Wigner function for SU(1,1)

U. Seyfarth1, A. B. Klimov2, H. de Guise3, G. Leuchs1,4, and L. L. Sanchez-Soto1,5

1Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany
2Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
3Department of Physics, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada
4Institute for Applied Physics, Russian Academy of Sciences, 630950 Nizhny Novgorod, Russia
5Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

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In spite of their potential usefulness, Wigner functions for systems with SU(1,1) symmetry have not been explored thus far. We address this problem from a physically-motivated perspective, with an eye towards applications in modern metrology. Starting from two independent modes, and after getting rid of the irrelevant degrees of freedom, we derive in a consistent way a Wigner distribution for SU(1,1). This distribution appears as the expectation value of the displaced parity operator, which suggests a direct way to experimentally sample it. We show how this formalism works in some relevant examples.

$\textbf{Dedication}$: While this manuscript was under review, we learnt with great sadness of the untimely passing of our colleague and friend Jonathan Dowling. Through his outstanding scientific work, his kind attitude, and his inimitable humor, he leaves behind a rich legacy for all of us. Our work on SU(1,1) came as a result of long conversations during his frequent visits to Erlangen. We dedicate this paper to his memory.

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[1] H. Weyl, Quantenmechanik und Gruppentheorie, Z. Phys. 46, 1–46 (1927).

[2] E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40, 749–759 (1932).

[3] H. J. Groenewold, On the principles of elementary quantum mechanics, Physica 12, 405–460 (1946).

[4] J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45, 99–124 (1949).

[5] F. E. Schroek, Quantum Mechanics on Phase Space, Kluwer, 1996.

[6] W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.

[7] C. K. Zachos, D. B. Fairlie, and T. L. Curtright, editors. Quantum Mechanics in Phase Space. World Scientific, 2005.

[8] J. Weinbub and D. K. Ferry, Recent advances in Wigner function approaches, Appl. Phys. Rev. 5, 041104 (2019).

[9] A. A. Kirillov, Lectures on the Orbit Method, American Mathematical Society, 2004.

[10] V. I. Tatarskii, The Wigner representation in quantum mechanics, Sov. Phys. Usp. 26, 311–327 (1983).

[11] N. L. Balazs and B. K. Jennings, Wigner's function and other distribution functions in mock phase spaces, Phys. Rep. 104, 347–391 (1984).

[12] M. Hillery, R. F. O' Connell, M. O. Scully, and E. P. Wigner, Distribution functions in physics: Fundamentals, Phys. Rep. 106, 121–167 (1984).

[13] H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys. Rep. 259, 147–211 (1995).

[14] R. J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131, 2766–2788 (1963).

[15] E. C. G. Sudarshan, Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams, Phys. Rev. Lett. 10, 277–279 (1963).

[16] K. Husimi, Some formal properties of the density matrix, Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).

[17] W.-M. Zhang, D. H. Feng, and R. Gilmore, Coherent states: Theory and some applications, Rev. Mod. Phys. 62, 867–927 (1990).

[18] R. L. Stratonovich, On distributions in representation space, JETP 31, 1012–1020 (1956).

[19] F. A. Berezin, General concept of quantization, Commun. Math. Phys. 40, 153–174 (1975).

[20] J. C. Varilly and J. M. Gracia-Bondía, The Moyal representation for spin, Ann. Phys. 190, 107–148 (1989).

[21] G. S. Agarwal, Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions, Phys. Rev. A 24, 2889–2896 (1981).

[22] J. P. Dowling, G. S. Agarwal, and W. P. Schleich, Wigner distribution of a general angular momentum state: application to a collection of two-level atoms Phys. Rev. A 49, 4101–4109 (1994).

[23] L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, Wigner distribution function for Euclidean systems, J. Phys. A: Math. Gen. 31, 3875–3895 (1998).

[24] S. Heiss and S. Weigert, Discrete Moyal-type representations for a spin, Phys. Rev. A 63, 012105 (2000).

[25] S. M. Chumakov, A. B. Klimov, and K. B. Wolf, Connection between two Wigner functions for spin systems, Phys. Rev. A 61, 034101 (2000).

[26] A. B. Klimov, J. L. Romero, and H. de Guise, Generalized SU(2) covariant Wigner functions and some of their applications, J. Phys. A: Math. Theor. 50, 323001 (2017).

[27] N. Mukunda, Wigner distribution for angle coordinates in quantum mechanics, Am. J. Phys. 47, 182–187 (1979).

[28] J. F. Plebański, M. Prazanowski, J. Tosiek, and F. K. Turrubiates, Remarks on deformation quantization on the cylinder, Acta Phys. Pol. B 31, 561–587 (2000).

[29] I. Rigas, L. L. Sánchez-Soto, A. B. Klimov, J. Řeháček, and Z. Hradil, Orbital angular momentum in phase space, Ann. Phys. 326, 426–439 (2011).

[30] H. A. Kastrup, Wigner functions for the pair angle and orbital angular momentum, Phys. Rev. A 94, 062113 (2016).

[31] G. Molina-Terriza, J. P. Torres, and Ll. Torner, Twisted photons, Nat. Phys. 3, 305–310 (2007).

[32] S. Franke-Arnold, L. Allen, and M. Padgett, Advances in optical angular momentum, Laser Photon. Rev. 2, 299–313 (2008).

[33] C. Brif and A. Mann, A general theory of phase-space quasiprobability distributions, J. Phys. A: Math. Gen. 31, L9–L17 (1998).

[34] N. Mukunda, G. Marmo, A. Zampini, S. Chaturvedi, and R. Simon, Wigner–Weyl isomorphism for quantum mechanics on Lie groups, J. Math. Phys. 46, 012106 (2005).

[35] A. B. Klimov and H. de Guise, General approach to $\mathfrak{SU}(n)$ quasi-distribution functions, J. Phys. A: Math. Theor. 43, 402001 (2010).

[36] T. Tilma, M. J. Everitt, J. H. Samson, W. J. Munro, and K. Nemoto, Wigner functions for arbitrary quantum systems, Phys. Rev. Lett. 117, 180401 (2016).

[37] W. K. Wootters, A Wigner-function formulation of finite-state quantum mechanics, Ann. Phys. 176, 1–21 (1987).

[38] D. Galetti and A. F. R. de Toledo-Piza, An extended Weyl-Wigner transformation for special finite spaces, Physica A 149, 267–282 (1988).

[39] D. Galetti and A. F. R. de Toledo-Piza, Discrete quantum phase spaces and the mod $N$ invariance, Physica A 186, 513–523 (1992).

[40] K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Discrete phase space based on finite fields, Phys. Rev. A 70, 062101 (2004).

[41] A. Vourdas, Quantum systems with finite Hilbert space: Galois fields in quantum mechanics, J. Phys. A: Math. Theor. 40, R285–R331 (2007).

[42] G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, The discrete Wigner function, Prog. Opt. 51, 469–516 (2008).

[43] A. B. Klimov, C. Muñoz, and L. L. Sánchez-Soto, Discrete coherent and squeezed states of many-qudit systems, Phys. Rev. A 80, 043836 (2009).

[44] A. Orłowski and K. Wódkiewicz, On the SU(1,1) phase-space description of reduced and squeezed quantum fluctuations, J. Mod. Opt. 37, 295–301 (1990).

[45] M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, Wigner functions for curved spaces. I. on hyperboloids J. Math. Phys. 43, 5857–5871 (2002).

[46] K. Wodkiewicz and J. H. Eberly, Coherent states, squeezed fluctuations, and the SU(2) and SU(1,1) groups in quantum-optics applications, J. Opt. Soc. Am. B 2, 458–466 (1985).

[47] C. C. Gerry, Dynamics of SU(1,1) coherent states, Phys. Rev. A 31, 2721–2723 (1985).

[48] C. C. Gerry, Correlated two-mode SU(1,1) coherent states: nonclassical properties, J. Opt. Soc. Am. B 8, 685–690 (1991).

[49] C. C. Gerry and R. Grobe, Two-mode intelligent SU(1,1) states, Phys. Rev. A 51, 423–4131 (1995).

[50] J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, Realization of a nonlinear interferometer with parametric amplifiers, Appl. Phys. Lett. 99, 011110 (2011).

[51] F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, Quantum metrology with parametric amplifier-based photon correlation interferometers, Nat. Commun. 5, 3049 (2014).

[52] B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033–4054 (1986).

[53] M. V. Chekhova and Z. Y. Ou, Nonlinear interferometers in quantum optics, Adv. Opt. Photon. 8, 104–155 (2016).

[54] M. Novaes, Some basics of SU(1,1), Rev. Bras. Ensino Fís. 26, 351–357 (2004).

[55] S. Chaturvedi, G. Marmo, N. Mukunda, R. Simon, and A. Zampini, The Schwinger representation of a group: Concept and applications, Rev. Math. Phys. 18, 887–912 (2006).

[56] A. Royer, Wigner function as the expectation value of a parity operator, Phys. Rev. A 15, 449–450 (1977).

[57] K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, Direct measurement of the Wigner function by photon counting, Phys. Rev. A 60, 674–677 (1999).

[58] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Direct measurement of the Wigner function of a one-photon Fock state in a cavity, Phys. Rev. Lett. 89, 200402 (2002).

[59] G. Harder, Ch. 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, 133601 (2016).

[60] E. Binz and S. Pods, The Geometry of Heisenberg Groups. American Mathematical Society, 2008.

[61] K. E. Cahill and R. J. Glauber, Density operators and quasiprobability distributions, Phys. Rev. 177, 1882–1902 (1969).

[62] W. B. Case, Wigner functions and Weyl transforms for pedestrians, Am. J. Phys. 76, 937–946 (2008).

[63] A. Perelomov, Generalized Coherent States and their Applications. Springer, 1986.

[64] K. Hasebe, Sp(4;$\mathbb{R}$) squeezing for Bloch four-hyperboloid via the non-compact Hopf map, J. Phys. A: Math. Theor. 53, 055303 (2020).

[65] R. Mosseri and R. Dandoloff, Geometry of entangled states, Bloch spheres and Hopf fibrations, J. Phys. A: Math. Gen. 34, 10243–10252 (2001).

[66] K. Hasebe, The split algebras and noncompact Hopf maps, J. Math. Phys. 51, 053524 (2010).

[67] C. C. Gerry and J. Mimih, The parity operator in quantum optical metrology, Contemp. Phys. 51, 497–511 (2010).

[68] J. J . Bollinger, Wayne M. Itano, D. J. Wineland, and D. J. Heinzen, Optimal frequency measurements with maximally correlated states, Phys. Rev. A 54 R4649–R4652 (1996).

[69] P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Quantum metrology with two-mode squeezed vacuum: Parity detection beats the Heisenberg limit, Phys. Rev. Lett. 104, 103602 (2010).

[70] W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, Parity detection in quantum optical metrology without number-resolving detectors, New J. Phys. 12, 113025 (2010).

[71] C. C. Gerry, R. Birrittella, A. Raymond, and R. Carranza, Photon statistics, parity measurements, and Heisenberg-limited interferometry: example of the two-mode SU(1,1)$\otimes$SU(1,1) coherent states, J. Mod. Opt. 58, 1509–1517 (2011).

[72] D. Li, B. T. Gard, Y. Gao, C.-H. Yuan, W. Zhang, H. Lee, and J. P. Dowling, Phase sensitivity at the Heisenberg limit in an SU(1,1) interferometer via parity detection, Phys. Rev. A 94, 063840 (2016).

[73] E. E. Hach, R. Birrittella, P. M. Alsing, and C. C. Gerry, SU(1,1) parity and strong violations of a Bell inequality by entangled Barut-Girardello coherent states, J. Opt. Soc. Am. B 35, 2433–2442 (2018).

[74] V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. Math. 48, 568–640 (1947).

[75] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, emphQuantum Theory of Angular Momentum, World Scientific, 1988.

[76] N. Ja. Vilenkin and A. U. Klimyk, Reperesentation of Lie Groups and Special Functions, volume 1. Springer, 1991.

[77] H. Ui, Clebsch-Gordan formulas of the SU(1,1) group, Prog. Theor. Phys. 44, 689–702 (1970).

[78] NIST Digital Library of Mathematical Functions. http:/​/​dlmf.nist.gov/​, Chap.16, 2019. URL http:/​/​dlmf.nist.gov/​. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

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