1. Precursors ................................................... 1
1.1. P.A.M. Dirac, The quantum theory of the emission
and absorption of radiation, Proc. R. Soc. Lond.
A 114, 243 (1927) ....................................... 7
1.2. W.H. Louisell, Amplitude and phase uncertainty
relations, Phys. Lett. 7, 60 (1963) .................... 30
1.3. D. Judge, On the uncertainty relation for Lz and
Ø, Phys. Lett. 5,189 (1963) ............................ 32
1.4. D. Judge and J.L. Lewis, On the commutator
[LZ/ Ø], Phys. Lett. 5,190 (1963) ...................... 33
1.5. L. Susskind and J. Glogower, Quantum mechanical
phase and time operator, Physics 1, 49 (1964) ......... 34
1.6. R. Loudon, The quantum theory of light, first
edition (Oxford University Press, Oxford, 1973)
chapter 7 .............................................. 47
1.7. S.M. Barnett and D.T. Pegg, Phase in quantum optics,
J. Phys. A: Math. Gen. 19, 3849 (1986) ................. 72
2. The Phase Operator .......................................... 87
2.1. D.T. Pegg and S.M. Barnett, Unitary phase operator
in quantum mechanics, Europhys. Lett. 6,483 (1988) ..... 95
2.2. D.T. Pegg and S.M. Barnett, Hermitian phase
operator Ø in the quantum theory of light Harwell
Laboratory Report TP1290 .............................. 100
2.3. S.M. Barnett and D.T. Pegg, On the Hermitian
optical phase operator, J. Mod. Opt. 36, 7 (1989) ..... 106
2.4. D.T. Pegg and S.M. Barnett, Phase properties of
the quantized single-mode electromagnetic field,
Phys. Rev. A 39,1665 (1989) ........................... 119
2.5. S.M. Barnett and D.T. Pegg, Quantum theory of
rotation angles, Phys. Rev. A 41, 3427 (1990) ......... 130
2.6. David T. Pegg and Stephen M. Barnett, Quantum
optical phase, J. Mod. Opt. 44, 225 (1997) ............ 139
3. Mathematical Elaborations .................................. 179
3.1. J.A. Vaccaro and D.T. Pegg, Wigner function for
number and phase, Phys. Rev. A 41, 5156 (1990) ........ 187
3.2. D.T. Pegg, J.A. Vaccaro and S.M. Barnett, Quantum
optical phase and canonical conjugation, J. Mod.
Opt. 28, 87 (1990) .................................... 195
3.3. S.M. Barnett and D.T. Pegg, Limiting procedures
for the optical phase operator, J. Mod. Opt. 39,
2121 (1992) ........................................... 203
3.4. J.A. Vaccaro and D.T. Pegg, Consistency of quantum
descriptions of phase, Physica Scripta T48, 29
(1993) ................................................ 212
3.5. U. Leonhardt, J.A. Vaccaro, B. Bohmer and H. Paul,
Canonical and measured phase distributions, Phys.
Rev. A 51, 84 (1995) .................................. 219
3.6. J. Vaccaro, Number-phase Wigner function on Fock
space, Phys. Rev. A 52, 3474 (1995) ................... 231
3.7. J.A. Vaccaro and Y. Ben Aryeh, Antinormal ordering
of phase operators and the algebra of weak limits,
Opt. Commun. 113,427 (1995) ........................... 245
3.8. J.A. Vaccaro and R.F. Bonner, Pegg-Barnett
operators of infinite rank, Phys. Lett. A 198, 167
(1995) ................................................ 251
3.9. J.A. Vaccaro, Phase operators on Hilbert space,
Phys. Rev. A 51, 3309 (1995) .......................... 259
4. Phase Dynamics and Uncertainties ........................... 269
4.1. J.A. Vaccaro and D.T. Pegg, Phase properties of
squeezed states of light, Opt. Comm. 70, 529 (1989) ... 273
4.2. S.M. Barnett, S. Stenholm and D.T. Pegg, A new
approach to optical phase diffusion, Opt. Commun.
73, 314 (1989) ........................................ 279
4.3. S.M. Barnett and D.T. Pegg, Quantum theory of
optical phase correlations, Phys. Rev. A 42, 6713
(1990) ................................................ 284
4.4. J.A. Vaccaro and D.T. Pegg, Physical number phase
intelligent and minimum uncertainty states of
light, J. Mod. Opt. 37,17 (1990) ...................... 292
4.5. G.S. Summy and D.T. Pegg, Phase optimized quantum
states of light, Opt. Commun. 77, 75 (1990) ........... 315
4.6. J.A. Vaccaro, S.M. Barnett and D.T. Pegg, Phase
fluctuations and squeezing, J. Mod. Opt. 39, 603
(1992) ................................................ 320
4.7. J.A. Vaccaro and D.T. Pegg, Phase properties of
linear optical amplifiers, Phys. Rev. A 49,4985
(1994) ................................................ 332
5. Theory of Phase Measurement ................................ 343
5.1. S.M. Barnett and D.T. Pegg, Phase measurements,
Phys. Rev. A 47, 4537 (1993) .......................... 346
5.2. J.A. Vaccaro and D.T. Pegg, On measuring extremely
small phase fluctuations, Opt. Comm. 105, 335
(1994) ................................................ 350
5.3. H.M. Wiseman, Adaptive phase measurements of
optical modes: going beyond the marginal Q
distribution, Phys. Rev. Lett. 75, 4587 (1995) ........ 356
5.4. S.M. Barnett and D.T. Pegg, Phase measurements
by projection synthesis, Phys. Rev. Lett. 76, 4148
(1996) ................................................ 360
5.5. D.T. Pegg, S.M. Barnett and L.S. Phillips, Quantum
phase distribution by projection synthesis, J. Mod.
Opt. 44, 2135 (1997) .................................. 363
5.6. K.L. Pregnell and D.T. Pegg, Measuring the phase
variance of light, J. Mod. Opt. 48,1293 (2001) ........ 377
5.7. K.L. Pregnell and D.T. Pegg, Quantum phase
distribution by operator synthesis, J. Mod. Opt.
49,1135 (2002) ........................................ 387
5.8. K.L. Pregnell and D.T. Pegg, Single-shot
measurement of quantum optical phase, Phys. Rev.
Lett. 89,173601 (2002) ................................ 398
5.9. K.L. Pregnell and D.T. Pegg, Binomial states and
the phase distribution measurement of weak optical
fields, Phys. Rev. A 67, 063814 (2003) ................ 402
6. Experimental Demonstrations ................................ 411
6.1. M. Beck, D.T. Smithey, J. Cooper and M. G. Raymer,
Experimental determination of number-phase
uncertainty relations, Opt. Lett. 18,1259 (1993) ...... 414
6.2. D.T. Smithey, M. Beck, J. Coper and M.G. Raymer,
Measurement of number-phase uncertainty relations
for optical fields, Phys. Rev. A 48, 3159 (1993) ...... 417
6.3. M.A. Armen, J.K. Au, J.K. Stockton, A.C. Doherty,
and H. Mabuchi, Adaptive homodyne measurement of
optical phase, Phys. Rev. Lett. 89,133602 (2002) ...... 426
6.4. S. Franke-Arnold, S.M. Barnett, E. Yao, J. Leach,
J. Courtial and M. Padgett, Uncertainty principle
for angular position and angular momentum, New J.
Phys. 6,103 (2004) .................................... 430
6.5. D.T. Pegg, S.M. Barnett, R. Zambrini, S. Franke-
Arnold and M. Padgett, Minimum uncertainty states
of angular momentum and angular position, New J.
Phys. 7, 62 (2005) .................................... 438
7. Time ....................................................... 459
7.1. D.T. Pegg, Time in a quantum mechanical world, J.
Phys. A: Math. Gen. 24,3031 (1991) .................... 462
7.2. D.T. Pegg, Complement of the Hamiltonian, Phys.
Rev. A 58,4307 (1998) ................................. 472
References .................................................... 479
Appendix: Publications by David Pegg .......................... 487
Index ......................................................... 497
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