This paper reassesses the old dilemma “compact vs. extended phase” in the quantum theory of the rotator and the Josephson junction and the analogy of these two systems with a particle moving in a periodic potential. This dilemma is in fact the dilemma of whether the states with the phases φ and φ + 2π are distinguishable, or not. In the past it was widely accepted that in the Josephson junction these states are distinguishable, as in the case of a particle moving in a periodic potential. This paper argues that the states with the phases φ and φ + 2π are indistinguishable as in a pendulum (a particular example of the quantum rotator). However, this does not lead to revision of the conclusions of the conventional theory predicting the transition from the superconducting to the insulating state in the small Josephson junction.
REFERENCES
1.
M. Y.
Azbel
and Per
Bak
, “Analytical results on the pe-riodically driven damped pendulum, application to sliding charge-density waves and Josephson junctions
,” Phys. Rev. B
30
, 3722
(1984
). 2.
M.
Büttiker
, Y.
Imry
, and M. Y.
Azbel
, “Quantum oscillations in one-dimensional normal-metal rings
,” Phys. Rev. A
30
, 1982
(1984
). 3.
Y.
Gefen
, Y.
Imry
, and M. Y.
Azbel
, “Quantum oscillations and the Aharonov–Bohm effect for parallel resistors
,” Phys. Rev. Lett.
52
, 129
(1984
). 4.
P. A. M.
Dirac
, Proc. Roy. Soc. (London) Ser. A
114
, 243
(1927
). 5.
D.
Judge
, “On the uncertainty relation for angle variables
,” Nuovo Cimento
31
, 332
(1964
). 6.
P.
Carruthers
and M.
Nieto
, Phase and angle variables in quantum mechanics,
Rev. Mod. Phys.
40
, 411
(1968
). 7.
S. M.
Barnett
and D.
Pegg
, “Quantum theory of rotation angles
,” Phys. Rev. A
41
, 3427
(1990
). 8.
L.
Susskind
and J.
Glogower
, “Quantum mechanical phase and time operator
,” Physica
1
, 49
(1964
). 9.
A.
Murani
, N.
Bourlet
, H.
le Sueur
, F.
Portier
, C.
Altimiras
, D.
Esteve
, H.
Grabert
, J.
Stockburger
, J.
Ankerhold
, and P.
Joyez
, “Absence of a dissipative quantum phase transition in Josephson junctions
,” Phys. Rev. X
10
, 021003
(2020
). 10.
P. J.
Hakonen
and E. B.
Sonin
, Comment on “Absence of a dissipative quantum phase transition in Josephson junctions
,” Phys. Rev. X
11
, 018001
(2021
). 11.
A.
Murani
, N.
Bourlet
, H.
le Sueur
, F.
Portier
, C.
Altimiras
, D.
Esteve
, H.
Grabert
, J.
Stockburger
, J.
Ankerhold
, and P.
Joyez
, Reply to “Comment on absence of a dissipative quantum phase transition in Josephson junctions
,” Phys. Rev. X
11
, 018002
(2021
). 12.
A.
Schmid
, “Diffusion and localization in a dissipative quantum system
,” Phys. Rev. Lett.
51
, 1506
(1983
). 13.
S. A.
Bulgadaev
, “Phase diagram of a dissipative quantum system
,” JETP Lett.
39
, 315
(1984
).14.
E. U.
Condon
, “The physical pendulum in quantum mechanics
,” Phys. Rev.
31
, 891
(1928
). 15.
M. P.
Silverman
, “Exact spectrum of the two-dimensional rigid rotator in external fields. I. stark effect
,” Phys. Rev. A
24
, 339
(1981
). 16.
M.
Bütiker
, Y.
Imry
, and R.
Landauer
, “Josephson behavior in small normal one-dimensional rings
,” Phys. Lett. A
96
, 365
(1983
). 17.
H.-F.
Cheung
, Y.
Gefen
, E. K.
Riedel
, and W.-H.
Shih
, “Persistent currents in small one-dimensional metal rings
,” Phys. Rev. B
37
, 6050
(1988
). 18.
D.
Loss
and K.
Mullen
, “Effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions
,” Phys. Rev. A
43
, 2129
(1991
). 19.
20.
A.
Davidson
and P.
Santhanam
, “Quantum rotors, phase-slip centers, and the Coulomb blockade
,” Phys. Lett. A
149
, 476
(1990
). 21.
E. M.
Lifshitz
and L. P.
Pitaevskii
, Statistical Physics, Part 2
, (Pergamon Press
, 1980
).22.
J.
Ziman
, Principles of the Theory of Solids
, 2nd ed., (Cambridge University Press
, 1972
). 23.
A.
Widom
, G.
Megaloudis
, T. D.
Clark
, J. E.
Mutton
, R. J.
Prance
, and H.
Prance
, The Josephson pendulum as a nonlinear capacitor
, J. Low Temp. Phys.
57
, 651
(1984
). 24.
D.
Averin
, A.
Zorin
, and K.
Likharev
, “Bloch oscillations in small Josephson junctions
,” Zh. Eksp. Teor. Fiz.
61
, 407
(1985
). [Sov. Phys.–JETP 88, 692 (1985)].25.
K. K.
Likharev
and A. B.
Zorin
, Theory of the Bloch-wave oscillations in small Josephson junctions
, J. Low Temp. Phys.
59
, 347
(1985
). 26.
G.
Schön
and A.
Zaikin
, “Quantum coherent effects, phase transitions, and the dissipative dynamics of ultra-small tunnel junctions
,” Phys. Rep.
198
, 237
(1990
). 27.
J. S.
Penttilä
, U.
Parts
, P. J.
Hakonen
, M. A.
Paalanen
, and E. B.
Sonin
, “Superconductor-insulator transition” in a single Josephson junction
,” Phys. Rev. Lett.
82
, 1004
(1999
). 28.
J. S.
Penttilä
, P. J.
Hakonen
, E. B.
Sonin
, and M. A.
Paalanen
, Experiments on dissipative dynamics of single Josephson junctions
, J. Low Temp. Phys.
125
, 89
(2001
). 29.
A.
Caldeira
and A.
Leggett
, “Quantum tunneling in a dissipative system
,” Ann. Phys. (N.Y.)
149
, 374
(1983
). 30.
31.
32.
C. P.
Herrero
and A. D.
Zaikin
, “Superconductor-insulator quantum phase transition in a single Josephson junction
,” Phys. Rev. B
65
, 104516
(2002
). 33.
D.
Rogovin
and J.
Nagel
, “Quantum theory of the dc Josephson effect: Static tunneling characteristics of ultra-small josephson junctions
,” Phys. Rev. B
26
, 3698
(1982
). 34.
K.
Mullen
, D.
Loss
, and H. T. C.
Stoof
, “Resonant phenomena in compact and extended systems
,” Phys. Rev. B
47
, 2689
(1993
). 35.
E.
Ben-Jacob
and Y.
Gefen
, “New quantum oscillations in current driven small junctions
,” Phys. Lett. A
108
, 289
(1985
). 36.
W.
Zwerger
, A. T.
Dorsey
, and M. P. A.
Fisher
, “Effects of the phase periodicity on the quantum dynamics of a resistively shunted Josephson junction
,” Phys. Rev. B
34
, 6518
(1986
). 37.
S.
Apenko
, “Environment-induced decompactification of phase in Josephson junctions
,” Phys. Lett. A
142
, 277
(1989
). 38.
T.
Morel
and C.
Mora
, “Double-periodic Josephson junctions in a quantum dissipative environment
,” Phys. Rev. B
104
, 245417
(2021
). 39.
A. D.
Zaikin
and D. S.
Golubev
, Dissipative Quantum Mechanics of Nanostructures. Electron Transport, Fluctuations, and Interactions
, (Taylor & Francis, 2019
).40.
Y. C.
Chen
, M. P. A.
Fisher
, and A. J.
leggett
, The return of a hysteretic Josephson junction to the zero-voltage state: I–V characteristic and quantum retrapping
, J. Appl. Phys.
64
, 3119
(1988
). 41.
D. V.
Averin
and C.
Bruder
, “Indistinguishability of quantum states and rotation counting
,” Phys. Rev. B
, 98
(2018
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
