Resonant excitation of the biexciton state in an InAsP quantum dot by a phase-coherent pair of picosecond pulses allows preparing time-bin entangled pairs of photons via the biexciton–exciton cascade. We show that this scheme can be implemented for a dot embedded in an InP nanowire. The underlying physical mechanisms can be represented and quantitatively analyzed by an effective three-level open system master equation. Simulation parameters including decay and intensity dependent dephasing rates are extracted from experimental data, which in turn let us predict the resulting entanglement and optimal operating conditions.

## INTRODUCTION

In a future quantum world, long-distance quantum communication will allow users to communicate in perfect privacy, and it will connect quantum computers for distributed and blind computation tasks. Quantum repeaters^{1} will be necessary in order to establish the required long-distance entanglement, and for building even the simplest quantum repeaters, we will need reliable, high-rate, and high-fidelity sources of entangled photon pairs, besides quantum memories and local quantum processing. The emitted photon pairs must propagate with low loss and low decoherence in order to cover as much distance as possible. While the propagation loss in optical fibers is limited by intrinsic material properties, the decoherence can be minimized by choosing a suitable quantum information encoding.^{2} Time-bin entanglement^{3,4} has emerged as the optimal encoding for optical fiber quantum communication because it is immune to residual fiber birefringence as well as thermal and mechanical fluctuations up to very high frequencies.

So far, all sources of time-bin entanglement have been probabilistic, even the ones that used single quantum dots.^{5,6} Much of the work on quantum dots as entanglement sources has concentrated on maximizing polarization entanglement, for which elaborate growth and tuning techniques have been developed.^{7} Polarization entanglement can be converted probabilistically to time-bin entanglement^{6} or by using ultra-high-speed optical modulators, which, however, are always very lossy and thus do not allow a near-deterministic source. Therefore, we consider the direct creation of single time-bin entangled photon pairs from semiconductor quantum dots an important goal. The only known way to achieve this involves using at least three energy levels in the quantum dot, one of which must be metastable.^{8} While research into deterministic time-bin entanglement from quantum dots is being carried out in our laboratory, in this letter, as an intermediate step, we present the realization of probabilistic time-bin entanglement from a quantum dot in an optimized photonic structure.

In the past two decades, a lot of work has gone into improving the out-coupling efficiency of photons from quantum dots,^{9} e.g., via the implementation of circular Bragg gratings,^{10} enhancing emission into a collectable mode. Alternatively, realizing quantum dots embedded in tapered nanowires turned out to be a promising platform for coherent photon emission.^{11–14} The tapered part of the nanowire acts as an antenna that matches the impedance of the nanowire waveguide mode to the vacuum and thus achieves efficient out-coupling.^{15} In the following, we report the generation of time-bin entangled photon pairs in indium arsenide phosphide (InAsP) quantum dots embedded in indium phosphide (InP) nanowires via a resonant two-photon excitation^{16,17} (see Fig. 1). Furthermore, we present an extension of our theoretical model from previous work^{18} that includes the density matrix of time-bin entangled photons, which allows suggesting optimal parameter values.

## EXPERIMENT

The core of our setup is constituted by a quantum dot embedded in a nanowire. Our samples were manufactured utilizing a selective-area vapor–liquid–solid epitaxy, which produced InAsP quantum dots embedded in defect-free wurtzite InP nanowires.^{19} A single electron–hole pair trapped in the quantum dot is referred to as an exciton (|*x*⟩), while the confinement of two pairs is called a biexciton (|*b*⟩). A recombination of a single pair leads to the emission of a photon at a characteristic wavelength, as depicted in the inset of Fig. 1. The biexciton–exciton photon cascade is used in order to operate the quantum dot as a source of correlated photon pairs. The emission spectrum of our quantum dot can be found in Fig. S1 of the supplementary material.

The basic principle of the time-bin encoding scheme relies on the ability to create a coherent superposition of two well-defined excitation processes. Its simplest realization relies on addressing the emitter with two pump pulses of very low excitation probability each and postselecting biexciton–exciton photon pair detection events. The two excitation pulses are created in an unbalanced Mach–Zehnder interferometer and denoted by *e* (early) and *l* (late). The phase between the two pulses Δ_{ϕ} can be modified via a phase plate and determines the phase of the entangled state. Denoting biexciton and exciton photons by *b* and *x*, respectively, the created state can be written as

Using two other unbalanced Mach–Zehnder interferometers that are phase stable with respect to the pump interferometer, we carry out projective measurements on the created entangled state. In order to perform quantum state tomography, we analyze the result of 16 different pairs of phase settings and use a maximum likelihood approach.^{20,21} For collecting the 16 different projections necessary for the quantum state tomography, we employ four different phase settings in the analysis interferometers each and detect photons at each of the four output ports. We collect time tags of the detected photons for the 3600 s/phase setting and identify coincident photon pairs by imposing a coincidence window of 400 ps. The integration time was chosen such that it would yield sufficient statistics for the maximum likelihood reconstruction method.^{22}

For the generation of biexciton–exciton photon pairs, we employ resonant pulsed two-photon excitation from |*g*⟩ to |*b*⟩ (see the inset in Fig. 1). In order to calibrate and characterize the system, we observe Rabi oscillations by measuring the photon counts as a function of the average laser power, as shown in Fig. 2(a). We see that it is critical to identify an appropriate polarization as well as a sensible pulse duration. Choosing a circular pump polarization violates optical selection rules and leads to incoherent excitations rather than to a two-photon coherent coupling of the ground and biexciton state. By comparing the oscillations resulting from a linearly polarized pump and pulse lengths of 25 and 85 ps, we find a significantly stronger coherence for the longer pulse. Similar slopes at low excitation power of the biexciton and exciton emission probabilities for a linearly polarized pump indicate the superior pair production efficiency of this excitation scheme.

For the creation of time-bin entangled photons, we thus use the optimized pulse duration of 85 ps,^{18} resulting in a substantial increase in the excitation coherence, and we determine that the energy of a *π*/15-pulse is adequate, yielding an excitation probability of about 7.5%/pulse, which reduces the probability of emitting at both time bins to below 0.6%. Our theoretical model (see below) underpins the feasibility of the chosen parameters and provides the basis for even further improvements in future work.

## THEORETICAL MODEL

We propose a quantum optical model in order to identify a suitable set of parameter values to enhance the quality of entanglement of the created photon pairs. This allows us to estimate the reconstructed density matrix as shown in Fig. 4. Extending our earlier work,^{18} where we used a model for the internal dynamics of the quantum dot, we include a procedure for obtaining the photons’ density matrix from the said quantum dot dynamics. Our strategy is outlined in Fig. 3. The conceptual procedure introduced here is not restricted to this particular experimental setup and thus can be seen as a more fundamental framework for a general setting of time-bin entangled photons from a quantum dot. The pulsed laser driving field couples the ground state to the biexciton via the ground state to exciton and exciton to biexciton transition dipoles. The Hamiltonian in the effective interaction picture reads (for the derivation, see Sec. S4 of the supplementary material)

Here, Δ_{x} denotes the detuning from the exciton level to the laser frequency, Δ_{b} is the detuning from the biexciton level to the two-photon transition, and Ω(*t*) is the Rabi frequency featuring a Gaussian time profile,

with amplitude Ω_{0}, pulse duration (FWHM) *τ*, and time offset *t*_{0}.

To simulate the dynamics, we solve the master equation in Lindblad form numerically, i.e.,

where *ρ* = *ρ*_{dot}(*t*) is the quantum dot density matrix. We consider six dissipative channels associated with six different Lindblad operators *R*_{j}, where

describe the radiative decay of the biexciton and exciton levels with rates *γ*_{b} and *γ*_{x}, respectively, while

introduce dephasing. The rates $\gamma bx=\gamma bxI0\Omega (t)\Omega Sn$ and $\gamma xg=\gamma xgI0\Omega (t)\Omega Sn$ comprise their amplitudes $\gamma bxI0$ and $\gamma xgI0$, respectively, as well as the scaled time-dependent Rabi frequency to the *n*th power. Throughout this work, we consider *n* = 2. This laser intensity dependent dephasing can be explained by phonons coupling with the quantum dot.^{23} The scaling factor Ω_{S} = 1 THz accounts for the correct numerical values and leads to a unitless expression for the Rabi frequency. A minor role is attributed to the rates of constant dephasing $\gamma xgconst$ and $\gamma bxconst$ by the fit in Fig. 2(b).

In order to account for the decrease in photon counts for higher laser power, as depicted in Fig. 2(b), we introduce dark states that model a laser power dependent loss mechanism as states outside the three-level approximation become more prominent for higher laser powers. Moreover, this additional dark state loss counteracts the increased exciton population via a single photon transition that appears at higher laser intensities based on the broadening of the spectral linewidth due to the laser dependent dephasing. For bookkeeping purposes, we introduce two dark states |*d*_{x}⟩ and |*d*_{b}⟩, which are populated by laser dependent exciton and biexciton decay, respectively, whereas in general, one dark state would suffice to constitute the same effect. The corresponding Lindblad operators are given by

with laser intensity dependent decay rates $\gamma xd=\gamma xdI0\Omega (t)\Omega Sn$ and $\gamma bd=\gamma bdI0\Omega (t)\Omega Sn$, decay amplitudes $\gamma xdI0$ and $\gamma bdI0$, and the same power *n* as the dephasing mechanism.

Exemplary dynamics of the quantum dot when driven by a laser pulse are depicted in Fig. S2, and numerical values for the system parameters can be found in Tables S2 and S3 of the supplementary material.

In order to obtain the density matrix of the emitted photons from the quantum dot dynamics, we mimic the measurement of the photon coincidence counts in the experiment: first, we calibrate our model by fitting the emission probabilities

for *i* ∈ {*x*, *b*} to the biexciton and exciton Rabi data [see Fig. 2(b)]. This comprehensive and numerical demanding fitting loop is outlined in Sec. S2 of the supplementary material. Then, the density matrix of the quantum dot as a function of time *ρ*_{dot}(*t*) is the result of numerically solving the master equation. Relating the density matrix of the photons *ρ*_{photons} to *ρ*_{dot}(*t*) is achieved by calculating the resulting photon coincidence counts, where we derive analytic expressions for the detection probabilities of all 16 projective measurements (see S2 in the supplementary material) and subsequently use this estimate as an input for the conventional state tomography. This procedure is depicted schematically in Fig. 3. The density matrix resulting from this approach is shown in Fig. 4 and compared to the density matrix obtained in the experiment. In order to quantitatively compare the experimental results to our simulation, we employ the following definition of the fidelity for two mixed states:^{24}

The density matrix from theory and experiment has the same structural appearance as they show similar values at the prominent matrix elements. While the remaining entries of the simulated density matrix appear to be rather flat, we observe additional small fluctuations in these entries for the density matrix from the experiment.

## ENTANGLEMENT QUALITY OF THE PHOTONS

Ultimately, our goal is to achieve two-photon emission in a perfect Bell state,

Therefore, we identify suitable values for the laser intensity *I* ∝ Ω^{2} and its pulse duration *τ* in our simulation, which can assist in maximizing the fidelity,

to a Bell state in the experiment. Figure 5(a) shows a scan of the fidelity $F|\Phi +\u3009$ over the corresponding parameter space spanned by Ω_{0} and *τ*. Here, we study the influence of the parameter Ω_{0} instead of the intensity *I* as this parameter is more natural to the theoretical model. Once the model is calibrated to the experimental data, Ω_{0} can be converted to the average laser power.

Similar to the Rabi oscillations in Fig. 2, we observe an oscillatory pattern, which becomes less pronounced toward regions of higher energy (upper right corner). This can mostly be attributed to the intensity-dependent dephasing. For lower energies (lower left corner), the pattern roughly follows areas of constant energy, indicated by the yellow dashed lines. The red dot indicates the values chosen in the measurements that yield the reconstructed time-bin encoded photonic density matrix in Fig. 4. We show simulated density matrices for the same pulse length but different average laser powers in Fig. S4 of the supplementary material, where we observe an increase in the diagonal entries of the density matrix toward regions of lower fidelity, which means that the photonic state is becoming more classical in low fidelity regions for this pulse length. Reaching the regime of maximal fidelity has to be deferred to a future experimental setup, where our theoretical model can prove even more useful in fine-tuning the experimental parameters.

For a source of entangled photons, it is desirable to not only achieve a high fidelity but also yield sufficient output. Figure 5(b) depicts the normalized number of total expected counts of all simulated projective measurements (see the supplementary material, S2). Again, we observe an oscillatory behavior where we find some degree of anti-correlation between the pattern of the counts and the fidelity, i.e., dark areas with less output correspond to a relatively high fidelity, whereas bright areas are connected to a smaller fidelity. However, these two patterns are not perfectly anti-correlated as we find slightly varying fidelity for contours of constant counts. For some applications, a minimum amount of photons is required. Consequently, one might be interested in the optimal fidelity for a given photon count rate. For instance, we observe the fidelity along a contour of constant counts in the inset of Fig. 5(a). For this particular contour, we find the highest fidelity for long pulses with a relatively low intensity. In cases where the rate of output photons is not an issue, our study suggests that the optimal parameter regime is that of low pulse energy (lower left corner).

## CONCLUSIONS

In this work, we have shown the coherent coupling of the ground to biexciton state of an InAsP quantum dot embedded in an InP nanowire via an optimized two-photon resonant excitation scheme. We have used this method to generate time-bin entangled photons, yielding a fidelity of $F|\Phi +\u3009\u22480.90$ [see Eq. (14)] with respect to the maximally entangled |Φ^{+}⟩ Bell state.

In addition, we have presented a quantum optical model for simulating the dynamics of the quantum dot. By making use of the experimental reconstruction method, we have introduced a scheme for predicting the density matrix of the emitted photons based on the simulation of the dynamics of the quantum dot. The results of the model have been compared to the outcome of the experiment. With this, we are able to identify optimal parameter regimes in order to further increase the fidelity of the photons’ density matrix to a Bell state and to provide a more general toolbox for the study of time-bin entangled photons from a quantum dot.

## SUPPLEMENTARY MATERIAL

The supplementary material shows the quantum dot’s emission spectrum and details of the experimental methods. It features exemplary dynamics of the quantum dot upon excitation by a laser pulse and provides an in-depth mathematical assessment of the reconstruction of the photons’ density matrix from the quantum dot’s density matrix. Furthermore, it contains a summary of the chosen values for the simulation parameters, including the fit of the decay rates and a derivation of the Hamiltonian in Eq. (2).

## ACKNOWLEDGMENTS

P.A. and W.L. were supported by the Austrian Science Fund (FWF) through a START Grant under Project No. Y1067-N27 and SFB BeyondC Project No. F7108, the Hauser-Raspe Foundation, and the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 817482. M.P., F.K., and G.W. acknowledge partial support by the Austrian Science Fund (FWF) through Project Nos. W1259 (DK-ALM), I4380 (AEQuDot), and F7114 (SFB BeyondC). This work was supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR001120C0068. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the DARPA. We want to thank Doris Reiter and her group for fruitful discussions.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

P.A. and M.P. contributed equally to this work.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*et al.*, “