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How to Project onto an Arbitrary Single-Photon Wavepacket

The time-frequency degree of freedom of the electromagnetic field is the final frontier for single-photon measurements. The temporal and spectral distribution a measurement retrodicts (that is, the state it projects onto) is determined by the detector's intrinsic resonance structure. In this paper, we construct ideal and more realistic positive operator-valued measures (POVMs) that project onto arbitrary single-photon wavepackets with high efficiency and low noise. We discuss applications to super-resolved measurements and quantum communication. In doing so we will give a fully quantum description of the entire photo detection process, give prescriptions for (in principle) performing single-shot Heisenberg-limited time-frequency measurements of single photons, and discuss fundamental limits and trade-offs inherent to single-photon detection.

Photo detection is at its core an information theoretic process; a measurement outcome-a click-reveals information about the outside world quantifiable in bits [1]. In the case of a single-photon detector (SPD), a click is correlated (imperfectly) with the presence of a particular type of photon, thus revealing information about the presence of photons of that type along with whatever else in the world such a photon is correlated with. The most general quantum description of this process is in terms of a positive operator-valued measure (POVM), a set of positive operatorsΠ k that sum to the identity, where each k corresponds to a different measurement outcome. Given an arbitrary input stateρ the probability to obtain outcome k is given by the Born Rule P (k) = Tr(ρΠ k ). (1) Generically, each POVM elementΠ k can be writen as a weighted sum over orthonormal quantum stateŝ reducing to an ideal Von Neumann measurement only when the sum contains a single term with its weight w (k) equal to 1 [2]. The weight w (k) i equals the conditional probability to obtain measurement outcome k given input i. The posterior conditional probability that, given an outcome k, we project onto input i is given by Bayes' theorem [3] with P (k) = i w (k) i P (i) and P (i) the a priori probabilities to get outcome k and for input i to be present, respectively [4]. Through Bayes' theorem, an experimentalist is able to retrodict-that is, update their probability distribution over possible inputs-but only if they know what measurement their detector actually performs.
Knowledge of the POVM is essential for both gaining information from a measurement device and characterizing detector performance, hence the experimental need for detector tomography [5][6][7][8][9][10]. Commercial photo detectors are characterized by industry-standard figures of merit [11], which can be calculated from a POVM (for an in-depth review, see Ref. [12]). Here we will concern ourselves mostly with two figures of merit, detection efficiency and time-frequency uncertainty: Efficiency.-The maximum efficiency with which an SPD outcome k (for instance, a single click) can be triggered by input single photon states is the maximum relative weight in (2) The maximum efficiency is achieved only when the input quantum state is one the measurement projects onto. This follows directly from the Born rule; P (k|i) = Tr[Π kρ ] → w Time-Frequency Uncertainty.-The spectral uncertainty and (input-independent) timing jitter are determined entirely by the spectral and temporal widths of the states projected onto by the measurement outcome k [13], which form a retrodictive probability distribution. For any continuous variable X (here either time t or frequency ω), we find it less convenient to use the variance as measure of uncertainty and instead define the uncertainty entropically [12][13][14][15][16] Here H (k) X is the Shannon entropy defined as with the sum over discretized X-bins of size δX. p(j|k) is the a posteriori probability for the detected photon to be in bin j given outcome k, and is calculated as where we have defined a normalized distribution over X given by the norm squared of the quantum state |φ i (X)| 2 (where φ i (X) ≡ X|φ i ). The conditional probability P (i|k) is precisely the one from Bayes theorem (3); P (i|k) reduces to w (k) i /Ω (k) in the case of a uniform prior [17], where Ω (k) = i w (k) i is the bandwidth [12]. Critically, ∆X (k) is independent of the bin size δX in the small-bin limit, even though the entropy H (k) X is strongly dependent on the bin size. One can verify that this definition of uncertainty yields a Heisenberg uncertainty relation [14] ∆ω∆t ≥ eπ.
The construction of measurements projecting onto arbitrary single-photon states is critical in quantum optical and quantum communication experiments. Mismatch between the single-photon state generated and the state projected onto by the measurement induces an irreversible degradation in efficiency. Furthermore, the capacity to efficiently project onto orthogonal singlephoton states enables a wide range of quantum information and quantum optical applications, as we will discuss in section IV. From a foundational perspective, a procedure to build measurements projecting onto minimumuncertainty Gaussian single-photon wavepackets paves the way for future tests of fundamental quantum theory.

II. SIMPLIFIED MEASUREMENTS PROJECTING ONTO ARBITRARY SINGLE-PHOTON STATES
We will now discuss how to construct a simple POVM that efficiently projects onto an arbitrary single-photon wavepacket. To aid us, we will now make four simplifying assumptions. First, we will consider only the timefrequency degree of freedom of the electromagnetic field, as the other degrees of freedom (e.g. polarization) can be efficiently sorted prior to detection in a pre-filtering process [18][19][20]. Second, we consider only a single excitation incident to the photo detector. Multiple photons can always be efficiently multiplexed to achieve a photon number resolution using SPD pixels [21]. Third, we will not model a continuous measurement (as briefly discussed in the appendix of [22]), but instead a discretized measurement where at a particular time T we ascertain whether or not a photon has interacted with the SPD, ending the measurement. Lastly, we will consider only a binary-outcome photo detector, "click" or "no click." This simplifies the POVM so that it only contains the two elementsΠ T andΠ 0 , both projecting onto the Hilbert space of single photon states and the vacuum state. Generalizations to non-binary-outcome SPDs are straightforward: one can concatenate binary-outcome POVMs to generate non-binary-outcome experiments.
We now begin construction of the POVM {Π T ,Π 0 } in earnest.
Consider a two-level system with time-dependent transition frequency ∆(t), with timedependent coupling to a Markovian external electromagnetic continua of states [23]. Experimentally, a timedependent decay rate κ(t) is induced by a rapid variation of density of states [24,25] and a time-dependent resonance ∆(t) can be varied with a time-dependent external electric field (Stark effect, [26]) or through a two-channel Raman transition [27].
The general state of the two-level system can be written in the Schrödinger picture |ψ(t) = C 0 (t) |0 + C 1 (t) |1 . In the quantum trajectory picture, there are two types of evolution of |ψ(t) : Schrödinger-like smooth evolution with a non-Hermitian effective Hamiltonian and quantum jumps (at random times) [28,29]. A quantum jump will always correspond to the excitation leaking out of the system and so, in the absence of a dark counts, we only need consider the Schrödinger-like evolution. In this picture, the quantum state of the twolevel system remains pure with the time-dependent excited state amplitude C 1 (t) having the form of a Langevin equatioṅ where f (t) is a normalized input photon wavepacket [30]. We can solve this equation with the result where and where T 0 is a time in the distant past where our photodetector was still off, so that κ(T 0 ) = 0 and C 1 (T 0 ) = 0. Our measurement consists in checking if the system is in the excited state at time t = T . The probability to obtain a positive result (corresponding to detecting the incident photon wavepacket) is |C 1 (T )| 2 . We can write Whereas f (t) is a normalized wave function, Ψ(t) is subnormalized for finite T 0 , since We can interpret Ψ(t) as a retrodictive probability amplitude (for simple examples, see Fig. 1 and Fig. 2), identifying at which times a photon likely entered the system given a detector "click" at t = T .  (13). Only constant κ(t) = κ0 yields non-zero probability amplitude at t = T (where it is maximum). This illustrates the two roles κ(t) plays: determining the probability κ(t)dt with which an excitation can enter the system at a time t, and the rate κ(t) at which the excited state will decay. This latter effect drives down the probability amplitude for the distant past. For κ(T ) = 0, the most likely time that a photon entered the system is now whereas if κ(T ) = 0, there is some time of maximum likelihood determined by competition between the two effects (absorption and decay). For no order n is the retrodictive probability distributions continuously differentiable; these simple polynomial couplings do not yield measurements projecting onto smooth wave packets unlike the couplings plotted in Fig. 3 and Fig. 6. Note that κ(t) is not normalized and diverges as t → −∞, yet it results in a well-behaved, normalized retrodictive probability amplitudes.
We can define a normalized single-photon state with the creation operatorâ † in (t) acting on the input continuum of states. The arbitrary input single-photon state (which may have been created long before our detector was turned on at T 0 or long after the measurement ended The commutator relation for the input field operator is The probability for an arbitrary input photon wavepacket f (t) to result in the system being found in the excited state at a time T is The measurement does not project onto times after we have checked if the system is in the excited state, nor onto times before the detector was turned on. n varying from 0 (violet, lowest peak) to 15 (red, highest peak) denoted by the color and time measured w.r.t. detection time T . Except for n = 0 (also in Fig. 1 and included here for reference), these decays become non-zero at T0 = −2.5σ (when the photo detector is turned on) so that W from (13) is less than unity. As in Fig. 1, for no order n is the retrodictive probability distributions continuously differentiable, and do not project onto smooth wave packets as in Fig. 3 and Fig.  6. Also as in Fig. 1, we observe a time of maximum likelihood determined by competition between the two effects (absorption and decay).
We rewrite this probability in terms of a POVM element containing a single element To the extent our detector has been open long enough, such that W → 1, our detector could act as a perfectly efficient detector for a specific single-photon wavepacket with temporal mode function Ψ(t) [31]. This wavepacket is the time reverse of the wavepacket that would be emitted by our system if it started in the state |1 [32]. For this simple system, the POVM element is both pure (containing just one term [33]) and (almost) maximally efficient (the weight W may approach unity as close as we wish).
Here we observe an obvious trade-off between efficiency and photon counting rate: one cannot project onto a long single-photon wavepacket in a short time interval without cutting off the tails, lowering the overall detection efficiency [34].
The two-level system described in Eq. (8) is a special case but an important one; the two-level system is often a very good approximation of more complicated systems near-resonance [35]. In this paper, we will focus on the simple time-dependent system (8) as it is sufficiently general to perform a measurement described by any time-independent system, and more [36]. Indeed, (8) is general enough to project onto a completely arbitrary single-photon wavepacket, a result we will now prove.
Proof.-Consider a photon with complex wavepacket Ψ * (t) = A(t)e iφ(t) , positive amplitude A(t), and phase φ(t). Inserting this into (12), we arrive at two separate expressions The second line is always solvable by ∆(t) =φ(t) up to a constant global phase shift provided φ(t) is everywhere differentiable (smooth). We now focus on the first line.
Taking the natural logarithm we arrive at an expression Taking the time derivative of both sides, we arrive at a Bernoulli differential equation [37] κ Provided 1 is continuous, this is solved by Here, κ(t) is given by the square of the electromagnetic field, divided by a correction factor accounting for the finite response time imposed by κ(t) itself [38]. From (20), we observe that the only condition imposed on A(t) is that A 2 (t) have an antiderivative. We simply require A 2 (t) be continuous, which in turn requires A(t) to be continuous. Thus, any wavepacket with smooth phase profile φ(t) and smooth amplitude A(t) is projected onto by some physically realizable single photon detection scheme. Special Case: Heisenberg-Limited Measurement A Heisenberg-limited simultaneous measurement of time and frequency is achieved with a Gaussian timefrequency distribution. We want a temporal wavepacket Ψ * (t) that is the complex square root a Gaussian distribution where σ is the temporal half-width, and t 0 and ω 0 are the central time frequency of the Gaussian distribution. We find that this wavepacket is projected onto by a timedependent system with constant resonance ∆(t) = ω 0 and a time-dependent coupling For times near t ≈ T the coupling κ(t) is approximately Gaussian as an excitation absorbed at this time will not have sufficient time to decay back out. For earlier times, κ(t) is strictly larger than A(t) and is skewed towards earlier times to incorporate the time it takes the system to respond (∼ 1/κ(t)). Near t = −T , we observe the coupling κ(t) rapidly drops to zero, which is a direct consequence of the temporal wavepacket's symmetry about t0. [From (12) one can verify that, if A(t) is time-symmetric around t = t0, the coupling κ(t) must satisfy as in Fig. 3. Note that the coupling κ(t) is T -dependent even though the projected state (21) is T -independent, in agreement with the general case (20).

III. REALISTIC MEASUREMENTS PROJECTING ONTO ARBITRARY SINGLE-PHOTON STATES
The model of a SPD as an isolated two-level system is highly idealized. In a more realistic system, photo detection is an extended process wherein a photon is transmitted into the detector, interacting with the system and triggering a macroscopic change of the photo detector state (amplification) which can then be measured classically. Many theories of single photon detection have been developed over the past century, [39][40][41][42][43][44][45][46][47][48][49][50] and indeed there are numerous implementations of SPD technology [51][52][53][54][55]. Across all systems, we identify these FIG. 4. A POVM description of the three-stage model of photo detection, where the chain of inference (left to right) moves opposite the arrow of time, connecting a macroscopic "click" outcome to the state of the input field. A "click" outcome represented by the POVM elementΠT indicates k ∈ K click excitations were measured post amplification with detection efficiency η. This suggests n ≈ k/η (and, strictly, n ≥ k) excitations were present after in the target mode for amplification. In turn, this indicates that m ≈ (n −n th )/G ≤ n excitations were likely incident to the amplification process trigger, withn th the expected number of thermal excitations already in the amplification target mode. If m is larger than the expected number of thermal excitations in the amplification trigger modem th , we conclude that one (or more) input photon of the form |T ΨT T ΨT | was likely present. In addition to the states written explicitly, there are other possible states where k, n, and m deviate from their most likely values. These states (denoted by parallelograms) contribute to the POVM element, as the internal state of the photo detector is in general highly mixed. Nonetheless, in the end a SPD POVM elementΠT only projects onto the two input states |T ΨT T ΨT | and |vac vac|, remaining relatively pure. three stages of transmission, amplification, and measurement as universal. In this section, we derive a POVM that incorporates all three stages quantum mechanically, at the end of the section extending the model to include fluctuations of system parameters. The time-dependent two-level system from the previous section enabling arbitrary wavepacket projection is incorporated into the three-stage model as the trigger for the amplification mechanism. We will assume in this analysis that the system is left on for a sufficient time such that the subnormalization of Ψ(t) is minimal and W ≈ 1 (13).
In the spirit of what a POVM does (connect present outcomes to probabilistic statements about quantum states in the past), we will begin our endeavor at the very end of the photo detection process following Fig. 4. Consider a macroscopic measurement performed at time T with a binary response triggered by k excitations measured in the amplified signal [56]. Such a POVM can be written as a projector onto Fock states in the Hilbert space internal to the systemΠ Here, we have defined a set K click that sets the threshold for how many amplification excitations must be measured to trigger a macroscopic detection event. At this stage, we can already see that the internal state the POVM projects onto is highly mixed, but this will not directly translate to an impure measurement on the Hilbert space of input photons. Indeed, this is what we would expect; we do not need to know precisely the internal state of the photo detector in order to use it to efficiently detect the presence of a single photon. The macroscopic measurement performed on the amplified signal will, in general, be inefficient. We model this in a standard way [57], using a beamsplitter with frequency independent transmission amplitude √ η. We can then rewrite the POVM element (23) so that it projects onto Fock states in the amplification target mode prior to the measurement where we have defined the probability to detect k excitations given that there were n excitations in the output mode of the amplification process. An inefficiency 1 − η affects photo detection by changing which post-amplification Fock states are projected onto: the larger 1 − η is, the more postamplification states |n, T n, T | will be projected onto intermediately by the same POVM elementΠ T , making it harder to distinguish between signal and noise. We now move one step further back in the chain of inference (Fig. 4) so the POVM elementΠ T projects onto the number of excitations m input to the amplification trigger. Amplification is a generic feature of photo detection; without a macroscopic change in the internal state of a photo detector, there is no way to correlate detector outcomes with the presence of a single photon [58][59][60] (that is, without invoking additional single-excitation detectors in an argument circulus in probando). There are many interesting methods for implementing amplification [47,[58][59][60][61][62][63][64], but the fundamental quantum limit to amplification of any bosonic Fock state is achieved by a Schrödinger picture transformation [65] |m trig |M res |N targ −→ |m trig |M − Gm res |N + Gm targ (26) such that exactly G excitations are transferred from the reservoir mode to the target mode for each excitation in the trigger mode. In using this expression we do impose a restriction that there must be M > Gn excitations in the reservoir mode, but restrictions of this type are to be expected (the energy for amplification must come from somewhere) and we will be most interested in few photons (n = 0, 1, 2) in this analysis. In most physical platforms G will fluctuate [66], as will other (classical) system parameters which we will return to at the end of this section. (Exceptions do exist; for Hamiltonians that implement deterministic amplification schemes [with small integer values for G] see Ref. [67].) However, even with a definite gain factor G and number of input excitations m, we will still not end up with exactly n = N + Gm excitations if the target mode is initially in a thermal state with mean occupation numberN (as opposed to a Fock state with exactly N excitations). We now assume this, writing the state of the target mode in the Fock basiŝ with the probability for N thermal excitations given by where ω and k B T are the frequency and the temperature of the target mode. Assuming the ideal amplification scheme in (26), we now write the POVM elementΠ k in terms of the number m excitations that trigger the application mechanism where we define |Ψ T Ψ T | ⊗0 = |vac vac| and P th N = 0 for N < 0. We can now see the benefit of having a large gain factor G; it shifts the probability distribution over n that corresponds to non-zero excitations in the trigger mode, minimizing its overlap with the probability distribution for zero excitations. In this way, one can dramatically reduce the background noise (dark counts) without decreasing signal by changing Min[K click ]. In (29) we have reintroduced the state |Ψ T defined in (14) as the state described by projected onto by the trigger mechanism. As we did in the previous section, we will assume a time-dependent resonance frequency ∆(t) and decay rate κ(t) so that arbitrary pulse-shaping is possible.
The POVM element in (29) now projects onto quantum states internal to the photo detector. We need to connect the internal continuum of states coupled to the amplification trigger to the external continuum containing the photons we wish to detect (the transmission stage in Fig. 4). This is accomplished by introducing an arbitrary two-sided quantum network [22]. This is completely described by a single complex frequencydependent transmission coefficient T (ω) (related to a reflection coefficient at each frequency via |T (ω)| 2 + |R(ω)| 2 = 1 and R(ω)T * (ω) + R * ω)T (ω) = 0). We now invoke the single-photon assumption so that there is at most a single excitation input to the quantum network. Any other excitations present in the internal continua will be from internally-generated thermal fluctuations reflected by the quantum network back to the trigger mechanism. In this way, we can construct a POVM element that projects onto product-states |ψ ex ψ ex | ⊗ |ψ in ψ in | of the external and internal continuâ The first line corresponds to dark counts generated from thermal excitations post-amplification and the second line corresponds to dark counts generated by thermal excitations that then trigger the amplification mechanism.
Only the third line contains a projection onto a photon to be detected. (The multiplicative factor m in the third line is combinatorial in origin: m total excitations in the trigger mode with m − 1 generated from thermal fluctu-ations.) In writing (30) we have defined transmitted and reflected normalized single-photon states and coefficients whereâ † andb † are the creation operators for the external and internal continua and we have defined a Fouriertransformed wavepacket for the amplification trigger modeΨ(ω) = FT[ κ(t)Ψ(t)]. We can now see how preamplification dark counts (the second line of (30)) can be suppressed: by reducing the overlap of |Ψ(ω)| 2 and |R(ω)| 2 , that is, by only amplifying the frequencies we wish to detect so that ρ 2 1. In this case, the POVM element (30) will be dominated by the m = 1 term of the third line (the signal to be detected with no thermal excitations), as well as potentially the first line. (To reiterate, these are dark counts post-amplification, but these can be reduced by amplifying at a high frequency such that ω k B T , where ω and T are the frequency and temperature of the target mode.) Finally, we trace over the internal continua, which we assume is in a thermal state with temperature k B T [68] so the POVM projects onto the external continua onlŷ where in the last line we have absorbed the sums in front of the two projectors into weights so that the POVM element has the form of (2) and with P th j the probability to have j excitations (now in the non-monochromatic reflected mode defined in (31)). For a finite detector ontime T 0 > −∞, the weights w 0 and w T will be slightly less than in (32) due to wavepacket sub-normalization (13). However, this deviation is negligible provided the detector is left on for a time comparable to the temporal mode's width.
We now reconsider the question of projecting onto an arbitrary wavepacket, including the full quantum description. We find that this is possible to do in principle, provisio T (ω) is nowhere zero (except at infinity), a result we will now prove. That is, we can ensure that the single-photon wavepacket |T Ψ T has any desired (smooth) shape and will be projected onto with a high-efficiency and high-purity measurement.
Proof.-Consider a photon with complex normalized spectral wavepacketf (ω). If detection is achieved with a time-dependent two-level system preceded by a quantum network with filtering transmission function T (ω), the system will project onto a state |T Ψ T as defined in (31). In the low-noise limit this will be the only state projected onto by the (pure) POVM element. From the Born rule, the probability of detection will be with w T the overall weight given by (32) and maximum possible detection efficiency, which can be arbitrarily close to unity. It is possible to achieve P T = w T (mode-matched detection) in (33) if and only if From (12), we know that it is possible to generate an arbitrary temporal wavepacket FT −1 [Ψ(ω)] = κ(t)Ψ(t) from a time-dependent two-level system. The Fourier transform of a continuous smooth function is itself smooth and continuous. Thus, if the right hand side of (34) is a well-defined spectral wavepacket (smooth and continuous), one can find functions κ(t) and ∆(t) such thatΨ(ω) has the form of (34).
Remark.-Arbitrary wavepacket detection (and thus Heisenberg-limited simultaneous measurements of time and frequency) is in principle possible only when there are no photonic band gaps induced by the filter; if T (ω ) = 0 for some frequency ω , there is simply no way to compensate for the lost information about ω . Photonic band gaps are a generic feature of parallel (and hybrid) quantum networks [22] as well as certain non-Markovian systems [69]. Network/reservoir engineering must be employed to ensure any ω where T (ω ) = 0 is not a frequency of interest.
The POVM {Π T ,Π 0 } withΠ T defined in (32) and Π 0 =1 −Π T provides a complete description of the single photon detection process that is fully quantum from beginning to end (Fig. 4). However, there is a final element that must be considered to make the description applicable to labratory systems: classical parameter fluctuations. For continuous parameter fluctuations over any system parameter or set of system parameters X, these are naturally incorporated (35) where we have assumed a (known) probability distribution Pr(X). In (35), the system parameter(s) X could be  such that only the weights w 0 and w T depend on X, or X could be such that the state |T Ψ T depends on X as well (for a summary, see Fig. 5). In the case of the latter, the POVM will become less pure and will need rediagonalization to determine which states are projected onto [70]. This final POVM not only includes ignorance about the internal state of the photo detector as was depicted in Fig. 4, but also classical ignorance about the state of the photo detector due to system-lab interactions.

IV. APPLICATIONS
Using the time-dependent two-level system, we are able to project onto orthogonal quantum states (Fig. 6). This enables efficient detection of photonic qubits, an essential component of any quantum internet [71,72]. More generally, temporal modes provide a complete framework for quantum information science [73], with efficient detection of orthogonal modes (and their superpositions to create mutually unbiased bases) a key ingredient. Fully manipulable temporal modes also play a key role in error-corrected quantum transduction [74], where a timereversed temporal mode can restore an unknown superposition in a qubit. Here, efficient detection of arbitrary temporal modes is essential so that quantum jumps out of the dark state are efficiently heralded.
High-purity measurements that project onto orthogonal single-photon wavepackets also enable super-resolved measurements [77]. Suppose we have two single-photon sources emitting almost identical pure states differing FIG. 6. The time-dependent coupling κ(t) that generates a wavepacket exactly orthogonal to the minimum uncertainty Gaussian wavepacket in Fig. 3 are plotted for times of detection T = 2σ (W = 0.72) and T = 7σ (W = 1 − 10 −5 ) and detector on-time T0 = −∞. The wavepacket's time-dependent amplitude A(t) (depicted in the rotating frame) is an approximate first-order Hermite-Gaussian pulse, where the singular region of zero as been expanded with half-width of z = 0.5σ. In this way, the wavepacket's phase (solid black line, not to scale) can go from 0 to π in a finite time, whereas the Heaviside phase-flip in the exact Hermite-Gaussian pulse requires an unphysical delta-function detuning (second line of (17). Both the phase and amplitude have been convolved with a triangular smoothing function with full-width s = 0.5σ, ensuring A(t) is continuously differentiable to first order [75,76], which is pre-requisite for solving the Bernoulli equation (19). For any finite z ≥ s > 0 the phase flip [here implemented with a triangular detuning ∆(t)] occurs while the amplitude is zero. This ensures exact orthogonality of the approximate first-order Hermite-Gaussian to the gaussian pulse. In the limit s, z → 0, an exact first-order Hermite-Gaussian is recovered. This smoothing procedure generalizes to higher order Hermite-Gaussian pulses, forming a mutually unbiased basis for efficient detection of higher dimensional qudits [73]. slightly in either emission time or central frequency with φ 1 |φ 2 real, 1, and φ 1 |φ 2 = 0. Alternatively, we may imagine a single source of light but the light we receive may have either been slightly Doppler-shifted or it may have been slightly delayed.
Suppose now that we receive one photon that could equally likely be from either source so that our input state isρ If we can measure bothΠ 1 = η |φ 1 φ 1 | andΠ 1 = η |φ 2 φ 2 | (that is, if we have separate photodetectors with these (pure) POVM elements, or a single nonbinary-outcome photo detector), then we find the probability of clicks so that the ratio of clicks gives a direct estimate of , even for low efficiency η. Here all that is needed for time-frequency domain super-resolved measurement of are SPDs with time-dependent couplings and resonance frequencies as opposed to nonlinear optics [78].
In traditional quantum key distribution (QKD) schemes (that is, not measurement device independent (MDI)-QKD), specification of the measurement POVM is essential to robust security proofs [79][80][81]. Here, we have verified several assumptions about an eves-dropper's capabilities common in security proofs: that high-purity measurements are possible, that high efficiency measurements are possible, and (for continuous-variable (CV)-QKD proofs) minimum time-frequency uncertainty measurements are possible. In particular for CV-QKD, an eavesdropper can perform measurements that project onto variable-width spectral modes, disrupting temporal correlations between Alice and Bob (who are assumed to use fixed time-frequency bins) [82]. Here, the capacity to adjust the width of the spectral modeΨ(ω) provides Alice and Bob a new strategy to mitigate Eve's attack and extract a secure key.
More generally, detector tomography is an important tool across implementations of single-photon and number-resolved photo detection [7,8,83,84]. Realtime tomography could be useful in QKD protocols resistant to "trojan-horse attacks" [85] or any SPD platform subject to time-dependent environmental parameter fluctuations: for instance, atmospheric turbulence in MDI-QKD [86] or interplanetary medium in deep space classical communications [87]. Recently tomography speedups have been achieved using machine learning assisted tomography protocols [88]. The POVMs derived in this paper provide priors which can further speed up detector tomography [89]. These include approximate effects of environmental fluctuations as outlined in Fig. 5 and a global optimum POVM for single photon detection (32) which can be used to incorporate detector calibration and optimization into in situ tomographic protocols.

V. CONCLUSIONS
Having gone through applications of our work, we return to the fundamental (as opposed to practial) limits to single photon detection and their implications.
Here we have constructed single-photon measurements that are Heisenberg-limited in two ways: the first is that they can project onto Gaussian time-frequency states as illustrated in Fig. 3, and the second is that the amplification scheme reaches a Heisenberg-limited (linear in the gain G) signal-to-noise ratio, surpassing the standard quantum limit (a signal-to-noise ratio going like of √ G) [65]. Achieving these simultaneously is possible in principle with no drawback. Indeed, the only stringent tradeoff we encounter in this analysis is between efficiency and photon counting rate, which becomes substantial when an SPD is reset at a faster rate than ∼ 1/∆t. (The photon does not have sufficient time to excite the two-level system with high probability before the system is reset.) For other figures of merit, we find that they are either independent, or deteriorate together [90]. While it does appear from (32) that improving efficiency also increases dark counts, these are decoupled by ensuring the coefficient ρ 1-that is, by making T (ω) broader thañ Ψ(ω). While it is commonsense that one should only amplify the frequencies they wish to detect, our work clarifies how enormously important this is. The dark counts produced in this way are insuperable; they cannot be removed post-amplification without removing the single-photon signal as well.
Another conclusion from this work is rather optimistic. Here we have given a quantum description of an entire single photon detection process projecting onto arbitrary single photon states and the only fundamental limitations encountered are Heisenberg limits. Incorporating realistic descriptions of amplification and a final measurement reduce efficiency and increase dark counts, but even so a Heisenberg-limited measurement is still achievable in principle. Similarly, incorporating the filtering of a first irreversible step does not impede implementation of Heisenberg-limited measurements provided no frequencies are completely blocked from entering the trigger mechanism. Even considering parameter fluctuations (35) in internal temperatures k B T and k B T , amplification frequency ω , and amplification gain factor G-which are unavoidable in any realistic system-Heisenberg limited time-frequency measurements are achieved. To the authors' knowledge, this is first proposed quantum procedure for reaching Heisenberg limited time-frequency measurements in a realistic quantum system. In addition to being a fundamental limit to SPD performance, probing Heisenberg limits paves the way for future experimental tests of foundational quantum theory. This work is supported by funding from DARPA under Contract No. W911NF-17-1-0267.