We address in this paper the notion of relative phase shift for mixed quantum systems. We study the Pancharatnam–Sjöqvist phase shift for metaplectic isotopies acting on Gaussian mixed states. We complete and generalize the previous results obtained by one of us, while giving rigorous proofs. The key actor in this study is the theory of the Conley–Zehnder index which is an intersection index related to the Maslov index.
REFERENCES
1.
Aharonov
, Y.
and Bohm
, D.
, “Significance of electromagnetic potentials in quantum theory
,” Phys. Rev.
115
, 485
–491
(1959
).2.
Aharonov
, Y.
and Anandan
, J.
, “Phase change during a cyclic quantum evolution
,” Phys. Rev. Let.
58
, 1593
–1596
(1987
).3.
Arnol’d
, V. I.
, Mathematical Methods of Classical Mechanics
, Graduate Texts in Mathematics, 2nd ed. (Springer-Verlag
, 1989
).4.
Banyaga
, A.
, “Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique
,” Commun. Math. Helv.
53
, 174
–227
(1978
).5.
Berry
, M. V.
, “Quantal phase factors accompanying adiabatic changes
,” Proc. R. Soc. A
392
, 45
–57
(1984
).6.
Brislawn
, C.
, “Kernels of trace class operators
,” Proc. Am. Math. Soc.
104
(4
), 1181
–1190
(1988
).7.
Burdet
, G.
, Perrin
, M.
, and Perroud
, M.
, “Generating functions for the affine symplectic group
,” Commun. Math. Phys.
58
(3
), 241
–254
(1978
).8.
Cappell
, S. E.
, Lee
, R.
, and Miller
, E. Y.
, “On the Maslov index
,” Commun. Pure Appl. Math.
47
(2
), 121
–186
(1994
).9.
Combescure
, M.
and Robert
, D.
, “Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow
,” Asymptotic Anal.
14
(4
), 377
–404
(1997
).10.
Conley
, C.
and Zehnder
, E.
, “Morse-type index theory for flows and periodic solutions of Hamiltonian equations
,” Commun. Pure Appl. Math.
37
, 207
–253
(1984
).11.
Cordero
, E.
, De Mari
, F.
, Nowak
, K.
, and Tabacco
, A.
, “Reproducing groups for the metaplectic representation
,” in Pseudo-Differential Operators and Related Topics
(Springer
, 2006
), Vol. 12, pp. 227
–244
.12.
de Gosson
, M.
, “Maslov indices on the metaplectic group Mp(n)
,” Ann. Inst. Fourier
40
(3
), 537
–555
(1990
).13.
de Gosson
, M.
, “On the Weyl representation of metaplectic operators
,” Lett. Math. Phys.
72
(2
), 129
–142
(2005
).14.
de Gosson
, M.
, Symplectic Geometry and Quantum Mechanics
(Springer Science & Business Media
, 2006
), Vol. 166.15.
de Gosson
, M.
, “Metaplectic representation, Conley–Zehnder index, and Weyl calculus on phase space
,” Rev. Math. Phys.
19
(10
), 1149
–1188
(2007
).16.
de Gosson
, M.
, “On the usefulness of an index due to Leray for studying the intersections of Lagrangian and symplectic paths
,” J. Math. Pures Appl.
91
, 598
(2009
).17.
de Gosson
, M.
and Luef
, F.
, “Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics
,” Phys. Rep.
484
, 131
–179
(2009
).18.
de Gosson
, M.
, Symplectic Methods in Harmonic Analysis and in Mathematical Physics
(Springer Science & Business Media
, 2011
), Vol. 7.19.
de Gosson
, M.
, “Paths of canonical transformations and their quantization
,” Rev. Math. Phys.
27
(6
), 1530003
(2015
).20.
de Gosson
, M.
, The Wigner Transform
, Series: Advanced Texts in Mathematics (World Scientific
, 2017
).21.
Deng
, Y.
and Xia
, Z.
, “Conley–Zehnder index and bifurcation of fixed points of Hamiltonian maps
,” Ergodic Theory Dyn. Syst.
1
–22
(2017
).22.
Dittrich
, W.
and Reuter
, M.
, Classical and Quantum Dynamics
, 2nd Corrected and Enlarged Edition (Springer
, 1996
).23.
Du
, J.
and Wong
, M. W.
, “A trace formula for Weyl transforms
,” Approx. Theory Appl.
16
(1
), 41
–45
(1999
).24.
Dutta
, B.
, Mukunda
, N.
, and Simon
, R.
, “The real symplectic groups in quantum mechanics and optics
,” Pramana
45
(6
), 471
–497
(1995
).25.
Estrada
, R.
, Gracia-Bondia
, J. M.
, and Várilly
, J.
, “On asymptotic expansions of twisted products
,” J. Math. Phys.
30
(12
), 2789
–2796
(1989
).26.
Faulhuber
, M.
, de Gosson
, M. A.
, and Rottensteiner
, D.
, “Gaussian distributions and phase space Weyl–Heisenberg frames
,” e-print arXiv:1708.01551v1 [math-ph].27.
Folland
, G. B.
, Harmonic Analysis in Phase Space
, Annals of Mathematics studies (Princeton University Press
, Princeton, NJ
, 1989
).28.
Grossmann
, A.
, “Parity operators and quantization of δ-functions
,” Commun. Math. Phys.
48
, 191
–193
(1976
).29.
Hannay
, J. H.
, “Angle variable holonomy in adiabatic excursion of an integrable Hamiltonian
,” J. Phys. A: Math. Gen.
18
, 221
–230
(1985
).30.
Hofer
, H.
, Wysocki
, K.
, and Zehnder
, E.
, “Properties of pseudoholomorphic curves in symplectizations II: Embedding controls and algebraic invariants
,” Geom. Funct. Anal.
5
(2
), 270
–328
(1995
).31.
Leray
, J.
, Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index
(MIT Press
, Cambridge, Massachusetts
, 1981
), translated from Analyse Lagrangienne RCP 25, Strasbourg Collège de France (1976–1977).32.
Littlejohn
, R. G.
, “The semiclassical evolution of wave packets
,” Phys. Rep.
138
(4-5
), 193
–291
(1986
).33.
Mehlig
, B.
and Wilkinson
, M.
, “Semiclassical trace formulae using coherent states
,” Ann. Phys.
10
, 541
–559
(2001
).34.
Mehta
, C. L.
, “Diagonal coherent-state representation of quantum operators
,” Phys. Rev. Lett.
18
, 752
–754
(1967
).35.
Meinrenken
, E.
, “Semiclassical principal symbols and Gutzwiller’s trace formula
,” Rep. Math. Phys.
31
, 279
–295
(1992
).36.
Meinrenken
, E.
, “Trace formulas and the Conley–Zehnder index
,” J. Geom. Phys.
13
, 1
–15
(1994
).37.
Mukunda
, N.
and Simon
, R.
, “Quantum kinematic approach to the geometric phase
,” Ann. Phys.
228
, 205
–268
(1993
).38.
Nicacio
, F.
, Maia
, R. N. P.
, Vallejos
, R. O.
and Toscano
, F.
, “Phase space structure of generalized Gaussian cat states
,” Phys. Lett. A
374
, 4385
–4392
(2010
).39.
Nicacio
, F.
, Valdés-Hernández
, A.
, Majtey
, A. P.
, and Toscano
, F.
, “Unified framework to determine Gaussian states in continuous-variable systems
,” Phys. Rev. A
96
, 042341
(2017
); e-print arXiv:1707.01966 [quant-ph].40.
Pancharatnam
, S.
, “Generalized theory of interference, and its applications
,” Proc. Indian Acad. Sci., Sect. A
44
(5
), 247
(1956
).41.
Royer
, A.
, “Wigner functions as the expectation value of a parity operator
,” Phys. Rev. A
15
, 449
–450
(1977
).42.
Shubin
, M. A.
, Pseudodifferential Operators and Spectral Theory
(Springer-Verlag
, 1987
) [original Russian edition in Nauka, Moskva, 1978].43.
44.
Sjöqvist
, E.
, Pati
, A. K.
, Ekert
, A.
, Anandan
, J. S.
, Ericsson
, M.
, Oi
, D. K. L.
, and Vedral
, V.
, “Geometric phases for mixed states in interferometry
,” Phys. Rev. Lett.
85
(14
), 2845
–2849
(2000
).45.
Sun
, S.
, “Gutzwiller’s semiclassical trace formula and Maslov-type index for symplectic paths
,” J. Fixed Point Theory Appl.
19
, 299
–343
(2017
).46.
Wang
, D.
, “Some aspects of Hamiltonian systems and symplectic algorithms
,” Phys. D
73
, 1
–16
(1994
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
