1. Introduction

The spectral dynamics of single emitters can broadly be categorized as fluctuations and relaxation. Fluctuations are temporally stochastic variations around the equilibrium configuration of any physical or chemical system interacting with its environment at non-zero temperature. Relaxation refers to the system’s return to the ground state equilibrium after preparation of a non-equilibrium state, for example, through a laser-driven creation of an excited state population. For single optical emitters, bath fluctuations on timescales faster than the single-emitter lifetime induce pure dephasing by randomizing the phase of the transition. Fluctuations on longer timescales can manifest as spectral diffusion, which is spectral jumping occurring from nanoseconds to seconds that is reflective of the microscopic interaction of the bath with the excited state [1,2]. Relaxation occurs via irreversible phonon- or spin-mediated dissipation or spontaneous emission of photons, processes typically observed from picoseconds to microseconds [3].

Simultaneous measurement of the full relaxation and fluctuation dynamics for single emitters requires a technique with high spectral and temporal resolution with an additionally high temporal dynamic range from picoseconds to seconds. No single best-suited technique exists and different approaches present their own strengths and weaknesses. Streak-cameras can resolve the spectral evolution along the photoluminescence lifetime trajectory with picosecond resolution, but fail to resolve any relaxation beyond a few nanoseconds or fast spectral diffusion [4,5]. CCD-based single-emitter emission spectroscopy can have millisecond temporal resolution, especially if auto-correlation of individual spectra recorded at high sweep-rates is performed, but fails to resolve even faster fluctuations and does not discriminate spectral dynamics along the lifetime trajectory [6,7]. Hong-Ou-Mandel (HOM) spectroscopy has been used to measure the coherence of single photons through two-photon interference. HOM can resolve energy fluctuations through a decrease in photon-coalescence efficiency on fast (picosecond to nanosecond) timescales and lifetime-resolved photon-presorting is straightforward [8,9]. However, HOM spectroscopy is exclusively suitable for highly-coherent single-photon emitters at low temperatures as it requires photon-coherences near the transform limit. While the dynamic range of HOM can in principle be extended to microseconds by using optical fibers as delay-lines, [10] most implementations exhibit dynamic ranges limited to a few nanoseconds delay-time between the two interfering photons. Cross-correlating spectrally-filtered photons provides higher temporal dynamic range, but is limited by the finite bandwidth of optical filters restricting the technique to broad lineshapes and spectral diffusion with large energetic spread [11]. Optical dispersion has recently been introduced to translate the spectral width of single-photons into temporal delays that can be mapped out with photon-correlation, providing an interesting pathway for measuring spectral diffusion with high temporal dynamic range [12].

The case of Fourier spectroscopy is not bound by limitations in spectral bandwidth or two-photon delay-times as each photon self-interferes. As a result, Fourier spectroscopy can readily be married with photon correlation to provide spectral readout with high temporal dynamic range and with arbitrarily high temporal resolution only limited by photon shot-noise [13]. This technique, photon correlation Fourier spectroscopy (PCFS), has now been established as a powerful tool for the study of optical dephasing and spectral fluctuations of single quantum emitters at low and room temperatures. [14–16]. Recently, the temporal resolution of PCFS for fluctuation characterization has been pushed down to a few nanoseconds by using pulsed laser excitation of a single quantum emitter [17]. However, despite the success in characterizing spectral fluctuations, so far PCFS was unable to resolve any spectral changes associated with relaxation of the system back to the ground state, for example during radiative decay, phonon-mediated relaxation, or energy transfer between different states.

Here, we propose a pulsed excitation-laser analog of PCFS that readily extracts relaxation and fluctuation dynamics from single quantum emitters on timescales faster than the radiative lifetime. The proposed technique is depicted in Fig. 1. PCFS works by encoding the spectral information into intensity anti-correlations recorded at the output arms of an interferometer. In analogy to linear Fourier spectroscopy, which measures the spectrum, PCFS measures the spectral correlation, the auto-correlation of the spectrum compiled from photon-pairs with a certain temporal separation of $\tau$ along the macrotime axis $t$ of the experiment. In conventional PCFS, all photons emitted under continuous-wave laser excitation are used to compile the spectral correlation (a). We define the macrotime as the total experimental time elapsed when the photon is detected, often referred to as the absolute clock. The macrotime is calculated as the number of the laser trigger-pulses elapsed since the start of the experiment $N$ times the laser repetition time $t_{rep}$ plus the microtime T of the photon: $Nt_{rep}+T$. In lifetime-resolved PCFS, photon-pairs are additionally correlated in the microtime $T$, the time elapsed between the pulsed laser excitation trigger and photon-detection in units of multiple integers of the laser repetition time.

Specifically, the photons are binned according to their microtime, sometimes referred to as the time-correlated single-photon counting (TCSPC) channel, and spectral correlations are calculated using these microtime-separated photons (b). The technique can be readily implemented using picosecond single-photon counting equipment as shown in Fig. 1(c) and requires high-throughput post-processing photon-correlation analysis. We show through numerical simulation that this lifetime-resolved PCFS technique can separate the lineshapes and spectral diffusion dynamics of systems with more than one emissive state as long as the relative weights of the emission from different states changes over the course of the photoluminescence lifetime. More broadly, lifetime-resolved PCFS can in principle extract spectral fluctuations and relaxation with temporal resolutions only limited by the IRF response time of single-photon counting modules.

 

Fig. 1. In conventional PCFS, the spectral correlation is compiled from photon-pairs irrespective of their microtime T, often under continuous wave excitation (a). In lifetime-resolved PCFS, photon-pairs with a given microtime $T$ and macrotime separation $\tau$ are spectrally correlated (b). Here, we adopt a time-binning approach to collect photons with different T in suitable microtime intervals as indicated by the color-shaded background. The proposed optical setup is shown in (c). The photon-stream from a single quantum emitter under pulsed excitation is directed into a variable path-length difference Michelson interferometer. All photon-counts at the output arms of the interferometer are recorded in time-tagged (T3) mode using picosecond single-photon counting electronics.

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2. Theoretical derivation and numerical simulation

2.1 Lifetime-resolved PCFS

In PCFS, the interferometer path-length difference is adjusted to discrete positions that lie withing coherence length of the emission and periodically dithered on second timescales and over a multiple of the emitter center wavelength [13]. The dither introduces anti-correlations in the intensity cross-correlation functions of the output arms that encode the degree of spectral coherence at a given center position. The lineshape dynamics is thus encoded in the intensity correlations as a function of the time-separation between photons $\tau$. We show in Supplement 1 that the PCFS equations can straightforwardly be expanded to include spectral dynamics along the microtime $T$. The central observable in lifetime-resolved PCFS for a spectrum $s(\omega ,t,T)$ dependent on the microtime $t$ and macrotime $T$ is given by the spectral correlation $p(\zeta ,\tau ,T)$ as

We first discuss the general form of the spectral correlation for a system undergoing spectral diffusion in section 2.2. We then discuss two universal systems that map onto many specific real-world scenarios. First, we consider a system of two uncoupled and lifetime-distinct radiating dipoles undergoing uncorrelated spectral diffusion in section 2.3. Second, we consider a system of two coupled radiating dipoles subject to population exchange and correlated spectral diffusion in section 2.4.

2.2 Effect of spectral fluctuations on the spectral correlation

We consider Eq. (1) for spectral fluctuations $\delta \omega (t,T)$ that present along the macrotime axis of the experiment around the center frequency $\omega _{0}$ of a spectrum. We can write for the spectrum $s(\omega ,t,T)=s(\omega ,T) \otimes \delta (\omega -\delta \omega (t,T))$, where $\otimes$ is the convolution, $s(\omega ,T)$ the undiffused spectrum, and $\delta \omega (t,T)$ the time-dependent shift from the center wavelength. Spectral fluctuations can be characterized by the correlation function $C(\tau )=\langle \delta \omega (t,T)\delta \omega (t+\tau , T)\rangle$. The canonical form of any spectral correlation can then be recast as

2.3 Lifetime-distinct doublet undergoing Gaussian spectral fluctuations

We discuss a system of two uncoupled and lifetime-distinct radiating dipoles undergoing uncorrelated spectral diffusion and involving states $|A\rangle$ and $|B\rangle$. The system’s energy diagram is shown in Fig. 2(a) (inset). The microscopic interpretation involves a system with two emissive states coupled to different bath fluctuations. We show how lifetime-resolved PCFS can separate the homogeneous lineshape and spectral diffusion parameters of the the two transitions.

 

Fig. 2. Simulation of two uncoupled radiating dipoles involving states $|A\rangle$ and $|B\rangle$. The two transitions are coupled to two different bath fluctuations and exhibit different lifetimes $T_{1}$, linewidths $\Gamma$ and spectral diffusion parameters $\tau _{c}$ and $\sigma _{A,B}$. The total fluorescence lifetime of the system exhibits a biexponential decay (a). The shaded panels (b) and (c) show the cross-correlation functions $g_{X}^{(2)}(\tau )$ for different optical path-length differences $\delta _{0}$ and microtimes $T$, where $|A\rangle$ and $|B\rangle$ are the dominant emissive states, respectively. The loss of coherence with increasing $\tau$ is evident from the reduction in anti-correlation. This coherence loss occurs at earlier $\tau$ for early-$T$ photons (emission predominantly from $|A\rangle$, (b)) compared to late-$T$ photons (emission predominantly from $|B\rangle$, (c)). The PCFS interferogram $G^{(2)}(\delta ,\tau )$ for early-T photons is shown in (d) and reflects the evolution from the exponential homogeneous dephasing at early $\tau$ to the spectrally-diffused Gaussian dephasing at late $\tau$.

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The different emission lifetimes result in microtime-dependent relative weights of emission intensity originating from states $|A\rangle$ and $|B\rangle$ after equal populations have been prepared through laser excitation. We decompose the overall dynamic spectrum of the system $s(\omega ,t,T)$ into microtime-dependent components as $s(\omega ,t,T)=a(T)s_{A}(\omega ,t)+b(T)s_{B}(\omega ,t)$, where $a(T)$ and $b(T)$ are the relative probabilities of a given photon originating from either state $|A\rangle$ or $|B\rangle$, and show that the spectral correlation expands as

The terms quadratic in $a(T)$ and $b(T)$ represent the spectral auto-correlations of the individual states $p_{AA}$ and $p_{BB}$, while the cross-terms involving $p_{AB}$ represent the cross-correlation of the spectra $s_{A}(\omega ,t,T)$ and $s_{B}(\omega ,t,T)$. The form of the spectral correlation can be understood intuitively because the spectral correlation is compiled from pairs of photons with origins drawn from the four possible combinations of $|A\rangle$ and $|B\rangle$. Importantly, the left- and right-sided correlations $p_{AB}$ and $p_{BA}$ are not identical unless $s_{A}(\omega )$ and $s_{B}(\omega )$ share the same center frequency $\omega _{0}$ and are symmetric in $\omega$.

Spectral diffusion is a ubiquitous process observed for many single-quantum emitters. Common descriptions of single-emitter spectral diffusion are the non-Markovian and discrete Poissonian Wiener process [18] or the mean-reverting Ornstein-Uhlenbeck process [19]. These processes describe spectral diffusion phenomenologically and for simplicity we consider a simple non-Markovian Poissonian Gaussian jumping model (GJM) [15]. The GJM process is characterized by a time-invariant probability density for discrete spectral jump occurrence to a new spectral position drawn from a Gaussian probability distribution function over $\omega$. For the two states $|A\rangle$ and $|B\rangle$ as denoted in the subscripts, we write $Prob(\delta \omega _{A,B})=\frac {1}{\sigma _{A,B}\sqrt {2\pi }}e^{-\frac {\delta \omega ^{2}}{2\sigma _{A,B}^{2}}}$ for the probability of a given spectral shift at a point in time. Here, we have introduced the spectral fluctuation term $\delta \omega _{A,B}$ from earlier. The microscopic interpretation of this process is the time-stochastic variation of the bath assuming discrete conformations coupling to the system. The corresponding fluctuation correlation function can be written as $C(\tau )=e^{-\tau /\tau _{c}}$ and is described by an exponential decay with a characteristic spectral jump time of $\tau _{c}$. When the two states diffuse independently of each other, no correlation is present and $C_{AB}(\tau )=0$. In this case, $\langle \delta \omega _{A}(t)\delta \omega _{B}(t+\tau )\rangle =\langle \delta \omega _{a}(t)\rangle \langle \delta \omega _{B}(t+\tau )\rangle =0$ because independently diffusing emissive states will not be correlated and the cross-terms $p_{AB}$ and $p_{BA}$ in Eq. (3) only reflect the cross-correlations of the inhomogeneous components $p_{AB/BA}(\zeta ,\tau \rightarrow \infty )$. Absent any memory of spectral fluctuations even at early $\tau$, the time average over the spectral-correlations of all random configurations is the cross-correlation of the inhomogeneously broadened (diffused) spectra

We numerically simulate the system of independently-diffusing optical transitions with parameters commensurate with typical experimental cryogenic single-emitter spectroscopy (see Supplement 1). The time-domain results of the simulation are discussed in Fig. 2.

The configuration of the system is shown in Fig. 2(a), where $\tau _c$ is the characteristic jump time of spectral diffusion, $T_{1}$ the population decay time, and $\Gamma$ is the optical coherence time of the transition, which is related to the Lorentzian linewidth as $\hbar /\Gamma$. The corresponding lifetime exhibits biexponential decay behavior as expected. In (b) and (c), we compare the cross-correlation functions for two different slices with microtime ranges of $T=0-100$ ps and $T=2000-7000$ ps, where $|A\rangle$ and $|B\rangle$ are the dominant emissive states, respectively. Unlike for the static doublet discussed in Supplement 1, the cross-correlations $g_{X}^{(2)}(\tau )$ indicate spectral dynamics evident from the loss of anti-correlation at longer $\tau$. As we specify different jumping rates for the two states, the decay of the spectral coherence evident in (b) and (c) occurs at different $\tau$. The PCFS interferogram derived from the cross-correlations (see Supplement 1 for the derivation) for photons emitted with a microtime of $<100ps$ is shown in (d) and informs on the loss of photon-coherence between $1\mu \textrm {s}$ and $1\textrm {ms}$ owing to the energy fluctuations of the photons emitted $<100$ ps after laser excitation.

In Fig. 3 we discuss the same simulation results in the spectral domain. In (a), we show the full-width-at-half-maximum (FWHM) of the spectral correlation for both $T$ and $\tau$, a representation that makes immediately obvious the differences in the homogeneous linewidths at early $\tau$ and the differences in spectrally-diffused linewidths at late $\tau$.

 

Fig. 3. Spectral results of the lifetime-resolved PCFS simulation of two uncoupled dipoles. (a) shows the full-width-at-half-maximum (FWHM) of $p(\zeta ,\tau ,T)$ along $T$ and $\tau$. The difference in the homogeneous linewidths of $|A\rangle$ and $|B\rangle$ at early $\tau$ and in the diffused linewidths at late $\tau$ are immediately obvious in this representation. We show the evolution of the weights of auto- and cross-correlations between states along $T$ in (b).The weights are derived from the relative amplitude of the two exponential components of the photoluminescence decay in (Fig. 2(a)).The orange-shaded panels (c) and (d) show the effect of spectral diffusion for early-microtime photons originating mostly from state $|A\rangle$. Taking $p_{AA}$,$p_{BB}$, and $p_{AB,BA}$ into account, we apply a global fit to the spectral correlation along $T$ to recover the lineshape parameters of the undiffused system as shown in (e),(f) and (g). The broad underlying Gaussian component in (f) reflects the cross-correlation of the diffused distributions of $|A\rangle$ and $|B\rangle$ and has a width of $\sigma \approx \sqrt {\sigma _{A}^2+\sigma _{B}^2}$.

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For completeness we also show $p(\zeta ,T)$ (b) and $p(\zeta ,\tau )$ (c) for fixed $\tau$ and $T$, respectively. These two representations inform on the spectral evolution owing to spectral diffusion and changing relative emission contributions from different states, respectively. (d) displays the evolution of the spectral correlation from the narrow homogeneous spectrum with a Lorentzian lineshape to the diffused Gaussian lineshape.

One capability of lifetime-resolved PCFS is the ability to extract the homogeneous linewidths of different lifetime-distinct states in the presence of fast spectral diffusion. We demonstrate this ability through a global fit to the T-dependent spectral correlation. We define a model for the fit as a linear combination of two Lorentzians and a Gaussian with floating linewidths parameters. The relative amplitudes $p_{AA}$,$p_{BB}$, and $p_{AB,BA}$ are calculated according to Eq. (3) taking the weights $a(T)$ and $b(T)$ from fits to the emission lifetime into account. $p_{AA}$,$p_{BB}$, and $p_{AB,BA}$ are also displayed in Fig. 3(d). We apply a global fit to the slices of the spectral correlation $p(\zeta ,\tau =60 \mu \textrm {s}, T)$ along $T$ as shown in Figs. 3(e),(f) and (g). The cross-correlation $p_{AB,BA}$ present as a broad Gaussian background superimposed with the homogeneous Lorentzian spectral correlations $p_{AA,BB}$ as introduced in Eq. (3). The width of this Gaussian component is $\sigma _{AB}\approx \sqrt {\sigma _{A}^{2}+\sigma _{B}^{2}}$. The homogeneous lineshape parameters parsed into the numerical model are extracted by the fit within photon shot-noise, thus validating the approach adapted herein. We note that in PCFS, the high temporal resolution achieved through photon-correlation comes at the cost of the loss of the absolute phase of the spectral information. In other words, both the asymmetry of the lineshape and the center frequency of $s(\omega )$ is lost in the spectral correlation $p(\zeta )$. The unambiguous reconstruction of $s(\omega )$ from $p(\zeta )$ is therefore impossible and the spectral correlation is typically fit with a model parametrizing a suitable form for the underlying emission spectrum, as we adapted herein [15,16].

2.4 Dynamic doublet with population transfer and spectral fluctuations

We now turn to a system of two coupled radiating dipoles undergoing population exchange and subject to correlated spectral diffusion. A specific example would be solid-state quantum emitters undergoing incoherent and phonon-mediated population transfer after non-resonant excitation [20]. In quantum emitters, disentangling the relaxation rate and coherence times of the different fine-structure states in the presence of spectral diffusion is important for a detailed understanding of the dephasing process as phonon-mediated population exchange constitutes an important dephasing process in the solid-state [3]. We depict the system’s energy diagram in Fig. 4(a), which exhibits two excited states with equal oscillator strengths and an irreversible relaxation rate $k_{relax}$ from the higher to the lower-lying state.

 

Fig. 4. Lifetime-resolved PCFS simulation of two coupled dipoles undergoing population transfer and interacting with the same bath resulting in collective spectral diffusion of the doublet (a). We introduce a phonon-mediated relaxation rate between the upper and lower state of $k_{relax}=1/80 \textrm {ps}^{-1}$. As the radiative rates of the two states are chosen to be equal, the emission lifetime follows a monoexponential decay behavior despite changing relative populations of $|A\rangle$ and $|B\rangle$ with the microtime (b). The spectral correlation irrespective for all photons irrespective of their microtime is shown in (c) and demonstrates the transition from a triplet at early $\tau$ to the spectrally-diffused distribution at late $\tau$. The fine-structure splitting $\Omega$, the linewidths $\hbar \Gamma _{A,B}$, and the relaxation rate $k_{relax}$ can be recovered through lifetime-resolved PCFS and a global fit of the slices along $T$ with a fixed macrotime correlation of $\tau =8 \mu s$ (d),(e) and (f).

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In this system, photon emission from the higher-lying state $|A\rangle$ will start immediately after population of the state. Emission of the lower-lying state $|B\rangle$ requires further relaxation and is often phonon-mediated [3]. The relative population of states $|A\rangle$ and $|B\rangle$, which we define as $P_{A}$,$P_{B}$, will thus change during the emission lifetime of the overall system as long as the relaxation rate $k_{relax}$ is faster than the radiative rate $1/T_{1}$ of both $|A\rangle$ an $|B\rangle$. The population dynamics of the system can be described by the following set of coupled equations:

We show the effect of the changing relative cross-correlation probabilities between states $|A\rangle$ and $|B\rangle$ ($p_{AA}$,$p_{BB}$) in Fig. 4(b). Despite the $T$-invariant exponential population decay constant leading to a monoexponential photoluminescence lifetime of the overall system, the relative weights of $p_{AA}$ and $p_{BB}$ are changing with $T$. We show the spectral correlation of the lifetime-resolved PCFS experiment with indiscriminate $T$ in (c). On timescales shorter than the spectral diffusion time $\tau$, the fine-structure states are well-separated. At late $\tau$, the broad diffused lineshape obfuscates the fine-structure splitting. We demonstrate that lifetime-resolved PCFS can recover the lineshape parameters of the homogeneous doublet by applying a least-squares fit of a suitable model to the T-dependent spectral correlation as shown in (d),(e),(f). The model consists of two Lorentzians with the floating linewidths $\hbar /\Gamma _{1}$, $\hbar /\Gamma _{2}$, an energy offset $\Omega$, and a relaxation rate $k_{relax}$, which determines the temporal change of the relative emission contributions of $|A\rangle$ and $|B\rangle$. We recover all model parameters within photon shot-noise thus validating the utility of lifetime-resolved PCFS to extract the coherences and relaxation rates of different emissive fine-structure states. We note that the observation of early-$\tau$ multiplets in the spectral correlation compared to the broad Gaussian background in section 2.3 is the signature of correlated spectral diffusion dynamics between the two states. Our simulations suggest that measuring the photon-coherences of quantum emitters exhibiting spectral fluctuations and different emissive fine-structure states will provide an avenue to study quantum emitter optical dephasing through both fluctuations and population exchange between different electronic states.

3. Conclusions

We propose a new photon-correlation spectroscopic technique that extracts spectral fluctuations along the lifetime-trajectory of single quantum emitters. The technique works through time-correlation of photons detected at the output arms of a variable path-length difference interferometer in both the microtime and macrotime domain and can be implemented using standard picosecond photon-counting electronics. We show that lineshape and fluctuation parameters can be extracted from the fits to the lifetime-resolved spectral correlations. Our technique opens up multiple frontiers in single-emitter spectroscopy. We emphasize that our technique is general, but point to its special utility in quantum emitter research enabled by the high spectral resolution required to resolve photon-coherences at low temperatures. Experimental efforts will be directed towards probing the fluctuation dynamics of non-stationary systems and investigation of the decoherence processes in quantum emitters. Specific materials are readily available such as emissive defects in diamond and emerging 2D materials as well as semiconductor nanostructures.