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also P. A. M. Dirac, Principles of Quantum Mechanics (Oxford University Press, London, 1958)]. The theoretical search for the quantum phase operator continued after Dirac. Heitler [W. Heitler, The Quantum Theory of Radiation (Academic, London, 1954)] used Dirac’s formulation in the quantization of the electromagnetic field. Perhaps Heitler’s work somewhat made a landmark in the search for a quantum phase in the context of the quantum properties of the electromagnetic field and the quantum phase became a dominant research topic of quantum optics
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Here the author believes that one should make a clear distinction between two different major perspectives dominating the field of quantum phase research. The first perspective is to consider the quantum phase as a property of quantum mechanics within a generalized quantum-phase-space formalism. This was the original stand taken by Dirac in his search for a correspondence between the classical canonical variables in the action-angle phase space and their quantum counterparts [see P. A. M. Dirac, Proc. R. Soc. London, Ser. A 114, 243 (1927);and also P. A. M. Dirac, Principles of Quantum Mechanics (Oxford University Press, London, 1958)].The theoretical search for the quantum phase operator continued after Dirac. Heitler [W. Heitler, The Quantum Theory of Radiation (Academic, London, 1954)] used Dirac’s formulation in the quantization of the electromagnetic field. Perhaps Heitler’s work somewhat made a landmark in the search for a quantum phase in the context of the quantum properties of the electromagnetic field and the quantum phase became a dominant research topic of quantum optics.
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(1927)
Proc. R. Soc. London, Ser. A
, vol.114
, pp. 243
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Dirac, P.A.M.1
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11
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33846112882
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The introduction of the photon coherent states by Glauber [R. J. Glauber, Phys. Rev. 130, 2529 (1963)
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(1963)
Phys. Rev.
, vol.130
, pp. 2529
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Glauber, R.J.1
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12
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25044456610
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Phys. Rev.R. J. Glauber131, 2766 (1963)] and the developments in the laser physics in the 1960s enhanced the role of quantum optics in the quantum-phase research.
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(1963)
, vol.131
, pp. 2766
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Glauber, R.J.1
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13
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0346268082
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Another course was taken by Carruthers and Nieto in their seminal work [P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40, 411 (1968)], which they wisely entitled Phase and Angle Variables in Quantum Mechanics, where a revival of the canonical phase-space perspective of the quantum-phase problem was initiated.
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(1968)
Rev. Mod. Phys.
, vol.40
, pp. 411
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Carruthers, P.1
Nieto, M.M.2
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14
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0039433388
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This line of canonical approach has been furthered (the author apologizes for not being able to include a complete list of important theoretical and experimental works) by the works of Rocca and Siruge [F. Rocca and M. Siruge, Commun. Math. Phys. 34, 111 (1973)]
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(1973)
Commun. Math. Phys.
, vol.34
, pp. 111
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Rocca, F.1
Siruge, M.2
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18
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18344362202
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In 1989 Pegg and Barnett introduced the most popular phase operator formalism [D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989)] based on a mathematical abstraction of the concept of truncated Hilbert spaces where they were able to define a Hermitian operator corresponding to a quantum phase. Although the Pegg-Barnett formalism is practical from the calculational point of view, it is not algebraic because of the arbitrary introduction of a truncation in the Hilbert space.
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(1989)
Phys. Rev. A
, vol.39
, pp. 1665
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Pegg, D.T.1
Barnett, S.M.2
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4243243578
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Phys. Rev. AOn the opposite side, Moshinsky and Seligman’s work has been recently followed by Luis and Sánchez-Soto [A. Luis and L. L. Sanchez-Soto, 47, 1492 (1993)
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(1993)
Phys. Rev. A
, vol.47
, pp. 1492
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Luis, A.1
Sanchez-Soto, L.L.2
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Phys. Rev. AA. LuisL. L. Sanchez-Soto48, 752 (1993)] to find a canonical path to the phase problem.
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(1993)
, vol.48
, pp. 752
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Luis, A.1
Sanchez-Soto, L.L.2
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0032555392
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More recently an algebraic canonical approach based on the generalized quantum-phase-space formalism was introduced by the present author [T. Hakioğlu, J. Phys. A 31, 6975 (1998)
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(1998)
J. Phys. A
, vol.31
, pp. 6975
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Hakioğlu, T.1
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J. Phys. AT. Hakioğlu31, 707 (1998)]. The second perspective is to understand the properties of the quantum phase by developing a theory of phase measurement. There is substantial literature on different classical and quantum measurement schemes allowing one to extract the measured properties of the phase. These schemes were mostly inherited from the recent developments in quantum optics, thus being confined to the phase properties of the electromagnetic field.
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(1998)
, vol.31
, pp. 707
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Hakioğlu, T.1
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In this stream, some important works are, in addition to the other references of this article, by Vogel and Schleich [W. Vogel and W. Schleich, Phys. Rev. A 44, 7642 (1991)] and
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(1991)
Phys. Rev. A
, vol.44
, pp. 7642
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Vogel, W.1
Schleich, W.2
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25
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who introduced the quantum-phase-space sampling by quadrature rotations, which was carried on successfully both experimentally and theoretically by Smithey et al. [D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1993)
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(1993)
Phys. Rev. Lett.
, vol.70
, pp. 1244
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Smithey, D.T.1
Beck, M.2
Raymer, M.G.3
Faridani, A.4
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D. T. Smithey, M. Beck, J. Cooper, and M. G. Raymer, Phys. Rev. A 48, 3159 (1993)] and
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(1993)
Phys. Rev. A
, vol.48
, pp. 3159
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Smithey, D.T.1
Beck, M.2
Cooper, J.3
Raymer, M.G.4
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0031104156
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Recently, an interesting approach to the radiation phase was also made in A. Shumovsky, Opt. Commun. 136, 219 (1997)
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(1997)
Opt. Commun.
, vol.136
, pp. 219
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Shumovsky, A.1
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). It must be stressed that these two distinct perspectives are not mutually exclusive but rather they should be considered, in the author’s view, as generating complementary information on the properties of quantum phase
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Opt. Commun.A. Shumovsky and Özgür Müstecaplioğlu, 146, 124 (1998)]. An impressive overview, from the quantum optics point of view, of the major historical developments in the 70-year-old theoretical and experimental search for the properties of the quantum phase and a substantial list of references are available in the recent books by Werner Vogel and Dirk-Gunnar Welsch, Quantum Optics (Akademia-Verlag, Berlin, 1994);L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). It must be stressed that these two distinct perspectives are not mutually exclusive but rather they should be considered, in the author’s view, as generating complementary information on the properties of quantum phase.
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(1998)
, vol.146
, pp. 124
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Shumovsky, A.1
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Bernard Yurke, Samuel L. McCall, and John R. Klauder, Phys. Rev. A 33, 4033 (1986).
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(1986)
Phys. Rev. A
, vol.33
, pp. 4033
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