## Abstract

We describe a method to probe the spectral fluctuations of a transition over broad ranges of frequencies and timescales with the high spectral resolution of Fourier spectroscopy, and a temporal resolution as high as the excited state lifetime, even in the limit of very low photocounting rates. The method derives from a simple relation between the fluorescence spectral dynamics of a single radiating dipole and its fluorescence intensity correlations at the outputs of a continuously scanning Michelson interferometer. These findings define an approach to investigate the fast fluorescence spectral dynamics of single molecules and other faint light sources beyond the time-resolution capabilities of standard spectroscopy experiments.

© 2006 Optical Society of America

## 1. Introduction

Chromophores embedded in a condensed medium inevitably exhibit time-dependent, fluctuating optical properties reporting on the dynamics of their nanoscale environment. Studied on ensembles of molecules [1], these fluctuations illuminate the complex dynamics of a broad range of disordered host systems such as low-temperature glasses, proteins and liquids, over timescales extending from femtoseconds to hours [2, 3]. Observed at the single emitter level, chromophore fluctuations reveal surprisingly varied and complex dynamical phenomena kept hidden in ensemble-averaged experiments [4], whose understanding is crucial for developing applications such as single molecule probes in biophysics or single photon sources in quantum information processing [5, 6].

Single molecule spectroscopy reaches its fullest potential when combined with high time resolution, so as to resolve the fast processes and temporal heterogeneities of any given isolated emitter. Obtained down to timescales shorter than the excited state lifetime for single emitter intensity fluctuations [7], high time resolution proves much more difficult to achieve when probing spectral fluctuations, due to the finite integration time (typically larger than milliseconds) necessary to collect enough photons to measure the spectrum of the emitter.

The first attempt to achieve fast single molecule spectroscopy was undertaken by Plakhotnik, who demonstrated that time resolution could be improved by averaging the autocorrelation functions of many individual fast scan spectra provided by single molecule laser spectroscopy [8, 9]. The time resolution of this intensity-time-frequency correlation (ITFC) method is however technically limited to the order of milliseconds by the finite scan rate of the laser, and no application followed this pionneering work. As a result, our current understanding of fast spectral fluctuations in single emitters is so far mostly inferred from inhomogenously broadened ensemble experiments.

In this article, we describe an approach overcoming these limitations to explore the spectral dynamics of a single transition at both high temporal and high spectral resolutions with standard photocounting equipment, and without requiring mode-locked laser sources for excitation as in ITFC. The method is based on the observation that spectral fluctuations under continuous excitation can be directly encoded in the intensity correlation functions measured at the output of a continuously scanning Michelson interferometer by a pair of photodiodes followed by a photon correlation counting board [Fig. 1]. The setup is investigated theoretically, simulated numerically, and compared to standard spectroscopy experiments.

## 2. Theoretical background

#### 2.1. Intensity correlation functions at the output of a scanning interferometer

The basis of our approach is the existence of an intimate relation between the distribution *p*
_{τ}(ζ) of spectral shifts ζ a transition undergoes over a duration τ and the intensity correlation function of the corresponding radiated field after transmission through a scanning Michelson interferometer. We first derive this relation in the case of a monochromatic transition showing random stationnary fluctuations *δ*
*w*(*t*) around a frequency *ω*
_{o}=〈*ω*(*t*)〉 over a mean-squared range σ^{2}=〈*δ*
*ω*(*t*=0)^{2}〉, and later generalize to fluctuating transitions with any lineshape.

Starting from an initial optical path difference *δ*
_{i}
at time *t*=0, the mirror travelling at a constant velocity *V* creates a time dependent optical path difference *δ*(*t*)=/*δ*
_{i}
+2*Vt* between the two arms of the interferometer. After entering the interferometer, frequency fluctuations propagate in the arms and recombine on the beamsplitter. This recombination can be described as instantaneous, as the retardation effects (of order *δ*
_{i}
/*c*) induced by optical path differences *δ*
_{i}
shorter than the maximum possible coherence length *cT*
_{1} of the emitter remain neglible over the timescales τ>*T*
_{1} considered throughout this paper (i.e. *δ*
_{i}
/*c*≪τ. See also [10]). Hence, we have :

These oscillating intensities feed a photon counting board integrating photons between *t*=0 up to a time *t*=*T* for computing the time-averaged intensity correlation function between the two outputs of the interferometer

where $\overline{\dots}$
denotes time-averaging from *t*=0 up to *t*=*T*. The integration time *T* is chosen so that the corresponding change in optical path difference Δ=*δ*(*T*)-*δ*(0)=2*VT* spans a large number of fringes, reducing the time-averages in Equation 2 to :

with *α*(*t*)=2*δω*(*t*+τ)*V*τ/*δ*
_{i}
+(1+2*Vt*/*δ*
_{i}
)ζ_{τ}(*t*), where ζ_{τ}(*t*)=*δω*(*t*+τ)-*δω*(*t*) denotes the random frequency jump observed between times *t* and *t*+τ.

This general expression of *g*(τ) simplifies with an appropriate choice of the scanning parameters *V* and *δ*
_{i}
, since the first term of *α*(*t*) becomes negligible in *g*(τ) at slow scanning velocities, when 2*δω*(*t*+τ)*V*τ/*δ*
_{i}
~2_{σ}
*V*τ/*δ*
_{i}
≪*c*/*δ*
_{i}
, i.e. as the change in the optical path 2*V*τ occuring over the timescale t under investigation remains small compared to the coherence length Λ=*c*/σ of the emitter. The quantity *α*(*t*) then reduces to *α*(*t*)=ζ_{τ} (*t*) in Eq. 3 when *δ*
_{i}
≫Δ, and the intensity correlation function becomes :

$g\left(\tau \right)=1-\frac{1}{2T}{\int}_{0}^{T}\mathrm{cos}(2{\omega}_{o}V\tau \u2044c+{\zeta}_{\tau}\left(t\right){\delta}_{i}\u2044c)dt.$

For integration times *T* much larger than the typical timescale over which spectral fluctuations occur, time averages can be replaced by the ensemble-average over the distribution *p*
_{τ}(ζ) of all the possible realizations of the random variable ζ=ζτ(*t*=0), so *g*(τ) now becomes :

$g\left(\tau \right)=1-\frac{1}{2}{\int}_{-\infty}^{+\infty}\mathrm{cos}(2{\omega}_{o}V\tau \u2044c+\zeta {\delta}_{i}\u2044c){p}_{\tau}\left(\zeta \right)d\zeta ,$

which can be recast in a more compact form as :

when the fluctuation process is time-reversal invariant, which imposes *p*
_{τ}(ζ)=*p*
_{τ}(-ζ).

Hence, provided the two conditions i) 2*V*τ≪*c*/σ and ii) *δ*
_{i}
≫Δ are fulfilled, the time-averaged intensity correlation function *g*(τ) measured at the output of a scanning Michelson interferometer oscillates with the frequency 2*ω*
_{o}
*V*/*c* at which fringes oscillate on the photodetectors, and with an amplitude given by the value of the Fourier transform FT[*p*
_{τ}(ζ)](*θ*) at *θ*=*δ*
_{i}
/*c*, where *p*
_{τ}(ζ) is the distribution of frequency shifts.

## 2.2. General case : narrow transition with arbitrary lineshape

The result obtained above holds for fluctuating transition of any (narrow) lineshape, integration over the finite linewidth simply changing the distribution of frequency shifts *p*
_{τ}(ζ) into the more general expression :

where 〈…〉 denotes ensemble averaging over all possible realizations of spectral fluctuations, and *s*
_{t}
(*ω*) is the time-resolved emission spectrum of the transition [2] :

where *T*
_{1} is the excited state lifetime of the transition. Remarkably, Eq. 5 identifies with the time-frequency spectrum provided by past ITFC experiments, and is the Fourier transform (in ζ) of the echo obtained in three-pulse photon echo spectroscopy [2, 9].

## 2.3. Photon-correlation Fourier spectroscopy

The key point is that at timescales τ shorter than the periodicity *πc*/*ω*
_{o}
*V* at which the fringes oscillate on the photodetectors, condition i) (cf. 2.1) is automatically fulfilled for any narrow spectral fluctuations (i.e. when σ≪*ω*
_{o}) and Eq. 4 can be reverted as :

The distribution of the spectral fluctuations *p*
_{τ}(ζ) of the emitter can therefore be determined directly from the measurement of its fluorescence intensity correlation function *g*(τ) at various optical path differences *δ*
_{i} of the scanning interferometer.

Determining the distribution *p*
_{τ}(ζ) of an emitter rather than its fluorescence spectrum *s*
_{t}
(*ω*) suggests an original approach to investigate its fast spectral dynamics, very much in the same way that measuring an intensity autocorrelation function instead of an intensity time trace transformed the study of photon statistics on short timescales in faint light sources such as distant stars or nanoscale emitters [11]. This can be understood by taking a closer look at some of the general properties of the distribution *p*
_{τ}(ζ). When τ→∞, the spectra *s*
_{t}
(*ω*) and *s*
_{t+τ}(*ω*) are statistically independent, and so *p*
_{τ}(ζ) reduces to the autocorrelation of the time-averaged (inhomogenous) linewidth *s*(*ω*-*ω*
_{o}) of the transition accessed by any standard fluorescence spectroscopy experiment. On the contrary, in the limit τ→ 0, *p*
_{τ}(ζ) is given by the autocorrelation of the time-resolved (homogenous) Lorentzian lineshape of purely radiative width ${T}_{1}^{-1}$, as no fluctuation has time to occur within delays τ=0. The distribution *p*
_{τ}(ζ) hence naturally bridges the gap between single molecule fluorescence trajectory analysis and ultra-fast spectroscopy ensemble experiments.

The time-frequency distribution *p*
_{τ}(ζ) also has the property of being time- independent for any stationnary distribution of fluctuations. The measurement of *p*
_{τ}(ζ) can therefore be made by integrating photon coincidences over long durations to im- prove signal-to-noise ratio without degrading temporal and spectral resolutions. Derived within the frame of classical electrodynamics, our analysis extends down to timescales τ_{min} as short as the excited state lifetime of the emitter *T*
_{1}, readily accessible by the modified Hanbury-Brown Twiss photon correlation detection setup described Fig. 1 [10]. This approach - which we call photon correlation Fourier spectroscopy (PCFS) - therefore combines the advantages of Fourier spectroscopy (no photon is lost, high spectral resolution ζ_{min}~*c*/*δ*
_{i}
) with a high temporal resolution down to the Fourier-transform limit τ_{min}ζ_{min}~1, beyond the capabilities of standard single molecule experiments.

## 3. Numerical simulations and discussion

Numerical simulations were performed to investigate the validity of PCFS for exploring fast spectral fluctuations and highlight some of its instrumental properties. All simulations were made assuming an excited state lifetime *T*
_{1}=1 ns, and fast random spectral fluctuations over a total range *δL*=2*λ*
_{o}σ/*ω*
_{o}=1 nm around an average wavelength *λ*
_{o}=600 nm, with an exponential frequency correlation function *C*(τ)=〈*δω*(*t*)*δω*(*t*+τ)〉 of correlation time τ
_{c}
=${\int}_{0}^{\infty}$
*C*(*t*)*dt*/σ^{2}=5 *µ*s. The choice of these numerical parameters corresponds to typical values as encountered in standard single molecule spectroscopy experiments for emitters such as molecules or semiconductor quantum dots.

PCFS measurements were numerically simulated with a speed of the translation stage set to *V*=30 *µ*m/s to fulfill condition i) (*V*≪1.8 mm/s) over timescales τ<100 ms. The photon arrival times at the entrance of the interferometer were drawn according to the Poissonian statistics of a light beam of intensity *I*=5×10^{4} photons/s. This corresponds to a situation where spectral fluctuations occur at timescales τ
_{c}
=5 *µ*s much faster than the average delay *I*
^{-1}=20 *µ*s between two successive photodetection events, and so are completely averaged out in standard single molecule spectroscopy experiments. The photons were statistically directed towards either photodiode depending on their wavelength and arrival time in the interferometer. Intensity correlation functions were calculated for various δ
_{i}
by integrating photons over 30 fringes over 500 scans, corresponding to a total acquisition time of 5 min per intensity correlation function.

## 3.1. PCFS and spectral fluctuation dynamics analysis

We first investigated the outcome of a PCFS experiment for a transition undergoing discrete spectral fluctuations, with a frequency *ω*(*t*) switching between two values *ω*
_{1} and *ω*
_{2}=*ω*
_{1}+Ω (here corresponding to wavelength jumps of ±1 nm) as a random telegraph signal *S*(*t*) [Fig. 2, Tab. 1] as encountered in the study of chromophores interacting with the two-level systems in glasses at low temperature [2]. The transition frequency fluctuations were generated with exponentially distributed waiting times of rate *k*
_{1} and *k*
_{2} with *k*
_{1}=*k*
_{2} (and τ
_{c}
=[*k*
_{1}+*k*
_{2}]^{-1}). Observed in standard spectroscopy, the transition spectrum would appear as predicted by the Anderson-Kubo lineshape theory [12], i.e. a doublet of separation Ω=2s centered in [*ω*
_{1}+*ω*
_{2}]/2, indiscernible from the spectrum of a static doublet of transitions at frequencies *ω*
_{1} and *ω*
_{2} with similar intensities.

Figure 2 shows the result of PCFS experiments on a static doublet (left), and on the switching transition (right). As seen in Fig. 2(a)
(d), in both cases, the shape of the intensity correlation function strongly depends of the delay *δ*
_{i}
where it was measured. Repeating the measurement of *g*(τ) over different optical path differences *δ*
_{i}
, we determined the evolution of *g*(τ) with *δ*
_{i}
[Fig. 2(b)
(e)], from which the distribution *p*
_{τ}(ζ) was extracted with Eq. 7 [Fig. 2(c)
(f)]. For a static spectrum, photons do not exhibit spectral correlation, and so *p*
_{τ}(ζ) reduces to the autocorrelation of the time-averaged spectrum measured in standard spectroscopy (here a doublet), i.e. a triplet of lines of intensities {1/4,1/2,1/4} at frequencies {-Ω,0,+Ω} independently of the timescale τ, as observed in the experimental results shown Fig. 2(c).

For the switching transition, this pattern is only preserved over timescales where fluctuations are uncorrelated, i.e. when τ>τ
_{c}
=5 *µ*s [Fig. 2(f)], and breaks down as soon as τ~τ
_{c}
, when the sidebands of *p*
_{τ}(ζ) in ζ=±Ω decay progressively as τ decreases, asymptotically leaving us with the autocorrelation of the time resolved spectrum of the transition as τ→0 - here a Lorentzian of width ${T}_{1}^{-1}$. Interestingly, the calculation of *g*(τ) from Eq. 4–6 (cf. Tab. 1) indicates that the correlation function *C*(τ) of the random telegraph signal governing the frequency switching process is directly encoded in the decay of the sidebands amplitude *A*
_{Ω} with τ.

The insets in Fig. 2(c)
(f) - showing the correlation functions *C*(τ) extracted from the decay of the measured sideband amplitudes *A*
_{Ω} - confirm this prediction. Indeed, they indicate respectively that the static and dynamic doublet exhibit null and exponentially decaying frequency correlation function, as expected. Illustrated here on binary spectral jumps, the ability of PCFS to investigate spectral fluctuations dynamics down to the excited state lifetime of the emitter can also be exploited on a transition coupled to a collection of flipping two-level systems, for example to determine the detailed physical properties of the latter (energy splittings Ω, switching timescales τ
_{c}
, etc.) [9].

## 3.2. High-resolution spectroscopy beyond temporal inhomogenous broadening

PCFS also opens a perspective for performing high resolution spectroscopy despite the presence of broad, fast, continuous spectral fluctuations, as reported on most emitters - often down to low temperatures [6]. This point is illustrated Fig. 3, showing the result of a PCFS experiment simulated for a doublet undergoing fast stationnary Gaussian fluctuations over a range σ much broader than the doublet separation Ω (here σ=5Ω), with an exponential correlation function *C*(τ) (Ornstein-Uhlenbeck fluctuation process) with τ
_{c}
=5 *µ*s.

Observed in conventional spectroscopy, the transition would appear as a Gaussian lineshape of width σ, and the existence of the underlying doublet would remain unnoticed. PCFS, in comparison, reveals a completely different pattern. Here again, the intensity correlation functions were found strongly dependent of *δ*
_{i}
[Fig. 3(a)], and, measured for different values of *δ*
_{i}
, provided the distribution *p*
_{τ}(ζ) [Fig. 3(b)
(c)]. As seen in Fig. 3(c), the distribution *p*
_{τ}(ζ) - broad (here a Gaussian of FWHM=2σ) over durations τ>τ
_{c}
- progressively narrows when τ<τ
_{c}
. This is consistent with the fact that for photons separated by delays τ shorter than the fluctuation correlation time τ
_{c}
, fluctuations are increasingly seen as “frozen” as τ→0.

Due to this fluorescence line-narrowing effect, the doublet can be resolved over short timescales, as seen from the quasi-periodic oscillations (of periodicity 2*πc*/Ω in *δ*
_{i}
) appearing in the intensity correlation function *g*(τ) when τ~5 ns [Fig. 3(b)], which translate into the lineshape of *p*
_{τ<5ns}(ζ) expected for a static doublet, i.e. a triplet of intensities (1/4,1/2,1/4) at frequencies (-Ω,0,Ω) [Fig. 3(c)]. A corollary of the fluorescence line-narrowing effect is an increased coherence length of the emitter, coherence in the transition emission (i.e. *g*(τ)<1 in Fig. 3(b)) remaining visible over short timescales τ≪τ
_{c}
although we have *δ*
_{i}
>5 mm≫Λ, i.e. even if the optical path difference is much larger than the bare coherence length Λ=120 *µ*m of the emitter, when no fringe would be observed in standard Fourier spectroscopy.

## 4. Experimental properties

From a technical standpoint, the intensity correlation measurements at the basis of PCFS can be considered as an intensity homodyne detection, where the oscillating intensity detected by a photodiode is demodulated by the oscillating intensity detected on the other photodiode. A first consequence of this observation is that PCFS directly provides the envelope of the Fourier transform interferogram of the radiation field, without the complex demodulation schemes usually involved in scanning Fourier spectroscopy.

Secondly, the self-demodulation process implies that the shape of *g*(τ) is robust against fluctuations in the scanning velocity *V* (as caused by stick-slip and vibrations in the translation stage), and is independent of the exact average frequency *ω*
_{o} of the transition, making PCFS intrinsically insensitive to rare, large spectral jumps which often limit the measurement time of laser spectroscopy experiments (e.g. the PCFS inversion formula (Eq. 8) is indeed independent of *V* and *ω*
_{o}). Numerical simulations (not shown) confirmed this analysis. For example, random velocity fluctuations of 30% were found to have no significant impact on any of the simulated results presented in Fig. 2 and Fig. 3.

Finally, we note that PCFS also offers high time and high spectral resolution over a broad range of frequencies and timescales, contrasting with laser spectroscopy, which - because of its very scanning nature - only provides high time and spectral resolutions simultaneously in the limit of vanishing spectral ranges.

## 5. Conclusion

Replacing the beamsplitter of a Hanbury-Brown Twiss detection system by a scanning Michelson interferometer allows the measurement of spectral fluctuations of a transition at high spectral resolution, down to timescales as short as the transition excited state lifetime, even for weak emitters.

In solid state physics, this photon-correlation method indicates how to transform standard single-molecule Fourier spectroscopy experiments [13] in order to explore the relation between lineshape broadening and decoherence in molecules and semiconductor quantum dots, as well as the dynamical interactions of these systems with optical fields and their nanoscale environment. Implemented in a fluorescence correlation spectroscopy (FCS) experiment, PCFS might also provide some insight into the spectral dynamics of nanoscale emitters under the influence of chemical reactions, conformational changes or intermolecular interactions in liquid environments.

## Acknowledgments

We are grateful to J. Enderlein for his help in the fast computation of intensity correlation functions [14]. This research was funded in part through the NSF-Materials Research Science and Engineering Center Program (DMR-0213282) and the Packard Foundation.

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