Quantum limits to resolution and discrimination of spontaneous emission lifetimes

Measurement of the spontaneous emission lifetime of one or more single-photon emitters is a technique of central importance across the chemical and physical sciences. In the context of single-molecule (SM) microscopy, fluorescence lifetime is routinely used as a sensitive indicator of molecular-scale changes in the probe’s environment [1, 2]. The degree of freedom that determines the time delay between excitation pulse and subsequent photodetection is thus a conduit of chemical and physical information. Moreover, since SM emitters typically exhibit a finite budget of photocycles before irreversibly photobleaching, and since nanoscale environmental fluctuations can occur on timescales comparable to the lifetime, this resource can be an especially precious one. A careful analysis is thus warranted as to how one might carry out the tasks of lifetime estimation and discrimination as efficiently as possible with respect to photon budget, and certain metrics from classical information theory are now routinely invoked in such SM analyses [3].

In classical treatments of fluorescence lifetime measurement one typically models the photon arrival time as an exponential random variable with probability density:

where $\tau$ is the fluorescence lifetime parameter and $H(t)$ is the Heaviside step function defined by $H(t)=0$ for $t<0$ and $H(t)=1$ for $t\geq 0$. Note that the time origin is arbitrarily set to account for the timing of the preceding excitation pulse as well as the finite distance between emitter and detector. Multi-exponential decay can be modelled classically via a statistical mixture of probability densities of the form in Eq. (1) with different lifetimes $\{\tau_{0},\tau_{1},...\}$. This parameterization allows one to compute the classical Fisher information (CFI) associated with estimation of $\{\tau_{0},\tau_{1},...\}$ (as well as the relative amplitudes of the mixture), as presented, e.g., in Ref. [4]. The inverse of the CFI gives the classical Cramér-Rao bound (CCRB), which sets the minimum variance of any unbiased estimator of these parameters. Thus the CFI and CCRB are useful metrics for assessing the efficiency of lifetime estimators given use of the Time-Correlated Single-Photon Counting (TCSPC) measurement apparatus depicted in Fig. 1, which we henceforth refer to as the “direct measurement” scheme. In practice estimator choice is only one optimization concern, and a more general optimization over all possible measurement schemes (i.e. other than direct measurement) provides more fundamental information bounds. Such a generalization of the CFI and CCRB are given by the quantum Fisher information (QFI) and quantum Cramér-Rao bound (QCRB), which enumerate the information contained within the quantum state of the system (in this case the quantum state of the emitted photon) about the estimanda of interest, irrespective of the choice of Positive Operator-Valued Measure (POVM) [5]. In this work we will present and analyze the QFI, QCRB, and related quantum information metrics for some illustrative, challenging tasks related to lifetime measurement of spontaneous emission.

In particular, the majority of our discussion will be devoted to the representative task of resolving a bi-exponential decay process via estimation of the “separation” (in a generalized sense, vide infra) of the lifetime parameters. This effort is inspired in part by recent theoretical and experimental work that demonstrates the power in applying quantum information metrics to the analogous problem of spatial resolution between mutually incoherent point sources [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Namely, the CCRB associated with estimation of the source separation as measured by direct imaging is known to diverge at small separations (deemed “Rayleigh’s Curse”), while the corresponding QCRB remains finite. The discrepancy can be bridged by implementing a more sophisticated measurement scheme, e.g. Spatial-Mode Demultiplexing (SPADE) [7, 13], in order to recover otherwise wasted information. The present study is also related to recent applications of SPADE-like measurement techniques to the temporal resolution of shaped ultrafast optical pulses [23, 24, 25]. Our work is distinguished by the facts that we consider the distinct pulse shapes (exponential) and timings ($\sim$nanoseconds) relevant for spontaneous optical emission, and by the fact that our resolution of lifetimes refers to the difference in temporal line widths rather than the offset between peaks of subsequent pulses.

We model the emitter as a two level system located at the origin with energy splitting $\hbar\omega_{0}$ and consider the evolution of the coupled emitter-electromagnetic field system after initially preparing the emitter in the excited state with the quantized electromagnetic field in the vacuum state. We invoke the Weisskopf-Wigner approximation and consider the long-time limit such that at some time $t$ later the emitter has returned to the ground state and the electromagnetic field is left in the one-photon state [26, 27]:

where the index $\lambda$ denotes the polarization of the field mode, $\mathbf{k}$ is the linear momentum, $g(\mathbf{k},\lambda)$ subsumes relevant constants as well as the orientational factor associated with electric dipole emission, and $a^{\dagger}_{\mathbf{k},\lambda}$ is the canonical bosonic creation operator for the corresponding mode. For our purposes we ignore the Lamb shift to the photon’s mean frequency. We can simplify the relevant one-photon state by imagining placing a detector with finite aperture some distance away such that only modes within some small solid angle avail themselves to photodetection. Let’s also consider just the one linear polarization mode that dominates in the direction of the detector given the dipolar Green’s tensor [27]. Tracing out the unimportant modes (and ignoring the information-less vacuum contribution to the state of the field) leaves the normalized one-photon state:

where we have made use of the fact that $\omega_{0}\tau\gg 0$ to extend the lower limit of the integral to $-\infty$. We can alternatively express ${|{\psi_{\tau}}\rangle}$ in terms of the Fourier-transformed creation operator [28] (up to a global phase):

wherein $[a(t),a^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$. The Fourier integral in going from Eq. (3) to Eq. (4) can be computed via complex contour integration. Define the function:

such that

We will use Eq. (6) as a starting point for the computation of quantum information metrics described in the remainder of the manuscript. Note that $\langle a^{\dagger}(t)a(t)\rangle=|\psi(t;\tau)|^{2}$, which coincides with the classical probability density function $f(t;\tau)$ as defined in Eq. (1).

If the two-level system is embedded in free space the relation between lifetime $\tau$ and center frequency $\omega_{0}$ is fixed via:

where $\epsilon_{0}$ is the permittivity of free space and $\mu$ is the magnitude of the emitter’s transition dipole moment [26, 27]. For an emitter embedded in a dielectric medium, however, the relationship is altered in a way that depends explicitly on the (possibly inhomogeneous) dielectric constant [27]. We focus our discussion by considering the case in which $\omega_{0}$ is fixed (and known) but $\tau$ is variable, e.g. in response to a changing local environment in the vicinity of the emitter.

The QFI matrix $\mathcal{K}\in\mathbb{R}^{k\times k}$, the CFI matrix $\mathcal{J}\in\mathbb{R}^{k\times k}$, and the covariance matrix $\Sigma\in\mathbb{R}^{k\times k}$ of an arbitrary unbiased estimator for a given set of $k$ estimanda are related via:

where the matrix inequalities are meant to indicate that $(\mathcal{J}^{-1}-\mathcal{K}^{-1})$ and $(\Sigma-\mathcal{J}^{-1})$ are positive semidefinite. We begin by considering an emitter decaying with a single lifetime $\tau$ to be estimated. In this case the CFI and QFI are both scalars. For calculation of the CFI we specify a direct measurement in which an excitation pulse that is much shorter than the lifetime illuminates the system, then a subsequent photodetection event is tagged by its arrival time. In practice the time-tagging must be binned, but we consider the limiting case of infinitesimally short time bins for the calculation of $\mathcal{J}$. If no photodetection event occurs then the datum is simply thrown out. The cycle may be repeated until $N_{\text{photons}}$ photodetections have been recorded. The per-photon CFI associated with estimation of $\tau$ from such a measurement can be computed:

The CFI associated with $N_{\text{photons}}$ such photodetections is then simply $N_{\text{photons}}\times\mathcal{J}_{\tau}^{\text{(direct)}}$.

Next we compute the QFI associated with estimating $\tau$ given the pure state ${|{\psi_{\tau}}\rangle}$, irrespective of the measurement scheme. Since the state is pure, we can compute the per-photon QFI directly via the Fubini-Study metric [29]:

with:

The calculation is straightforward and yields:

and so we conclude that in this case the CFI associated with direct measurement saturates the QFI. Direct measurement is optimal for this particular estimation task. More interesting cases are encountered when we consider multi-exponential decays due to incoherent mixtures of single-exponential processes.

Next we turn our attention to computing the QFI with respect to $\bar{\tau}$ and $\varepsilon$. Since $\rho$ represents a mixed state we must compute the elements of $\mathcal{K}$ via the more general [30]:

where $\mathcal{L}_{i}$ is the symmetric logarithmic derivative (SLD) operator associated with the parameter $\theta_{i}\in\{\bar{\tau},\varepsilon\}$, defined implicitly via:

Details of the calculation of $\mathcal{K}$ can be found in Appendix A. The result is:

Note that $\mathcal{K}_{\bar{\tau}\bar{\tau}}\rightarrow 1/\bar{\tau}^{2}$ as $\varepsilon\rightarrow 1$. More importantly, we find that $\mathcal{K}_{\varepsilon\varepsilon}$ not only remains nonzero as $\varepsilon\rightarrow 1$, it actually takes on it’s maximum value in this limit. We compare the direct measurement CFI with the measurement-agnostic QFI in Fig. 3. The square-root QCRBs are defined by:

and are plotted vs. the direct measurement square-root CCRBs in Fig. 4. Evidently direct measurement leaves an increasing fraction of available information on lifetime separation on the table as the two time constants $\tau_{0}$ and $\tau_{1}$ approach one another. In the regime where direct measurement has the most difficulty discerning the decay constants, we next show that there exists some alternative measurement scheme that performs optimally.

The discussion to this point has focused on the specific task of estimating the square-root ratio of time constants given an incoherent sum of two exponential decays of equal likelihood. This exact scenario may be seldom if ever encountered in any real experiment. It serves, however, as a basic example demonstrating the potential power in measurement and inference schemes that more fully exploit the finite temporal coherence available when quantum sources spontaneously emit. Here we present an additional example of this fact by considering a binary hypothesis test in which one aims to discriminate between the following two states: A) a single exponential decay and B) that same single exponential decay incoherently mixed with a second weaker decay rate. To facilitate some quantification, we set the main decay channel’s time constant to $\tau_{A}=1$ ns and the second channel’s constant to $\tau_{B}=1.25$ ns. State A is defined by a unit probability of decay via the main channel; State B is defined by a $p_{A}=0.9$ probability of decay via the main channel and a $p_{B}=0.1$ probability of decay via the second channel. The one-photon density operators representing these two states are:

with

and $\psi(t;\tau_{A})$ and $\psi(t;\tau_{B})$ defined as in Eq. (5). If a total of $N_{\text{photons}}$ identically prepared photons are available for detection before a decision is made, the probability of error in the discrimination task improves exponentially with asymptotic scaling:

where the factor $\xi$ is given by the Chernoff bound, for which there exist both classical [37] and quantum [38] definitions. The classical Chernoff bound is appropriate when discriminating between two classical probability distributions. For the specific case of discriminating States A and B by implementation of a direct lifetime decay measurement, the task is equivalent to distinguishing between the two classical probability density functions (Fig. 9):

Assuming either state is equally likely, the classical Chernoff bound associated with direct measurement can then be computed according to:

Direct measurement is one of infinitely many possible measurement schemes that can distinguish States A and B with some finite probability. A general measurement’s ability to discriminate $\rho_{A}$ and $\rho_{B}$ (again assuming equal a priori likelihood) is governed by the quantum Chernoff bound [38]:

Since $\rho_{A}$ is pure in the case under consideration the above expression simplifies to:

In such a case it’s been established that a projection measurement $\{\rho_{A},\mathbb{I}-\rho_{A}\}$ would saturate the quantum limit to hypothesis discrimination [39]. In Fig. 10 we plot the approximate probability of error associated with the direct measurement classical Chernoff bound and the quantum Chernoff bound as a function of number of detected photons. Evidently the error rate can be improved by orders of magnitude for a number of photons feasibly detected from a single molecule.

For the same task we also analyzed the performance of measurements based on interferometers iC and iS as described in Section II.3.2, but with the substitution $\bar{\tau}\mapsto\tau_{A}$. We computed the classical Chernoff bounds associated with these two schemes, and plot the associated probabilities of error in Fig. 10. These schemes do not saturate the quantum bound, but do offer significant improvement over direct measurement for sufficiently many photons.

Detection of a weak decay path as described in this section is just one example of a symmetric binary hypothesis test related to spontaneous emission lifetime that one might conceivably encounter. In this case the optimal measurement is straightforward to derive (at least on paper). For more general symmetric binary hypothesis tests in which both possible states are mixed, the optimal measurement scheme is not necessarily clear. Our preliminary calculations indicate that projection measurements onto WL modes with appropriately chosen reference lifetime at least offer significant improvement over direct measurement. This would be in line with the previously reported observation that Hermite-Gaussian projection saturates the quantum Chernoff bound for certain binary discrimination tasks related to sub-diffraction imaging [40, 41]. A more thorough investigation of the performance of WL projection measurements for generalized binary discrimination tasks will be the subject of future work.