The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). For coherent light, this condition is determined by the average photon number. The small-noise approximation relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: For a coherent input, how many photons must be present in the input to a quantum linear amplifier for the phase noise at the output to be amenable to a small-noise analysis? We address these questions by showing how the phase uncertainty can be obtained without recourse to the small-noise approximation. It is shown that for an ideal linear amplifier (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for a nonideal amplifier.
 M. Sargent, M. Scully, and W. Lamb. Laser Physics. Westview Press, Boulder, CO, United States, 1978.
 M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri. True random numbers from amplified quantum vacuum. Optics Express, 19 (21): 20665–20672, 2011. 10.1364/OE.19.020665.
 F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo. Ultrafast quantum random number generation based on quantum phase fluctuations. Optics Express, 20 (11): 12366–12377, 2012. 10.1364/OE.20.012366.
 C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. W. Mitchell. Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode. Optics Express, 22 (2): 1645–1654, 2014. 10.1364/OE.22.001645.
 B. Septriani, O. de Vries, and M. Gräfe. New insights in phase diffusion process in a gain-switched semiconductor laser for quantum random number generation (QRNG). In Quantum Information and Measurement, pages F5A–44. Optical Society of America, 2019. 10.1364/QIM.2019.F5A.44.
 R. Loudon and T. J. Shepherd. Properties of the optical quantum amplifier. Optica Acta: International Journal of Optics, 31 (11): 1243–1269, 1984. 10.1080/713821446.
 S. M. Barnett, S. Stenholm, and D. T. Pegg. A new approach to optical phase diffusion. Optics Communications, 73 (4): 314–318, 1989. 10.1016/0030-4018(89)90224-1.
 A. Bandilla. The broadening of the phase distribution due to linear amplification. Optics Communications, 80 (3-4): IN1–274, 1991. 10.1016/0030-4018(91)90264-E.
 L. Thylén, M. Gustavsson, A. Karlsson, and T. K. Gustafson. Phase noise in traveling-wave inverted-population-type optical amplifiers. JOSA B, 9 (3): 369–373, 1992. 10.1364/JOSAB.9.000369.
 I. Bialynicki-Birula, M. Freyberger, and W. Schleich. Various measures of quantum phase uncertainty: a comparative study. Physica Scripta, 1993 (T48): 113, 1993. 10.1088/0031-8949/1993/t48/017.
 B. Huttner, N. Imoto, N. Gisin, and T. Mor. Quantum cryptography with coherent states. Physical Review A, 51 (3): 1863, 1995. 10.1103/physreva.51.1863.
 Y. Zhang, I. B. Djordjevic, and M. A. Neifeld. Weak-coherent-state-based time-frequency quantum key distribution. Journal of Modern Optics, 62 (20): 1713–1721, 2015. 10.1080/09500340.2015.1075616.
 M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov. Invited review article: Single-photon sources and detectors. Review of Scientific Instruments, 82 (7): 071101, 2011. 10.1063/1.3610677.
 Chao Wang, Fang-Xiang Wang, Hua Chen, Shuang Wang, Wei Chen, Zhen-Qiang Yin, De-Yong He, Guang-Can Guo, and Zheng-Fu Han. Realistic device imperfections affect the performance of hong-ou-mandel interference with weak coherent states. Journal of Lightwave Technology, 35 (23): 4996–5002, 2017. 10.1109/JLT.2017.2764140.
 E. Moschandreou, J. I. Garcia, B. J. Rollick, B. Qi, R. Pooser, and G. Siopsis. Experimental study of hong–ou–mandel interference using independent phase randomized weak coherent states. Journal of Lightwave Technology, 36 (17): 3752–3759, 2018. 10.1109/JLT.2018.2850282.
 A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf. Introduction to quantum noise, measurement, and amplification. Reviews of Modern Physics, 82 (2): 1155, 2010. 10.1103/RevModPhys.82.1155.
 C. M. Caves, J. Combes, Z. Jiang, and S. Pandey. Quantum limits on phase-preserving linear amplifiers. Physical Review A, 86 (6): 063802, 2012. 10.1103/PhysRevA.86.063802.
 A. Chia, M. Hajdušek, R. Nair, R. Fazio, L.-C. Kwek, and V. Vedral. Noise-induced amplification: Parametric amplifiers cannot simulate all phase-preserving linear amplifiers. arXiv:1903.09370, 2019.
 P. Lähteenmäki, V. Vesterinen, J. Hassel, G. S. Paraoanu, H. Seppä, and P. Hakonen. Advanced concepts in josephson junction reflection amplifiers. Journal of Low Temperature Physics, 175 (5-6): 868–876, 2014. 10.1007/s10909-014-1170-0.
 A. Roy and M. Devoret. Quantum-limited parametric amplification with josephson circuits in the regime of pump depletion. Physical Review B, 98 (4): 045405, 2018. 10.1103/PhysRevB.98.045405.
 S. Friberg and L. Mandel. Coherence properties of the linear photon amplifier. Optics Communications, 46 (2): 141–148, 1983. 10.1016/0030-4018(83)90394-2.
 C. K. Hong, S. Friberg, and L. Mandel. Conditions for nonclassical behavior in the light amplifier. JOSA B, 2 (3): 494–496, 1985. 10.1364/JOSAB.2.000494.
 P. Marian, T. A. Marian, and H. Scutaru. Quantifying nonclassicality of one-mode gaussian states of the radiation field. Physical Review Letters, 88 (15): 153601, 2002. 10.1103/PhysRevLett.88.153601.
 H. Huang, S.-Y. Zhu, and M. S. Zubairy. Preservation of nonclassical character during the amplification of a schrödinger cat state. Physical Review A, 53: 1027–1030, Feb 1996. 10.1103/PhysRevA.53.1027.
 N. Spagnolo, C. Vitelli, T. De Angelis, F. Sciarrino, and F. De Martini. Wigner-function theory and decoherence of the quantum-injected optical parametric amplifier. Physical Review A, 80: 032318, Sep 2009. 10.1103/PhysRevA.80.032318.
 H. Nha, G. J. Milburn, and H. J. Carmichael. Linear amplification and quantum cloning for non-gaussian continuous variables. New Journal of Physics, 12 (10): 103010, 2010. 10.1088/1367-2630/12/10/103010.
 H. Jeong and T. C. Ralph. Schrödinger Cat States for Quantum Information Processing, pages 159–179. Imperial College Press, 2007. 10.1142/9781860948169_0009.
 A. Gilchrist, K. Nemoto, W. J. Munro, T. C. Ralph, S. Glancy, S. L. Braunstein, and G. J. Milburn. Schrödinger cats and their power for quantum information processing. Journal of Optics B: Quantum and Semiclassical Optics, 6 (8): S828, 2004. 10.1088/1464-4266/6/8/032.
 H. J. Carmichael. Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations. Springer Science & Business Media, 2013. 10.1007/978-3-662-03875-8.
 C. W. Gardiner. Handbook of Stochastic Methods, volume 3. Springer Berlin, 1985.
 H. Paul. Phase of a microscopic electromagnetic field and its measurement. Fortschritte der Physik, 22 (11): 657–689, 1974. 10.1002/prop.19740221104.
 A. Chia, M. Hajdusek, R. Nair, R. Fazio, L. C. Kwek, and V. Vedral, "Noise-induced amplification: Parametric amplifiers cannot simulate all phase-preserving linear amplifiers", arXiv:1903.09370.
The above citations are from SAO/NASA ADS (last updated successfully 2020-04-03 19:38:32). The list may be incomplete as not all publishers provide suitable and complete citation data.
On Crossref's cited-by service no data on citing works was found (last attempt 2020-04-03 19:38:30).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.