Open Access
Add to BibSonomy
Abstract
Photon number resolving detectors are the ultimate measurement of quantum optics, which is the reason why developing the technology is getting significant attention in recent years. With this arises the question of how to evaluate the performance of the detectors. We suggest that performance of a photon number detector can be evaluated by comparing it to a multiplex of on-off detectors in a practical scenario: conditional preparation of a photon number state. Here, both the quality of the prepared state and the probability of the preparation are limited by the number of on-off detectors in the multiplex, which allows us to set benchmarks that can be achieved or surpassed by the photon number resolving detectors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Photon number resolving detectors are the most sought measurement of quantum optics. By counting the individual photons of the signal they are the ultimate refinement of intensity detectors, which are currently the dominant way of observing light. Even the trademarks of modern quantum optics, the homodyne detectors capable of directly measuring the electric intensity of the field, rely on continuous detection of intensity of light. Photon number resolving detectors (PNRD) are also crucial from the perspective of contemporary quantum optics and its focus on the transition from Gaussian to non-Gaussian resources. The non-Gaussian resources, be it states, operations, or measurements, are essential or at least advantageous for many applications, such as quantum computation [1,2], quantum metrology [3–5], or quantum communication [6,7]. PNRDs will unambiguously discriminate between states with different photon numbers and thus can be used for non-Gaussian quantum state preparation [8,9] as well as for deterministic implementation of important non-Gaussian operations [10–12].
The first successful attempts to detect individual photons came in the form of avalanche photodiodes capable of discerning presence of at least one photon in the field. These on-off detectors were instrumental for preparing isolated photons which in turn enabled the entire field of quantum optics with individual photons [13]. The avalanche photodiodes were also used to develop the first approximations of photon number resolving detectors based on multiplexing the optical field into separate spatial or temporal modes and counting separate detection events. Such detectors, which we will call Multiplexed Single Photon Detectors (MSPD), can be used for measuring photon number distributions [14–16] and point values of Wigner functions [17–20], and they were even used in preparation of higher order non-Gaussian states [21] and implementation of non-Gaussian operations [22–24]. The quality of these detectors depends on their architecture as well as on the number of individual on-off detectors [25,26].
The most apparent feature of the MSPDs arises from the physical limit of $M$ individual detectors: the detector cannot detect photon numbers above this value. Furthermore, the probability of receiving an erroneous result increases with the number of photons in the detected signal. This would not be the case for an ideal PNRD that implements perfect projections on individual Fock states $|n\rangle$. Such the detector is an idealized construct, but it can be asymptotically achieved. The approach which, at the present time holds most promise for achieving it, employs superconducting transition edge sensors [27–33], which, instead of using a number of individual detectors, take advantage of physical processes that have a measurable inherent response to extremely low changes of intensity of the incident light.
However, at the current state of the art, the ideal photon number resolving capability remains a distant goal for both MSPDs and realistic PNRDs. This comes with an interesting question, though: how can we quantify how far from the goal are we? While a natural parameter for MSPDs is the number of detectors used in their composition, for PNRDs the situation is more complicated. In this paper we suggest that even PNRDs can be evaluated in this way. We show that for conditional preparation of photon number states, which is a practical task, schematically depicted in Fig. 1, that tests both the nature of the detectors and their integrability with the standard tools of quantum optics [34], there is no qualitative difference between imperfect PNRDs and MSPDs with a specific number of detectors. Consequently, performance of PNRDs can be quantified by the number of detectors forming a MSPD that provides equivalent results.
2. State preparation by detection
In general, quantum physics represents detectors by positive operator valued measures (POVM). For each possible measurement outcome, there is an non-negative hermitian operator POVM element which can be used to determine the probability of this outcome and its action on the rest of the physical system. For the ideal single mode PNRD, these POVM elements are simple. An outcome related to observing $n$ photons has POVM element which is just projector on the respective Fock state, $\Pi _{PNRD} = |n\rangle \langle n|$. For ideal MSPDs, which are constructed by multiplexing the measured mode into $M$ spatial or temporal modes, measuring each of them by a single on-off detector, and taking $n$ as the number of positive events, the POVM can be derived to be:
where the conditional probabilities are
Fig. 1. (a) Schematic depiction of experiment for preparation and verification of photon number states. Two modes of the entangled two-mode squeezed states are separated by mode separator (MS) and one of the modes is fed into the conditioning measurement realized by MSPD (b) or PNRD (c). The properties of the state thus prepared in the remaining mode can be verified by MSPD, PNRD, or homodyne detection (d). Virtual beam splitters with transmission coefficients $\eta _M$ and $\eta _S$ represent effective quantum efficiencies of the detectors.
Fig. 2. POVM elements for detecting exactly five photons by MSPD composed of $M$ individual APD detectors. For comparison, Ideal PNR detector has $p(k|5) = \delta _{k5}$.
Fig. 3. The number of detectors $M$ required for $p(n|n)$ larger than a certain threshold: green diamonds - 0.5, blue triangles - 0.8, magenta squares - 0.9, red diamonds - 0.95.
Photon number states of light can be prepared by the setup sketched in Fig. 1(a) by measuring one mode of a Gaussian two-mode squeezed state by a detector capable of discerning different Fock states and conditioning the remaining system when the desired outcome has been detected. Formally we can represent generated quantum state in mode $S$ as:
where $P_S = \mathrm {Tr}[ \rho _{S,M} \Pi _M]$ is the success rate, $\rho _{S,M}$ is the state of the initial two-mode squeezed state, and $\Pi _M$ is the positive operator valued measure (POVM) element corresponding to the desired outcome of measuring modes $M$. In the ideal scenario, the two-mode squeezed state is pure and can be represented by state vector $|\psi _{TMS}\rangle _{S,M} = \sqrt {1-\lambda ^{2}}\sum _{n = }^{\infty } \lambda ^{n} |n,n\rangle _{S,M}$. In practice, various experimental inefficiencies will manifest as losses causing the resource sate to be mixed. The losses, separate for each of the subsystems, can be represented with help of the decoherence mapWithout any losses, $\eta _M = \eta _S = 1$, PNRD always produces the perfect state $|n\rangle$ with probability $P_S = (1-\lambda ^{2})\lambda ^{2n}$, which can be maximized with respect to the entanglement of the shared state parameterized by $\lambda$. On the other hand, MSPD can achieve the same Fock state only in the limit of large enough number of detectors $M \rightarrow \infty$ or in the limit of negligible entanglement $\lambda \rightarrow 0$. For a realistic number of detectors, MSPDs will always need to make concessions between high probability of success and high quality of the state.
The situation is not as clearly cut in the practical scenario in which the detectors are no longer perfect, $\eta _M \neq \eta _S = 1$, as in this case even PNRD cannot always produce state with perfect quality. In this case we can characterize the produced quantum state by two main parameters: the success rate $P_S = \mathrm {Tr}[ \rho _{S,M} \Pi _M]$ and the fidelity $F = \langle n| \rho _S| n \rangle$. For the case of preparing quantum states $|3\rangle$ and $|5\rangle$, the comparison of the detectors is shown in Fig. 4. We can see that while both detectors show a trade-off between the fidelity and the probability of success, the PNRD, even with lower efficiency, still perform significantly better than even the ideal version of the sample MSPD. That is, MSPD can still generate states with fidelity as large as that of the states produced by PNRD, but their probability of success is significantly lower.
Fig. 4. Fidelity relative to success probability for preparation of Fock states $|3\rangle$ (a), and $|5\rangle$ (b). Dashed lines represent PNRD detectors with colors ranging from green ($\eta _M = 1$) to blue ($\eta _M = 0.5$). Solid lines represent MSPD detectors with $M=n$ with colors ranging from yellow ($\eta _M = 1$) to red ($\eta _M = 0.5$). The respective efficiencies for all lines are also marked in the figures.
However, fidelity is not the only measure of quality of quantum states. Photon number states states are, by their very nature, highly nonclassical and non-Gaussian states. For odd photon number states fidelity higher than $F > 0.5$ guarantees negative Wigner function and thus both nonclassicality and nongaussianity, but using a witness with more general application, such as [36] is more prudent. To this end we apply hierarchy of witnesses of genuine quantum non-Gaussianity that has recently been derived and experimentally tested [37]. In the hierarchy, witness of $n$-th order discerns whether the investigated state is truly incompatible with a state that could be created as a mixture of photon number superposition states $\sum _{k=0}^{n-1}c_k|k\rangle$ transformed by arbitrary Gaussian operations.
One of the telling features of Gaussian states is that, with the exception of the vacuum state, their photon number representation spans an infinite dimensional Hilbert space. As a consequence, for Gaussian states the probability of detecting any particular photon number state $|n\rangle$, which can be represented by fidelity $F_n = \langle n|\rho _S|n\rangle$, is always bounded from above [38]. This remains true even if, instead of Gaussian states, we consider an arbitrary finite superpositions of photon number states $|k\rangle$ with $k<n$, which are affected by arbitrary Gaussian operations. This upper bound can be further made tighter by deriving it under specific constraints. One such constraint is limiting the cumulative probability of detecting photon number states with numbers larger than $n$, $p_{n+} = \mathrm {Tr}[ \rho _S \sum _{k=n+1}^{\infty }|k\rangle \langle k|]$. Value $B_n(p_{n+})$ can be then defined as the maximal fidelity $F_n$ that can be, for the specific value of $p_{n+}$, achieved by state $\sum _{k=0}^{n-1}c_k|k\rangle$ after it undergoes an arbitrary Gaussian operation [37]. The values of the bound can be numerically obtained and compared to the fidelities of the prepared states. As a numerical quantifier for the witness of $n$-th order we can take the difference $W_{NG}(n) = F_n - B_n(p_{n+})$, where $F_n$ is the fidelity of the observed state, $p_{n+}$ is calculated from the photon number distribution of the observed state, and $B_n(p_{n+})$ is obtained by numerical optimization for the particular value [37]. If this value is positive it signifies that the state could not have been created from states with purely lower photon number components, even with help of arbitrary Gaussian operations. Such state is therefore genuinely $n$-photon quantum non-Gaussian state according to the definition in [37]. The values of the witnesses with appropriate orders are shown in Fig. 5. We can see that both kinds of detectors are capable of producing genuine quantum non-Gaussian states, both show similar trade-offs, and, again, PNRD visibly outperforms the MSPD. This is most prominent for the displayed scenario with $M=n$. Increasing $M$ would move the MSPD curves towards PNRD curves and eventually, for $M\rightarrow \infty$, they would merge.
Fig. 5. Genuine quantum nongaussianity witnesses $W_{NG}(3)$ for Fock states $|3\rangle$ (a) and $W_{NG}(5)$ for $|5\rangle$ (b). Dashed lines represent PNRD detectors with colors ranging from green ($\eta _M = 1$) to blue ($\eta _M = 0.5$). Solid lines represent MSPD detectors with $M=n$ with colors ranging from yellow ($\eta _M = 1$) to red ($\eta _M = 0.5$). The respective efficiencies for all lines are also marked in the figures. Genuine quantum non-Gaussian states have values of the witness above zero, which is marked by the black dotted line.
Comparison of Fig. 4 and Fig. 5 reveals strong similarity in the trends of the trade-offs with respect to detection efficiencies $\eta _M$. In both situations, photon number state $|5\rangle$ is more difficult to prepare than Fock state $|3\rangle$, meaning that the same quality is achieved at the cost of lower success rate. The minimal efficiency of PNRD required to overcome perfect MSPD also increases. Interestingly, in between the two figures of merit, the actual values of efficiency in which imperfect PNRD performs as perfect MSPD are equal. These efficiencies are are $\eta _M = 0.54$ for $M = 3$ and three photon state, and $M = 5$ and $\eta _M = 0.50$ for five photon state. For each MSPD we can therefore define effective collection efficiency $\eta _{M,\mathrm {eff}}$ as the efficiency of PNRD leading to effectively identical trade-off between the fidelity of a prepared Fock state and the success probability of the procedure. These effective efficiencies for some composite detectors are shown in Fig. 6. These values start high and diminish with increasing $n$ as the performance of MSPDs worsens and PNRD with lower efficiency become competitive enough. The effective efficiencies depend on the ratio between the number of the detectors and the required $n$ of the Fock state. Numerical analysis reveals that the effective efficiencies approach $\eta _{M,\mathrm {eff}} \rightarrow 0.47$ for when the number of detectors is equal to the number of detected photons, then $\eta _{M,\mathrm {eff}} \rightarrow 0.75$ when the number of detectors is two-times higher and $\eta _{M,\mathrm {eff}}\rightarrow 0.84$ when it is three times higher. We can thus see that the number of detectors forming a MSPD can play a similar role as efficiency of PNRD.
Fig. 6. Effective efficiency with which imperfect PNRD operates at the level of ideal MSPDs for preparation of fock states $|n\rangle$. The total numbers of APDs forming the detectors are $M = n$ (red circles), $M = 2n$ (blue squares), and $M = 3n$ (green diamonds).
3. Benchmarking tools
We have established that for preparation of a specific photon number states imperfect PNRDs can behave similarly to MSPDs with some number of ideal detectors $M$. We can use this to experimentally characterize the quality of an unknown PNRD. The experiment, sketched in Fig. 1, attempts to conditionally prepare the photon number state by suitable detection. We can always calculate the performance of MSPDs composed of $M$ on-off detectors and compare it to the experimental results. In the first step we can consider using the same detector for both preparation and subsequent verification. In this case the suitable figure of merit is the joint probability of detecting a selected photon number state $|n\rangle$, which can be obtained as $P_{nn} = \max _{\lambda } \mathrm {Tr}[\rho _S \Pi _M\otimes \Pi _M]$, where the POVM elements $\Pi _M$ correspond to the desired detection result. Figure 7 shows the maximal joint probabilities that can be, for MSPD with given number of detectors $M$, obtained when preparing photon number states $|3\rangle$, $|5\rangle$, and $|7\rangle$. Experimentally obtaining higher joint probability then guarantees that detector performs better than the respective PNRD.
Fig. 7. Joint probability benchmark for simultaneous detection of fock state $|3\rangle$ (red circles), $|5\rangle$ (blue squares), and $|7\rangle$ (green diamonds) by two ideal MSPDs with variable $M$. The dashed lines with the respective colors represent maximal joint probability achievable by perfect PNRD.
The biggest drawback of the direct approach is that the bound is optimized over the entanglement of the resource state. Failure to overcome the value for a given MSPD can be therefore caused by lower quality of the detector, but also by inability to experimentally prepare the most suitable two mode squeezed states. This issue can be avoided by refining the benchmarking procedure and compare not only the success probabilities, but also the fidelity of the state, which can be obtained, for example, by quantum state estimation. For this, however, the MSPD bound is represented not by a single value, but by a trade-off between the two quantities. Sample benchmarks for preparation of quantum states $|3\rangle$ and $|5\rangle$ are plotted in Fig. 8. Any experimental result above the line formed by the trade-off then confirms superiority of the tested detector over the respective MSPD.
Fig. 8. Maximal fidelity $F_{\mathrm {id}}$ of states $|3\rangle$ (a) and $|5\rangle$ (b) prepared by MSPDs with $M = 5, 10, 20, 50$.
The benchmarks introduced so far are relevant for PNRDs with high quantum efficiency experiments capable of producing entangled states with high purity. When this is not the case, the benchmarks are still valid, but may be too strict to be surpassed. However, in these situations it is possible to derive new, more forgiving, benchmarks that already take the lowered detector efficiency $\eta _M<1$ in the account, similarly to [39]. Such benchmarks can be obtained by straightforward numerical evaluation of the fidelity with respect to probability of success in a system with detector efficiency $\eta _M$. However, there is another issue of reduced purity of the shared entangled state. Such states can be equivalently represented by pure entangled states with losses in both modes of the state. The losses of the measured mode can be subsumed into the detector efficiency $\eta _M$ and treated in the same way. The losses in the state preparation mode, represented by reduced efficiency $\eta _S$, can be resolved in a similar manner, by deriving new benchmarks with values of $\eta _S$ and $\eta _M$.
Another way to approach the issue of reduced efficiency $\eta _S$ is by numerically compensating the losses on the measurement data, which is a tool occasionally employed in analyzing experimental results [40,41]. In our case, in which data is in the form of photon number distribution, the losses can be compensated by the inverse decoherence map [42]
4. Conclusion
To fully describe a realistic photon number resolving detector we need to estimate its numerous POVM elements. These elements depend both on the detector’s quantum efficiency and on its inherent response to different photon number states. The most serious drawback of this approach is that each of these, in principle, infinitely many elements is an operator of infinite rank, which clashes with the practical need for simple characterization.
We have shown that there is a single parameter that can be used as a short descriptor of the overall behavior of an indeterminate photon number resolving detector. It is the number of ideal on-off detectors that need to be used in a chosen practical scenario to obtain the same results as the tested PNRD. We suggest that the most suitable scenario is that of conditional preparation of photon number states, in which we can evaluate not only the quality of the detectors but also their compatibility with the quantum optics toolbox.
The benchmarking itself can be realized in three steps. In the first one we can use the unknown detector as both the conditioning and the verification measurement. We experimentally try to maximize the joint probability of success and compare it to respective benchmarks for different MSPDs. In the second step the unknown detector is used only for conditioning and the prepared state is estimated in order to obtain fidelity. The fidelity and success probability are then compared to the benchmarks. Finally, estimation can be used to obtain the full photon number distribution which can be then employed to compensate for the inherent losses in the system.
Appendix A: Proof that MSPD approaches PNRD
This limit hold when $\lim _{M\rightarrow \infty } p(n|k) = \delta _{n,k}$. First, let us show that $\lim _{M\rightarrow \infty } p(n|n) = 1$. Directly evaluating (2) we get
Similarly, for $\lim _{M\rightarrow \infty } p(n|k) = 0$ when $n\neq k$,
Funding
Grantová Agentura České Republiky (GA18-21285S); Horizon 2020 Framework Programme (731473, project 8C20002 ShoQC); Palacký University (IGA-PrF-2019-010, IGA-PrF-2020-009).
Acknowledgments
We also acknowledge national funding from the MEYS.
Disclosures
The authors declare no conflicts of interest.
References
1. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001). [CrossRef]
2. T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68(4), 042319 (2003). [CrossRef]
3. J. C. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, “Towards practical quantum metrology with photon counting,” npj QI 2(1), 16023 (2016). [CrossRef]
4. J.-Y. Wu, N. Toda, and H. F. Hofmann, “Quantum enhancement of sensitivity achieved by photon-number-resolving detection in the dark port of a two-path interferometer operating at high intensities,” Phys. Rev. A 100(1), 013814 (2019). [CrossRef]
5. M. von Helversen, J. Bahm, M. Schmidt, M. Gschrey, J.-H. Schulze, A. Strittmatter, S. Rodt, J. Beyer, T. Heindel, and S. Reitzenstein, “Quantum metrology of solid-state single-photon sources using photon-number-resolving detectors,” New J. Phys. 21(3), 035007 (2019). [CrossRef]
6. F. E. F. E. Becerra, J. Fan, and A. Migdall, “Photon number resolution enables quantum receiver for realistic coherent optical communications,” Nat. Photonics 9(1), 48–53 (2015). [CrossRef]
7. M. T. DiMario, L. Kunz, K. Banaszek, and F. E. Becerra, “Optimized communication strategies with binary coherent states over phase noise channels,” npj QI 5(1), 65 (2019). [CrossRef]
8. J. Tiedau, T. J. Bartley, G. Harder, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “On the scalability of parametric down-conversion for generating higher-order fock states,” Phys. Rev. Lett. 100(4), 041802 (2019). [CrossRef]
9. K. R. Motes, R. L. Mann, N. M. Studer, E. A. Bergeron, A. Gilchrist, J. P. Dowling, D. W. Berry, and P. P. Rohde, “Efficient recycling strategies for preparing large Fock states from single-photon sources: Applications to quantum metrology,” Phys. Rev. A. 94, 012344 (2016). [CrossRef]
10. D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Phys. Rev. A 64(1), 012310 (2001). [CrossRef]
11. S. Ghose and B. C. Sanders, “Non-gaussian ancilla states for continuous variable quantum computation via gaussian maps,” J. Mod. Opt. 54(6), 855–869 (2007). [CrossRef]
12. F. Arzani, N. Treps, and G. Ferrini, “Polynomial approximation of non-gaussian unitaries by counting one photon at a time,” Phys. Rev. A 95(5), 052352 (2017). [CrossRef]
13. J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777–838 (2012). [CrossRef]
14. D. Achilles, C. Silberhorn, and I. A. Walmsley, “Direct, loss-tolerant characterization of nonclassical photon statistics,” Phys. Rev. Lett. 97(4), 043602 (2006). [CrossRef]
15. M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn, “Photon number statistics of multimode parametric down-conversion,” Phys. Rev. Lett. 101(5), 053601 (2008). [CrossRef]
16. J. Peřina, M. Hamar, V. Michálek, and O. c. v. Haderka, “Photon-number distributions of twin beams generated in spontaneous parametric down-conversion and measured by an intensified ccd camera,” Phys. Rev. A 85(2), 023816 (2012). [CrossRef]
17. K. Laiho, M. Avenhaus, K. N. Cassemiro, and C. Silberhorn, “Direct probing of the wigner function by time-multiplexed detection of photon statistics,” New J. Phys. 11(4), 043012 (2009). [CrossRef]
18. K. Laiho, K. N. Cassemiro, D. Gross, and C. Silberhorn, “Probing the negative wigner function of a pulsed single photon point by point,” Phys. Rev. Lett. 105(25), 253603 (2010). [CrossRef]
19. M. Bohmann, J. Tiedau, T. Bartley, J. Sperling, C. Silberhorn, and W. Vogel, “Incomplete detection of nonclassical phase-space distributions,” Phys. Rev. Lett. 120(6), 063607 (2018). [CrossRef]
20. G. Harder, C. Silberhorn, J. Rehacek, Z. Hradil, L. Motka, B. Stoklasa, and L. L. Sánchez-Soto, “Local sampling of the wigner function at telecom wavelength with loss-tolerant detection of photon statistics,” Phys. Rev. Lett. 116(13), 133601 (2016). [CrossRef]
21. M. Yukawa, K. Miyata, T. Mizuta, H. Yonezawa, P. Marek, R. Filip, and A. Furusawa, “Generating superposition of up-to three photons for continuous variable quantum information processing,” Opt. Express 21(5), 5529–5535 (2013). [CrossRef]
22. P. Marek and R. Filip, “Coherent-state phase concentration by quantum probabilistic amplification,” Phys. Rev. A 81(2), 022302 (2010). [CrossRef]
23. M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nat. Phys. 6(10), 767–771 (2010). [CrossRef]
24. C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Probabilistic cloning of coherent states without a phase reference,” Phys. Rev. A 86(1), 010305 (2012). [CrossRef]
25. M. Jönsson and G. Björk, “Evaluating the performance of photon-number-resolving detectors,” Phys. Rev. A 99(4), 043822 (2019). [CrossRef]
26. Y. S. Teo, H. Jeong, J. Řeháček, Z. Hradil, L. L. Sánchez-Soto, and C. Silberhorn, “On the prospects of multiport devices for photon-number-resolving detection,” Quantum Reports 1(2), 162–180 (2019). [CrossRef]
27. A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16(5), 3032 (2008). [CrossRef]
28. B. Calkins, P. L. Mennea, A. E. Lita, B. J. Metcalf, W. S. Kolthammer, A. Lamas-Linares, J. B. Spring, P. C. Humphreys, R. P. Mirin, J. C. Gates, P. G. R. Smith, I. A. Walmsley, T. Gerrits, and S. W. Nam, “High quantum-efficiency photon-number-resolving detector for photonic on-chip information processing,” Opt. Express 21(19), 22657–22670 (2013). [CrossRef]
29. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013). [CrossRef]
30. G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116(14), 143601 (2016). [CrossRef]
31. I. A. Burenkov, A. K. Sharma, T. Gerrits, G. Harder, T. J. Bartley, C. Silberhorn, E. A. Goldschmidt, and S. V. Polyakov, “Full statistical mode reconstruction of a light field via a photon-number-resolved measurement,” Phys. Rev. A 95(5), 053806 (2017). [CrossRef]
32. J. Sperling, W. R. Clements, A. Eckstein, M. Moore, J. J. Renema, W. S. Kolthammer, S. W. Nam, A. Lita, T. Gerrits, W. Vogel, G. S. Agarwal, and I. A. Walmsley, “Detector-independent verification of quantum light,” Phys. Rev. Lett. 118(16), 163602 (2017). [CrossRef]
33. J. Tiedau, T. J. Bartley, G. Harder, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Scalability of parametric down-conversion for generating higher-order Fock states,” Phys. Rev. A. 100(16), 041802 (2019). [CrossRef]
34. U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Discrete and continuous variable quantum, information,” Nat. Phys. 11(9), 713–719 (2015). [CrossRef]
35. H. Paul, P. Torma, T. Kiss, and I. Jex, “Photon chopping: New way to measure the quantum state of light,” Phys. Rev. Lett. 76(14), 2464–2467 (1996). [CrossRef]
36. I. Straka, L. Lachman, J. Hloušek, M. Miková, M. Mičuda, M. Ježek, and R. Filip, “Quantum non-gaussian multiphoton light,” npj Quantum Information 4(1), 4 (2018). [CrossRef]
37. L. c. v. Lachman, I. Straka, J. Hloušek, M. Ježek, and R. Filip, “Faithful hierarchy of genuine n-photon quantum non-gaussian light,” Phys. Rev. Lett. 123(4), 043601 (2019). [CrossRef]
38. R. Filip and L. Mišta, “Detecting quantum states with a positive wigner function beyond mixtures of gaussian states,” Phys. Rev. Lett. 106(20), 200401 (2011). [CrossRef]
39. R. Filip and L. c. v. Lachman, “Hierarchy of feasible nonclassicality criteria for sources of photons,” Phys. Rev. A 88(4), 043827 (2013). [CrossRef]
40. E. Bimbard, R. Boddeda, N. Vitrant, A. Grankin, V. Parigi, J. Stanojevic, A. Ourjoumtsev, and P. Grangier, “Homodyne tomography of a single photon retrieved on demand from a cavity-enhanced cold atom memory,” Phys. Rev. Lett. 112(3), 033601 (2014). [CrossRef]
41. K. Huang, H. Le Jeannic, J. Ruaudel, V. B. Verma, M. D. Shaw, F. Marsili, S. W. Nam, E. Wu, H. Zeng, Y.-C. Jeong, R. Filip, O. Morin, and J. Laurat, “Optical synthesis of large-amplitude squeezed coherent-state superpositions with minimal resources,” Phys. Rev. Lett. 115(2), 023602 (2015). [CrossRef]
42. P. Marek, J. Lee, and M. S. Kim, “Vacuum as a less hostile environment to entanglement,” Phys. Rev. A 77(3), 032302 (2008). [CrossRef]
