S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018).

[Crossref]

E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018).

[Crossref]

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26, 12499–12505 (2018).

[Crossref]
[PubMed]

S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Comm. 1, 045001 (2017).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

A. Metelmann and A. Clerk, “Quantum-Limited Amplification via Reservoir Engineering,” Phys. Rev. Lett. 112, 133904 (2014).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

B. Laikhtman, “Are excitons really bosons?” J. Phys.: Condens. Matter 19, 295214 (2007).

M. Combescot, O. Betbeder-Matibet, and R. Combescot, “Exciton-exciton scattering: Composite boson versus elementary boson,” Phys. Rev. B 75, 174305 (2007).

[Crossref]

U. Gavish, B. Yurke, and Y. Imry, “Generalized constraints on quantum amplification,” Phys. Rev. Lett. 93, 250601 (2004).

[Crossref]

A. Imamoḡlu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002).

[Crossref]

M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature 406, 1039–1046 (2000).

[Crossref]
[PubMed]

G. Björk, J. Söderholm, and A. Karlsson, “Superposition-preserving photon-number amplifier,” Phys. Rev. A 57, 650–658 (1998).

[Crossref]

H. P. Yuen, “Quantum amplifiers, quantum duplicators and quantum cryptography,” Quantum Semiclass. Opt.: J. Eur. Opt. Soc. Part B 8, 939 (1996).

[Crossref]

G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996).

[Crossref]

M. S. Ünlü and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).

[Crossref]

G. D’Ariano, “Hamiltonians for the photon-number-phase amplifiers,” Phys. Rev. A 45, 3224–3227 (1992).

[Crossref]

D. Pegg and S. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).

[Crossref]

J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

[Crossref]
[PubMed]

H. P. Yuen, “Generation, detection, and application of high-intensity photon-number-eigenstate fields,” Phys. Rev. Lett. 56, 2176–2179 (1986).

[Crossref]
[PubMed]

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).

[Crossref]

H. G. Dehmelt, “Proposed 1014 δv<v laser fluorescence spectroscopy on Tl + ion mono-oscillator II,” Bull. Am. Phys. Soc. 20, 60 (1975).

L. V. Keldysh and A. N. Kozlov, “Collective properties of excitons in semiconductors,” Sov. Phys. JETP 27, 521–528 (1968).

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

D. Pegg and S. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).

[Crossref]

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26, 12499–12505 (2018).

[Crossref]
[PubMed]

R. Nehra, C.-H. Chang, A. Beling, and O. Pfister, “Photon-number-resolving segmented avalanche-photodiode detectors,” e-print arXiv:1708.09015 [physics.ins-det] (2017).

J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

[Crossref]
[PubMed]

D. Wineland, J. Bergquist, W. M. Itano, and R. Drullinger, “Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions,” Opt. Lett. 5, 245–247 (1980).

[Crossref]
[PubMed]

M. Combescot, O. Betbeder-Matibet, and R. Combescot, “Exciton-exciton scattering: Composite boson versus elementary boson,” Phys. Rev. B 75, 174305 (2007).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

G. Björk, J. Söderholm, and A. Karlsson, “Superposition-preserving photon-number amplifier,” Phys. Rev. A 57, 650–658 (1998).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).

[Crossref]

R. Nehra, C.-H. Chang, A. Beling, and O. Pfister, “Photon-number-resolving segmented avalanche-photodiode detectors,” e-print arXiv:1708.09015 [physics.ins-det] (2017).

A. Metelmann and A. Clerk, “Quantum-Limited Amplification via Reservoir Engineering,” Phys. Rev. Lett. 112, 133904 (2014).

[Crossref]
[PubMed]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

M. Combescot, O. Betbeder-Matibet, and R. Combescot, “Exciton-exciton scattering: Composite boson versus elementary boson,” Phys. Rev. B 75, 174305 (2007).

[Crossref]

M. Combescot, O. Betbeder-Matibet, and R. Combescot, “Exciton-exciton scattering: Composite boson versus elementary boson,” Phys. Rev. B 75, 174305 (2007).

[Crossref]

G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996).

[Crossref]

G. D’Ariano, “Hamiltonians for the photon-number-phase amplifiers,” Phys. Rev. A 45, 3224–3227 (1992).

[Crossref]

H. G. Dehmelt, “Proposed 1014 δv<v laser fluorescence spectroscopy on Tl + ion mono-oscillator II,” Bull. Am. Phys. Soc. 20, 60 (1975).

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature 406, 1039–1046 (2000).

[Crossref]
[PubMed]

E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018).

[Crossref]

J. Dowling. Private communication.

U. Gavish, B. Yurke, and Y. Imry, “Generalized constraints on quantum amplification,” Phys. Rev. Lett. 93, 250601 (2004).

[Crossref]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

[Crossref]
[PubMed]

A. Imamoḡlu, “High efficiency photon counting using stored light,” Phys. Rev. Lett. 89, 163602 (2002).

[Crossref]

U. Gavish, B. Yurke, and Y. Imry, “Generalized constraints on quantum amplification,” Phys. Rev. Lett. 93, 250601 (2004).

[Crossref]

J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

[Crossref]
[PubMed]

D. Wineland, J. Bergquist, W. M. Itano, and R. Drullinger, “Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions,” Opt. Lett. 5, 245–247 (1980).

[Crossref]
[PubMed]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

G. Björk, J. Söderholm, and A. Karlsson, “Superposition-preserving photon-number amplifier,” Phys. Rev. A 57, 650–658 (1998).

[Crossref]

L. V. Keldysh and A. N. Kozlov, “Collective properties of excitons in semiconductors,” Sov. Phys. JETP 27, 521–528 (1968).

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

L. V. Keldysh and A. N. Kozlov, “Collective properties of excitons in semiconductors,” Sov. Phys. JETP 27, 521–528 (1968).

B. Laikhtman, “Are excitons really bosons?” J. Phys.: Condens. Matter 19, 295214 (2007).

E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018).

[Crossref]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996).

[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018).

[Crossref]

A. Metelmann and A. Clerk, “Quantum-Limited Amplification via Reservoir Engineering,” Phys. Rev. Lett. 112, 133904 (2014).

[Crossref]
[PubMed]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

R. Nehra, C.-H. Chang, A. Beling, and O. Pfister, “Photon-number-resolving segmented avalanche-photodiode detectors,” e-print arXiv:1708.09015 [physics.ins-det] (2017).

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

D. Pegg and S. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).

[Crossref]

R. Nehra, C.-H. Chang, A. Beling, and O. Pfister, “Photon-number-resolving segmented avalanche-photodiode detectors,” e-print arXiv:1708.09015 [physics.ins-det] (2017).

Tz. B. Propp and S. J. van Enk, “POVMs for photo detection,” in preparation.

Tz. B. Propp and S. J. van Enk, “Quantum networks for single photon detection,” e-print arXiv:1901.09974 [quant-ph] (2019).

S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018).

[Crossref]

A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155–1208 (2010).

[Crossref]

M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature 406, 1039–1046 (2000).

[Crossref]
[PubMed]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

G. Björk, J. Söderholm, and A. Karlsson, “Superposition-preserving photon-number amplifier,” Phys. Rev. A 57, 650–658 (1998).

[Crossref]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996).

[Crossref]

M. S. Ünlü and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).

[Crossref]

M. S. Ünlü and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).

[Crossref]

S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Comm. 1, 045001 (2017).

[Crossref]

Tz. B. Propp and S. J. van Enk, “POVMs for photo detection,” in preparation.

Tz. B. Propp and S. J. van Enk, “Quantum networks for single photon detection,” e-print arXiv:1901.09974 [quant-ph] (2019).

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

J. Bergquist, R. G. Hulet, W. M. Itano, and D. Wineland, “Observation of quantum jumps in a single atom,” Phys. Rev. Lett. 57, 1699–1702 (1986).

[Crossref]
[PubMed]

D. Wineland, J. Bergquist, W. M. Itano, and R. Drullinger, “Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions,” Opt. Lett. 5, 245–247 (1980).

[Crossref]
[PubMed]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018).

[Crossref]

G. D’Ariano, C. Macchiavello, N. Sterpi, and H. Yuen, “Quantum phase amplification,” Phys. Rev. A 54, 4712–4718 (1996).

[Crossref]

H. P. Yuen, “Quantum amplifiers, quantum duplicators and quantum cryptography,” Quantum Semiclass. Opt.: J. Eur. Opt. Soc. Part B 8, 939 (1996).

[Crossref]

S.-T. Ho and H. P. Yuen, “Scheme for realizing a photon number amplifier,” Opt. Lett. 19, 61–63 (1994).

[Crossref]
[PubMed]

H. P. Yuen, “Generation, detection, and application of high-intensity photon-number-eigenstate fields,” Phys. Rev. Lett. 56, 2176–2179 (1986).

[Crossref]
[PubMed]

U. Gavish, B. Yurke, and Y. Imry, “Generalized constraints on quantum amplification,” Phys. Rev. Lett. 93, 250601 (2004).

[Crossref]

H. G. Dehmelt, “Proposed 1014 δv<v laser fluorescence spectroscopy on Tl + ion mono-oscillator II,” Bull. Am. Phys. Soc. 20, 60 (1975).

M. S. Ünlü and S. Strite, “Resonant cavity enhanced photonic devices,” J. Appl. Phys. 78, 607–639 (1995).

[Crossref]

S. J. van Enk, “Photodetector figures of merit in terms of POVMs,” J. Phys. Comm. 1, 045001 (2017).

[Crossref]

B. Laikhtman, “Are excitons really bosons?” J. Phys.: Condens. Matter 19, 295214 (2007).

F. Marsili, V. Verma, J. Stern, S. Harrington, A. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7, 210–214 (2013).

[Crossref]

M. H. Devoret and R. J. Schoelkopf, “Amplifying quantum signals with the single-electron transistor,” Nature 406, 1039–1046 (2000).

[Crossref]
[PubMed]

Q. Yu, K. Sun, Q. Li, and A. Beling, “Segmented waveguide photodetector with 90% quantum efficiency,” Opt. Express 26, 12499–12505 (2018).

[Crossref]
[PubMed]

L.-P. Yang and Z. Jacob, “Quantum critical detector: amplifying weak signals using discontinuous quantum phase transitions,” Opt. Express 27, 10482–10494 (2019).

[Crossref]
[PubMed]

E. E. Wollman, V. B. Verma, A. D. Beyer, R. M. Briggs, B. Korzh, J. P. Allmaras, F. Marsili, A. E. Lita, R. Mirin, and S. Nam, “UV superconducting nanowire single-photon detectors with high efficiency, low noise, and 4K operating temperature,” Opt. Express 25, 26792–26801 (2017).

[Crossref]
[PubMed]

D. Wineland, J. Bergquist, W. M. Itano, and R. Drullinger, “Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions,” Opt. Lett. 5, 245–247 (1980).

[Crossref]
[PubMed]

S.-T. Ho and H. P. Yuen, “Scheme for realizing a photon number amplifier,” Opt. Lett. 19, 61–63 (1994).

[Crossref]
[PubMed]

D. Pegg and S. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665–1675 (1989).

[Crossref]

C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving linear amplifiers,” Phys. Rev. A 86, 063802 (2012).

[Crossref]

E. S. Matekole, H. Lee, and J. P. Dowling, “Limits to atom-vapor-based room-temperature photon-number-resolving detection,” Phys. Rev. A 98, 033829 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “General modeling framework for quantum photodetectors,” Phys. Rev. A 98, 063835 (2018).

[Crossref]

S. M. Young, M. Sarovar, and F. Léonard, “Fundamental limits to single-photon detection determined by quantum coherence and backaction,” Phys. Rev. A 97, 033836 (2018).

[Crossref]

G. D’Ariano, “Hamiltonians for the photon-number-phase amplifiers,” Phys. Rev. A 45, 3224–3227 (1992).

[Crossref]

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Note the dimension of both input and output mode Hilbert spaces are s + 1; they necessarily match in the Heisenberg picture.

One way around this limitation is for the incident photons to only interact with a single symmetrized collective degree of freedom of many fermions, on which a measurement is then made. In this idealized case, this collective degree of freedom plays the role of a single bosonic mode and amplification could still be described by Eq. (5) and photon number amplification is improved past the limit for linear fermionic amplification [5].

The signal-to-noise ratios Eqs. (22) and (23) for linear amplification become infinite at G = 1 simply because there is no noise when both G = 1 and Δna = 0.

We find the linear dependence on G resulting from single-shot single-mode amplification holds for transformations describing higher-order amplification of photon number operator, again subject to the constraints of [23].

The space is further filled in by considering nonlinear amplification where G excitations are distributed into G′ > G modes so that ancillary modes contribute only to the noise and not to the signal. In this case, we find that the SNR goes to 0 as G′ → ∞; the effect of additional noise modes is always to reduce the SNR and move us away from the optimal SNR in Eq. (24).

Photon-number resolved photo detection can be achieved by multiplexing an n-photon signal to many (N ≫ n) single photon detectors [28], each satisfying Eq. (28) independently. However, this means an additional noise mode will be added with each splitting of the signal, decreasing the integrated signal-to-noise ratio. To avoid added noise a nonlinear multi-photon filtering process could be used, but for this a full S-matrix treatment must be used, see [45–47].

See, for example, [48]. The result is that, instead of certain frequencies, it is certain spectral “Schmidt modes” that are detected perfectly.

Tz. B. Propp and S. J. van Enk, “Quantum networks for single photon detection,” e-print arXiv:1901.09974 [quant-ph] (2019).

J. Dowling. Private communication.

Tz. B. Propp and S. J. van Enk, “POVMs for photo detection,” in preparation.

This could be a mode internal to the detector.

The construction of physically implementable transformations such as Eq. (3) (and transformations including higher powers of the input photon number operator) are highly constrained by two conditions: that the spectrum of the operator-representation be ℕ0 (the natural numbers including zero) and that the commutator be preserved [14]. From these conditions, in Eq. (3) we need at least one term with a number operator on the right with a prefactor of one. Additional reservoirs with arbitrary prefactors are allowed but they will carry additional noise and decrease the SNR.

Phase randomization is necessary for optimal amplification and measurement of photon number due to number-phase uncertainty. Indeed, amplification of photon number deamplifies phase and vice versa, see [17].

The transformation in Eq. (2) can be realized only when M ≥ Gn. There is always such a restriction on amplification relations; the energy transferred to reservoir 2 must come from somewhere.