Abstract
Quantum phenomena such as entanglement can improve fundamental limits on the sensitivity of a measurement probe. In optical interferometry, a probe consisting of N entangled photons provides up to a \(\sqrt{N}\) enhancement in phase sensitivity compared to a classical probe of the same energy. Here, we employ highgain parametric downconversion sources and photonnumberresolving detectors to perform interferometry with heralded quantum probes of sizes up to N = 8 (i.e. measuring up to 16photon coincidences). Our probes are created by injecting heralded photonnumber states into an interferometer, and in principle provide quantumenhanced phase sensitivity even in the presence of significant optical loss. Our work paves the way toward quantumenhanced interferometry using large entangled photonic states.
Introduction
Optical interferometry provides a means to sense very small changes in the path of a light beam. These changes may be induced by a wide range of phenomena, from pressure and temperature variations that impact refractive index, to modifications of the spacetime metric that characterize gravitational waves. In its simplest form, interferometry measures distortions via the phase difference ϕ between the two paths of the interferometer. The uncertainty Δϕ in a measurement of this phase difference is limited fundamentally by the quantum noise of the illuminating light beams. This noise can be reduced by employing light exhibiting nonclassical properties such as entanglement and squeezing in order to improve the sensitivity of an interferometer beyond classical limits^{1}. Quantum states of light are most effective when it is desirable to maximize the phase sensitivity per photon inside an interferometer, such as in gravitational wave detectors^{2,3} or when characterizing delicate photosensitive samples^{4,5,6,7,8}.
In principle, Nphoton quantum states of light such as the highly entangled N00N state can provide up to a \(\sqrt{N}\) precision enhancement over a classical state of equal energy^{9,10,11,12,13,14,15,16,17}. Unfortunately these highly entangled states are vulnerable to decoherence, especially at large photon numbers. In practice, their enhanced sensitivity disappears in the presence of loss which may originate from interactions inside the interferometer (e.g. absorption in a sample) as well as external losses in the state preparation and detection^{18}.
Although a \(\sqrt{N}\) enhancement is not achievable in the presence of loss, one can engineer states that tradeaway sensitivity for losstolerance in order to still achieve some advantage over classical limits^{19,20}. For example, squeezed light^{21,22,23,24} and nonmaximally entangled states such as HollandBurnett states^{25,26,27,28,29,30,31} can surpass classical limits despite some losses. Importantly, the precision enhancement achievable with such states can grow with N, even in the presence of loss^{19}. Experimental demonstrations have prepared unheralded N = 6^{29,30} (or heralded N = 2^{28}) HollandBurnett states, but further increase of N is constrained by source brightness as well as detector efficiency and numberresolution. This motivates developing experimental protocols that can produce and detect losstolerant states with larger photon numbers.
In this work, we address a number of key challenges in order to scaleup quantumenhanced interferometry using definite photonnumber states of light. Firstly, we introduce probe states that are prepared by combining two photonnumber states on a beam splitter similarly to HollandBurnett states. However, unlike the latter, we allow the initial photonnumber states to be unequal. We show that these generalized HollandBurnett states are more sensitive than both HollandBurnett and N00N states in the presence of loss and approximate the performance of the optimal probe^{19}. Secondly, we experimentally implement our scheme using highgain parametric downconversion sources^{32,33} and stateoftheart photonnumberresolving detectors^{34} in order to access a large photonnumber regime. We herald entangled probes of sizes up to N = 8 and measure up to 16photon coincidences, thereby further increasing the scale of experimental multiphoton quantum technologies^{35,36,37}.
The idea is illustrated in Fig. 1a. Two typeII parametric downconversion (PDC) sources each produce pairs of light beams that are quantumcorrelated in photon number, i.e. a twomode squeezed vacuum state
Here, λ is a parameter that determines the average number of photons in each beam, 〈n〉 = λ^{2}/(1 − λ^{2}). Measuring one of the beams with an ideal lossless photonnumberresolving detector projects the second beam onto a known photonnumber state \(\left{h}_{1}\right\rangle\). Duplicating this procedure with a second independent source and detector, we herald pairs of photonnumber states that are not necessarily identical, i.e. the probe \(\left{h}_{1},{h}_{2}\right\rangle\). When these states are combined on the first beam splitter, multiphoton interference generates a pathentangled probe inside the interferometer^{38}.
We quantify the phasesensitivity of the probe inside the interferometer by calculating the quantum Fisher information \({\mathcal{Q}}\). The quantity \({\mathcal{Q}}\) provides a lower limit on the best achievable phase uncertainty via the quantum CramerRao bound, \(\Delta \phi \ge 1/\sqrt{{\mathcal{Q}}}\). The bound can be saturated using the optimal measurement strategy, which in the absence of loss is photon counting for the probes considered here^{39,40}.
In Fig. 1b, we plot \({\mathcal{Q}}\) for several probes with the same total photon number N = h_{1} + h_{2} = 8, but different Δ = ∣h_{1} − h_{2}∣, as a function of the signal transmissivity η_{s} which we assume to be equal in both interferometer modes. Probes with a small Δ provide a greater advantage over the classical shotnoise limit but are more sensitive to losses. Since the probe is heralded in our scheme, one can choose the optimal Δ for a given η_{s}.
Also shown in Fig. 1b is \({\mathcal{Q}}\) for the optimal state that maximizes this parameter for a given N and η_{s}. This state has been found in ref. ^{19}; the derivation is reproduced in the Supplementary Method 1. For the lossfree case (η_{s} = 1), the optimal state is the N00N state. However, for efficiencies below ~90%, our probes significantly surpass the N00N state in terms of \({\mathcal{Q}}\), exhibiting performance close to optimal. Moreover, in contrast to the N00N and HollandBurnett states, our probe performs at least as well as the shotnoise limit for any amount of loss.
We now turn to the experiment. Both PDC sources are periodically poled potassium titanyl phosphate (ppKTP) waveguides pumped with ~0.5 ps long pulses from a modelocked laser at a repetition rate of 100 kHz. The four detectors are superconducting transition edge sensors which we use to count up to 10 photons with a detection efficiency exceeding 95%^{34}. The interferometer is a fiberbased device in which we can control the distance between two evanescently coupled fibers using a micrometer to vary ϕ, much like changing the path length difference between two arms of an interferometer. Further details on the experimental setup can be found in the Methods.
We measure interference fringes given by \({{\rm{pr}}}_{{s}_{1},{s}_{2},{h}_{1},{h}_{2}}(\phi )\), the joint photonnumber probability per pump pulse to obtain the herald outcome (h_{1}, h_{2}) and measure (s_{1}, s_{2}) at the output of the interferometer when the phase difference is ϕ. We will refer to this as the (s_{1}, s_{2}, h_{1}, h_{2}) rate. To quantify the phase sensitivity of the rates measured with a particular herald outcome (h_{1}, h_{2}), we calculate the Fisher information:
where ∂_{ϕ} denotes the partial derivative with respect to ϕ, and \({\tilde{{\rm{pr}}}}_{{s}_{1},{s}_{2},{h}_{1},{h}_{2}}(\phi )\) is a model fitted to the measured rates (see Supplementary Method 2). Note that \({{\mathcal{F}}}_{{h}_{1},{h}_{2}}(\phi )\) quantifies the amount of information about ϕ in our measurement results, i.e. for a specific measurement strategy, and so \({{\mathcal{F}}}_{{h}_{1},{h}_{2}}(\phi )\le {\mathcal{Q}}\). We compare the performance of our photon counting strategy to the optimal measurement strategy in the Supplementary Discussion 1.
Our primary figure of merit is the Fisher information per detected signal photon conditioned on measuring (h_{1}, h_{2}) at the heralding detectors,
where
is the total number of detected signal photons. Injecting a coherent state into our interferometer would in principle yield the Fisher information \({\mathcal{F}}=\langle \tilde{n}\rangle\) when the detected mean photon number is \(\langle \tilde{n}\rangle\)^{18}. Thus, our figure of merit can be easily compared to the shotnoise limit which corresponds to \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )=1\).
We measured the total efficiency of both the heralding and signal modes to be between 47–55% (see Supplementary Method 3). This includes ~90% waveguide transmission, ~70% mode coupling efficiency into fibers, 90% interferometer transmission, and ≥95% detector efficiency. Due to the latter two losses, the detected \(\langle \tilde{n}\rangle\) is 10–15% smaller than the mean photon number inside the interferometer. As such, the Fisher information per photon inside the interferometer (which is the relevant resource when e.g. probing a delicate sample) is ~10–15% smaller than \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\).
Results
Low gain regime
We begin with low pump power to test our setup in the weak gain regime (λ ~ 0.25, 10 μW per source). In Fig. 2, we show results for two different probes, (a) \(\left1,1\right\rangle\), the wellstudied N = 2 N00N or HollandBurnett state, and (b) \(\left2,1\right\rangle\), a probe studied here for the first time. We calculate \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) using two methods. In the first, we discard events in which we know photons were lost by only including rates where s_{1} + s_{2} = h_{1} + h_{2} in the sums of Eqs. (2) and (3). These rates are shown in the top panels of Fig. 2. Using this first method, \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) [green curves] surpasses the shotnoise limit by 0.09 ± 0.01 for \(\left1,1\right\rangle\) and 0.10 ± 0.04 for \(\left2,1\right\rangle\) at its highest point. In the second method, we include all measured events. Note that this may include events where s_{1} + s_{2} < h_{1} + h_{2} due to loss in the signal modes, but also s_{1} + s_{2} > h_{1} + h_{2} due to loss in the herald modes. Conditioned on obtaining the herald outcome (h_{1}, h_{2}), the probability of the latter occurring can be minimized by reducing the pump power and hence λ. This increases the purity of the probe at the cost of reducing its heralding rate. Without postselection, \(\tilde{{\mathcal{F}}}(\phi )\) [red curves] drops below the shotnoise limit mainly due to losses.
In addition to loss, the spectral purity and distinguishability of our photons are also sources of imperfection that reduce the contrast of the fringes and hence diminish \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\)^{41}. Consider the probe \(\left1,1\right\rangle\), for example. For ϕ = ±π/2, the whole interferometer acts as a balanced beam splitter, in which case HongOuMandel interference should lead to a complete suppression in coincidences at its output. However, as can be seen in the orange (1, 1, 1, 1) fit in Fig. 2a, the visibility of this interference effect is ~75%. This visibility exceeds \(\sqrt{0.5}\), which is the minimum required for demonstrating postselected quantumenhanced sensitivity with the probe \(\left1,1\right\rangle\)^{12,28,42}. In addition to spectral mismatch between the signal modes, the visibility is degraded by uncorrelated background photons (~5% of detected photons) and the slight multimode nature of our sources, both of which reduce the purity of our heralded photons. We discuss source imperfections in more detail in the Supplementary Discussion 2. The finite detector energy resolution also plays a small role as the detectors have a ~1% chance to mislabel an event by ±1 photon^{43}.
High gain regime
Next, we increase the pump power to reach a high gain regime (λ ~ 0.75, 135 μW per source) in which we can herald large photon numbers. We detect 16photon events at a rate of roughly 7 per second, which is much higher than the stateoftheart achievable with bulk crystal PDC sources^{36} or quantum dots^{37}. In Fig. 3a, we plot \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) calculated without postselection for all probes with N = 8. As expected given the amount of loss in our experiment, probes with larger Δ are more phase sensitive due to their increased robustness to loss [Fig. 1b]. In particular, the sensitivity of the Δ = N probe should be shotnoise limited regardless of losses^{44}. However, in practice, the heralded detection of 0 photons could occur due to photon loss in the corresponding herald mode, resulting in the contamination of the signal with states for which Δ ≠ N. This degrades the performance of the Δ = 8 probe [orange curve]. In the Supplementary Discussion 3, we show that shotnoise limited performance with the Δ = N probe is recovered by blocking one of the sources.
Discussion
The fringes produced by our probes exhibit a number of different features compared to those measured with N00N or HollandBurnett states. For example, with these two states, the expected signature of Nphoton interference are fringe oscillations that vary as \(\cos (N\phi )\). While our measured fringes do not exhibit such oscillations in the high gain regime, they do exhibit sharper features than classical fringes. We show this explicitly by comparing our rates to those measured with distinguishable photons. This is achieved by temporally delaying photons coming from the top source with respect to photons coming from the bottom source by more than their coherence time. As an example, we consider the probe \(\left3,2\right\rangle\) in Fig. 4. When the photons are injected inside the interferometer at the same time, the fringe contrast is significantly higher than when they are temporally delayed [Fig. 4a]. Likewise, when we calculate \({\tilde{{\mathcal{F}}}}_{3,2}(\phi )\) without postselection, we find an improvement in the probe’s sensitivity in the former case [Fig. 4b]. This demonstrates that the probe sensitivity derives from multiphoton interference even at high photon numbers.
With any finite amount of loss, \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) vanishes when all fringes share a common turning point such as at ϕ = 0. In the case of HollandBurnett (Δ = 0) and N00N states, there are also common turning points at ϕ = ±π/2 which causes the reduction in \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) around these phase values [Fig. 3c]. In contrast, the probes with Δ = 4, 6, 8 do not have a dip in \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\pm\! \pi /2)\). The origin of this effect for Δ = 8 can be seen directly in the rates shown in Fig. 3b. The region of the fringe with high sensitivity to ϕ (i.e. large gradient) is different for different values of ∣s_{1} − s_{2}∣. This feature of \({\tilde{{\mathcal{F}}}}_{{h}_{1},{h}_{2}}(\phi )\) allows estimating ϕ without prior knowledge of the range in which it lies, as is required for N00N or HollandBurnett states, and thus provides a means for global phase estimation without using an adaptive protocol^{27,45}.
Finally, we briefly compare our results to other works reporting Fisher information per detected photon. The highest achieved here is ~1.1 using the herald outcome (2, 1), i.e. a N = 3 probe. Reference ^{31} and ref. ^{17}, respectively, report ~1.25 and ~1.2 using a N = 2 probe. The latter work also achieves a Fisher information per photon inside the interferometer (i.e. accounting for undetected photons) of ~1.15 which thus far is the only experiment demonstrating an unconditional improvement to the shotnoise limit. In the Supplementary Discussion 4, we estimate that an efficiency of 80% (in all four modes) and quantum interference visibility of 85% would be sufficient to demonstrate an improvement to the shotnoise limit with N = 8 photons without postselection. Although we do not attain these parameters in our experiment, our results do demonstrate the robustness of our probes to losses despite their large size. For example, the Fisher information per photon calculated without postselection for the N = 8 probe with Δ = 6 [Fig. 3a] is slightly higher than that of the N = 2 N00N state [Fig. 2a]. This contradicts the usual expectation that large entangled probes will necessarily be more fragile to noise and loss.
In summary, we proposed and experimentally demonstrated a scheme for quantumenhanced interferometry that exploits bright twomode squeezed vacuum sources and photonnumberresolving detectors. We measured interference fringes involving up to 16 photons which is significantly higher than the previous stateoftheart^{35,36}. Crucially, our scheme prepares probes that are nearly optimally robust to losses and hence addresses one of the principal challenges when scalingup to large entangled photonic states. With further improvements in the quality (e.g. coupling efficiency into optical fiber and purity) of bright twomode squeezed vacuum sources compatible with transition edge sensors^{33,46}, we believe our losstolerant scheme provides a promising route towards achieving quantumenhanced resolution using large entangled photonic states.
Methods
Sources
We pick 150fs pulses from a modelocked Ti:Sapphire laser (Coherent MiraHP) at a rate of 100 kHz using a Pockelscellbased pulse picker having a 50 dB extinction ratio. This repetition rate is chosen to accommodate the recovery time of the transition edge sensor detectors. The pump pulses are filtered to 783 ± 2 nm [fullwidth at half maximum] using a pair of angletuned bandpass filters. We split the pulses into two paths that are matched in length using a translation stage. In each path, we pump a 8 mm long ppKTP waveguide that is phasematched for typeII parametric downconversion. At the exit of the waveguide, the pump light is rejected with a longpass filter, and the orthogonallypolarized downconverted modes are separated using a polarizing beam splitter. Each downconverted mode is filtered with a bandpass filter whose bandwidth is chosen to transmit the main feature of the downconverted spectrum but reject its sidelobes. The herald modes (1566 ± 7 nm) are coupled into singlemode fibers and sent directly to the detectors. The signal modes (1567 ± 7 nm) are coupled into polarizationmaintaining singlemode fibers and sent into the interferometer. Details on the coupling efficiency and the spectral indistinguishability of the signal modes are provided in the Supplementary Discussion 2.
Interferometer
The interferometer is a fiberbased variable beam splitter (Newport FCPL1550PFP). The splitting ratio is adjusted by controlling the distance between two evanescently coupled fibers using a micrometer, which is analogous to changing the path length difference between two arms of an interferometer. In fact, any variable beam splitter that coherently splits light into two modes can be described by the same transformation as a MachZendertype interferometer^{47}.
During data acquisition, we scan the distance x between the two evanescently coupled fibers. To display our data as a function of the interferometer phase, we first calculate the transmission coefficient T(x) of the variable beam splitter using the measured (1, 0, 1, 0) and (0, 1, 1, 0) rates:
At low powers, we find that the quantity T(x) typically varies within [0.02, 0.98]. To obtain the corresponding phase, we correct for the imperfect visibility:
such that T_{corr}(x) varies between [0, 1]. For a single photon injected into a MachZender type interferometer with phase difference ϕ between its two arms, one expects \({T}_{{\rm{corr}}}(x)=[1\cos (\phi )]/2\). Solving for ϕ, we find:
Detectors
Our detectors are superconducting transition edge sensor detectors that operate at a temperature of 85 mK inside a dilution refrigerator. Details on their physical operation can be found in ref. ^{34}. An electrical trigger signal from the pump laser begins a 6 μs time window of data acquisition during which the detector outputs are amplified and recorded with an analoguetodigital converter. We use a matchedfilter technique in realtime to convert each detector’s output trace into a scalar value^{48}. The scalar value is then converted into a photon number using bins that are set during an initial calibration run prior to data acquisition.
Data availability
The data sets generated and/or analyzed during this study are available from the corresponding author on reasonable request. Correspondence and requests for materials should be addressed to G.S.T.
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Acknowledgements
We thank B. Vlastakis for his assistance with the operation of the dilution refrigerator. This work was supported by the following: the Natural Sciences and Engineering Research Council of Canada (NSERC); the Networked Quantum Information Technologies Hub (NQIT) as part of the UK National Quantum Technologies Programme GrantEP/N509711/1) and within “First Team” project No. POIR.04.04.0000220E/1600 (originally: FIRST TEAM/20162/17) of the Foundation for Polish Science cofinanced by the European Union under the European Regional Development Fund.
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Both G.S.T. and M.E.M. contributed equally. G.S.T. performed the experiment with assistance from B.A.B, C.G.W, A.E, D.S.P.; M.E.M., and G.S.T. performed numerical calculations with assistance from A.B.; A.E.L., T.G., and S.W.N. developed the detectors; R.B.P., M.S., A.I.L., and I.A.W. initiated and/or supervised the project; G.S.T. and M.E.M. wrote the paper with input from all authors.
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Thekkadath, G.S., Mycroft, M.E., Bell, B.A. et al. Quantumenhanced interferometry with large heralded photonnumber states. npj Quantum Inf 6, 89 (2020). https://doi.org/10.1038/s4153402000320y
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DOI: https://doi.org/10.1038/s4153402000320y
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