum coherence (DQC) frequency combination is $\Omega_{\text{s}}=\Omega_{\text{III}}=+\Omega_{1}+\Omega_{2}-\Omega_{3}$. The signal has two contributing ESA pathways, cf. Fig. S2(b).

The rephasing PE one-quantum signal in frequency domain is calculated by Fourier transforming the third-order heterodyne-detected signal with respect to the time intervals $t_{1}$ and $t_{3}$ between the pulses. $t_{2}$ is fixed (In particular, in the two-pulse FWM it is vanishing). The three contributions have the form AbramaviciusChemRev09

[TABLE]

where we defined $\xi_{ab}\equiv\omega_{ab}-i\gamma_{ab}$. $\bm{d}_{ab}$ denotes the dipole moment, $\omega_{ab}$ the resonance energy and $\gamma_{ab}$ the homogeneous broadening of the $b\rightarrow a$ transition. The total rephasing PE signal is the sum of the three contributions:

[TABLE]

Similarly, the double-quantum signal is obtained by Fourier transforming the signal function with respect to the delay times $t_{2}$ and $t_{3}$. It is composed of the two ESA pathways:

[TABLE]

with

[TABLE]

These signal functions enable us to calculate 2D maps of the rephasing and non-rephasing FWM pathways depending on the Fourier transformed pulse delays.

I.3 Model parameters

The parameters used to calculate the rephasing and non-rephasing 2D FWM spectra were chosen in agreement with the experimental data (see Supplementary Tab. 1).

The homogeneous linewidth is typically in the order of few $\mu$eV, which corresponds to a dephasing time of few hundred picoseconds StockPRB11 ; OstapenkoPRB12 . Here, we choose $\gamma=1/(500\,{\rm ps})$ BorriPRL01 . Moreover, the spectrometer resolution of $\sim 25\,\mu$eV was incorporated into the homogeneous linewidth for a quick estimation of its effect. The transition dipole moments of both QDs are chosen equally. Spectral wandering induces an inhomogeneous broadening MermillodOptica16 of $10\,\mu$eV. This is included in the calculations by averaging the contributions to the signal functions for normally distributed values of the system resonances.

I.4 Evaluation of the spectra

Fig. 5 and of the main text shows the calculated one-quantum rephasing (a) and two-quantum non-rephasing (b) 2D spectra of the QD molecule specified by the parameters given in Sec. I.3. As mentioned before, Coulomb interactions between the excited states cause that the doubly excited states are energetically shifted compared to the sum of the energies of the (isolated) single exciton constituents. This breaks the symmetry between the $G\rightarrow X$ and $X\rightarrow B$ transitions and leads to off-diagonal peaks in the rephasing 2D spectrum. In the measured spectrum in Fig. 5 (a) of the main text, multiple off-diagonal peaks appear, indicating multiple exciton couplings. The spin-orbit coupling splits the exciton resonances of every QD, leading to four-peak clusters in the rephasing spectrum. Moreover, the upper $X_{2}X_{1}$ $(\omega_{3}=E_{2};\omega_{1}=E_{1})$ and lower $X_{1}X_{2}$ $(\omega_{3}=E_{1};\omega_{1}=E_{2})$ cross peaks form a square with the two $GX_{1}$ and $GX_{2}$ resonances at energies $E_{1}$ = 1359.7 meV and $E_{2}$ = 1358.95 meV on the diagonal. This pattern indicates an interdot exciton-exciton coupling between QD 1 and QD 2. (Without this coupling, these cross-peaks interfere destructively and cancel.) The electrostatic interdot coupling $\Delta_{12}$ = 0.09 meV is small compared to the linewidth and spectrometer resolution. Thus, the corresponding interaction-shifted resonances cannot be identified as separate peaks in the spectrum, but as blueshifted high-energy shoulders of the exciton peaks, as explicitly depicted in Supplementary Fig. X. Finally, the off-diagonal $X_{1}B_{1}$ and $X_{2}B_{2}$ peaks at one-photon frequencies $E_{1}$ and $E_{2}$ are redshifted along the FWM axis by the intradot biexciton binding energies $\Delta_{1}$ = -3.3 meV in QD 1 and $\Delta_{2}$ =-3.6 meV in QD 2. In the two-quantum spectrum, the coupling of the two QDs leads to a peak pair labeled $X_{2}B_{12}$ and $X_{1}B_{12}$ at the interaction-shifted two-exciton transition $GB_{12}$ at energy $\omega_{2}=E_{1}+E_{2}+\Delta_{12}$ with FWM frequencies $\omega_{3}=E_{1}$ and $\omega_{3}=E_{2}$, respectively. Moreover, QD 1 (QD 2) forms a fine-structure split exciton-biexciton complex at two-photon frequency $2E_{1}-\lvert\Delta_{1}\rvert$ ($2E_{2}-\lvert\Delta_{2}\rvert$). The corresponding peak pairs are labeled $GX_{i}$ and $X_{i}B_{i}$ with $i\in 1,2$.

A comparison of the measured and calculated spectra also shows that exciton energy transfer processes between the two quantum dots such as dipole-induced (Förster) interaction and Dexter-type coupling via wave-function overlap are negligible in the considered system: First, the corresponding coupling strengths in the order of $\mu$eV KasprzakNPho11 are too small to be detected for the achieved spectrometer resolution. Second, the exciton transfer between the two QDs would lead to additional peaks at spectral positions where the second pulse creates a $GB$ coherence involving an intradot biexciton in one QD before the third pulse induces a $GX$ or $XB$ coherence involving a single exciton in the other QD. To illustrate this, Suppl. Fig. S3 shows the calculated spectra including a (large) Förster coupling of 0.1 meV between the two QDs. The additional peaks arising due to interdot exciton transfer are highlighted by white circles.

This way, the analysis of the measured and calculated spectra enables us to identify the coupling type and strength, providing evidence of a QD molecule consisting of two electrostatically coupled QDs.

II Auxiliary Experimental Results

II.1 Implementation of the phase-referencing protocol

In this section, we present the protocol which we established to reference the phase of the FWM signal by using the excitation pulses ${\cal E}_{1}$ and ${\cal E}_{2}$. For all measured delays $\tau_{12}$, before and after acquiring FWM signal at $2\Omega_{2}-\Omega_{1}$ frequency, we measure heterodyne spectral interferences at $\Omega_{1}$ and $\Omega_{2}$, allowing to measure the phase of ${\cal E}_{1}$ and ${\cal E}_{2}$: $\varphi_{1}$ and $\varphi_{2}$, by performing spectral interferometry. Passive stabilization of the setup turns out to be sufficient to maintain the phase stability for each $\tau_{12}$ during a typical acquisition time of ten seconds, as required here. If however phase drift is detected, the supplementary phase-reference data are interleaved in between the FWM acquisition to monitor phase variation and to correct for it.

In Fig. S4 a and b we present the spectrally resolved real part of the signal heterodyned at $\Omega_{1}$ and $\Omega_{2}$, respectively. Here, $\tau_{R2}=3\,$ps and remains fixed during the measurement, whereas $\tau_{12}$ (and thus also delay between ${\cal E}_{1}$ and ${\cal E}_{\rm R}$, $\tau_{R1}$) is scanned. As a result, $\varphi_{1}$ shows strong fluctuations, while $\varphi_{2}$ displays a slope of only 2$\pi$ through the whole sequence.

In the next step, we test the performance of the phase correction routine described in the main text. In panels d) and e) we see that $\varphi_{1}$ and $\varphi_{2}$ can be aligned with respect to a chosen frequency of around 1360.1 meV.

We will now impose this externally determined correction onto the FWM data shown c). Specifically, the FWM is corrected by phase factors -$\varphi_{1}$ and $2\varphi_{2}$, which are retrieved from d) and e), respectively. The real part of the FWM, phase-corrected using such external referencing is shown in panel h). The FWM phase is directly linked with the one of ${\cal E}_{1}$ and ${\cal E}_{2}$, which are now measured and provide the reference frame at which we synchronize the FWM. The accuracy of the phase-referencing is confirmed by formation of narrow peaks in 2D FWM displayed in panel k). For comparison, panel i) shows 2D FWM obtained after correcting the phase directly on the FWM data.

In Fig. S5 we present phase-referenced 2D FWM, obtained in the non-rephasing configuration, $\tau_{12}<0$.

II.2 Single and double quantum 2D FWM of individual

quantum dots in a low excitation regime

In Fig. S6 we present single (a) and double quantum (b) 2D FWM spectroscopy of individual QDs obtained under low excitation $\chi^{(3)}$ regime. The spectra reveal exciton-biexciton pairs, occurring in different QDs. In (a), the biexcitons are shifted from the excitons — placed at the diagonal (white line) $\omega_{3}=\omega_{1}$ — by their binding energies of -0.7 meV (white dotted line), -3.3 meV (orange dashed line), -3.55 meV (green dash-dotted line). We note that, in contrast to the data shown in Fig. 4 a of the main manuscript, the signatures of biexcitons do not show up on the diagonal. This means the ground state-to-biexciton transitions are now not excited by ${\cal E}_{1}$, confirming the low excitation regime. By inspecting the exciton-biexciton pair at (1360.67, 1357.1) meV we observe that, in the rephasing configuration, the FWM amplitude of the biexciton is an order of magnitude weaker than the exciton’s one. Instead, in the non-rephasing situation the amplitude of both transitions is equal. Such a suppression of biexciton transition in the rephasing case is characteristic for transitions exhibiting a large inhomogeneous broadening (here around 0.2 meV, an order of magnitude larger then typically observed in this sample), and has been previously noticed in FWM of GaAs quantum wells LangbeinPSSA02b . Two other transitions, with an order of magnitude lower inhomogeneous broadening, show a textbook behavior with the exciton’s amplitude twice stronger than (equal to) the biexciton in the rephasing (non-rephasing) configuration KasprzakNJP13 ; MermillodOptica16 .

II.3 Experimental insight into a QD molecule: FWM amplitude and

phase, and hyperspectral imaging

To better illustrate coherent coupling measured at the QD molecule investigated in Fig. 5 of the main manuscript, in Fig. S7 we present FWM amplitude (black traces) and phase (orange traces), retrieved from intersections through the 2D FWM spectrum in the rephasing configuration. The diagonal peaks display fine-structure splitting and overall phase shift of $\pi$ over the resonances. The amplitude of off-diagonals show additional contributions, in agreement with simulations shown in Fig S8. Each off-diagonal peak pair exhibits a high-energy shoulder that is blueshifted by the interdot biexciton binding energy $\Delta_{12}=90\,\mu$eV. This lifts the degeneracy of the $GX_{1}$ ($GX_{2}$) and $X_{2}B_{12}$ ($X_{1}B_{12}$) contributions, such that they do not coincide spectrally and therefore do not cancel each other out. We also observe a 2$\pi$ phase shift across the off-diagonals, indicating electrostatic coupling resulting in the biexciton blueshift KasprzakNPho11 ; KasprzakJOSAB12 . Note that the phase shifts across the off-diagonals in the theoretical spectrum differ quantitatively from the measured ones, since here the FSS is resolved better than in the measured spectrum. On the other hand, the overlap with neighboring peaks has a stronger impact in the calculated spectrum (due to the Lorentzian form of the resonances), leading to phase interferences. As expected, both QDs are spatially co-localized with a few hundreds of nm, as confirmed by the FWM hyperspectral imaging. They are simultaneously excited by the laser spot (focussed down to the diffraction limit of around 0.85$\,\mu$m) positioned at the crossing point of blue lines.