Phase diffusion and the small-noise approximation in linear amplifiers: Limitations and beyond

Andy Chia1, Michal Hajdušek1, Rosario Fazio2,3, Leong-Chuan Kwek1,4,5, and Vlatko Vedral1,6

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
2ICTP, Strada Costiera 11, I-34151 Trieste, Italy
3Dipartemento di Fisica, Università di Napoli "Federico II", Monte S. Angelo, I-80126, Italy
4MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore
5National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
6Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK

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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.

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► References

[1] H. A. Haus and J. A. Mullen. Quantum noise in linear amplifiers. Physical Review, 128 (5): 2407, 1962. 10.1103/​PhysRev.128.2407.

[2] H. Heffner. The fundamental noise limit of linear amplifiers. Proceedings of the IRE, 50 (7): 1604–1608, 1962. 10.1109/​JRPROC.1962.288130.

[3] C. M Caves. Quantum limits on noise in linear amplifiers. Physical Review D, 26 (8): 1817, 1982. 10.1103/​PhysRevD.26.1817.

[4] A. L Schawlow and C. H. Townes. Infrared and optical masers. Physical Review, 112 (6): 1940, 1958. 10.1103/​PhysRev.112.1940.

[5] Melvin Lax. Classical noise. v. noise in self-sustained oscillators. Physical Review, 160 (2): 290, 1967. 10.1103/​PhysRev.160.290.

[6] Charles Henry. Theory of the linewidth of semiconductor lasers. IEEE Journal of Quantum Electronics, 18 (2): 259–264, 1982. 10.1109/​JQE.1982.1071522.

[7] M. Sargent, M. Scully, and W. Lamb. Laser Physics. Westview Press, Boulder, CO, United States, 1978.

[8] M. O. Scully and M. S. Zubairy. Quantum Optics. AAPT, 1999. 10.1017/​CBO9780511813993.

[9] 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.

[10] 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.

[11] 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.

[12] 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.

[13] S. Cialdi, E. Suerra, S. Olivares, S. Capra, and M. G. A. Paris. Squeezing phase diffusion. arXiv:1905.13158, 2019.

[14] 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.

[15] 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.

[16] N. Lu. Phase-diffusion coefficients and phase-diffusion rate. Physical Review A, 42 (9): 5641, 1990. 10.1103/​PhysRevA.42.5641.

[17] 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.

[18] 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.

[19] J. A. Vaccaro and D. T. Pegg. Phase properties of optical linear amplifiers. Physical Review A, 49 (6): 4985, 1994. 10.1103/​physreva.49.4985.

[20] A. Bandilla. Strong linear amplification of quantum fields. Annalen der Physik, 504 (2): 117–124, 1992. 10.1002/​andp.19925040208.

[21] 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.

[22] M. J. W. Hall. Phase resolution and coherent phase states. Journal of Modern Optics, 40 (5): 809–824, 1993. 10.1080/​09500349314550841.

[23] 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.

[24] 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.

[25] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden. Quantum cryptography. Reviews of Modern Physics, 74 (1): 145, 2002. 10.1103/​RevModPhys.74.145.

[26] 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.

[27] 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.

[28] 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.

[29] 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.

[30] 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.

[31] A. Chia, M. Hajdušek, R. Fazio, L.-C. Kwek, and V. Vedral. Quantum nonlinear dynamics: From noise-induced amplification to strong nonlinearity. arXiv:1711.07376, 2017.

[32] 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.

[33] G. S. Agarwal. Quantum Optics. Cambridge University Press, 2012. 10.1017/​CBO9781139035170.

[34] 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.

[35] 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.

[36] L. Mandel and E. Wolf. Optical Coherence and Quantum Optics. Cambridge university press, 1995. 10.1017/​CBO9781139644105.

[37] 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.

[38] 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.

[39] 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.

[40] B. R. Mollow and R. J. Glauber. Quantum theory of parametric amplification. i. Physical Review, 160: 1076–1096, Aug 1967. 10.1103/​PhysRev.160.1076.

[41] 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.

[42] 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.

[43] 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.

[44] H. Jeong and T. C. Ralph. Schrödinger Cat States for Quantum Information Processing, pages 159–179. Imperial College Press, 2007. 10.1142/​9781860948169_0009.

[45] 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.

[46] 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.

[47] H. Risken. Fokker-planck equation. In The Fokker-Planck Equation, pages 63–95. Springer, 1996. 10.1007/​978-3-642-61544-3.

[48] C. W. Gardiner. Handbook of Stochastic Methods, volume 3. Springer Berlin, 1985.

[49] H. Paul. Phase of a microscopic electromagnetic field and its measurement. Fortschritte der Physik, 22 (11): 657–689, 1974. 10.1002/​prop.19740221104.

[50] S. M. Barnett and J. A. Vaccaro. The Quantum Phase Operator: A Review. CRC Press, 2007. 10.1201/​b16006.

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