Lifetime-resolved Photon-Correlation Fourier Spectroscopy
LL IFETIME - RESOLVED P HOTON -C ORRELATION F OURIER S PECTROSCOPY
Hendrik Utzat
Department of Materials Science and EngineeringStanford UniversityStanford, CA 94305 [email protected]
Moungi G. Bawendi
Department of ChemistryMassachusetts Institute of TechnologyCambridge, MA 02139 [email protected]
February 9, 2021 A BSTRACT
The excited state population of single solid-state emitters is subjected to energy fluctuations aroundthe equilibrium driven by the bath and relaxation through the emission of phonons or photons.Simultaneous measurement of the associated spectral dynamics requires a technique with a highspectral and temporal resolution with an additionally high temporal dynamic range. We propose apulsed excitation-laser analog of Photon-Correlation Fourier Spectroscopy (PCFS), which extractsthe lineshape and spectral diffusion dynamics along the emission lifetime trajectory of the emitter,effectively discriminating spectral dynamics from relaxation and bath fluctuations . This lifetime-resolved PCFS correlates photon-pairs at the output arm of a Michelson interferometer in both theirtime-delay between laser-excitation and photon-detection T and the time-delay between two photons τ . We propose the utility of the technique for systems with changing relative contributions to theemission from multiple states, for example, quantum emitters exhibiting phonon-mediated exchangebetween different fine-structure states. The spectral dynamics of single emitters can broadly be categorized as fluctuations and relaxation. Fluctuations aretemporally stochastic variations around the equilibrium configuration of any chemical system interacting with itsenvironment at non-zero temperature. Relaxation refers to the system’s return to the ground state equilibrium afterpreparation of a non-equilibrium state, for example, through a laser-driven creation of an excited state population.For single optical emitters, fluctuations can manifest as spectral diffusion, which is spectral jumping occurring fromnanoseconds to seconds that is reflective of the microscopic interaction of the bath with the excited state.[1, 2] Relaxationoccurs via irreversible phonon- or spin-mediated dissipation or spontaneous emission of photons, processes typicallyobserved from picoseconds to microseconds.[3]Simultaneous measurement of the full relaxation and fluctuation dynamics for single emitters requires a technique withhigh spectral and temporal resolution with an additionally high temporal dynamic range from picoseconds to seconds.No single such technique exists and different approaches present their own strengths and weaknesses. Streak-camerascan resolve the spectral evolution along the photoluminescence lifetime trajectory with picosecond resolution, butfail to resolve any dynamics beyond a few nanoseconds and typically lack single-molecule sensitivity.[4] CCD-basedsingle-molecule emission spectroscopy can have millisecond temporal resolution, especially if auto-correlation ofindividual spectra recorded at high sweep-rates is performed, but fails to resolve even faster fluctuations and doesnot discriminate spectral dynamics along the lifetime trajectory.[5, 6] Hong-Ou-Mandel (HOM) spectroscopy hasbeen used to measure the coherence of single photons through two-photon interference. HOM can resolve energyfluctuations through a decrease in photon-coalescence efficiency on fast (picosecond to nanosecond) timescales andlifetime-resolved photon-presorting is straightforward.[7, 8] However, HOM spectroscopy is exclusively suitable forsingle-photon emitters at low temperatures as it requires photon-coherences near the transform limit. Moreover, thedynamic range of HOM is practically limited to a few nanoseconds delay-time between the two interfering photons,rendering HOM of limited utility for measuring fluctuations occurring over many orders of magnitude in time. Cross- a r X i v : . [ qu a n t - ph ] F e b Y - F EBRUARY
9, 2021correlating spectrally-filtered photons provides higher temporal dynamic range, but is limited by the finite bandwidth ofoptical filters restricting the technique to broad lineshapes and spectral diffusion with large energetic spread.[9]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 readoutwith high temporal dynamic range and with arbitrarily high temporal resolution only limited by photon shot-noise.[10]This technique, Photon Correlation Fourier Spectroscopy (PCFS), has now been established as a powerful tool for thestudy of optical dephasing and spectral fluctuations of single emitters at low and room temperatures.[11, 12, 13]Despitethe success in characterizing spectral fluctuations, so far PCFS was not able to resolve any spectral changes associatedwith relaxation of the system back to the ground state, for example phonon-mediated relaxation or energy transferbetween different states.Here, we propose a pulsed excitation-laser analog of PCFS that readily extracts relaxation and fluctuation dynamicsfrom single emitters. The proposed technique is shown in Figure 1. In conventional PCFS, all photons emitted after thecontinuous-wave laser excitation are used to compile the spectral correlation (a), the auto-correlation of the spectrumcompiled from photon pairs with a temporal separation of τ along the macrotime axis t of the experiment and anenergy difference of ζ . In lifetime-resolved PCFS, photon-pairs are additionally correlated in the microtime T afterpulsed laser excitation. Specifically, the photons are binned according to their microtime, sometimes referred toas the TCSPC channel, and spectral correlations are calculated using these microtime-separated photons (b). Thetechnique can be readily implemented using picosecond single-photon counting equipment as shown in Fig. 1c andrequires high-throughput post-processing photon-correlation analysis. We show through numerical simulation that thislifetime-resolved PCFS technique can separate the lineshapes and spectral diffusion dynamics of systems with morethan one emissive state as long as the relative weights of the emission from different states changes over the course ofthe photoluminescence lifetime. More broadly, lifetime-resolved PCFS can in principle extract spectral fluctuations andrelaxation with temporal resolutions only limited by the IRF response time of single-photon counting modules. In PCFS, the interferometer path-length difference is adjusted to discrete positions inside the coherence length of theemission and periodically dithered on second timescales and over a multiple of the emitter center wavelength.[10] Thedither introduces anti-correlations in the intensity cross-correlation functions of the output arms that encode the degreeof spectral coherence at a given center position. The lineshape dynamics is thus encoded in the intensity correlations as afunction of the time-separation between photons τ . We show in the Supplementary Information that the PCFS equationscan straightforwardly be expanded to include spectral dynamics along the microtime T . The central observable inlifetime-resolved PCFS for a spectrum s ( ω, t, T ) dependent on the microtime t and macrotime T is given by the spectralcorrelation p ( ζ, τ, T ) as p ( ζ, τ, T ) = (cid:104) (cid:90) ∞−∞ s ( ω, t, T ) ∗ s ( ω + ζ, t + τ, T ) dω (cid:105) , (1)where (cid:104) ... (cid:105) represents the time average. Equation 1 describes the central observable in lifetime-resolved PCFS and canbe understood intuitively as a histogram of photon-pairs with a shared microtime T , a macrotime separation of τ , andan energy separation ζ . The form and interpretation of p ( ζ, τ, T ) depend on the dynamics of the emissive system andwill be discussed in the following sections.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 oftwo 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 diffusionin section 2.4. We consider equation 14 for spectral fluctuations δω ( t, T ) that present along the macrotime axis of the experimentaround the center frequency ω of a spectrum. We can write for the spectrum s ( ω, t, T ) = s ( ω, T ) ⊗ δ ( ω − δω ( t, T )) ,where ⊗ is the convolution, s ( ω, T ) the undiffused spectrum, and δω ( t, T ) the time-dependent shift from the centerwavelength. Spectral fluctuations can be characterized by the correlation function C ( τ ) = (cid:104) δω ( t, T ) δω ( t + τ, T ) (cid:105) . Thecanonical form of any spectral correlation can then be recast as2 Y - F EBRUARY
9, 2021 p ( ζ, τ, T ) = (cid:104) (cid:90) ∞−∞ s ( ω, t, T ) s ( ω + ζ, t + τ, T ) dω (cid:105) = C ( τ ) p ( ζ, τ → , T ) + [1 − C ( τ )] p ( ζ, τ → ∞ , T ) , (2)reflecting the transition from the undiffused spectral correlation (absent any fluctuations p ( ζ, τ → , T ) ) to the diffusedspectral correlation p ( ζ, τ → ∞ , T ) with the evolution of C ( τ ) . Note that for τ → , δω ( t , T ) = δω ( t , T ) andthe spectral correlation thus reduces to the homogeneous spectral correlation p ( ζ, τ → , T ) = (cid:104) (cid:82) ∞−∞ s ( ω, T ) s ( ω − ζ, T ) dω (cid:105) . We discuss a system of two uncoupled and lifetime-distinct radiating dipoles undergoing uncorrelated spectral diffusionand involving states | A (cid:105) and | B (cid:105) . The system’s energy diagram is shown in figure 2a (inset). The microscopicinterpretation 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.The different emission lifetimes result in microtime-dependent relative weights of emission intensity originating fromstates | A (cid:105) and | B (cid:105) after equal populations have been prepared through laser excitation. We decompose the overalldynamic spectrum of the system s ( ω, t, T ) into microtime-dependent components as s ( ω, t, T ) = a ( T ) s A ( ω, t ) + b ( T ) s B ( ω, t ) , where a ( T ) and b ( T ) are the relative probabilities of a given photon originating from either state | A (cid:105) or | B (cid:105) , and show that the spectral correlation expands as p ( ζ, τ, T ) = a ( T ) p AA ( ζ, τ )+ a ( T ) b ( T )( p AB ( ζ, τ ) + p BA ( ζ, τ ))+ b ( T ) p BB ( ζ, τ ) (3)(see Supplementary Information). The terms quadratic in a ( T ) and b ( T ) represent the spectral auto-correlations ofthe individual states p AA and p BB , while the cross-terms involving p AB represent the cross-correlation of the spectra s A ( ω, t, T ) and s B ( ω, t, T ) . The form of the spectral correlation can be understood intuitively because the spectralcorrelation is compiled from pairs of photons with origins drawn from the four possible combinations of | A (cid:105) and | B (cid:105) .Importantly, the left- and right-sided correlations p AB and p BA are not identical unless s A ( ω ) and s B ( ω ) share thesame center frequency ω and are symmetric in ω .Spectral diffusion is a ubiquitous process observed for many single emitters. Common descriptions of single-emitterspectral diffusion are the non-Markovian and discrete Poissonian Wiener process[14] or the mean-reverting Ornstein-Uhlenbeck process.[15] These processes describe spectral diffusion phenomenologically and for simplicity we considera simple non-Markovian Poissonian Gaussian jumping model (GJM).[12] The GJM process is characterized by atime-invariant probability density for discrete spectral jump occurrence to a new spectral position drawn from aGaussian probability distribution function over ω . For the two states | A (cid:105) and | B (cid:105) as denoted in the subscripts, wewrite P rob ( δω A,B ) = σ A,B √ π e − δω σ A,B for the probability of a given spectral shift at a point in time. Here, wehave introduced the spectral fluctuation term δω A,B from earlier. The microscopic interpretation of this process isthe time-stochastic variation of the bath assuming discrete conformations coupling to the system. The correspondingfluctuation correlation function can be written as C ( τ ) = e − τ/τ c and is described by an exponential decay with acharacteristic spectral jump time of τ c . When the two states diffuse independently of each other, no correlation ispresent and C AB ( τ ) = 0 . In this case, (cid:104) δω A ( t ) δω B ( t + τ ) (cid:105) = (cid:104) δω a ( t ) (cid:105)(cid:104) δω B ( t + τ ) (cid:105) = 0 because independentlydiffusing emissive states will not be correlated and the cross-terms p AB and p BA in equation 3 only reflect the cross-correlations of the inhomogeneous components p AB/BA ( ζ, τ → ∞ ) . Absent any memory of spectral fluctuations evenat early τ , the time average over the spectral-correlations of all random configurations is the cross-correlation of theinhomogeneously broadened (diffused) spectra p AB ( ζ ) = (cid:104) (cid:90) ∞−∞ e − δω σ A e − ( δω + ζ )22 σ B dδω (cid:105) , (4)where σ A and σ B are the widths of the Gaussian probability envelopes of the diffused distributions of states | A (cid:105) and | B (cid:105) .We numerically simulate the system of independently-diffusing optical transitions with parameters commensurate withtypical experimental cryogenic single-molecule spectroscopy (see Supplementary Information). The time-domain results3 Y - F EBRUARY
9, 2021of the simulation are discussed in Figure 2. The configuration of the system is shown in Figure 2(a). The correspondinglifetime exhibits biexponential decay behavior as expected. In (b) and (c), we compare the cross-correlation functionsfor two different slices with microtime ranges of T = 0 − ps and T = 2000 − ps, where | A (cid:105) and | B (cid:105) are thedominant emissive states, respectively. Unlike for the static doublet discussed in the Supplementary Information, thecross-correlations g (2) X ( τ ) indicate spectral dynamics evident from the loss of anti-correlation at longer τ . As we specifydifferent jumping rates for the two states, the decay of the spectral coherence evident in (b) and (c) occurs at different τ .The PCFS interferogram derived from the cross-correlations (see Supplementary Information for the derivation) forphotons emitted with a time constant of < ps is shown in (d) and informs on the loss of photon-coherence between µ s and ms owing to the energy fluctuations of the photons emitted < ps after laser excitation.In Figure 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 τ , a representation that makes immediately obvious thedifferences in the homogeneous linewidths at early τ and the differences in spectrally-diffused linewidths at late τ .For completeness we also show p ( ζ, T ) (d) and p ( ζ, τ ) (b) for fixed τ and T , respectively. These two representationsinform on the spectral evolution owing to spectral diffusion and changing relative emission contributions from differentstates, respectively. (c) displays the evolution of the spectral correlation from the narrow homogeneous spectrum with aLorentzian 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-dependentspectral correlation. We define a model for the fit as a linear combination of two Lorentzians and a Gaussian withfloating linewidths parameters. The relative amplitudes p AA , p BB , and p AB,BA are calculated according to 3 taking theweights 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 (d).We apply a global fit to the slices of the spectral correlation p ( ζ, τ = 60 µ s , T ) along T as shown in (e),(f) and (g). Thecross-correlation p AB,BA present as a broad Gaussian background superimposed with the homogeneous Lorentzianspectral correlations p AA,BB as introduced in equation 3. The width of this Gaussian component is σ AB ≈ (cid:112) σ A + σ B .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 throughphoton-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 ( ω ) is lost in the spectral correlation p ( ζ ) . Theunambiguous reconstruction of s ( ω ) from p ( ζ ) is therefore impossible and the spectral correlation is typically fit with amodel parametrizing a suitable form for the underlying emission spectrum, as we adapted herein. [13, 12] We now turn to a system of two coupled radiating dipoles undergoing population exchange and subject to correlatedspectral diffusion. A specific example would be solid-state quantum emitters undergoing incoherent and phonon-mediated population transfer after non-resonant excitation. [16] In quantum emitters, disentangling the relaxation rateand coherence times of the different fine-structure states in the presence of spectral diffusion is important for a detailedunderstanding of the dephasing process as phonon-mediated population exchange constitutes an important dephasingprocess in the solid-state.[3] We depict the system’s energy diagram in Figure 4(a), which exhibits two excited stateswith equal oscillator strengths and an irreversible relaxation rate k from the higher to the lower-lying state. In thissystem, photon emission from the higher-lying state | A (cid:105) will start immediately after population of the state. Emissionof the lower-lying state | B (cid:105) requires further relaxation and is often phonon-mediated.[3] The relative population ofstates | A (cid:105) and | B (cid:105) will thus change during the emission lifetime of the overall system as long as the relaxation rate k isfaster than the radiative rate /T of both | A (cid:105) an | B (cid:105) . The population dynamics of the system can be described by thefollowing set of coupled equations: d | A (cid:105) dt = − ( k + 1 /T ) | A (cid:105) (5) d | B (cid:105) dt = k | A (cid:105) − /T | B (cid:105) (6)with the solutions: | A (cid:105) ( t ) = | A (cid:105) e − ( k +1 /T ) t (7) | B (cid:105) ( t ) = − e − (1 /T + k ) t + Ce − /T t . (8)4 Y - F EBRUARY
9, 2021We show the effect of the changing relative cross-correlation probabilities between states | A (cid:105) and | B (cid:105) ( p AA , p BB ) inFigure 4 (b). Despite the T -invariant exponential population decay constant leading to a monoexponential photolumi-nescence lifetime of the overall system, the relative weights of p AA and p BB are changing with T .We show the spectralcorrelation of the lifetime-resolved PCFS experiment with indiscriminate T in (c). On timescales shorter than thespectral diffusion time τ , the fine-structure states are well-separated. At late τ , the broad diffused lineshape obfuscatesthe fine-structure splitting.We demonstrate that lifetime-resolved PCFS can recover the lineshape parameters of the homogeneous doublet byapplying a least-squares fit of a suitable model to the T-dependent spectral correlation as shown in (d),(e),(f). Themodel consists of two Lorentzians. with the floating linewidths Γ , Γ , an energy offset Ω and a relaxation rate k ,which determines the temporal change of the relative emission contributions of (cid:104) A (cid:105) and (cid:104) B (cid:105) . We recover all modelparameters within photon shot-noise thus validating the utility of lifetime-resolved PCFS to extract the coherences andrelaxation rates of different emissive fine-structure states. We note that the observation of early- τ multiplets in thespectral correlation compared to the broad Gaussian background in section 2.3 is the signature of correlated spectraldiffusion dynamics between the two states. Our simulations suggest that measuring the photon-coherences of quantumemitters exhibiting spectral fluctuations and different emissive fine-structure states will provide an avenue to studyquantum emitter optical dephasing through both fluctuations and population exchange between different electronicstates. We propose a new photon-correlation spectroscopic technique that extracts spectral fluctuations along the lifetime-trajectory of single emitters. The technique works through time-correlation of photons detected at the output arms of avariable path-length difference interferometer in both the microtime and macrotime domain and can be implementedusing standard picosecond photon-counting electronics. We show that lineshape and fluctuation parameters can beextracted 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 emitterresearch enabled by the high spectral resolution required to resolve photon-coherences at low temperatures. Experimentalefforts will be directed towards probing the fluctuation dynamics of non-stationary systems and investigation of thedecoherence processes in quantum emitters. Specific materials are readily available such as emissive defects in diamondand emerging 2D materials as well as semiconductor nanostructures.
The lead author of this study (H.U., study conception, derivation, modeling and interpretation) was initially fundedby the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering(award no. DE-FG02-07ER46454) and funded by Samsung Inc. (SAIT) during the completion of the study. Wethank Weiwei Sun, David Berkinsky, Alex Kaplan, Andrew Proppe, and Matthias Ginterseder for critically reading themanuscript and their feedback. 5 Y - F EBRUARY
9, 2021Figure 1: In conventional PCFS, the spectral correlation is compiled from photon-pairs irrespective of their microtimeT, often under continuous wave excitation (a). In lifetime-resolved PCFS, photon-pairs with a given microtime T and macrotime separation τ are spectrally correlated (b). Here, we adopt a time-binning approach to collect photonswith different T in suitable microtime intervals as indicated by the color-shaded background. The proposed opticalsetup is shown in (c). The photon-stream from a single emitter under pulsed excitation is directed into a variablepath-length difference Michelson interferometer. All photon-counts at the output arms of the interferometer are recordedin time-tagged (T3) mode using picosecond single-photon counting electronics.6 Y - F EBRUARY
9, 2021Figure 2: Simulation of two uncoupled radiating dipoles involving states | A (cid:105) and | B (cid:105) . The two transitions are coupled totwo different bath fluctuations and exhibit different lifetimes T , linewidths Γ and spectral diffusion parameters k jump and σ 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 (2) X ( τ ) for different optical path-length differences δ and microtimes T , where | A (cid:105) and | B (cid:105) are the dominant emissive states, respectively. The loss of coherence with increasing τ is evident from thereduction in anti-correlation. This coherence loss occurs at earlier τ for early- T photons (emission predominantly from | A (cid:105) , (b)) compared to late- T photons (emission predominantly from | B (cid:105) , (c)). The PCFS interferogram G (2) ( δ, τ ) forearly-T photons is shown in (d) and reflects the evolution from the exponential homogeneous dephasing at early τ tothe spectrally-diffused Gaussian dephasing at late τ . 7 Y - F EBRUARY
9, 2021Figure 3: Spectral results of the lifetime-resolved PCFS simulation of two uncoupled dipoles. (a) shows the full-width-at-half-maximum (FWHM) of p ( ζ, τ, T ) along T and τ . The difference in the homogeneous linewidths of | A (cid:105) and | B (cid:105) at early τ and in the diffused linewidths at late τ are immediately obvious in this representation. The orange-shadedpanels (b) and (c) show the effect of spectral diffusion for early-microtime photons originating mostly from state | A (cid:105) .We show the evolution of the weights of auto- and cross-correlations between states along T in (d).The weights arederived from the relative amplitude of the two exponential components of the photoluminescence decay in (Fig.2a).Taking p AA , p BB , and p AB,BA into account, we apply a global fit to the spectral correlation along T to recover thelineshape parameters of the undiffused system as shown in (e),(f) and (g). The broad underlying Gaussian componentin (f) reflects the cross-correlation of the diffused distributions of | A (cid:105) and | B (cid:105) and has a width of σ ≈ (cid:112) σ A + σ B .8 Y - F EBRUARY
9, 2021Figure 4: Lifetime-resolved PCFS simulation of two coupled dipoles undergoing population transfer and interacting withthe same bath resulting in collective spectral diffusion of the doublet (a). We introduce a phonon-mediated relaxationrate between the upper and lower state of k relax = 1 / ps − . As the radiative rates of the two states are chosen to beequal, the emission lifetime follows a monoexponential decay behavior despite changing relative populations of | A (cid:105) and | B (cid:105) with the microtime (b). The spectral correlation irrespective for all photons irrespective of their microtime isshown in (c) and demonstrates the transition from a triplet at early τ to the spectrally-diffused distribution at late τ . Thefine-structure splitting Ω , the linewidths Γ A,B , and the relaxation rate k can be recovered through lifetime-resolvedPCFS and a global fit of the slices along T with a fixed macrotime correlation of τ = 8 µs (d),(e) and (f).9 Y - F EBRUARY
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Supplementary Information: Lifetime-Resolved Photon-Correlation Fourier Spectroscopy
Derivation of Lifetime-Resolved PCFS
We derive PCFS classically, that is using the field rather than the photon description of light. In practical PCFSexperiments, the finite integration time renders the photon anti-bunching timescales outside of reasonable signal-to-noise regimes, justifying our classical approach adapted herein.[SI1, SI2] We denote the microtime as T , the macrotimeof the absolute experimental clock as t , and the time-separation between photons as τ . We show how the experiment canaccess the spectral correlation p ( ζ, τ, T ) , a histogram of photon pairs with a temporal separation τ along the macrotimeaxis, a separation from the laser excitation pulse T , and an energy separation of ζ .The intensity distribution at the two interferometer outputs a and b for an arbitrary spectrum s ( ω, t, T ) evolving over themicrotime T after laser excitation and the macrotime t of the absolute experimental clock at a given center path-lengthdifference δ is given by I a,b ( δ , t, T ) = 12 I ( t )[1 ± F T { s ( ω, t, T ) } δ ( t ) ] , (9)where δ ( t ) represents a time-dependent path-length difference defined by the chosen scanning trajectory of theinterferometer and exhibiting the form δ ( t ) = δ + ∆ δ ( t ) , where δ is the center path-length difference and ∆ δ ( t ) is a time-dependent term introduced experimentally through periodic dithering of one of the interferometer arms. F T { s ( ω, t, T ) } δ ( t ) is the Fourier transform of the time-dependent single-emitter spectrum. Substitution of 9 into thecanonical form for the intensity cross-correlation function g (2) × ( τ ) g (2) × ( τ ) = (cid:104) I a ( t ) I b ( t + τ ) (cid:105)(cid:104) I a ( t ) (cid:105)(cid:104) I b ( t + τ ) (cid:105) , (10)where (cid:104) ... (cid:105) denotes the time average and I a,b ( t, T ) are the time-dependent intensities at the two detector outputs a and b , gives g (2) × ( δ , τ, T ) = (cid:104) I ( t, T ) I ( t + τ, T )[1 + F T { s ( ω, t, T ) } δ ( t ) ][1 − F T { s ( ω, t + τ, T ) } δ ( t + τ ) ] (cid:105)(cid:104) I ( t, T )[1 + F T { s ( ω, t, T ) } δ ( t ) ] (cid:105)(cid:104) I ( t + τ, T )[1 − F T { s ( ω, t + τ, T ) } δ ( t + τ )] (cid:105) . (11)The exact functional form of ∆ δ ( t ) can be chosen arbitrarily as long as it is experimentally ensured that the path-length difference for each recorded correlation function is scanned symmetrically around the center position δ : (cid:82) T ac ∆ δ ( t ) dt = 0 , where T ac is the data acquisition time at each δ . Under this condition, terms linear in the Fouriertransforms of the form (cid:104) F T { s ( ω, t, T ) } δ ( t ) (cid:105) or (cid:104) F T { s ( ω, t + τ, T ) } δ ( t + τ ) (cid:105) vanish and the expansion of equation 11can be recast as g (2) × ( δ , τ, T ) = (cid:104) I ( t, T ) I ( t + τ, T ) (cid:105)(cid:104) I ( t, T ) (cid:105)(cid:104) I ( t + τ, T ) (cid:105) (1 − (cid:104) F T { s ( ω, t, T ) } δ ( t ) × F T { s ( ω, t + τ, T ) } δ ( t + τ ) (cid:105) ) . (12)The first term is just the intensity auto-correlation function g (2) || ( τ, T ) . We then introduce the PCFS interferogram G (2) ( δ , τ, T ) defined as: G (2) ( δ , τ, T ) = 1 − g (2) × ( δ o , τ, T ) g (2) || ( δ , τ, T ) = (cid:104) F T { s ( ω, t, T ) } δ ( t ) × F T { s ( ω, t + τ, T ) } δ ( t + τ ) (cid:105) = (cid:104) F T { p ( ζ, τ, T ) }(cid:105) , (13)where we have used the spectral correlation p ( ζ, τ, T ) , the auto-correlation of the spectrum compiled from photon pairswith a separation τ along the macrotime axis and a separation T from the laser excitation pulse. We note that equation13 is only valid for δ ( t ) ≈ δ ( t + τ ) , i.e. for inter-photon lag-times much shorter than the dither period. Explicitly, for amicrotime- and macrotime-variant spectrum s ( ω, t, T ) , the spectral correlation p ( ζ, τ, T ) can be written as1 Y - F EBRUARY
9, 2021 p ( ζ, τ, T ) = (cid:28)(cid:90) ∞−∞ s ( ω, t, T ) ∗ s ( ω + ζ, t + τ, T ) dω (cid:29) . (14)Equation 14 describes the central observable in lifetime-resolved PCFS and can be understood intuitively as a histogramof photon-pairs with a microtime T , a macrotime separation of τ , and an energy separation ζ . The form and interpretationof p ( ζ, τ, T ) are dependent on the dynamics of the emissive system and are discussed in the manuscript. The lifetime-distinct spectral correlation
One instructive system conducive to investigation with lifetime-resolved PCFS is a system with two transition dipolesradiating with distinct oscillator strengths for the states | A (cid:105) and | B (cid:105) . The different radiative lifetimes result in microtime-dependent relative weights of emission intensity originating from states | A (cid:105) and | B (cid:105) under δ -like non-resonant excitation.Substituting s ( ω, t, T ) as s ( ω, t, T ) = a ( T ) s A ( ω, t ) + b ( T ) s B ( ω, t ) , where a ( T ) and b ( T ) are the relative probabilitiesof a given photon originating from either state | A (cid:105) or | B (cid:105) , and separating into the sum of time-averaged terms (cid:104) .... (cid:105) yields p ( ζ, τ, T ) = (cid:104) (cid:90) ∞−∞ a ( T ) s A ( ω, t ) ∗ s A ( ω + ζ, t + τ ) dω (cid:105) + (cid:104) (cid:90) ∞−∞ a ( T ) b ( T ) s A ( ω, t ) ∗ s B ( ω + ζ, t + τ ) dω (cid:105) + (cid:104) (cid:90) ∞−∞ b ( T ) a ( T ) s B ( ω, t ) ∗ s A ( ω + ζ, t + τ ) dω (cid:105) + (cid:104) (cid:90) ∞−∞ b ( T ) s B ( ω, t ) ∗ s B ( ω + ζ, t + τ ) dω (cid:105) , (15)for the spectral correlation. We can write more compactly: p ( ζ, τ, T ) = a ( T ) p AA ( ζ, τ )+ a ( T ) b ( T )( p AB ( ζ, τ ) + p BA ( ζ, τ ))+ b ( T ) p BB ( ζ, τ ) . (16)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 ( ω, t, T ) and s B ( ω, t, T ) . Theform of the spectral correlation can be understood intuitively because the spectral correlation is compiled from pairs ofphotons with origins drawn from the four possible combinations of | A (cid:105) and | B (cid:105) . Importantly, the left- and right-sidedcorrelations p AB and p BA are not identical unless s A ( ω ) and s B ( ω ) share the same center frequency ω and aresymmetric in ω . A lifetime-distinct static doublet
We now consider the expected form of the spectral correlation for a static doublet, where s A,B ( ω, t ) = s A,B ( ω, t ) for all t , in other words time-invariant spectral lines for each state | A (cid:105) and | B (cid:105) . Per definition, C ( τ ) = 0 as δω ( t ) = 0 for all t . The corresponding terms of the spectral correlation are invariant in τ and equation 3 reduces to p ( ζ, T ) = a ( T ) p AA ( ζ )+ a ( T ) b ( T )[ p AB ( ζ ) + p BA ( ζ )]+ b ( T ) p BB ( ζ ) , (17)reflecting the absence of any dynamic spectral changes.We numerically simulate the experiment (see Methods section) to showcase the technique and resulting spectralcorrelation for the static doublet as shown in Figure 1. We introduce two states with 100 and 1000 ps radiativelifetimes T and different coherence times T of and ps with exponential decoherence (Lorentzian lineshape).2 Y - F EBRUARY
9, 2021The corresponding fluorescence lifetime as compiled from the numerical photon-arrival time data is shown in (a) andreflects the biexponential decay corresponding to the emission from the two states. (b) shows the cross-correlationfunction g (2) X ( τ ) for all photons irrespective of the microtime T . The individual correlation functions are τ -invariant asexpected in the absence of spectral fluctuations. The degree of anti-correlation increases towards a maximum for δ = 0 .The spectral correlation p ( ζ, τ ) is shown in the contourplot in (c). The higher noise-level at early τ in (b) and (c) canbe attributed to photon-shot noise. The lower panel in (c) shows the lifetime-resolved PCFS data for τ = 60 µ s. Theevolution from the narrow (long coherence, fast lifetime) to broad (shorter coherence, slower lifetime) is consistent withthe evolution of the emission probabilities from state | A (cid:105) and | B (cid:105) . We also plot the relative cross-correlation probabilityof photons originating from | A (cid:105) and | B (cid:105) . At early T , a significant fraction of photon-correlation counts stem from thecross-correlation between | A (cid:105) and | B (cid:105) as indicated by the side-peaks in the spectral correlation. These side-peaks aremaximal around T ≈ ps, where the relative weights of emission from states | A (cid:105) and | B (cid:105) are equal.In PCFS, the high temporal resolution achieved through photon-correlation comes at the cost of the loss of the absolutephase of the spectral information. In other words, both the asymmetry of the lineshape (not not apply for the systemsconsidered in this work) and the center frequency of s ( ω ) is lost in the spectral correlation p ( ζ ) . The unambiguousreconstruction of s ( ω ) from p ( ζ ) is therefore impossible and the spectral correlation is typically fit with a modelparametrizing a suitable form for the underlying emission spectrum.[SI3, SI2] Using a similar approach for lifetime-resolved PCFS, the lineshape parameters for the two different states can be extracted by applying a global fit tothe spectral correlation of a spectral doublet with relative weights of the emission lines of a ( T ) and b ( T ) . For thelifetime-distinct doublet, a ( T ) and b ( T ) can be extracted as the relative weights of the exponential fits describing thebiexponential decay dynamics as shown in Figure 1 (c). We validate this approach by applying a global fit to the slicesof the spectral correlation p ( ζ, τ = 60 µ s , T ) along T using equation 11 and assigning a Lorentzian lineshape to | A (cid:105) and | B (cid:105) as shown in Figure 2e,f and g. Numerical Simulation Methods
We use a custom MATLAB library to numerically simulate the relaxation-resolved PCFS experiments. All code ismade publicly available on github.[SI4] The numerical simulations have three broad components. i) the simulation ofthe photon-emission of a given system, ii) the transcription of these photon-streams into time-tagged single-photondata and iii) the PCFS analysis of the time-tagged single-photon data. We first model the photon-streams with thePhotonSimulator.m class by writing the photon-emission stream into arrays containing the macrotime information t ,microtime information T , center wavelength λ and a lineshape function g ( t ) .For any given time increment of the experiment /f rep , where f rep is the laser repetition frequency, the probabilityof photon detection is drawn from a Poissonian distribution for reasonable total average detection count rates ofaround photons/second and laser repetition rates of f rep = 10 − M Hz : p ( λ, k ) = λ k e − λ k ! with k = 1 and λ = 1 x cps/f rep . By omitting higher-order (multi-photon) detection events after the same laser excitation pulsewith k ≥ , the photon-stream reflects the quantum behavior of the single emitter. The total time of photon emissionsimulated at each stage position was chosen to be 30-40 s, commensurate with a total integration time for the experimentof around 1-2 h. For the lifetime-distinct doublet, each photon is randomly assigned a state | A (cid:105) or | B (cid:105) and thecorresponding center wavelength λ A,B and g ( t ) are chosen respectively. For the coupled dipole system, the populationis drawn from the microtime-dependent population distribution that changes owing to the relaxation from state | A (cid:105) to | B (cid:105) . The microtime T is drawn from an exponential distribution with the respective emission lifetime T A,B as p ( T ) ∝ e − T/T A,B . A single-mode photon exhibits a Lorentzian lineshape, and the corresponding lineshape function g ( t ) will present as exponential dephasing in the time domain. Our simulations are limited to Lorentzian lineshapes. Theadaptation to non-pure single-photons i.e. photons in superposition states of multiple spectral modes, is straightforwardthrough the choice of a Gaussian form for g ( t ) . We introduce Gaussian spectral diffusion by drawing the probability forjump occurrence from the boolean set { , } with a time-invariant probability for each laser pulse. For each jump, anenergy offset δω is drawn from a Gaussian distribution and added to the color λ of the photon of the respective state.The action of the PCFS experiment is to assign photons of a given color, coherence time, and lineshape a detector outputflag. This assignment is performed with the function PhotonAnalyer.m, which draws the exit flag from the boolean set ∈ { , } with the respective probabilities p , given by: p , ( δ ( t ) , λ ) g ( x ) = 1 / ± g ( δ ( t ) /c ) cos ( 2 πδ ( t ) λ ) . (18)The resulting photon arrival-time data is presorted according to the microtime interval of choice and the differentmicrotime-sorted photon-streams are cross-correlated using a MATLAB implementation of the multi-tau algorithm toperform the PCFS analysis.[SI5] The details of the PCFS analysis are described in Utzat et al. [SI2].3 Y - F EBRUARY
9, 2021Figure 1: Example system of a static doublet of lifetime-distinct states with different optical coherence times (a,inset). The corresponding photoluminescence lifetime exhibits biexponential behavior (a). The cross-correlationfunctions g (2) X ( τ ) at different stage positions transcribing the degree of spectral coherence into anti-correlations areshown in (b). The invariance of the correlation functions indicates the absence of any spectral fluctuations. The spectralcorrelation p ( ζ, τ ) is shown in (c) for all T . A slice along T for a given τ is shown in (d). (e) shows the evolution of theweights p AA , p BB and p AB/BA of the different contributions to the spectral correlations, highlighting the possibilityof extracting the lineshape parameters from lifetime-resolved PCFS data. The corresponding interferogram is shownin (f) for three different T , where p AA , p AB,BA , and p BB are dominant, respectively. Taking the respective weightsinto account, the evolution of p ( ζ, T ) with T can be fit with two Lorentzian peaks with widths Γ , Γ and their energyseparation Ω , (g-i). 4 Y - F EBRUARY
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Supporting References [SI1] X. Brokmann, M. Bawendi, L. Coolen, and J.-P. Hermier, “Photon-correlation Fourier spectroscopy.,”
Opticsexpress , vol. 14, no. 13, pp. 6333–6341, 2006.[SI2] H. Utzat, “Spectroscopy and Rational Chemical Design of Single Emitters for Quantum Photonics,”
MIT PhDThesis , pp. 1–175, 2019.[SI3] J. Cui, A. P. Beyler, T. S. Bischof, M. W. B. Wilson, and M. G. Bawendi, “Deconstructing the photon streamfrom single nanocrystals: from binning to correlation.,”
Chemical Society reviews , pp. 1287–1310, nov 2013.[SI4] H. Utzat, “photon.m - A MATLAB Class for Photon Counting,https://https://github.com/hutzat/lifetime_resolved_PCFS.”[SI5] D. Magatti and F. Ferri, “Fast multi-tau real-time software correlator for dynamic light scattering,”