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Resonant and phonon-assisted ultrafast coherent control of a single hBN color center

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Abstract

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 hexagonal boron nitride (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 be performed not only 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 in the coherent control signal when traveling phonon wave packets are emitted. Furthermore, we demonstrate that the quantum superposition between a phonon-assisted process and 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.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. INTRODUCTION

The ultrafast optical coherent manipulation of individual quantum systems is a key challenge in the development of quantum technologies [1]. Over the last decades, several different solid-state systems, such as color centers in diamond [2], perovskite [3], and colloidal [4] and self-assembled quantum dots [5], have been realized, improved, and utilized [6]. Recently, the family of quantum emitters has been joined by single-photon sources in layered van der Waals materials [7,8]. Compared to bulk materials, these anisotropic crystals with their two-dimensional character are particularly promising due to their structural flexibility [9]. Emitters have been found not only in semiconducting transition metal dichalcogenides [1014] but also in insulating hexagonal boron nitride (hBN) in the form of color centers [15]. The latter have attracted massive attention because of their efficient single-photon emission at room temperature, emission over a wide spectral range spanning the near-infrared [16], visible [17] and ultraviolet [18], and spectral tunability via external electric fields [19] or strain [20]. Furthermore, their characteristic coupling to phonons is remarkable. Due to the polar character, the color centers couple to both acoustic and optical phonon modes, giving rise to strong phonon sidebands at room temperature [21,22]. The bright emitters are found not only in deliberately grown samples [23] but also in commercially available nanocrystals, which makes their processability particularly easy [24]. Although some progress has been made regarding resonant optical excitation by demonstrating Rabi oscillations in ${g^{(2)}}$-measurement [25] or the detection of lifetime-limited linewidths [26], the microscopic impact of phonons still requires deeper understanding. Studies indicate that there might be a connection between the phonon coupling and the emission linewidth of the emitter [27], but a consistent study in the spectral and temporal domains, taking also phonon-assisted processes into account, has been pending. Here, we present the ultrafast coherent optical control of a single hBN color center. We demonstrate the role of different spectral jitter environments and reveal the impact of different phononic excitations. By combining experiment and theory, we explicitly quantify the emitter’s pure and phonon-assisted coherence and at the same time access the impact of spectral noise. By performing a systematic study and thorough simulations, we reach a consistent picture of the emitter’s coherence properties in both spectral and temporal domains and unravel the role of phonons in this context. To achieve an excellent agreement between the measurements and simulation, we need to go beyond the calculation of correlation functions in the independent boson model [22] and also take into account the exact optical excitation and detection conditions, as well as phonon decay dynamics.

 figure: Fig. 1.

Fig. 1. Light emission from an individual hBN color center. (a) Photoluminescence (PL) and photoluminescence excitation (PLE) spectra in red and blue, respectively. Measurement (dark red) and simulation (light red) with green shaded areas marking the different phonon sidebands (PSBs). The zero-phonon line (ZPL) energy is ${E_{{\rm{ZPL}}}} = 2.036\,\,{\rm{eV}}$. (b) Schematic drawing of the two-level structure (solid lines) including ZPL (violet), phonon-assisted processes via virtual excited states (dashed lines) as PSBs (green), and PLE (blue).

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2. RESULTS

A. Optical Emission

For our experiments, we use commercially available hBN nanopowder crystals, which host bright and stable single-photon emitters [22,24,28]. Details about sample preparation, the optical setup, and data acquisition are given in Appendix A and Supplement 1, Sec. S1. Figure 1(a) displays the light emission from an individual color center. Single-photon emission is confirmed by a photon correlation measurement using a Hanbury Brown and Twiss setup presented in Supplement 1, Sec. S1.C. We measure (dark lines) and calculate (bright lines) photoluminescence (PL, red) and PL excitation spectra (PLE, blue). The PL spectrum (dark red) is measured at a temperature of $T = 80\,\,{\rm{K}}$ under pulsed excitation at 2.20 eV with a pulse duration of 230 fs. We find the characteristic sharp zero-phonon line (ZPL) at ${E_{{\rm{ZPL}}}} = 2.036\,\,{\rm{eV}}$, i.e., 160 meV below the excitation energy. The ZPL is accompanied by typical peaked longitudinal optical (LO) phonon sidebands (PSBs) between 150 meV and 200 meV below the ZPL and a broad, continuous PSB stemming from longitudinal acoustic (LA) modes, exceeding $E - {E_{{\rm{ZPL}}}} = - 100\,\,{\rm{meV}}$. We also detect second order PSBs for the sharp LO PSBs in the range of twice the LO phonon energies below the ZPL, i.e., between ${-}300\,\,{\rm{meV}}$ and ${-}400\,\,{\rm{meV}}$. To measure the PLE spectrum, we scan the excitation laser energy above the ZPL and detect the PL integrated over ZPL and PSBs. The respective data in dark blue show prominent maxima between $E - {E_{{\rm{ZPL}}}} = 150\,\,{\rm{meV}}$ and 200 meV, which excellently agrees with the LO phonon energies. This demonstrates that the emitter is efficiently excited via a LO-phonon-assisted process [22,29,30]. Such a phonon-assisted excitation was, for example, combined with stimulated emission depletion experiments that, when compared to PLE measurements, revealed a redshift of one LO mode in the excited state [31]. While at room temperature, two prominent optical PSBs ${{\rm{LO}}_1}$ at 165 meV and ${{\rm{LO}}_2}$ at 200 meV are typically observed [21,22], at cryogenic temperatures, the ${{\rm{LO}}_1}$ emission exhibits two additional lines (see, e.g., Fig. 6 in Ref. [22]), marked as ${\rm{L}}{{\rm{O^\prime}}_1}$ and ${\rm{L}}{\rm{O_{1}^{\prime \prime}}}$ in Fig. 1(a). We assume that resonances ${{\rm{LO}}_1}$ and ${{\rm{LO}}_2}$ stem from the phonon dispersion maxima at the $\Gamma$ point, and additional resonances ${\rm{L}}{{\rm{O^\prime}}_1}$ and ${{\rm{LO_{1}^{\prime \prime}}}}$ from Van Hove singularities at non-vanishing lattice momenta [32,33]. The second LO-PSBs between $E - {E_{{\rm{ZPL}}}} = - 300$ and ${-}400\,\,{\rm{meV}}$ can be derived from the four lines ${{\rm{LO}}_1}$, ${\rm{L}}{{\rm{O^\prime}}_1}$, ${{\rm{LO_{1}^{\prime \prime}}}}$, and ${{\rm{LO}}_2}$ by summing all 10 possible combinations of them [Fig. 1(a)].

The model, schematically displayed in Fig. 1(b), includes an optically driven two-level system (TLS) with ground $| G \rangle$ and excited $| X \rangle$ states, the spectrum of the laser pulse, the detection process, and the coupling to phonons giving rise to the characteristic sidebands [see Eqs. (A2) and (A6) for details]. The coupling to phonons is considered via the independent boson model [34] taking into account a continuous distribution of LA phonons via a standard super-ohmic spectral density with Lorentz–Drude cutoff and four discrete LO modes via the Einstein model [35]. Anharmonic decay of the LO modes into lower energy phonons and pure dephasing of the emitter, both leading to a finite width of spectral lines, are included via Lindblad dissipators [36] [see Eq. (A1) and Supplement 1, Sec. S2.A]. The phonon parameters remain unchanged throughout this work and are summarized together with all other parameters used in the model in Supplement 1, Sec. S3.E. The simulated spectrum of the second LO-PSBs agrees well with the measured one, which indicates that all PSBs marked by green areas in Fig. 1(a) belong to the same ZPL. The narrow peaks at $E - {E_{{\rm{ZPL}}}} \approx - 40\,\,{\rm{meV}}$ and ${-}340\,\,{\rm{meV}}$ (marked by $*$) do not belong to the investigated emitter (see Supplement 1, Sec. S1.E) and are therefore not considered in the following.

Overall, we find an excellent agreement between the measured spectroscopy data and the theoretical model considering one single bright state coupled to phonons [Fig. 1(b)]. All aspects of the measured time-integrated optical spectra of the emitter, namely, all peak positions and their intensities and widths in both PL and PLE, are reproduced by our model. The applicability of the model is further confirmed by measurements and simulations at a temperature of $T = 8\,\,{\rm{K}}$, which are shown in Supplement 1, Sec. S3.C. In the following, we demonstrate that ultrafast coherent control of the emitter is consistently reproduced and explained in the considered model.

 figure: Fig. 2.

Fig. 2. Ultrafast coherent control experiment on a single-photon emitter in hBN. (a) Schematic of the coherent control experiment: (top) constructive timing of the second pulse, increasing the emitter occupation; (bottom) destructive timing decreasing the occupation. (b) Measured variations of PL intensity for a small coarse delay ($\Delta t = 1\,\,{\rm{ps}}$, left) and large coarse delay ($\Delta t = 33\,\,{\rm{ps}}$, right) for resonant excitation on the ZPL at $T = 8\,\,{\rm{K}}$. The upper $x$ axis marks the relative phase between emitter and second pulse. (c) PL visibility dynamics for resonant excitation on the ZPL. Dark circles represent the experiment, and the solid curve represents the simulation.

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B. Coherent Control Experiment

The time-integrated PL spectrum of a quantum emitter provides, however, only very limited insight into its coherence properties and the impact of specific environments such as spectral noise or lattice vibrations. To unravel these aspects, we perform ultrafast coherent control experiments on a single quantum emitter. We apply a double-pulse sequence, where the two laser pulses have equal intensities and a fixed phase relation, as schematically shown in Fig. 2(a) [3743]. To gain a general understanding of the coherent control signal, it is instructive to first consider the TLS without any phonon coupling. The first laser pulse creates a coherent superposition $| \psi \rangle = {\alpha _G}| G \rangle + {\alpha _X}| X \rangle$ of ground and excited states of the emitter, characterized by its occupation $f = \langle {| X \rangle} {\langle X |} \rangle = |{\alpha _X}{|^2}$ and coherence $p = \langle {| G \rangle} {\langle X |} \rangle = |\alpha _G^*{\alpha _X}|{e^{- i{\phi _0}}}$, as depicted as blue and red curves in Fig. 2(a), respectively. The occupation typically decays within a few nanoseconds [22] and can therefore be assumed constant during the considered time scales in the picosecond range. The initial phase ${\phi _0}$ of the quantum coherence is determined by the phase of the first laser pulse. During the following dynamics, it rotates with the optical transition frequency of the emitter $\phi (t) = {\phi _0} + {\omega _{{\rm{ZPL}}}}t$, with ${\omega _{{\rm{ZPL}}}} = {E_{{\rm{ZPL}}}}/\hbar = 2\pi /{T_{{\rm{ZPL}}}} \approx 3\;{\rm{f}}{{\rm{s}}^{- 1}}$, and its amplitude $|p|$ is damped with the pure dephasing rate ${\gamma _{{\rm{pd}}}}$. After a delay time $\Delta t$, the second laser pulse interacts with the quantum emitter. The final occupation of the quantum system depends on the relative phase between the second pulse and the quantum coherence. Because the applied pulse pair has a fixed phase relation, the phase difference between pulse and quantum state can be varied by tuning the time delay between the pulses $\Delta t$. This is visualized in Fig. 2(a) by the two distinct timings for the second pulse (green Gaussians). When the second pulse arrives with a delay $\Delta {t_{{\rm{constr.}}}}$ such that the emitter’s occupation gets efficiently enhanced [Fig. 2(a), top], we call this “constructive.” In contrast, as shown in the bottom of Fig. 2(a) for a delay of half a coherence period later, $\Delta {t_{{\rm{constr.}}}} + {T_{{\rm{ZPL}}}}/2 = \Delta {t_{{\rm{destr.}}}}$, the impact of the second pulse is “destructive,” and the occupation is decreased. The occupation after the second pulse will therefore oscillate with the frequency ${\omega _{{\rm{ZPL}}}}$ as a function of the pulse delay $\Delta t$. If the coherence has decayed entirely, the impact of the second pulse does not depend on the delay anymore, because the system is in a statistical mixture of $| G \rangle$ and $| X \rangle$, with $|p| = 0$.

After the second laser pulse, the occupation of the emitter decays fully into the ground state within its nanosecond radiative lifetime by emitting a photon. The next pulse pair hits the emitter at the repetition time of the laser (ns), which is long enough to guarantee the full decay of the emitter. Note that a potential decay into a long-lived intermediate state with a lifetime longer than the repetition time does not disturb the coherent control experiment. Such a state would not be affected by the next pulse pair and therefore does not contribute to the detected signal. The measured PL intensity is proportional to the occupation of the emitter, and we can compare the PL between constructive and destructive action of the second pulse within one period ${T_{{\rm{ZPL}}}}$. With this, we define the visibility of the coherent control experiment via

$${v_{{\rm{PL}}}} = \frac{{{\max}({I_{{\rm{PL}}}}) - {\min}({I_{{\rm{PL}}}})}}{{{\max}({I_{{\rm{PL}}}}) + {\min}({I_{{\rm{PL}}}})}} ,$$
which is a measure for the quantum coherence of the color center. In the experiment, the visibility is retrieved from the data by a more involved procedure to reduce the impact of noise, as discussed in Supplement 1, Sec. S1.D. Details on the simulation of the visibility are given in Supplement 1, Sec. S2.D.3.

In the experiment, we use laser pulses (duration $1/{\alpha _{{l}}} = 230\,\,{\rm{fs}}$) provided by a femtosecond Er:fiber laser system with nonlinear conversion stages (see Appendix A). The visibility of the coherent control signal is detected by varying the delay on a femtosecond time scale (rotation of the phase), while the typical decay of the signal due to decoherence is on a picosecond time scale. Therefore, we use a Mach–Zehnder interferometer to create two pulses with a coarse delay varying on a picosecond time scale and a fine delay scanning over a few femtoseconds [see Fig. 2(a) and Supplement 1, Sec. S1.B]. While changing the fine delay $\Delta {t_{\rm{f}}}$, we detect the PL of the emitter via the PSBs and observe the oscillating signal depicted in Fig. 2(b). The dynamics cover several oscillations of the quantum coherence as marked by the relative phase between the emitter and the second pulse on the upper $x$ axis. For small coarse delay times ($\Delta t = 1\,\,{\rm{ps}}$), the oscillation amplitude is close to unity [Fig. 2(b), left panel] due to the almost perfect constructive and destructive interaction for short delays, implying ${v_{{\rm{PL}}}} \approx 1$ as discussed in Supplement 1, Sec. S1.D. As mentioned earlier, for longer delays ($\Delta t = 33\,\,{\rm{ps}}$, right panel) the decoherence of the system before the second pulse results in a reduced oscillation amplitude of the detected PL intensity and consequently a smaller coherent control visibility.

 figure: Fig. 3.

Fig. 3. Jitter dynamics and time scales for coherent control. (a) Illustration of the spectral jitter changing the transition energy in time. (b) Illustration of the repetition of the coherent control experiment. (c), (d) Fast and slow jitter components with corresponding homogeneous and inhomogeneous decays of the PL visibility. (e) Illustration of time scales in the experiment. Quantities marked in blue (homogeneous decoherence) originate from fast jitter and those in green [random telegraph noise (RTN), inhomogeneous decoherence] from slow jitter.

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C. Resonant Excitation

Figure 2(c) displays measurements (dark circles) and corresponding simulations (solid curve) of the coherent control PL visibility for delays between $\Delta t = 0$ and 35 ps at a temperature of $T = 8\,\,{\rm{K}}$. The optical excitation is in resonance with the ZPL and results in a steady decay of visibility over time. The first crucial observation is that this decay cannot be fitted by a single mono-exponential or Gaussian function (see Supplement 1, Sec. S3.D). Instead, the measured decoherence dynamics are excellently reproduced by the theory when considering a combination of an exponential (homogeneous/pure) and a Gaussian (inhomogeneous) dephasing contribution via [see Eq. (A5) and discussion after Eq. (A7) in Appendix A]

$$\begin{split}{v_{{\rm{PL}}}}(\Delta t) &= \exp \left[{- \frac{{{\gamma _{{\rm{pd}}}}}}{2}\Delta t - \frac{{\sigma _{\rm{j}}^2{{(\Delta t)}^2}}}{2}} \right] ,\quad {\rm{with}}\\ 2/{\gamma _{{\rm{pd}}}} &= 55\;{\rm{ps}}\quad {\rm{and}}\quad 1/{\sigma _{\rm{j}}} = 19\;{\rm{ps}}.\end{split}$$

As illustrated in Fig. 3, these two dephasing contributions can be identified with spectral jitter components (also often referred to as spectral noise or spectral diffusion) [27], which act on different characteristic time scales. The critical value for this time scale is the repetition of the two-pulse coherent control experiment, which is ${T_{{\rm{rep}}}} = 25\,\,{\rm{ns}}$ in our case [Fig. 3(b)]. If the considered spectral jitter component is faster than the delay between pulses, the transition energy of the emitter is likely to change between the two pulses in a single run of the experiment [Fig. 3(c)]. This disturbs the quantum phase and results in an exponential loss of coherence [44], which is usually referred to as homogeneous dephasing. If the jitter is slower than the repetition time of the experiment, it is more likely that the transition energy of the emitter changes between single runs of the experiment [Fig. 3(d)] [45]. In this situation, the successive repetition of the coherent control experiment can be treated as an ensemble average over the spectral distribution representing the jitter (see Supplement 1, Sec. S2.D.5 for details). In our case, the Gaussian dephasing dynamics correspond to a Gaussian spectral distribution of the jitter-induced energy shifts, which is usually called inhomogeneous broadening. The combined spectral jitter is schematically shown in Fig. 3(a). Figure 3(e) and Table 1 provide an overview of various time scales including fast and slow spectral jitter. In addition, the table lists the determined dephasing times and their corresponding widths in optical spectra and in which figures the respective time scales are found.

Tables Icon

Table 1. List of the Different Time Scales in the Experimenta

From homogeneous and inhomogeneous dephasing, we estimate an effective emitter coherence time, where Eq. (2) has dropped to $1/e$ of the initial amplitude, to

$${T_{{\rm{eff}}}} = \frac{{{\gamma _{{\rm{pd}}}}}}{{2\sigma _{\rm{j}}^2}}\left({\sqrt {8\sigma _{\rm{j}}^2/\gamma _{{\rm{pd}}}^2 + 1} - 1} \right) \approx 21\;{\rm{ps}} .$$

This value is of the same order of magnitude as previously reported coherence times of around 80 ps in a hBN emitter [46]. ${T_{{\rm{eff}}}} \approx 21 {\rm{ps}}$ indicates that the investigated emitter experiences strong spectral jitter on different time scales. The jitter might stem from laser-induced charge fluctuations in hBN or the substrate [47], which influence the transition energy of the quantum emitter, e.g., via the Stark effect.

 figure: Fig. 4.

Fig. 4. Coherent control experiment with telegraph noise. (a) PL spectra as a function of detection time covering 100 s showing spectral jumps between two energies. (b) Time-integrated PL spectrum retrieved from (a). ZPL energy of ${E_{{\rm{ZPL}}}} = 2.036\,\,{\rm{eV}}$ is defined as the mean value of two peak positions. (c) Coherent control visibility dynamics of double-peaked ZPL. Dark circles represent the experiment, and solid curves represent the simulation.

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D. Telegraph Noise

Interestingly, in the course of the experiments after cooling from room temperature to cryogenic temperatures, the same emitter sometimes performed random jumps between two distinct emission energies, i.e., random telegraph noise (RTN), on a millisecond time scale as shown in Fig. 4(a) [48]. When assuming that the spectral jitter is induced by variations of the local surrounding of the emitter, heating and cooling cycles can likely result in varying noise situations. For each time step in this measurement, the PL was integrated over 10 ms. As explained in the previous section (see also Table 1), this spectral jitter is slow and can therefore be treated as a spectral ensemble. The RTN character is evident because one of the emission lines is dim when the other is bright. We also find slight variations in the peak positions, hinting towards an additional fluctuation of peak energies with a smaller amplitude on the order of the spectral linewidth. The amplitude of this jitter is in the range of a few tens of µeV (see Supplement 1, Sec. S1.F for details), which is in reasonable agreement with the value determined from inhomogeneous decoherence $\hbar {\sigma _{\rm{j}}} = 35\,\,\unicode{x00B5}{\rm{eV}}$ in the coherent control experiment in the previous section. After time integration, the RTN results in the double-peak structure of the ZPL’s PL spectrum in Fig. 4(b). The corresponding coherence dynamics in Fig. 4(c) exhibit a characteristic beat, given by the energy splitting of the RTN of $\Delta {E_{{\rm{ZPL}}}} = 0.7\,\,{\rm{meV}}$. Also, this behavior is reproduced by our model when additionally considering the RTN as an ensemble average [see discussion after Eq. (A7) in Appendix A]. These results demonstrate that even in the presence of different sources of spectral jitter, i.e., fast jitter leading to homogeneous decoherence, slow Gaussian noise resulting in inhomogeneous decoherence, and RTN introducing a beat, coherent control of the quantum state of an individual hBN color center is possible. The respective measurements of the coherence dynamics allow us to determine the present noise characteristics with high precision.

 figure: Fig. 5.

Fig. 5. Coherent control of optical phonon sidebands (PSBs). (a) Coherent control visibility dynamics for LO-phonon-assisted excitation in blue: ${{\rm{LO}}_1}$ mode, solid blue curve; ${{\rm{LO}}_2}$ mode, dashed blue curve; circles represent the experiment, and curves represent the simulation. In addition, the visibility for resonant excitation is shown in purple and laser autocorrelation is shown in green. The respective PLE spectra of the LO PSBs are shown in the inset. (b) Illustration of phonon-assisted coherence decay. The harmonic potentials represent phonon excitations in ground $| G \rangle$ and excited $| X \rangle$ states. The coherence is indicated by the orange ellipse. Phonon-induced dephasing (green arrow) results in the loss of coherence (dashed ellipse).

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E. Optical Phonons

Phonon-assisted excitation and emission have a remarkable strength in hBN color centers [21,22,29,49,50], with PSBs reaching peak intensities of almost 10% of the ZPL at $T = 80\,\,{\rm{K}}$, as discussed for the PL and PLE spectra in Fig. 1. We now detune the carrier frequency of the applied laser pulses to study the emitter’s coherence when phonon-assisted excitations are addressed. As discussed throughout Supplement 1 and in Appendix A, the experimental results crucially depend on the spectral widths of the laser and the emitter. To reduce the impact of slow spectral jitter, we increase the temperature to $T = 80\,\,{\rm{K}}$, which results in a broadening of spectral features, especially the ZPL. This generates a more robust situation, as the slow jitter has a reduced relative impact on broadening. Note that these experiments were performed when the emitter did not show any RTN, i.e., only a single ZPL was present. The violet data in Fig. 5(a) (dark squares: experiment, bright line: simulation) shows the coherent control visibility for resonant ZPL excitation, which exhibits a dephasing time of $2/{\gamma _{{\rm{pd}}}} = 3.5\,\,{\rm{ps}}$ [see Eq. (A5)]. This more than 10-fold increase in the decoherence rate compared to $T = 8\,\,{\rm{K}}$ confirms the temperature-dependent broadening of the ZPL [see Eq. (A2) in Appendix A]. We now tune the pulsed optical excitation to the LO PSBs, ${{\rm{LO}}_1}$ (filled circles, solid line) and ${{\rm{LO}}_2}$ (open circles, dashed line), as depicted in the inset of Fig. 5(a). Both resonances of the PSBs in the PLE spectrum stem from flat parts of the phonon dispersion relation with a vanishing group velocity [32]. Consequently, the created phonons remain localized in the region of the color center’s wave function. In Fig. 5(a), we plot the measured (dark blue circles) and simulated (bright blue lines) visibility of the coherent control signals, i.e., the coherence dynamics, for optical excitations on ${{\rm{LO}}_1}$ and ${{\rm{LO}}_2}$ PSBs above the ZPL. As a reference, we additionally display the autocorrelation signal of the laser (green, dark diamonds: experiment, bright line: simulation; see Supplement 1, Sec. S3.B). It is measured by directly interfering the two laser pulses without interacting with the sample and calculated via Eq. (A8). We find that for LO-assisted excitations, the coherence decays significantly faster than for the resonant excitation on the ZPL (violet), but decays slower than the autocorrelation of the laser pulses.

As schematically shown in Fig. 5(b), the first laser pulse (blue arrow) creates a phonon-assisted coherence between states $| {G,0} \rangle$ and $| {X,1} \rangle$, where the second entry describes the number of excited phonons (orange ellipse). Any phonon-induced dephasing (PID) then leads to the destruction of the quantum coherence among the optical phonon, ground state, and excited state of the quantum emitter (dashed ellipse). This process is naturally faster than decoherence without phonons. One obvious source of PID is the finite lifetime of optical phonons, which typically decay anharmonically into lower energy modes [51]. This process is given by the transition $| {X,1} \rangle \to | {X,0} \rangle$. As shown in Supplement 1, Sec. S2.E.3, a finite LO lifetime has a similar impact on the coherent control signal as a non-vanishing dispersion of the optical modes, which is another possible source of decoherence. To reproduce the measured visibility dynamics in Fig. 5(a), we find phonon decay times of $1/{\gamma _{{{\rm{LO}}_{{1}}}}} = 0.3\,\,{\rm{ps}}$ and $1/{\gamma _{{{\rm{LO}}_{{2}}}}} = 0.4\,\,{\rm{ps}}$ within our model. LO phonon lifetimes in hBN can also be measured by Raman scattering and are of the order of a few ps [51], which is longer than the values found here. Therefore, additional decoherence mechanisms apart from their decay, e.g., a non-vanishing dispersion, are likely.

F. Acoustic Phonons

Next, we reveal the impact of LA phonons on the coherence of the hBN color center. To confirm that LA-phonon-assisted excitations are feasible [30,50], we perform a PLE scan at $T = 80\,\,{\rm{K}}$ on a meV range around the ZPL as depicted in Fig. 6(a), where the experiment is shown as dark circles and the simulation as bright lines. We find that the PLE spectrum is asymmetric with a flatter slope on the low energy side and a plateau-like region on the high energy side. This asymmetry has two reasons: (i) at $T = 80\,\,{\rm{K}}$, the LA phonon emission is more likely than LA phonon absorption, which results in a stronger LA PSB at positive detuning forming the plateau. To verify this, the dashed line in Fig. 6(a) depicts the calculated PLE spectrum without phonon coupling. (ii) The measured PLE spectrum is a convolution of the intrinsic absorption spectrum of the emitter and the laser spectrum. The asymmetry of the laser spectrum has a non-trivial impact on the final PLE spectrum, as seen in the dashed curve and discussed in detail in Supplement 1, Sec. S4.B.

 figure: Fig. 6.

Fig. 6. Coherent control of acoustic PSBs. (a) Measured and simulated PLE data are shown as dark circles and bright curves, respectively. The colored vertical lines mark the detunings considered in (b). (b) Visibility dynamics for increasing detunings from top to bottom with negative detunings (left) and positive detunings (right). Experiment is shown as dark circles and simulation as bright curves. Laser autocorrelation is shown in green (dashed curves). Vertical dashed lines mark half a period corresponding to the frequency detuning between laser and ZPL. (c) Illustration of the laser-pulse-induced generation of a phonon wave packet.

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The coherent control visibility with LA-phonon-assisted optical excitation is shown in Fig. 6(b) for $T = 80\,\,{\rm{K}}$. The experiment is depicted as dark circles, the simulation as bright lines, and the detuning between laser energy and transition energy $\hbar \delta = {E_{{\rm{laser}}}} - {E_{{\rm{ZPL}}}}$ increases from top to bottom, with negative detunings (red) in the left and positive (blue) in the right column. The resonant case (violet) is the same as in Fig. 5(a), and the considered detunings ($\hbar \delta = \pm 4$, ${\pm}5$, and ${\pm}6\,\,{\rm{meV}}$) are additionally marked as red (negative) and blue (positive) vertical lines in the PLE spectrum in Fig. 6(a). As a reference, we again plot the laser autocorrelation as green dashed curves in Fig. 6(b). For detuned (off-resonant) excitation, all coherent control signals exhibit a rapid initial visibility drop within 0.5 ps, which is followed by a slower decoherence or, interestingly, an intermediate recovery of the coherence. We find a remarkable agreement between the experimental data and theoretical simulations. Notably, the asymmetric spectral shape of the femtosecond laser pulses has to be accurately considered in the theoretical model to achieve this almost perfectly consistent match between the coherent control signal dynamics and the PL spectrum in Fig. 1(a) (see Supplement 1, Sec. S4.A, for details). According to our model, the observed coherence dynamics for delays below ${\approx} 1\,\,{\rm{ps}}$ have two contributions [see Eq. (A9) and Supplement 1, Sec. S2.D.4]: (i) an initial loss of coherence resulting in the rapid drop of coherent control visibility, followed by a quasi-stationary coherence for delays larger than ${\approx} 1\,\,{\rm{ps}}$; (ii) a quantum beat resulting in an oscillation for delays below ${\approx} 1\,\,{\rm{ps}}$.

Finding (i) is specific for the coupling to acoustic phonons and well known from self-assembled semiconductor quantum dots as a PID effect [52,53]. Its physical origin is the formation of a polaron, i.e., a deformed lattice configuration in the emitter region, which is created on the time scale of the optical excitation. As illustrated in Fig. 6(c), this rapid lattice deformation after the first laser pulse is accompanied by the emission of a phonon wave packet due to the linear dispersion of the LA phonons. As this phonon pulse is emitted from the color center with the speed of sound (several nm/ps in hBN), it carries part of the coherence with it, which is lost for the quantum state of the emitter. This loss of coherence is then probed by the second laser pulse in the coherent control experiment.

The amplitude of the visibility at $\Delta t = 2\,\,{\rm{ps}}$ reflects the coherence of the polaron state, which is generated by the first pulse. In each line of Fig. 6(b), positive (blue, right) and negative (red, left) detunings with the same absolute values are directly compared. From the visibility values at $\Delta t = 2\,\,{\rm{ps}}$, we find that the PID is always more pronounced for positive detunings caused by the asymmetric laser spectrum and a mismatch between phonon emission and absorption due to the thermal occupation of phonons ${n_{{\rm{ph}}}}$ at $T = 80\,\,{\rm{K}}$ (${n_{{\rm{ph}}}} \approx 1$ for modes with ${E_{{\rm{ph}}}} \approx 5\,\,{\rm{meV}}$). For positive detunings, phonons must be generated to occupy the polaron, while for negative detunings, phonons need to be absorbed. The two processes scale with $({n_{{\rm{ph}}}} + 1) \approx 2$ and ${n_{{\rm{ph}}}} \approx 1$, respectively. This mismatch results in the more pronounced PID drop in visibility for positive detunings. The quasi-stationary signal at $\Delta t \approx 2\,\,{\rm{ps}}$ is a measure for the overlap of the laser spectrum with the ZPL. The visibility is generally smaller for larger absolute detunings, e.g., ${v_{{\rm{PL}}}}(\Delta t = 2\,\,{\rm{ps}}) \approx 0.5$ at $\hbar \delta = + 4\,\,{\rm{meV}}$ compared to $\approx 0.25$ at $\hbar \delta = + 6\,\,{\rm{meV}}$, as the polaron is excited less directly and more via phonon-assisted processes. This leads to a stronger loss of coherence induced by the creation of the phonon wave packet when the absolute detuning $|\delta |$ is larger.

The additional oscillation of the signal, i.e., effect (ii), results in dynamics that are even faster than the laser autocorrelation [green curves in Fig. 6(b)]. We can understand these oscillations when considering the theoretical expression for the coherent control signal for short delays in Eq. (A9) in Appendix A. We find that the detuning of the laser and the remaining overlap with the ZPL effectively select two spectral components: the pure emitter coherence and the phonon-assisted coherence at the carrier frequency of the laser pulse. These two contributions result in a quantum beat with the frequency ${\omega _{{\rm{beat}}}} = |\delta |$, which shows up as an oscillation in the coherent control signal. This directly explains why the oscillations get faster for larger detunings. The expected minimum from such a beat at ${T_{{\rm{beat}}}}/2 = \pi /|\delta |$ is marked by vertical dashed lines in Fig. 6(b) and matches well with the measured minima in the visibility dynamics. This beat is a clear indication of the coherent coupling between the acoustic phonon modes and the emitter.

3. CONCLUSION

By performing optical coherent control experiments with ultrafast laser pulses, we have revealed the coherence dynamics of a single hBN color center. We found an excellent agreement with simulations, being consistent for spectrally resolved time-integrated PL and the time-resolved coherent control signal. We highlighted the impact of different sources of spectral jitter: Fast jitter on or below the picosecond time scale resulted in homogenous/exponential decoherence dynamics, while slow jitter on a nanosecond and longer time scale led to inhomogeneous/Gaussian decoherence dynamics. The presence of RTN was revealed by a characteristic beat of the coherent control signal. Furthermore, we demonstrated the impact of phonons on the emitter coherence by using phonon-assisted excitation. In the case of optical phonons, we found that the increased decoherence rates are due to dephasing processes of the phonon state, which partially stem from its anharmonic decay. When the phonon-assisted excitation involved acoustic phonon modes, we found that (i) the generation of the phonon wave packet led to a characteristic drop in the coherent control signal and (ii) the simultaneous resonant and phonon-assisted excitation resulted in the selection of two energetically detuned contributions, which manifested as beat dynamics in the signal.

Hence, the coherence of hBN color centers is limited by phonon-induced decoherence and spectral jitter. While the phonon impact could be controlled by deliberately shaped laser pulses [5456], control over the crystal environment could reduce spectral jitter. An overall increased coherence time will enable advanced coherent control schemes, e.g., employing coherent optoelectronics [57].

Our demonstration of phonon-assisted coherent control of individual hBN color centers is a key step towards hybrid quantum technologies, which combine electronic and phononic excitations [36]. We envision that the impact of phonons can further be controlled by external acoustic driving [58], which will allow to deterministically address photon emission properties.

APPENDIX A: METHODS

Experiment

Commercially available hBN multilayer nanopowder (Sigma-Aldrich, diameter ${\lt}150\,\,{\rm{nm}}$) is annealed at ${850^ \circ}$ C to increase the number of single-photon emitters and is randomly scattered on a clean quartz substrate. The sample is then placed in a variable temperature helium bath cryostat with a low-temperature transmission microscope inside the sample chamber. The microscope consists of two objective lenses with a numerical aperture (NA) of 0.82, enabling efficient photon collection. The sample, as well as the microscope, is cooled by evaporating the coolant in the sample chamber. The cryostat is operated using liquid helium or nitrogen, resulting in sample temperatures of 8 K and 80 K, respectively. Spectrally tunable laser pulses with durations of $230\,\,{\rm{fs}}$ are provided by a frequency-doubled supercontinuum from a femtosecond Er:fiber laser. To obtain two identical pulses, a single laser pulse is split at a 50:50 plate beam splitter. The path of the split pulses can be controlled by a combination of a 204 mm long translation stage and a 100 µm closed-loop piezo scanner, resulting in a coarse and fine delay of pulses in the range of ps and sub-fs, respectively. The beam path of the two pulses is rejoined by an identical 50:50 plate beam splitter and then focused on the single-photon emitter. The PL is detected by either a spectrometer and cryogenically cooled CCD for spectroscopy and the dynamics measurements or a single-mode fiber-coupled Hanbury Brown and Twiss setup for photon correlation measurements. Further details can be found in the Supplement 1, Sec. S1.

Theory

In Supplement 1, Sec. S2, we derive all required equations to model the depicted optical signals in detail. Here, we summarize the most important final results for each measurement. We model the emitter as a TLS and consider the coupling to phonons in the independent boson model and a semi-classical coupling to light leading to the Hamiltonian

$$\begin{split}H& = \hbar {\omega _{{\rm{TLS}}}}{X^\dagger}X + \frac{1}{2}\left[{{\cal E}(t){X^\dagger} + {\cal E^*}{{(t)}}X} \right] + \sum\limits_n \hbar {\omega _n}b_n^\dagger {b_n} \\[-4pt]&\quad+ {X^\dagger}X\sum\limits_n \hbar \left({{g_n}b_n^\dagger + g_n^*{b_n}} \right) + {H_{{\rm{pd}}}} + {H_{{\rm{anh}}}}.\end{split}$$

Here, $\hbar {\omega _{{\rm{TLS}}}}$ is the transition energy of the emitter in the absence of phonon coupling. The ZPL energy additionally includes the polaron shift $\hbar {\omega _{{\rm{ZPL}}}} = \hbar {\omega _{{\rm{TLS}}}} - \hbar \sum\nolimits_n |{g_n}{|^2}/{\omega _n}$, depending on the phonon frequencies ${\omega _n}$ and their coupling strengths ${g_n}$. ${X^{(\dagger)}}$ and $b_n^{(\dagger)}$ are the TLS and phonon annihilation (creation) operators, respectively, and ${\cal E}(t)$ describes the laser pulses, being the projection of the electric field of the laser onto the emitter’s dipole moment. We also include ${H_{{\rm{pd}}}}$ and ${H_{{\rm{anh}}}}$ to describe the pure dephasing (pd) of the TLS and the anharmonic decay (anh) of the LO phonons, which are treated in terms of Lindblad dissipators in the model.

The PL spectrum of the emitter is calculated via

$${I_{{\rm{PL}}}}(\omega) = \int\limits_{- \infty}^\infty {\rm{d}}\Omega R(\omega ,\Omega)\int\limits_{- \infty}^\infty {\rm{d}}t {G_{- +}}(t){e^{- i(\Omega - {\omega _{{\rm{ZPL}}}})t}}{e^{- {\gamma _{{\rm{pd}}}}|t|/2}} ,$$
where $R$ is the spectrometer response, and ${\gamma _{{\rm{pd}}}}$ is the pure dephasing rate. For the PL presented in Fig. 1(a), the spectrometer response is given by a normal distribution with mean $\omega$ and standard deviation $\omega \Delta \lambda /\lambda (\omega)$, where $\lambda (\omega)$ is the corresponding wavelength, and $\Delta \lambda = 0.3\,\,{\rm{nm}}$ is the resolution of the spectrometer. We emphasize that the spectrometer response is crucial for accurately modeling PL spectra, as the resolution influences the relative height of ZPL and LA-PSB. For some phonon parameters, this implies that they cannot be accurately determined from the PL spectra, unless the spectrometer response is well-known, as discussed in Supplement 1, Sec. S3.C.

${G_{- +}}(\tau)$ in Eq. (A2) is the phonon correlation function, which captures the information on the emitter–phonon coupling. It is given by

$${G_{- +}}(\tau) = \exp [\phi (\tau) - \phi (0)],$$
$$\begin{split}\phi (\tau) &= \int\limits_0^\infty {\rm{d}}\omega \frac{{{J_{{\rm{LA}}}}(\omega)}}{{{\omega ^2}}}\left\{{\left[{n(\omega) + 1} \right]{e^{- i\omega \tau}} + n(\omega){e^{i\omega \tau}}} \right\} \\[-4pt]&\quad+ \int\limits_0^\infty {\rm{d}}\omega \frac{{{J_{{\rm{LO}}}}(\omega)}}{{{\omega ^2}}}\exp \left[{- i\omega \tau - \frac{{\gamma (\omega)}}{2}|\tau |} \right] .\end{split}$$

Here, ${J_{{\rm{LA}}}}(\omega) = \sum\nolimits_{n \in \{{\rm{LA}}\}} \!|{g_n}{|^2}\delta ({\omega _n} - \omega)$ is the LA spectral density, where the sum runs over all LA modes. The LO spectral density is defined in an analogous way. $n(\omega)$ is the thermal occupation of the phonon mode with frequency $\omega$. In the case of LO modes, we assume $n(\omega) = 0$ since they have energies ${\gt}150\,\,{\rm{meV}}$, and we consider temperatures well below 300 K. $\gamma (\omega)$ is the decay rate of the LO mode with frequency $\omega$. Equation (A4) shows that the central quantity determining the phonon correlation function is the spectral density. All details on the structure of the couplings ${g_n}$, i.e., its dependence on the quantum number $n$, are lost in this effective description, and we can in principle choose a spectral density for our modeling that fits best with the experimental observation. For LA phonons, we choose a super-ohmic spectral density with a Lorentz–Drude cutoff. The LO spectral density is modeled as a sum of four $\delta$ functions, i.e., it describes four dispersionless modes in the Einstein model.

The PLE spectrum and the visibility dynamics of the coherent control experiment can be calculated from the same expression:

$$\begin{split}{I_0}({\omega _l},\Delta t) &= \int\limits_{- \infty}^\infty {\rm{d}}t \int\limits_{- \infty}^\infty {\rm{d}}\tau {\tilde {\cal E}^*}(t)\tilde {\cal E}(t + \tau)\\&\quad\times{e^{- i{\omega _l}\tau}}{G_{- +}}(\Delta t - \tau){e^{- i{\omega _{{\rm{ZPL}}}}(\Delta t - \tau)}}{e^{- {\gamma _{{\rm{pd}}}}|\Delta t - \tau |/2}} ,\end{split}$$
where $\tilde {\cal E}(t)$ is the envelope of a single laser pulse, and ${\omega _l}$ its carrier frequency. In this paper, we define the carrier frequency as the maximum of the detected laser spectrum, as this can be conveniently identified in the experiment. The temporal shape of the laser pulse is derived from an asymmetric spectrum, modeled by a Pearson-IV distribution (see Supplement 1, Sec. S3.B for details). The PLE is then given by
$${I_{{\rm{PLE}}}}({\omega _l}) = {I_0}({\omega _l},\Delta t = 0),$$
and the visibility for a fixed carrier frequency, depending on the delay $\Delta t$ between pulses, is
$$v(\Delta t) = \frac{{|{I_0}({\omega _l},\Delta t)|}}{{{I_0}({\omega _l},\Delta t = 0)}} ,$$
i.e., it is the absolute value of ${I_0}$, normalized by the PLE signal, which satisfies $v(\Delta t) \le 1$ with equality for $\Delta t = 0$.

Slow spectral jitter, i.e., jumps of the ZPL frequency on a time scale longer than a single repetition of the experiment, can be incorporated via an ensemble average. This is done by replacing $\exp [- i{\omega _{{\rm{ZPL}}}}(\Delta t - \tau)]$ under the integral in Eq. (A5) by the weighted average over a probability distribution $p({\omega _{{\rm{ZPL}}}})$. We show in Supplement 1, Sec. S2.D.5, that Gaussian jitter on a spectral scale well below the width of the laser spectrum simply leads to a prefactor $\exp [- \sigma _{\rm{j}}^2{(\Delta t)^2}/2]$ in Eq. (A5), with $1/{\sigma _{\rm{j}}} \approx 19\,\,{\rm{ps}}$ throughout the paper. Telegraph noise jitter, as seen in Fig. 4, is modeled by two $\delta$- functions in $p({\omega _{{\rm{ZPL}}}})$.

As discussed in Supplement 1, Sec. S2.D.7, the visibility of direct laser interference, measured in the same coherent control setup, is given by the autocorrelation of the electric field of a single pulse. Here, we deal with linearly polarized light, such that the amplitude of the electric field is proportional to ${\cal E}(t)$. Therefore, the visibility of the laser interference is modeled by normalizing

$$\int\limits_{- \infty}^\infty {\rm{d}}t {\tilde {\cal E}^*}(t)\tilde {\cal E}(t + \Delta t){e^{- i{\omega _l}\Delta t}} .$$

In Supplement 1, Sec. S2.D.4, we discuss an approximation for the visibility dynamics in Eq. (A7) when exciting close to the ZPL and considering short time scales where pure dephasing can be neglected. In this case, we arrive at the expression

$$v(\Delta t) \approx \frac{{|a(\Delta t){e^{- i\delta \Delta t}} + b|}}{{a(\Delta t = 0) + b}} ,$$
where $\delta = {\omega _l} - {\omega _{{\rm{ZPL}}}}$ is the detuning between the laser and ZPL. The function $a(\Delta t)$ is given by
$$a(\Delta t) = {I_{{\rm{abs,PSB}}}}({\omega _l})\int\limits_{- \infty}^\infty {\rm{d}}t {\tilde {\cal E}^*}(t)\tilde {\cal E}(t + \Delta t) ,$$
i.e., it is the envelope of the pulse autocorrelation, weighted by the LA-PSB contribution to the absorption spectrum at the position of the carrier frequency ${\omega _l}$, as defined in Supplement 1, Sec. S2.D.4. The constant $b$ is determined by
$$b = {B^2}I_{{\rm{laser}}}^{{\rm{ideal}}}({\omega _{{\rm{ZPL}}}}) ,$$
where ${B^2} = \exp [- \phi (0)]$, with $\phi (\tau)$ from Eq. (A4) as a measure for the phonon coupling strength. For strong phonon coupling or large temperatures, ${B^2}$ tends towards zero. In the PL spectra, ${B^2}$ is also the prefactor of the pure ZPL contribution. For weak phonon coupling, ${B^2}$ tends towards unity, i.e., the spectrum consists of only a ZPL and contains no PSBs. $I_{{\rm{laser}}}^{{\rm{ideal}}}({\omega _{{\rm{ZPL}}}})$ is the ideal (perfect resolution) laser spectrum at the position of the ZPL, as defined in Supplement 1, Sec. S2.D.2. Thus, $b$ measures the overlap of the laser pulse spectrum with the ZPL.

$a(\Delta t)$ measures the strength of the LA-PSB at the carrier frequency for $\Delta t = 0$ and decays along with the pulse autocorrelation. Thus, we expect dynamics that decay on the time scale of the laser autocorrelation towards a final value of $b/(a(0) + b)$, which is a measure of how efficiently we excite the system resonantly on the ZPL compared to off-resonant excitation via PSBs. The visibility dynamics exhibit a beat with detuning frequency $|\delta |$, in addition to the decay, which leads to dynamics faster than the pulse autocorrelation in Fig. 6, as discussed in the main text.

As a final note on the visibility dynamics on short time scales, we want to emphasize that the derivation was made for the case of ${\gamma _{{\rm{pd}}}} = 0$, i.e., no pure dephasing. This leads to a $\delta$-shaped ZPL and strong dependence of parameter $b$ on the ZPL frequency ${\omega _{{\rm{ZPL}}}}$ via the overlap with the laser spectrum. Therefore, in this regime (${\gamma _{{\rm{pd}}}} \to 0$), spectral jitter can have a significant impact and makes the experimental identification of the discussed phenomenon challenging. Thus, rather counterintuitively, it is more instructive to perform the coherent control experiment with detuned excitation at more elevated temperatures. In that case, ${\gamma _{{\rm{pd}}}}$ is larger, i.e., the system dephases more quickly, leading to a broader ZPL and therefore less sensitivity to spectral jitter.

Funding

Narodowa Agencja Wymiany Akademickiej (NAWA) (PPN/ULM/2019/1/00064).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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49. M. A. Feldman, A. Puretzky, L. Lindsay, E. Tucker, D. P. Briggs, P. G. Evans, R. F. Haglund, and B. J. Lawrie, “Phonon-induced multicolor correlations in hBN single-photon emitters,” Phys. Rev. B 99, 020101 (2019). [CrossRef]  

50. P. Khatri, A. J. Ramsay, R. N. E. Malein, H. M. Chong, and I. J. Luxmoore, “Optical gating of photoluminescence from color centers in hexagonal boron nitride,” Nano Lett. 20, 4256–4263 (2020). [CrossRef]  

51. R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018). [CrossRef]  

52. T. Jakubczyk, V. Delmonte, S. Fischbach, D. Wigger, D. E. Reiter, Q. Mermillod, P. Schnauber, A. Kaganskiy, J.-H. Schulze, A. Strittmatter, S. Rodt, W. Langbein, T. Kuhn, S. Reitzenstein, and J. Kasprzak, “Impact of phonons on dephasing of individual excitons in deterministic quantum dot microlenses,” ACS Photon. 3, 2461–2466 (2016). [CrossRef]  

53. D. Wigger, V. Karakhanyan, C. Schneider, M. Kamp, S. Höfling, P. Machnikowski, T. Kuhn, and J. Kasprzak, “Acoustic phonon sideband dynamics during polaron formation in a single quantum dot,” Opt. Lett. 45, 919–922 (2020). [CrossRef]  

54. D. Wigger, S. Lüker, D. Reiter, V. M. Axt, P. Machnikowski, and T. Kuhn, “Energy transport and coherence properties of acoustic phonons generated by optical excitation of a quantum dot,” J. Phys. Condens. Matter 26, 355802 (2014). [CrossRef]  

55. T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017). [CrossRef]  

56. T. Kaldewey, S. Lüker, A. V. Kuhlmann, S. R. Valentin, J.-M. Chauveau, A. Ludwig, A. D. Wieck, D. E. Reiter, T. Kuhn, and R. J. Warburton, “Demonstrating the decoupling regime of the electron-phonon interaction in a quantum dot using chirped optical excitation,” Phys. Rev. B 95, 241306 (2017). [CrossRef]  

57. S. Michaelis de Vasconcellos, S. Gordon, M. Bichler, T. Meier, and A. Zrenner, “Coherent control of a single exciton qubit by optoelectronic manipulation,” Nat. Photonics 4, 545–548 (2010). [CrossRef]  

58. D. Wigger, K. Gawarecki, and P. Machnikowski, “Remote phonon control of quantum dots and other artificial atoms,” Adv. Quantum Technol. 4, 2000128 (2021). [CrossRef]  

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    [Crossref]
  51. R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018).
    [Crossref]
  52. T. Jakubczyk, V. Delmonte, S. Fischbach, D. Wigger, D. E. Reiter, Q. Mermillod, P. Schnauber, A. Kaganskiy, J.-H. Schulze, A. Strittmatter, S. Rodt, W. Langbein, T. Kuhn, S. Reitzenstein, and J. Kasprzak, “Impact of phonons on dephasing of individual excitons in deterministic quantum dot microlenses,” ACS Photon. 3, 2461–2466 (2016).
    [Crossref]
  53. D. Wigger, V. Karakhanyan, C. Schneider, M. Kamp, S. Höfling, P. Machnikowski, T. Kuhn, and J. Kasprzak, “Acoustic phonon sideband dynamics during polaron formation in a single quantum dot,” Opt. Lett. 45, 919–922 (2020).
    [Crossref]
  54. D. Wigger, S. Lüker, D. Reiter, V. M. Axt, P. Machnikowski, and T. Kuhn, “Energy transport and coherence properties of acoustic phonons generated by optical excitation of a quantum dot,” J. Phys. Condens. Matter 26, 355802 (2014).
    [Crossref]
  55. T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
    [Crossref]
  56. T. Kaldewey, S. Lüker, A. V. Kuhlmann, S. R. Valentin, J.-M. Chauveau, A. Ludwig, A. D. Wieck, D. E. Reiter, T. Kuhn, and R. J. Warburton, “Demonstrating the decoupling regime of the electron-phonon interaction in a quantum dot using chirped optical excitation,” Phys. Rev. B 95, 241306 (2017).
    [Crossref]
  57. S. Michaelis de Vasconcellos, S. Gordon, M. Bichler, T. Meier, and A. Zrenner, “Coherent control of a single exciton qubit by optoelectronic manipulation,” Nat. Photonics 4, 545–548 (2010).
    [Crossref]
  58. D. Wigger, K. Gawarecki, and P. Machnikowski, “Remote phonon control of quantum dots and other artificial atoms,” Adv. Quantum Technol. 4, 2000128 (2021).
    [Crossref]

2022 (1)

S. Michaelis de Vasconcellos, D. Wigger, U. Wurstbauer, A. W. Holleitner, R. Bratschitsch, and T. Kuhn, “Single-photon emitters in layered van der Waals materials,” Phys. Status Solidi B 259, 2100566 (2022).
[Crossref]

2021 (6)

N. Mendelson, L. Morales-Inostroza, C. Li, R. Ritika, M. A. P. Nguyen, J. Loyola-Echeverria, S. Kim, S. Götzinger, M. Toth, and I. Aharonovich, “Grain dependent growth of bright quantum emitters in hexagonal boron nitride,” Adv. Opt. Mater. 9, 2001271 (2021).
[Crossref]

J. A. Preuß, E. Rudi, J. Kern, R. Schmidt, R. Bratschitsch, and S. Michaelis de Vasconcellos, “Assembly of large hBN nanocrystal arrays for quantum light emission,” 2D Mater. 8, 035005 (2021).
[Crossref]

S. White, C. Stewart, A. S. Solntsev, C. Li, M. Toth, M. Kianinia, and I. Aharonovich, “Phonon dephasing and spectral diffusion of quantum emitters in hexagonal boron nitride,” Optica 8, 1153–1158 (2021).
[Crossref]

R. N. E. Malein, P. Khatri, A. J. Ramsay, and I. J. Luxmoore, “Stimulated emission depletion spectroscopy of color centers in hexagonal boron nitride,” ACS Photon. 8, 1007–1012 (2021).
[Crossref]

D. Groll, T. Hahn, P. Machnikowski, D. Wigger, and T. Kuhn, “Controlling photoluminescence spectra of hBN color centers by selective phonon-assisted excitation: a theoretical proposal,” Mater. Quantum Technol. 1, 015004 (2021).
[Crossref]

D. Wigger, K. Gawarecki, and P. Machnikowski, “Remote phonon control of quantum dots and other artificial atoms,” Adv. Quantum Technol. 4, 2000128 (2021).
[Crossref]

2020 (9)

D. Wigger, V. Karakhanyan, C. Schneider, M. Kamp, S. Höfling, P. Machnikowski, T. Kuhn, and J. Kasprzak, “Acoustic phonon sideband dynamics during polaron formation in a single quantum dot,” Opt. Lett. 45, 919–922 (2020).
[Crossref]

B. Spokoyny, H. Utzat, H. Moon, G. Grosso, D. Englund, and M. G. Bawendi, “Effect of spectral diffusion on the coherence properties of a single quantum emitter in hexagonal boron nitride,” J. Phys. Chem. Lett. 11, 1330–1335 (2020).
[Crossref]

P. Khatri, A. J. Ramsay, R. N. E. Malein, H. M. Chong, and I. J. Luxmoore, “Optical gating of photoluminescence from color centers in hexagonal boron nitride,” Nano Lett. 20, 4256–4263 (2020).
[Crossref]

M. K. Boll, I. P. Radko, A. Huck, and U. L. Andersen, “Photophysics of quantum emitters in hexagonal boron-nitride nano-flakes,” Opt. Express 28, 7475–7487 (2020).
[Crossref]

G. Grosso, H. Moon, C. J. Ciccarino, J. Flick, N. Mendelson, L. Mennel, M. Toth, I. Aharonovich, P. Narang, and D. R. Englund, “Low-temperature electron–phonon interaction of quantum emitters in hexagonal boron nitride,” ACS Photon. 7, 1410–1417 (2020).
[Crossref]

M. Hoese, P. Reddy, A. Dietrich, M. K. Koch, K. G. Fehler, M. W. Doherty, and A. Kubanek, “Mechanical decoupling of quantum emitters in hexagonal boron nitride from low-energy phonon modes,” Sci. Adv. 6, eaba6038 (2020).
[Crossref]

A. Dietrich, M. Doherty, I. Aharonovich, and A. Kubanek, “Solid-state single photon source with Fourier transform limited lines at room temperature,” Phys. Rev. B 101, 081401 (2020).
[Crossref]

R. Camphausen, L. Marini, S. A. Tawfik, T. T. Tran, M. J. Ford, and S. Palomba, “Observation of near-infrared sub-Poissonian photon emission in hexagonal boron nitride at room temperature,” APL Photon. 5, 076103 (2020).
[Crossref]

N. Mendelson, M. Doherty, M. Toth, I. Aharonovich, and T. T. Tran, “Strain-induced modification of the optical characteristics of quantum emitters in hexagonal boron nitride,” Adv. Mater. 32, 1908316 (2020).
[Crossref]

2019 (8)

Y. Xia, Q. Li, J. Kim, W. Bao, C. Gong, S. Yang, Y. Wang, and X. Zhang, “Room-temperature giant Stark effect of single photon emitter in van der Waals material,” Nano Lett. 19, 7100–7105 (2019).
[Crossref]

H. Utzat, W. Sun, A. E. Kaplan, F. Krieg, M. Ginterseder, B. Spokoyny, N. D. Klein, K. E. Shulenberger, C. F. Perkinson, M. V. Kovalenko, and M. G. Bawendi, “Coherent single-photon emission from colloidal lead halide perovskite quantum dots,” Science 363, 1068–1072 (2019).
[Crossref]

M. Bayer, “Bridging two worlds: colloidal versus epitaxial quantum dots,” Ann. Phys. 531, 1900039 (2019).
[Crossref]

M. Toth and I. Aharonovich, “Single photon sources in atomically thin materials,” Annu. Rev. Phys. Chem. 70, 123–142 (2019).
[Crossref]

D. Wigger, R. Schmidt, O. Del Pozo-Zamudio, J. A. Preuß, P. Tonndorf, R. Schneider, P. Steeger, J. Kern, Y. Khodaei, J. Sperling, S. Michaelis de Vasconcellos, R. Bratschitsch, and T. Kuhn, “Phonon-assisted emission and absorption of individual color centers in hexagonal boron nitride,” 2D Mater. 6, 035006 (2019).
[Crossref]

K. Konthasinghe, C. Chakraborty, N. Mathur, L. Qiu, A. Mukherjee, G. D. Fuchs, and A. N. Vamivakas, “Rabi oscillations and resonance fluorescence from a single hexagonal boron nitride quantum emitter,” Optica 6, 542–548 (2019).
[Crossref]

T. T. Tran, C. Bradac, A. S. Solntsev, M. Toth, and I. Aharonovich, “Suppression of spectral diffusion by anti-Stokes excitation of quantum emitters in hexagonal boron nitride,” Appl. Phys. Lett. 115, 071102 (2019).
[Crossref]

M. A. Feldman, A. Puretzky, L. Lindsay, E. Tucker, D. P. Briggs, P. G. Evans, R. F. Haglund, and B. J. Lawrie, “Phonon-induced multicolor correlations in hBN single-photon emitters,” Phys. Rev. B 99, 020101 (2019).
[Crossref]

2018 (1)

R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018).
[Crossref]

2017 (3)

B. Sontheimer, M. Braun, N. Nikolay, N. Sadzak, I. Aharonovich, and O. Benson, “Photodynamics of quantum emitters in hexagonal boron nitride revealed by low-temperature spectroscopy,” Phys. Rev. B 96, 121202 (2017).
[Crossref]

T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
[Crossref]

T. Kaldewey, S. Lüker, A. V. Kuhlmann, S. R. Valentin, J.-M. Chauveau, A. Ludwig, A. D. Wieck, D. E. Reiter, T. Kuhn, and R. J. Warburton, “Demonstrating the decoupling regime of the electron-phonon interaction in a quantum dot using chirped optical excitation,” Phys. Rev. B 95, 241306 (2017).
[Crossref]

2016 (5)

T. Jakubczyk, V. Delmonte, S. Fischbach, D. Wigger, D. E. Reiter, Q. Mermillod, P. Schnauber, A. Kaganskiy, J.-H. Schulze, A. Strittmatter, S. Rodt, W. Langbein, T. Kuhn, S. Reitzenstein, and J. Kasprzak, “Impact of phonons on dephasing of individual excitons in deterministic quantum dot microlenses,” ACS Photon. 3, 2461–2466 (2016).
[Crossref]

T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, “Quantum emission from hexagonal boron nitride monolayers,” Nat. Nanotechnol. 11, 37–41 (2016).
[Crossref]

T. Vuong, G. Cassabois, P. Valvin, A. Ouerghi, Y. Chassagneux, C. Voisin, and B. Gil, “Phonon-photon mapping in a color center in hexagonal boron nitride,” Phys. Rev. Lett. 117, 097402 (2016).
[Crossref]

T. T. Tran, C. Elbadawi, D. Totonjian, C. J. Lobo, G. Grosso, H. Moon, D. R. Englund, M. J. Ford, I. Aharonovich, and M. Toth, “Robust multicolor single photon emission from point defects in hexagonal boron nitride,” ACS Nano 10, 7331–7338 (2016).
[Crossref]

R. Bourrellier, S. Meuret, A. Tararan, O. Stéphan, M. Kociak, L. H. Tizei, and A. Zobelli, “Bright UV single photon emission at point defects in h-BN,” Nano Lett. 16, 4317–4321 (2016).
[Crossref]

2015 (5)

P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. van der Zant, S. M. de Vasconcellos, and R. Bratschitsch, “Single-photon emission from localized excitons in an atomically thin semiconductor,” Optica 2, 347–352 (2015).
[Crossref]

C. Chakraborty, L. Kinnischtzke, K. M. Goodfellow, R. Beams, and A. N. Vamivakas, “Voltage-controlled quantum light from an atomically thin semiconductor,” Nat. Nanotechnol. 10, 507–511 (2015).
[Crossref]

Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu, and J.-W. Pan, “Single quantum emitters in monolayer semiconductors,” Nat. Nanotechnol. 10, 497–502 (2015).
[Crossref]

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotechnol. 10, 503–506 (2015).
[Crossref]

A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A. Kis, and A. Imamoğlu, “Optically active quantum dots in monolayer WSe2,” Nat. Nanotechnol. 10, 491 (2015).
[Crossref]

2014 (2)

K. Liu, Q. Yan, M. Chen, W. Fan, Y. Sun, J. Suh, D. Fu, S. Lee, J. Zhou, S. Tongay, J. Ji, J. B. Neaton, and J. Wu, “Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures,” Nano Lett. 14, 5097–5103 (2014).
[Crossref]

D. Wigger, S. Lüker, D. Reiter, V. M. Axt, P. Machnikowski, and T. Kuhn, “Energy transport and coherence properties of acoustic phonons generated by optical excitation of a quantum dot,” J. Phys. Condens. Matter 26, 355802 (2014).
[Crossref]

2011 (1)

I. Aharonovich, S. Castelletto, D. Simpson, C. Su, A. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
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2010 (2)

S. Michaelis de Vasconcellos, S. Gordon, M. Bichler, T. Meier, and A. Zrenner, “Coherent control of a single exciton qubit by optoelectronic manipulation,” Nat. Photonics 4, 545–548 (2010).
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A. J. Ramsay, “A review of the coherent optical control of the exciton and spin states of semiconductor quantum dots,” Semicond. Sci. Technol. 25, 103001 (2010).
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2007 (2)

R. S. Kolodka, A. J. Ramsay, J. Skiba-Szymanska, P. W. Fry, H. Y. Liu, A. M. Fox, and M. S. Skolnick, “Inversion recovery of single quantum-dot exciton based qubit,” Phys. Rev. B 75, 193306 (2007).
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J. Serrano, A. Bosak, R. Arenal, M. Krisch, K. Watanabe, T. Taniguchi, H. Kanda, A. Rubio, and L. Wirtz, “Vibrational properties of hexagonal boron nitride: inelastic x-ray scattering and ab initio calculations,” Phys. Rev. Lett. 98, 095503 (2007).
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2005 (1)

S. Stufler, P. Ester, A. Zrenner, and M. Bichler, “Quantum optical properties of a single InxGa1-xAs-GaAs quantum dot two-level system,” Phys. Rev. B 72, 121301 (2005).
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2003 (1)

J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Philos. Trans. R. Soc. A 361, 1655–1674 (2003).
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2002 (1)

H. Htoon, T. Takagahara, D. Kulik, O. Baklenov, A. L. Holmes, and C. K. Shih, “Interplay of Rabi oscillations and quantum interference in semiconductor quantum dots,” Phys. Rev. Lett. 88, 087401 (2002).
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1999 (1)

G. Kern, G. Kresse, and J. Hafner, “Ab initio calculation of the lattice dynamics and phase diagram of boron nitride,” Phys. Rev. B 59, 8551 (1999).
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1998 (1)

N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
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1997 (1)

S. A. Empedocles and M. G. Bawendi, “Quantum-confined stark effect in single CdSe nanocrystallite quantum dots,” Science 278, 2114–2117 (1997).
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1996 (1)

A. P. Heberle, J. J. Baumberg, E. Binder, T. Kuhn, K. Kohler, and K. H. Ploog, “Coherent control of exciton density and spin,” IEEE J. Sel. Top. Quantum Electron. 2, 769–775 (1996).
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1995 (1)

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75, 2598 (1995).
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Aharonovich, I.

N. Mendelson, L. Morales-Inostroza, C. Li, R. Ritika, M. A. P. Nguyen, J. Loyola-Echeverria, S. Kim, S. Götzinger, M. Toth, and I. Aharonovich, “Grain dependent growth of bright quantum emitters in hexagonal boron nitride,” Adv. Opt. Mater. 9, 2001271 (2021).
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S. White, C. Stewart, A. S. Solntsev, C. Li, M. Toth, M. Kianinia, and I. Aharonovich, “Phonon dephasing and spectral diffusion of quantum emitters in hexagonal boron nitride,” Optica 8, 1153–1158 (2021).
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G. Grosso, H. Moon, C. J. Ciccarino, J. Flick, N. Mendelson, L. Mennel, M. Toth, I. Aharonovich, P. Narang, and D. R. Englund, “Low-temperature electron–phonon interaction of quantum emitters in hexagonal boron nitride,” ACS Photon. 7, 1410–1417 (2020).
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A. Dietrich, M. Doherty, I. Aharonovich, and A. Kubanek, “Solid-state single photon source with Fourier transform limited lines at room temperature,” Phys. Rev. B 101, 081401 (2020).
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N. Mendelson, M. Doherty, M. Toth, I. Aharonovich, and T. T. Tran, “Strain-induced modification of the optical characteristics of quantum emitters in hexagonal boron nitride,” Adv. Mater. 32, 1908316 (2020).
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M. Toth and I. Aharonovich, “Single photon sources in atomically thin materials,” Annu. Rev. Phys. Chem. 70, 123–142 (2019).
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T. T. Tran, C. Bradac, A. S. Solntsev, M. Toth, and I. Aharonovich, “Suppression of spectral diffusion by anti-Stokes excitation of quantum emitters in hexagonal boron nitride,” Appl. Phys. Lett. 115, 071102 (2019).
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B. Sontheimer, M. Braun, N. Nikolay, N. Sadzak, I. Aharonovich, and O. Benson, “Photodynamics of quantum emitters in hexagonal boron nitride revealed by low-temperature spectroscopy,” Phys. Rev. B 96, 121202 (2017).
[Crossref]

T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, “Quantum emission from hexagonal boron nitride monolayers,” Nat. Nanotechnol. 11, 37–41 (2016).
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T. T. Tran, C. Elbadawi, D. Totonjian, C. J. Lobo, G. Grosso, H. Moon, D. R. Englund, M. J. Ford, I. Aharonovich, and M. Toth, “Robust multicolor single photon emission from point defects in hexagonal boron nitride,” ACS Nano 10, 7331–7338 (2016).
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I. Aharonovich, S. Castelletto, D. Simpson, C. Su, A. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
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Allain, A. V.

A. Srivastava, M. Sidler, A. V. Allain, D. S. Lembke, A. Kis, and A. Imamoğlu, “Optically active quantum dots in monolayer WSe2,” Nat. Nanotechnol. 10, 491 (2015).
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Andersen, U. L.

Antón, C.

T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
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Arenal, R.

J. Serrano, A. Bosak, R. Arenal, M. Krisch, K. Watanabe, T. Taniguchi, H. Kanda, A. Rubio, and L. Wirtz, “Vibrational properties of hexagonal boron nitride: inelastic x-ray scattering and ab initio calculations,” Phys. Rev. Lett. 98, 095503 (2007).
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Arora, A.

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotechnol. 10, 503–506 (2015).
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Artús, L.

R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018).
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Auffèves, A.

T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
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Axt, V. M.

D. Wigger, S. Lüker, D. Reiter, V. M. Axt, P. Machnikowski, and T. Kuhn, “Energy transport and coherence properties of acoustic phonons generated by optical excitation of a quantum dot,” J. Phys. Condens. Matter 26, 355802 (2014).
[Crossref]

Baklenov, O.

H. Htoon, T. Takagahara, D. Kulik, O. Baklenov, A. L. Holmes, and C. K. Shih, “Interplay of Rabi oscillations and quantum interference in semiconductor quantum dots,” Phys. Rev. Lett. 88, 087401 (2002).
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Bao, W.

Y. Xia, Q. Li, J. Kim, W. Bao, C. Gong, S. Yang, Y. Wang, and X. Zhang, “Room-temperature giant Stark effect of single photon emitter in van der Waals material,” Nano Lett. 19, 7100–7105 (2019).
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Baumberg, J. J.

A. P. Heberle, J. J. Baumberg, E. Binder, T. Kuhn, K. Kohler, and K. H. Ploog, “Coherent control of exciton density and spin,” IEEE J. Sel. Top. Quantum Electron. 2, 769–775 (1996).
[Crossref]

A. P. Heberle, J. J. Baumberg, and K. Köhler, “Ultrafast coherent control and destruction of excitons in quantum wells,” Phys. Rev. Lett. 75, 2598 (1995).
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Bawendi, M. G.

B. Spokoyny, H. Utzat, H. Moon, G. Grosso, D. Englund, and M. G. Bawendi, “Effect of spectral diffusion on the coherence properties of a single quantum emitter in hexagonal boron nitride,” J. Phys. Chem. Lett. 11, 1330–1335 (2020).
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H. Utzat, W. Sun, A. E. Kaplan, F. Krieg, M. Ginterseder, B. Spokoyny, N. D. Klein, K. E. Shulenberger, C. F. Perkinson, M. V. Kovalenko, and M. G. Bawendi, “Coherent single-photon emission from colloidal lead halide perovskite quantum dots,” Science 363, 1068–1072 (2019).
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S. A. Empedocles and M. G. Bawendi, “Quantum-confined stark effect in single CdSe nanocrystallite quantum dots,” Science 278, 2114–2117 (1997).
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Bayer, M.

M. Bayer, “Bridging two worlds: colloidal versus epitaxial quantum dots,” Ann. Phys. 531, 1900039 (2019).
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Beams, R.

C. Chakraborty, L. Kinnischtzke, K. M. Goodfellow, R. Beams, and A. N. Vamivakas, “Voltage-controlled quantum light from an atomically thin semiconductor,” Nat. Nanotechnol. 10, 507–511 (2015).
[Crossref]

Benson, O.

B. Sontheimer, M. Braun, N. Nikolay, N. Sadzak, I. Aharonovich, and O. Benson, “Photodynamics of quantum emitters in hexagonal boron nitride revealed by low-temperature spectroscopy,” Phys. Rev. B 96, 121202 (2017).
[Crossref]

Bichler, M.

S. Michaelis de Vasconcellos, S. Gordon, M. Bichler, T. Meier, and A. Zrenner, “Coherent control of a single exciton qubit by optoelectronic manipulation,” Nat. Photonics 4, 545–548 (2010).
[Crossref]

S. Stufler, P. Ester, A. Zrenner, and M. Bichler, “Quantum optical properties of a single InxGa1-xAs-GaAs quantum dot two-level system,” Phys. Rev. B 72, 121301 (2005).
[Crossref]

Binder, E.

A. P. Heberle, J. J. Baumberg, E. Binder, T. Kuhn, K. Kohler, and K. H. Ploog, “Coherent control of exciton density and spin,” IEEE J. Sel. Top. Quantum Electron. 2, 769–775 (1996).
[Crossref]

Boll, M. K.

Bonadeo, N. H.

N. H. Bonadeo, J. Erland, D. Gammon, D. Park, D. S. Katzer, and D. G. Steel, “Coherent optical control of the quantum state of a single quantum dot,” Science 282, 1473–1476 (1998).
[Crossref]

Bosak, A.

J. Serrano, A. Bosak, R. Arenal, M. Krisch, K. Watanabe, T. Taniguchi, H. Kanda, A. Rubio, and L. Wirtz, “Vibrational properties of hexagonal boron nitride: inelastic x-ray scattering and ab initio calculations,” Phys. Rev. Lett. 98, 095503 (2007).
[Crossref]

Bourrellier, R.

R. Bourrellier, S. Meuret, A. Tararan, O. Stéphan, M. Kociak, L. H. Tizei, and A. Zobelli, “Bright UV single photon emission at point defects in h-BN,” Nano Lett. 16, 4317–4321 (2016).
[Crossref]

Bradac, C.

T. T. Tran, C. Bradac, A. S. Solntsev, M. Toth, and I. Aharonovich, “Suppression of spectral diffusion by anti-Stokes excitation of quantum emitters in hexagonal boron nitride,” Appl. Phys. Lett. 115, 071102 (2019).
[Crossref]

Bratschitsch, R.

S. Michaelis de Vasconcellos, D. Wigger, U. Wurstbauer, A. W. Holleitner, R. Bratschitsch, and T. Kuhn, “Single-photon emitters in layered van der Waals materials,” Phys. Status Solidi B 259, 2100566 (2022).
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J. A. Preuß, E. Rudi, J. Kern, R. Schmidt, R. Bratschitsch, and S. Michaelis de Vasconcellos, “Assembly of large hBN nanocrystal arrays for quantum light emission,” 2D Mater. 8, 035005 (2021).
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D. Wigger, R. Schmidt, O. Del Pozo-Zamudio, J. A. Preuß, P. Tonndorf, R. Schneider, P. Steeger, J. Kern, Y. Khodaei, J. Sperling, S. Michaelis de Vasconcellos, R. Bratschitsch, and T. Kuhn, “Phonon-assisted emission and absorption of individual color centers in hexagonal boron nitride,” 2D Mater. 6, 035006 (2019).
[Crossref]

P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. van der Zant, S. M. de Vasconcellos, and R. Bratschitsch, “Single-photon emission from localized excitons in an atomically thin semiconductor,” Optica 2, 347–352 (2015).
[Crossref]

Braun, M.

B. Sontheimer, M. Braun, N. Nikolay, N. Sadzak, I. Aharonovich, and O. Benson, “Photodynamics of quantum emitters in hexagonal boron nitride revealed by low-temperature spectroscopy,” Phys. Rev. B 96, 121202 (2017).
[Crossref]

Bray, K.

T. T. Tran, K. Bray, M. J. Ford, M. Toth, and I. Aharonovich, “Quantum emission from hexagonal boron nitride monolayers,” Nat. Nanotechnol. 11, 37–41 (2016).
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Briggs, D. P.

M. A. Feldman, A. Puretzky, L. Lindsay, E. Tucker, D. P. Briggs, P. G. Evans, R. F. Haglund, and B. J. Lawrie, “Phonon-induced multicolor correlations in hBN single-photon emitters,” Phys. Rev. B 99, 020101 (2019).
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Buscema, M.

Camphausen, R.

R. Camphausen, L. Marini, S. A. Tawfik, T. T. Tran, M. J. Ford, and S. Palomba, “Observation of near-infrared sub-Poissonian photon emission in hexagonal boron nitride at room temperature,” APL Photon. 5, 076103 (2020).
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Cassabois, G.

R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018).
[Crossref]

T. Vuong, G. Cassabois, P. Valvin, A. Ouerghi, Y. Chassagneux, C. Voisin, and B. Gil, “Phonon-photon mapping in a color center in hexagonal boron nitride,” Phys. Rev. Lett. 117, 097402 (2016).
[Crossref]

Castellanos-Gomez, A.

Castelletto, S.

I. Aharonovich, S. Castelletto, D. Simpson, C. Su, A. Greentree, and S. Prawer, “Diamond-based single-photon emitters,” Rep. Prog. Phys. 74, 076501 (2011).
[Crossref]

Chakraborty, C.

K. Konthasinghe, C. Chakraborty, N. Mathur, L. Qiu, A. Mukherjee, G. D. Fuchs, and A. N. Vamivakas, “Rabi oscillations and resonance fluorescence from a single hexagonal boron nitride quantum emitter,” Optica 6, 542–548 (2019).
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C. Chakraborty, L. Kinnischtzke, K. M. Goodfellow, R. Beams, and A. N. Vamivakas, “Voltage-controlled quantum light from an atomically thin semiconductor,” Nat. Nanotechnol. 10, 507–511 (2015).
[Crossref]

Chassagneux, Y.

T. Vuong, G. Cassabois, P. Valvin, A. Ouerghi, Y. Chassagneux, C. Voisin, and B. Gil, “Phonon-photon mapping in a color center in hexagonal boron nitride,” Phys. Rev. Lett. 117, 097402 (2016).
[Crossref]

Chauveau, J.-M.

T. Kaldewey, S. Lüker, A. V. Kuhlmann, S. R. Valentin, J.-M. Chauveau, A. Ludwig, A. D. Wieck, D. E. Reiter, T. Kuhn, and R. J. Warburton, “Demonstrating the decoupling regime of the electron-phonon interaction in a quantum dot using chirped optical excitation,” Phys. Rev. B 95, 241306 (2017).
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Chen, M.

K. Liu, Q. Yan, M. Chen, W. Fan, Y. Sun, J. Suh, D. Fu, S. Lee, J. Zhou, S. Tongay, J. Ji, J. B. Neaton, and J. Wu, “Elastic properties of chemical-vapor-deposited monolayer MoS2, WS2, and their bilayer heterostructures,” Nano Lett. 14, 5097–5103 (2014).
[Crossref]

Chen, M.-C.

Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu, and J.-W. Pan, “Single quantum emitters in monolayer semiconductors,” Nat. Nanotechnol. 10, 497–502 (2015).
[Crossref]

Cherkez, V.

M. Koperski, K. Nogajewski, A. Arora, V. Cherkez, P. Mallet, J.-Y. Veuillen, J. Marcus, P. Kossacki, and M. Potemski, “Single photon emitters in exfoliated WSe2 structures,” Nat. Nanotechnol. 10, 503–506 (2015).
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Chong, H. M.

P. Khatri, A. J. Ramsay, R. N. E. Malein, H. M. Chong, and I. J. Luxmoore, “Optical gating of photoluminescence from color centers in hexagonal boron nitride,” Nano Lett. 20, 4256–4263 (2020).
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Ciccarino, C. J.

G. Grosso, H. Moon, C. J. Ciccarino, J. Flick, N. Mendelson, L. Mennel, M. Toth, I. Aharonovich, P. Narang, and D. R. Englund, “Low-temperature electron–phonon interaction of quantum emitters in hexagonal boron nitride,” ACS Photon. 7, 1410–1417 (2020).
[Crossref]

Clark, G.

Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu, and J.-W. Pan, “Single quantum emitters in monolayer semiconductors,” Nat. Nanotechnol. 10, 497–502 (2015).
[Crossref]

Coppola, G.

T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
[Crossref]

Cuscó, R.

R. Cuscó, L. Artús, J. H. Edgar, S. Liu, G. Cassabois, and B. Gil, “Isotopic effects on phonon anharmonicity in layered van der Waals crystals: isotopically pure hexagonal boron nitride,” Phys. Rev. B 97, 155435 (2018).
[Crossref]

De Santis, L.

T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lematre, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics,” Phys. Rev. Lett. 118, 253602 (2017).
[Crossref]

de Vasconcellos, S. M.

Del Pozo-Zamudio, O.

D. Wigger, R. Schmidt, O. Del Pozo-Zamudio, J. A. Preuß, P. Tonndorf, R. Schneider, P. Steeger, J. Kern, Y. Khodaei, J. Sperling, S. Michaelis de Vasconcellos, R. Bratschitsch, and T. Kuhn, “Phonon-assisted emission and absorption of individual color centers in hexagonal boron nitride,” 2D Mater. 6, 035006 (2019).
[Crossref]

Delmonte, V.

T. Jakubczyk, V. Delmonte, S. Fischbach, D. Wigger, D. E. Reiter, Q. Mermillod, P. Schnauber, A. Kaganskiy, J.-H. Schulze, A. Strittmatter, S. Rodt, W. Langbein, T. Kuhn, S. Reitzenstein, and J. Kasprzak, “Impact of phonons on dephasing of individual excitons in deterministic quantum dot microlenses,” ACS Photon. 3, 2461–2466 (2016).
[Crossref]

Dietrich, A.

M. Hoese, P. Reddy, A. Dietrich, M. K. Koch, K. G. Fehler, M. W. Doherty, and A. Kubanek, “Mechanical decoupling of quantum emitters in hexagonal boron nitride from low-energy phonon modes,” Sci. Adv. 6, eaba6038 (2020).
[Crossref]

A. Dietrich, M. Doherty, I. Aharonovich, and A. Kubanek, “Solid-state single photon source with Fourier transform limited lines at room temperature,” Phys. Rev. B 101, 081401 (2020).
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Ding, X.

Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, C.-Y. Lu, and J.-W. Pan, “Single quantum emitters in monolayer semiconductors,” Nat. Nanotechnol. 10, 497–502 (2015).
[Crossref]

Doherty, M.

N. Mendelson, M. Doherty, M. Toth, I. Aharonovich, and T. T. Tran, “Strain-induced modification of the optical characteristics of quantum emitters in hexagonal boron nitride,” Adv. Mater. 32, 1908316 (2020).
[Crossref]

A. Dietrich, M. Doherty, I. Aharonovich, and A. Kubanek, “Solid-state single photon source with Fourier transform limited lines at room temperature,” Phys. Rev. B 101, 081401 (2020).
[Crossref]

Doherty, M. W.

M. Hoese, P. Reddy, A. Dietrich, M. K. Koch, K. G. Fehler, M. W. Doherty, and A. Kubanek, “Mechanical decoupling of quantum emitters in hexagonal boron nitride from low-energy phonon modes,” Sci. Adv. 6, eaba6038 (2020).
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Although in our model we do not specify the character of the optical and acoustic modes coupled to the emitter, for convenience, we will use the labels longitudinal optical (LO) and longitudinal acoustic (LA) here, since at least in the region close to the center of the Brillouin zone, these are often the most strongly coupled ones [22].

Supplementary Material (1)

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Supplement 1       Supplemental document

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Light emission from an individual hBN color center. (a) Photoluminescence (PL) and photoluminescence excitation (PLE) spectra in red and blue, respectively. Measurement (dark red) and simulation (light red) with green shaded areas marking the different phonon sidebands (PSBs). The zero-phonon line (ZPL) energy is ${E_{{\rm{ZPL}}}} = 2.036\,\,{\rm{eV}}$. (b) Schematic drawing of the two-level structure (solid lines) including ZPL (violet), phonon-assisted processes via virtual excited states (dashed lines) as PSBs (green), and PLE (blue).
Fig. 2.
Fig. 2. Ultrafast coherent control experiment on a single-photon emitter in hBN. (a) Schematic of the coherent control experiment: (top) constructive timing of the second pulse, increasing the emitter occupation; (bottom) destructive timing decreasing the occupation. (b) Measured variations of PL intensity for a small coarse delay ($\Delta t = 1\,\,{\rm{ps}}$, left) and large coarse delay ($\Delta t = 33\,\,{\rm{ps}}$, right) for resonant excitation on the ZPL at $T = 8\,\,{\rm{K}}$. The upper $x$ axis marks the relative phase between emitter and second pulse. (c) PL visibility dynamics for resonant excitation on the ZPL. Dark circles represent the experiment, and the solid curve represents the simulation.
Fig. 3.
Fig. 3. Jitter dynamics and time scales for coherent control. (a) Illustration of the spectral jitter changing the transition energy in time. (b) Illustration of the repetition of the coherent control experiment. (c), (d) Fast and slow jitter components with corresponding homogeneous and inhomogeneous decays of the PL visibility. (e) Illustration of time scales in the experiment. Quantities marked in blue (homogeneous decoherence) originate from fast jitter and those in green [random telegraph noise (RTN), inhomogeneous decoherence] from slow jitter.
Fig. 4.
Fig. 4. Coherent control experiment with telegraph noise. (a) PL spectra as a function of detection time covering 100 s showing spectral jumps between two energies. (b) Time-integrated PL spectrum retrieved from (a). ZPL energy of ${E_{{\rm{ZPL}}}} = 2.036\,\,{\rm{eV}}$ is defined as the mean value of two peak positions. (c) Coherent control visibility dynamics of double-peaked ZPL. Dark circles represent the experiment, and solid curves represent the simulation.
Fig. 5.
Fig. 5. Coherent control of optical phonon sidebands (PSBs). (a) Coherent control visibility dynamics for LO-phonon-assisted excitation in blue: ${{\rm{LO}}_1}$ mode, solid blue curve; ${{\rm{LO}}_2}$ mode, dashed blue curve; circles represent the experiment, and curves represent the simulation. In addition, the visibility for resonant excitation is shown in purple and laser autocorrelation is shown in green. The respective PLE spectra of the LO PSBs are shown in the inset. (b) Illustration of phonon-assisted coherence decay. The harmonic potentials represent phonon excitations in ground $| G \rangle$ and excited $| X \rangle$ states. The coherence is indicated by the orange ellipse. Phonon-induced dephasing (green arrow) results in the loss of coherence (dashed ellipse).
Fig. 6.
Fig. 6. Coherent control of acoustic PSBs. (a) Measured and simulated PLE data are shown as dark circles and bright curves, respectively. The colored vertical lines mark the detunings considered in (b). (b) Visibility dynamics for increasing detunings from top to bottom with negative detunings (left) and positive detunings (right). Experiment is shown as dark circles and simulation as bright curves. Laser autocorrelation is shown in green (dashed curves). Vertical dashed lines mark half a period corresponding to the frequency detuning between laser and ZPL. (c) Illustration of the laser-pulse-induced generation of a phonon wave packet.

Tables (1)

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Table 1. List of the Different Time Scales in the Experimenta

Equations (14)

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v P L = max ( I P L ) min ( I P L ) max ( I P L ) + min ( I P L ) ,
v P L ( Δ t ) = exp [ γ p d 2 Δ t σ j 2 ( Δ t ) 2 2 ] , w i t h 2 / γ p d = 55 p s a n d 1 / σ j = 19 p s .
T e f f = γ p d 2 σ j 2 ( 8 σ j 2 / γ p d 2 + 1 1 ) 21 p s .
H = ω T L S X X + 1 2 [ E ( t ) X + E ( t ) X ] + n ω n b n b n + X X n ( g n b n + g n b n ) + H p d + H a n h .
I P L ( ω ) = d Ω R ( ω , Ω ) d t G + ( t ) e i ( Ω ω Z P L ) t e γ p d | t | / 2 ,
G + ( τ ) = exp [ ϕ ( τ ) ϕ ( 0 ) ] ,
ϕ ( τ ) = 0 d ω J L A ( ω ) ω 2 { [ n ( ω ) + 1 ] e i ω τ + n ( ω ) e i ω τ } + 0 d ω J L O ( ω ) ω 2 exp [ i ω τ γ ( ω ) 2 | τ | ] .
I 0 ( ω l , Δ t ) = d t d τ E ~ ( t ) E ~ ( t + τ ) × e i ω l τ G + ( Δ t τ ) e i ω Z P L ( Δ t τ ) e γ p d | Δ t τ | / 2 ,
I P L E ( ω l ) = I 0 ( ω l , Δ t = 0 ) ,
v ( Δ t ) = | I 0 ( ω l , Δ t ) | I 0 ( ω l , Δ t = 0 ) ,
d t E ~ ( t ) E ~ ( t + Δ t ) e i ω l Δ t .
v ( Δ t ) | a ( Δ t ) e i δ Δ t + b | a ( Δ t = 0 ) + b ,
a ( Δ t ) = I a b s , P S B ( ω l ) d t E ~ ( t ) E ~ ( t + Δ t ) ,
b = B 2 I l a s e r i d e a l ( ω Z P L ) ,
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