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Excitons in twisted bilayers of transition metal dichalcogenides have strongly modified dispersion relations due to the formation of periodic moir\'e potentials. The strong coupling to a light field in an optical cavity leads to the appearance of moir\'e polaritons. In this paper, we derive a theoretical model for the linear absorption spectrum of the coupled moir\'e polariton-phonon system based on the time-convolutionless (TCL) approach. Results obtained by numerically solving the TCL equation are compared to those obtained in the Markovian limit and from a perturbative treatment of non-Markovian corrections. A key quantity for the interpretation of the findings is the generalized phonon spectral density. We discuss the phonon impact on the spectrum for realistic moir\'e exciton dispersions by varying twist angle and temperature. Key features introduced by the coupling to phonons are broadenings and energy shifts of the upper and lower polariton peak and the appearance of phonon sidebands between them. We analyze these features with respect to the role of Markovian and non-Markovian effects and find that they strongly depend on the twist angle. We can distinguish between the regimes of large, small, and intermediate twist angles. In the latter phonon effects are particularly pronounced due to dominating phonon transitions into regions which are characterized by van Hove singularities in the density of states.

The probing of coherent lattice vibrations in solids has been conventionally carried out using time-resolved transient spectroscopy where only the relative oscillation amplitude can be obtained. Using time-resolved X-ray techniques, absolute electron-phonon coupling strength could be extracted. However, the complexity of such an experiment renders it impossible to be carried out in conventional laboratories. Here we demonstrate that the electron-phonon, anharmonic phonon-phonon coupling, and their relaxation dynamics can be probed in real-time using high-harmonic spectroscopy. Our technique is background-free and has extreme sensitivity directly in the energy domain. In combination with the optical deformation potential calculated from density functional perturbation theory and the absolute energy modulation depth, our measurement reveals the maximum displacement of neighboring oxygen atoms in {\alpha}-quartz crystal to tens of picometers in real space. By employing a straightforward and robust time-windowed Gabor analysis for the phonon-modulated high-harmonic spectrum, we successfully observe channel-resolved four-phonon scattering processes in such highly nonlinear interactions. Our work opens a new realm for accurate measurement of coherent phonons and their scattering dynamics, which allows for potential benchmarking ab-initio calculations in solids

The negatively charged nitrogen-vacancy (NV$^{-}$) center in diamond has shown great potential in nanoscale sensing and quantum information processing due to its rich spin physics. An efficient coupling with light, providing strong luminescence, is crucial for realizing these applications. Laser-written waveguides in diamond promote NV$^{-}$ creation and improve their coupling to light but, at the same time, induce strain in the crystal. The induced strain contributes to light guiding but also affects the energy levels of NV$^{-}$ centers. We probe NV$^{-}$ spin states experimentally with the commonly used continuous-wave zero-field optically detected magnetic resonance (ODMR). In our waveguides, the ODMR spectra are shifted, split, and consistently asymmetric, which we attribute to the impact of local strain. To understand these features, we model ensemble ODMR signals in the presence of strain. By fitting the model results to the experimentally collected ODMR data, we determine the strain tensor components at different positions, thus determining the strain profile across the waveguide. This shows that zero-field ODMR spectroscopy can be used as a strain imaging tool. The resulting strain within the waveguide is dominated by a compressive axial component transverse to the waveguide structure, with a smaller contribution from vertical and shear strain components., Comment: 12 pages, 7 figures + Supplementary Information (6 pages, 3 figures)

Chiral coupling, which allows directional interactions between quantum dots (QDs) and photonic crystal waveguide modes, holds promise for enhancing the functionality of quantum photonic integrated circuits. Elliptical polarizations of QD transitions offer a considerable enhancement in directionality. However, in epitaxial QD fabrication, the lack of precise control over lateral QD positions still poses a challenge in achieving efficient chiral interfaces. Here, we present a theoretical analysis in which we propose to optimize the polarization of a QD emitter against the spatially averaged directionality and demonstrate that the resulting emitter offers a considerable technological advantage in terms of the size and location of high-directionality areas of the waveguide as well as their overlap with the regions of large Purcell enhancement, thereby improving the scalability of the device. Moreover, using $\mathbf{\mathit{k}}\cdot\mathbf{\mathit{p}}$ modeling, we demonstrate that the optimal elliptical polarization can be achieved for neutral exciton transitions in a realistic QD structure. Our results present a viable path for efficient chiral coupling in QD-based photonic integrated circuits, to a large extent overcoming the challenges and limitations of the present manufacturing technology., Comment: Some text modifications in the Introduction, references added, typos corrected, Fig. 7 updated, and the title changed

In today's development of quantum technologies a hybrid integration of phononic excitations becomes increasingly attractive. As natural quasi-particle excitations in solid state systems, phonons couple to virtually any other excitation and therefore constitute a useful interaction channel between different building blocks in hybrid quantum systems. This work explores how the efficient light-scattering properties of a single-photon emitter and the appearance of characteristic sidebands in resonance fluorescence spectra, when interfaced with an arbitrary phonon quantum state, can be utilized for acousto-optical transduction. Within reasonable approximations, an analytical description for the optical spectra in the low excitation limit is developed which can be used to read the number statistics of the initial phonon state from a given spectrum. It is shown that the readout is faulty in situations where relevant resonant transitions are forbidden due to vanishing Franck-Condon factors, especially when considering spectra with a noisy background. Two possible solutions to this problem are presented: (A) changing the detuning of the laser relative to the single-photon emitter which modifies the relevant resonant transitions, or (B) increasing dissipation of the single-photon emitter to promote off-resonant transitions.

Semiconductor quantum dot molecules are considered as promising candidates for quantum technological applications due to their wide tunability of optical properties and coverage of different energy scales associated with charge and spin physics. While previous works have studied the tunnel-coupling of the different excitonic charge complexes shared by the two quantum dots by conventional optical spectroscopy, we here report on the first demonstration of a coherently controlled inter-dot tunnel-coupling focusing on the quantum coherence of the optically active trion transitions. We employ ultrafast four-wave mixing spectroscopy to resonantly generate a quantum coherence in one trion complex, transfer it to and probe it in another trion configuration. With the help of theoretical modelling on different levels of complexity we give an instructive explanation of the underlying coupling mechanism and dynamical processes.

Quantifying and controlling the coherent dynamics and couplings of optically active excitations in solids is of paramount importance in fundamental research in condensed matter optics and for their prospective optoelectronic applications in quantum technologies. Here, we perform ultrafast coherent nonlinear spectroscopy of a charge-tunable MoSe$_2$ monolayer. The experiments show that the homogeneous and inhomogeneous line width and the population decay of exciton complexes hosted by this material can be directly tuned by an applied gate bias, which governs the Fermi level and therefore the free carrier density. By performing two-dimensional spectroscopy, we also show that the same bias-tuning approach permits us to control the coherent coupling strength between charged and neutral exciton complexes.

Single-photon emitters in solid-state systems are important building blocks for scalable quantum technologies. Recently, quantum light emitters have been discovered in the wide-gap van der Waals insulator hBN. These color centers have attracted considerable attention due to their quantum performance at elevated temperatures and wide range of transition energies. Here, we demonstrate coherent state manipulation of a single hBN color center with ultrafast laser pulses and investigate in our joint experiment-theory study the coupling between the electronic system and phonons. We demonstrate that coherent control can not only be performed resonantly on the optical transition giving access to the decoherence but also phonon-assisted, which reveals the internal phonon quantum dynamics. In the case of optical phonons we measure their decoherence, stemming in part from their anharmonic decay. Dephasing induced by the creation of acoustic phonons manifests as a rapid decrease of the coherent control signal when traveling phonon wave packets are emitted. Furthermore, we demonstrate that the quantum superposition between a phonon-assisted process and the resonant excitation causes ultrafast oscillations of the coherent control signal. Our results pave the way for ultrafast phonon quantum state control on the nanoscale and open up a new promising perspective for hybrid quantum technologies.

We calculate the resonance fluorescence signal of a two-level system coupled to a quantized phonon mode. By treating the phonons in the independent boson model and not performing any approximations in their description, we also have access to the state evolution of the phonons. We confirm the validity of our model by simulating the limit of an initial quasi-classical coherent phonon state, which can be compared to experimentally confirmed results in the semiclassical limit. In addition we predict photon scattering spectra in the limit of purely quantum mechanical phonon states by approaching the phononic vacuum. Our method further allows us to simulate the impact of the light scattering process on the phonon state by calculating Wigner functions. We show that the phonon mode is brought into characteristic quantum states by the optical excitation process.

For future quantum technologies the combination of a long quantum state lifetime and an efficient interface with external optical excitation are required. In solids, the former is for example achieved by individual spins, while the latter is found in semiconducting artificial atoms combined with modern photonic structures. One possible combination of the two aspects is reached by doping a single quantum dot, providing a strong excitonic dipole, with a magnetic ion, that incorporates a characteristic spin texture. Here, we perform four-wave mixing spectroscopy to study the system's quantum coherence properties. We characterize the optical properties of the undoped CdTe quantum dot and find a strong photon echo formation which demonstrates a significant inhomogeneous spectral broadening. Incorporating the Mn$^{2+}$ ion introduces its spin-5/2 texture to the optical spectra via the exchange interaction, manifesting as six individual spectral lines in the coherent response. The random flips of the Mn-spin result in a special type of spectral wandering between the six transition energies, which is fundamentally different from the quasi-continuous spectral wandering that results in the Gaussian inhomogeneous broadening. Here, the discrete spin-ensemble manifests in additional dephasing and oscillation dynamics.

Monolayers of transition metal dichalcogenides display a strong excitonic optical response. Additionally encapsulating the monolayer with hexagonal boron nitride allows to reach the limit of a purely homogeneously broadened exciton system. On such a MoSe$_{2}$-based system we perform ultrafast six-wave mixing spectroscopy and find a novel destructive photon echo effect. This process manifests as a characteristic depression of the nonlinear signal dynamics when scanning the delay between the applied laser pulses. By theoretically describing the process within a local field model we reach an excellent agreement with the experiment. We develop an effective Bloch vector representation and thereby demonstrate that the destructive photon echo stems from a destructive interference of successive repetitions of the heterodyning experiment.

Quantum technologies that rely on photonic qubits require a precise controllability of their properties. For this purpose hybrid approaches are particularly attractive because they offer a large flexibility to address different aspects of the photonic degrees of freedom. When combining photonics with other quantum platforms like phonons, quantum transducers have to be realized that convert between the mechanical and optical domain. Here, we realize this interface between phonons in the form of surface acoustic waves (SAWs) and single photons, mediated by a single semiconductor quantum dot exciton. In this combined theoretical and experimental study, we show that the different sidebands exhibit characteristic blinking dynamics that can be controlled by detuning the laser from the exciton transition. By developing analytical approximations we gain a better understanding of the involved internal dynamics. Our specific SAW approach allows us to reach the ideal frequency range of around 1 GHz that enables simultaneous temporal and spectral phonon sideband resolution close to the combined fundamental time-bandwidth limit.

Using a novel approach to ultrafast resonant pump-probe spectroscopy we investigate the spectral shape and dynamics of absorption features related to the A exciton in an hBN/MoSe$_2$/hBN van der Waals heterostructure. While in a pure two-level system a pump-probe experiment measures the occupation or the polarization dynamics, depending on the time ordering of the pulse pair, in the transition metal dichalcogenide (TMD) system both quantities get thoroughly mixed by strong exciton-exciton interaction. We find that for short positive delays the spectral lines experience pronounced changes in their shape and energy and they relax to the original situation on a picosecond time scale. For negative delays distinctive spectral oscillations appear indicating the first-time observation of perturbed free induction decay for a TMD system. The comparison between co-circular and cross-circular excitation schemes further allows us to investigate the rapid inter-valley scattering. By considering a three-level system as a minimal model including the local field effect, excitation induced dephasing and scattering between the excited states we explain all phenomena observed in the experiment with excellent consistency. Our handy model can be even further reduced to two levels in the case of a co-circular excitation, for which we derive analytic expressions to describe the detected signals. This allows us to trace back the spectral shapes and shifts to the impact of local field effect and excitation induced dephasing thus fully reproducing the complex behavior of the observed effects.

In recent years the physics of two-dimensional semiconductors was revived by the discovery of the class of transition metal dichalcogenides. In these systems excitons dominate the optical response in the visible range and open many perspectives for nonlinear spectroscopy. To describe the coherence and polarization dynamics of excitons after ultrafast excitation in these systems, we employ the Bloch equation model of a two-level system extended by a local field describing the exciton-exciton interaction. We calculate four-wave mixing signals and analyze the dependence of the temporal and spectral signals as a function of the delay between the exciting pulses. Exact analytical results obtained for the case of ultrafast ($\delta$-shaped) pulses are compared to numerical solutions obtained for finite pulse durations. If two pulses are used to generate the nonlinear signal, characteristic spectral line splittings are restricted to short delays. When considering a three-pulse excitation the line splittings, induced by the local field effect, persist for long delays. All of the found features are instructively explained within the Bloch vector picture and we show how the exciton occupation dynamics govern the different four-wave mixing signals.

Color centers in hexagonal boron nitride (hBN) show stable single photon emission even at room temperature, making these systems a promising candidate for quantum information applications. Besides this remarkable property, also their interaction with longitudinal optical (LO) phonons is quite unique because they lead to dominant phonon sidebands (PSBs), well separated from the zero phonon line (ZPL). In this work we utilize this clear spectral separation to theoretically investigate the influence of phonon decay dynamics on time-dependent photoluminescence (PL) signals. Our simulations show, that by using tailored optical excitation schemes it is possible to create a superposition between the two LO modes, leading to a phonon quantum beat that manifests in the time-dependent PL signal.

The Jaynes-Cummings (JC) model represents one of the simplest ways in which single qubits can interact with single photon modes, leading to profound quantum phenomena like superpositions of light and matter states. One system, that can be described with the JC model, is a single quantum dot embedded in a micropillar cavity. In this joint experimental and theoretical study we investigate such a system using four-wave mixing (FWM) micro-spectroscopy. Special emphasis is laid on the dependence of the FWM signals on the number of photons injected into the microcavity. By comparing simulation and experiment, which are in excellent agreement with each other, we infer that up to ~20 photons take part in the observed FWM dynamics. Thus we verify the validity of the JC model for the system under consideration in this non-trivial regime. We find that the inevitable coupling between the quantum dot exciton and longitudinal acoustic phonons of the host lattice influences the real time FWM dynamics and has to be taken into account for a sufficient description of the quantum dot-microcavity system. Performing additional simulations in an idealized dissipation-less regime, we observe that the FWM signal exhibits quasi-periodic dynamics, analog to the collapse and revival phenomenon of the JC model. In these simulations we also see that the FWM spectrum has a triplet structure, if a large number of photons is injected into the cavity.

In quantum physics, two prototypical model systems stand out due to their wide range of applications. These are the two-level system (TLS) and the harmonic oscillator. The former is often an ideal model for confined charge or spin systems and the latter for lattice vibrations, i.e., phonons. Here, we couple these two systems, which leads to numerous fascinating physical phenomena. Practically, we consider different optical excitations and decay scenarios of a TLS, focusing on the generated dynamics of a single phonon mode that couples to the TLS. Special emphasis is placed on the entropy of the different parts of the system, predominantly the phonons. While, without any decay, the entire system is always in a pure state, resulting in a vanishing entropy, the complex interplay between the single parts results in non-vanishing respective entanglement entropies and non-trivial dynamics of them. Taking a decay of the TLS into account leads to a non-vanishing entropy of the full system and additional aspects in its dynamics. We demonstrate that all aspects of the entropy's behavior can be traced back to the purity of the states and are illustrated by phonon Wigner functions in phase space.

When an electron-hole pair is optically excited in a semiconductor quantum dot the host crystal lattice needs to adapt to the presence of the generated charge distribution. Therefore the coupled exciton-phonon system has to establish a new equilibrium, which is reached in the form of a quasiparticle called polaron. Especially, when the exciton is abruptly generated on a timescale faster than the typical lattice dynamics, the lattice displacement cannot follow adiabatically. Consequently, a rich dynamics on the picosecond timescale of the coupled system is expected. In this study we combine simulations and measurements of the ultrafast, coherent, nonlinear optical response, obtained by four-wave mixing spectroscopy, to resolve the formation of this polaron. By detecting and investigating the phonon sidebands in the four-wave mixing spectra for varying pulse delays and different temperatures we have access to the influence of phonon emission and absorption processes which finally result in the emission of an acoustic wave packet out from the quantum dot., Comment: accepted in Optics Letters

Wave mixing is an archetypical phenomenon in bosonic systems. In optomechanics, the bi-directional conversion between electromagnetic waves or photons at optical frequencies and elastic waves or phonons at radio frequencies is building on precisely this fundamental principle. Surface acoustic waves provide a versatile interconnect on a chip and, thus, enable the optomechanical control of remote systems. Here, we report on the coherent nonlinear three-wave mixing between the coherent fields of two radio frequency surface acoustic waves and optical laser photons via the dipole transition of a single quantum dot exciton. In the resolved sideband regime, we demonstrate fundamental acoustic analogues of sum and difference frequency generation between the two SAWs and employ phase matching to deterministically enhance or suppress individual sidebands. This bi-directional transfer between the acoustic and optical domains is described by theory which fully takes into account direct and virtual multi-phonon processes. Finally, we show that the precision of the wave mixing is limited by the frequency accuracy of modern radio frequency electronics., Comment: Submitted Manuscript

The coupling between single-photon emitters and phonons opens many possibilities to store and transmit quantum properties. In this paper we apply the independent boson model to describe the coupling between an optically driven two-level system and a discrete phonon mode. Tailored optical driving allows not only to generate coherent phonon states, but also to generate coherent superpositions in the form of Schr\"odinger cat states in the phonon system. We analyze the influence of decay and dephasing of the two-level system on these phonon preparation protocols. We find that the decay transforms the coherent phonon state into a circular distribution in phase space. Although the dephasing between two exciting laser pulses leads to a reduction of the interference ability in the phonon system, the decay conserves it during the transition into the ground state. This allows to store the phonon quantum state properties in the ground state of the single-photon emitter.

Defect centers in hexagonal boron nitride represent room-temperature single-photon sources in a layered van der Waals material. These light emitters appear with a wide range of transition energies ranging over the entire visible spectrum, which renders the identification of the underlying atomic structure challenging. In addition to their eminent properties as quantum light emitters, the coupling to phonons is remarkable. Their photoluminescence exhibits significant side band emission well separated from the zero phonon line (ZPL) and an asymmetric broadening of the ZPL itself. In this combined theoretical and experimental study we show that the phonon side bands can be well described in terms of the coupling to bulk longitudinal optical (LO) phonons. To describe the ZPL asymmetry we show that in addition to the coupling to longitudinal acoustic (LA) phonons also the coupling to local mode oscillations of the defect center with respect to the entire host crystal has to be considered. By studying the influence of the emitter's wave function dimensions on the phonon side bands we find reasonable values for size of the wave function and the deformation potentials. We perform photoluminescence excitation measurements to demonstrate that the excitation of the emitters is most efficient by LO-phonon assisted absorption., Comment: Accepted, Including Supplementary Material, Licence: https://creativecommons.org/licenses/by/3.0/ (preprint published online: https://iopscience.iop.org/article/10.1088/2053-1583/ab1188)

While the advanced coherent control of qubits is now routinely carried out in low frequency (GHz) systems like single spins, it is far more challenging to achieve for two-level systems in the optical domain. This is because the latter evolve typically in the THz range, calling for tools of ultrafast, coherent, nonlinear optics. Using four-wave mixing micro-spectroscopy, we here measure the optically driven dynamics of a single exciton quantum state confined in a semiconductor quantum dot. In a combined experimental and theoretical approach, we reveal the intrinsic Rabi oscillation dynamics by monitoring both central exciton quantities, i.e., its occupation and the microscopic coherence, as resolved by the four-wave mixing technique. In the frequency domain this oscillation generates the Autler-Townes splitting of the light-exciton dressed states, directly seen in the four-wave mixing spectra. We further demonstrate that the coupling to acoustic phonons strongly influences the FWM dynamics on the picosecond timescale, because it leads to transitions between the dressed states., Comment: accepted for publication in Optica

The exact optical response of quantum few-level systems depends crucially on the exact choice of the incoming pulse areas. We use four-wave mixing (FWM) spectroscopy to infer the coherent response and dynamics of single InAs quantum dots (QDs) and study their pulse area dependence. By combining atomic force microscopy with FWM hyperspectral imaging, we show that the retrieved FWM signals originate from individual QDs enclosed in natural photonic defects. The optimized light-matter coupling in these defects allows us to perform our studies in a wide range of driving field amplitudes. When varying the pulse areas of the exciting laser pulses the so-called Rabi rotations can be resolved by the two-pulse FWM technique. We investigate these Rabi rotations within two- and three-level systems, both theoretically and experimentally, and explain their damping by the coupling to acoustic phonons. To highlight the importance of the pulse area influence, we show that the phonon-induced dephasing of QD excitons depends on the pulse intensity.

A picosecond acoustic pulse can be used to control the lasing emission from semiconductor nanostructures by shifting their electronic transitions. When the active medium, here an ensemble of (In,Ga)As quantum dots, is shifted into or out of resonance with the cavity mode, a large enhancement or suppression of the lasing emission can dynamically be achieved. Most interesting, even in the case when gain medium and cavity mode are in resonance, we observe an enhancement of the lasing due to shaking by coherent phonons. In order to understand the interactions of the non-linearly coupled photon-exciton-phonon subsystems, we develop a semiclassical model and find an excellent agreement between theory and experiment.

The study of the fundamental properties of phonons is crucial to understand their role in applica- tions in quantum information science, where the active use of phonons is currently highly debated. A genuine quantum phenomenon associated with the fluctuation properties of phonons is squeezing, which is achieved when the fluctuations of a certain variable drop below their respective vacuum value. We consider a semiconductor quantum dot in which the exciton is coupled to phonons. We review the fluctuation properties of the phonons, which are generated by optical manipulation of the quantum dot, in the limiting case of ultra short pulses. Then we discuss the phonon properties for an excitation with finite pulses. Within a generating function formalism we calculate the corre- sponding fluctuation properties of the phonons and show that phonon squeezing can be achieved by the optical manipulation of the quantum dot exciton for certain conditions even for a single pulse excitation where neither for short nor for long pulses squeezing occurs. To explain the occurrence of squeezing we employ a Wigner function picture providing a detailed understanding of the induced quantum dynamics.