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Quantum effective action for dissipative semiclassical dynamics

Cesare Vianello1,2,*, Andrea Bardin1,2, and Luca Salasnich1,2,3

  • *Contact author: cesare.vianello@phd.unipd.it
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Phys. Rev. B 114, 144510 – Published 22 September, 2026

DOI: https://doi.org/10.1103/m2gf-nrsn

Abstract

Using the quantum effective action in the Schwinger-Keldysh formalism, we derive quantum-thermal corrections to the semiclassical dynamics of a dissipative system described by a macroscopic degree of freedom. We discuss the connection with the Ehrenfest theorem and, within the local potential approximation, obtain the corrections in closed form for arbitrary temperature and damping, showing that as the temperature increases, they cross over from purely quantum to purely thermal. In the low-temperature and weak-damping regime, corrections are set by the zero-point energy of fluctuations evaluated at the classical underdamped frequency, closely paralleling the conservative case, which allows us to go to higher order in the derivative expansion. We apply these general results to the resistively and capacitively shunted superconducting Josephson junction and to an elongated bosonic junction, where corrections can reach the percent level under realistic conditions.

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References (107)

  1. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics. Part 2: Theory of the Condensed State, Course of Theoretical Physics Vol. 9 (Butterworth-Heinemann, New York, 1980).
  2. B. V. Svistunov, E. S. Babaev, and N. V. Prokof'ev, Superfluid States of Matter (CRC Press, Boca Raton, 2015).
  3. B. D. Josephson, Possible new effects in superconductive tunneling, Phys. Lett. 1, 251 (1962).
  4. P. W. Anderson and J. M. Rowell, Probable observation of the Josephson superconducting tunneling effect, Phys. Rev. Lett. 10, 230 (1963).
  5. A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (John Wiley & Sons, New York, 1982).
  6. A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates, Phys. Rev. Lett. 79, 4950 (1997).
  7. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential, Phys. Rev. A 55, 4318 (1997).
  8. F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, and M. Inguscio, Josephson junction arrays with Bose-Einstein condensates, Science 293, 843 (2001).
  9. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. Rev. Lett. 95, 010402 (2005).
  10. R. Gati and M. K. Oberthaler, A bosonic Josephson junction, J. Phys. B: At. Mol. Opt. Phys. 40, R61 (2007).
  11. S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, The a.c. and d.c. Josephson effects in a Bose-Einstein condensate, Nature (London) 449, 579 (2007).
  12. K. Sakmann, A. I. Streltsov, O. E. Alon, and L. S. Cederbaum, Exact quantum dynamics of a bosonic Josephson junction, Phys. Rev. Lett. 103, 220601 (2009).
  13. G. Spagnolli, G. Semeghini, L. Masi, G. Ferioli, A. Trenkwalder, S. Coop, M. Landini, L. Pezzè, G. Modugno, M. Inguscio, A. Smerzi, and M. Fattori, Crossing over from attractive to repulsive interactions in a tunneling bosonic Josephson junction, Phys. Rev. Lett. 118, 230403 (2017).
  14. V. P. Singh, J. Polo, L. Mathey, and L. Amico, Shapiro steps in driven atomic Josephson junctions, Phys. Rev. Lett. 133, 093401 (2024).
  15. E. Bernhart, M. Röhrle, V. P. Singh, L. Mathey, L. Amico, and H. Ott, Observation of Shapiro steps in an ultracold atomic Josephson junction, Science 390, 1130 (2025).
  16. A. K. Saha, L. F. C. de Brito, C. Vianello, R. Roy, R. Dubessy, B. Chakrabarti, and A. Gammal, Dynamics of one-dimensional Bose-Josephson junction in a box trap: From coherent oscillations to many-body dephasing and dynamical freezing, arXiv:2604.18165.
  17. Yu. Kagan and A. J. Leggett, Quantum Tunnelling in Condensed Media, Modern Problems in Condensed Matter Sciences Vol. 34 (North Holland, Amsterdam, 1992).
  18. A. Larkin and A. Varlamov, Theory of Fluctuations in Superconductors (Oxford University Press, Oxford, 2005).
  19. K. Furutani and L. Salasnich, Quantum and thermal fluctuations in the dynamics of a resistively and capacitively shunted Josephson junction, Phys. Rev. B 104, 014519 (2021).
  20. I. Marino, S. Raghavan, S. Fantoni, S. R. Shenoy, and A. Smerzi, Bose-condensate tunneling dynamics: Momentum-shortened pendulum with damping, Phys. Rev. A 60, 487 (1999).
  21. F. Meier and W. Zwerger, Josephson tunneling between weakly interacting Bose-Einstein condensates, Phys. Rev. A 64, 033610 (2001).
  22. R. A. Barankov and S. N. Burmistrov, Dissipative dynamics of a Josephson junction in the Bose gases, Phys. Rev. A 67, 013611 (2003).
  23. M. Pigneur, T. Berrada, M. Bonneau, T. Schumm, E. Demler, and J. Schmiedmayer, Relaxation to a phase-locked equilibrium state in a one-dimensional bosonic Josephson junction, Phys. Rev. Lett. 120, 173601 (2018).
  24. M. Pigneur and J. Schmiedmayer, Analytical pendulum model for a bosonic Josephson junction, Phys. Rev. A 98, 063632 (2018).
  25. J. Polo, V. Ahufinger, F. W. J. Hekking, and A. Minguzzi, Damping of Josephson oscillations in strongly correlated one-dimensional atomic gases, Phys. Rev. Lett. 121, 090404 (2018).
  26. D. Stefanatos and E. Paspalakis, Relaxation dynamics in a stochastic bosonic Josephson junction, Phys. Lett. A 383, 2370 (2019).
  27. A. K. Saha, D. S. Ray, and B. Deb, Parametric oscillations in a dissipative bosonic Josephson junction, J. Phys. B: At. Mol. Opt. Phys. 53, 135301 (2020).
  28. K. Xhani, L. Galantucci, C. F. Barenghi, G. Roati, A. Trombettoni, and N. P. Proukakis, Dynamical phase diagram of ultracold Josephson junctions, New J. Phys. 22, 123006 (2020).
  29. F. Binanti, K. Furutani, and L. Salasnich, Dissipation and fluctuations in elongated bosonic Josephson junctions, Phys. Rev. A 103, 063309 (2021).
  30. J.-F. Mennemann, I. E. Mazets, M. Pigneur, H. P. Stimming, N. J. Mauser, J. Schmiedmayer, and S. Erne, Relaxation in an extended bosonic Josephson junction, Phys. Rev. Res. 3, 023197 (2021).
  31. K. Xhani and N. P. Proukakis, Dissipation in a finite-temperature atomic Josephson junction, Phys. Rev. Res. 4, 033205 (2022).
  32. A. K. Saha, D. S. Ray, and B. Deb, Phase diffusion and fluctuations in a dissipative Bose-Josephson junction, Phys. Rev. E 107, 034141 (2023).
  33. A. K. Saha, Vibrational resonance in a one-dimensional dissipative Bose-Josephson junction, arXiv:2606.02157.
  34. R. H. Koch, D. J. Van Harlingen, and J. Clarke, Quantum-noise theory for the resistively shunted Josephson junction, Phys. Rev. Lett. 45, 2132 (1980).
  35. R. H. Koch, D. J. Van Harlingen, and J. Clarke, Observation of zero-point fluctuations in a resistively shunted Josephson tunnel junction, Phys. Rev. Lett. 47, 1216 (1981).
  36. R. H. Koch, D. J. Van Harlingen, and J. Clarke, Measurements of quantum noise in resistively shunted Josephson junctions, Phys. Rev. B 26, 74 (1982).
  37. J. M. Martinis, M. H. Devoret, and J. Clarke, Energy-level quantization in the zero-voltage state of a current-biased Josephson junction, Phys. Rev. Lett. 55, 1543 (1985).
  38. M. H. Devoret, J. M. Martinis, and J. Clarke, Measurements of macroscopic quantum tunneling out of the zero-voltage state of a current-biased Josephson junction, Phys. Rev. Lett. 55, 1908 (1985).
  39. J. Clarke, A. N. Cleland, M. H. Devoret, D. Esteve, and J. M. Martinis, Quantum mechanics of a macroscopic variable: The phase difference of a Josephson junction, Science 239, 992 (1988).
  40. C. Vianello, M. Ferraretto, and L. Salasnich, Finite-temperature entanglement and coherence in asymmetric bosonic Josephson junctions, Phys. Rev. A 111, 063310 (2025).
  41. C. Vianello, G. Mazzarella, and L. Salasnich, Equilibrium and dynamical quantum phase transitions in dipolar atomic Josephson junctions, Phys. Rev. A 113, 063318 (2026).
  42. K. Furutani, J. Tempere, and L. Salasnich, Quantum effective action for the bosonic Josephson junction, Phys. Rev. B 105, 134510 (2022).
  43. C. Vianello, S. Salvatore, and L. Salasnich, Quantum action of the Josephson dynamics, Int. J. Theor. Phys. 64, 315 (2025).
  44. L. Salasnich, Effective action approach to quantum and thermal effects: From one particle to Bose-Einstein condensates, Atoms 13, 95 (2025).
  45. J. Goldstone, A. Salam, and S. Weinberg, Broken symmetries, Phys. Rev. 127, 965 (1962).
  46. G. Jona-Lasinio, Relativistic field theories with symmetry-breaking solutions, Nuovo Cim. 34, 1790 (1964).
  47. S. Coleman and E. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7, 1888 (1973).
  48. S. Coleman, R. Jackiw, and H. D. Politzer, Spontaneous symmetry breaking in the O(N) model for large N, Phys. Rev. D 10, 2491 (1974).
  49. D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose-Bose mixture, Phys. Rev. Lett. 115, 155302 (2015).
  50. R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-state phase diagram of a dipolar condensate with quantum fluctuations, Phys. Rev. A 94, 033619 (2016).
  51. L. Salasnich, Self-consistent derivation of the modified Gross-Pitaevskii equation with Lee-Huang-Yang correction, Appl. Sci. 8, 1998 (2018).
  52. E. Aybar and M. O. Oktel, Temperature-dependent density profiles of dipolar droplets, Phys. Rev. A 99, 013620 (2019).
  53. N. Guebli and A. Boudjemâa, Quantum self-bound droplets in Bose-Bose mixtures: Effects of higher-order quantum and thermal fluctuations, Phys. Rev. A 104, 023310 (2021).
  54. A. Bardin, F. Lorenzi, and L. Salasnich, Quantum fluctuations in atomic Josephson junctions: The role of dimensionality, New J. Phys. 26, 013021 (2024).
  55. J. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2, 407 (1961).
  56. L. V. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP 20, 1018 (1965).
  57. H. B. Callen and T. A. Welton, Irreversibility and generalized noise, Phys. Rev. 83, 34 (1951).
  58. R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29, 255 (1966).
  59. P. S. Bauer, Dissipative dynamical systems, Proc. Natl. Acad. Sci. USA 17, 311 (1931).
  60. H. Bateman, On dissipative systems and related variational principles, Phys. Rev. 38, 815 (1931).
  61. P. Caldirola, Forze non conservative nella meccanica quantistica, Nuovo Cim. 18, 393 (1941).
  62. E. Kanai, On the quantization of the dissipative systems, Prog. Theor. Phys. 3, 440 (1948).
  63. F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E 53, 1890 (1996).
  64. F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E 55, 3581 (1997).
  65. A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, Cambridge, 2011).
  66. M. Uguccioni, Quantum transport in mesoscopic systems: A non-equilibrium field theory approach, Ph.D. thesis, Università di Padova, 2025.
  67. P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dynamics of classical systems, Phys. Rev. A 8, 423 (1973).
  68. H.-K. Janssen, On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties, Z. Phys. B 23, 377 (1976).
  69. C. De Dominicis, Techniques de renormalisation de la théorie des champs et dynamique des phénomènes critiques, J. Phys. Colloques 37, C1-247 (1976).
  70. K. Furutani and L. Salasnich, Fokker-Planck equations for a trapped particle in a quantum-thermal Ohmic bath: General theory and applications to Josephson junctions, AAPPS Bull. 33, 19 (2023).
  71. C. P. Burgess, Introduction to Effective Field Theory (Cambridge University Press, Cambridge, 2021).
  72. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (World Scientific, Singapore, 2009).
  73. R. Jackiw, Functional evaluation of the effective potential, Phys. Rev. D 9, 1686 (1974).
  74. J. Iliopoulos, C. Itzykson, and A. Martin, Functional methods and perturbation theory, Rev. Mod. Phys. 47, 165 (1975).
  75. F. Cametti, G. Jona-Lasinio, C. Presilla, and F. Toninelli, Comparison between quantum and classical dynamics in the effective action formalism, in Proceedings of the International School of Physics “Enrico Fermi”, Course CXLIII, edited by G. Casati, I. Guarnieri, and U. Smilansky (IOS Press, Amsterdam, 2000), pp. 431–448.
  76. H. Metiu and G. Schön, Description of quantum noise by a Langevin equation, Phys. Rev. Lett. 53, 13 (1984).
  77. G. W. Ford and M. Kac, On the quantum Langevin equation, J. Stat. Phys. 46, 803 (1987).
  78. T. Dittrich, P. Hänggi, G.-L. Ingold, B. Kramer, G. Schön, and W. Zwerger, Quantum Transport and Dissipation (Wiley-VCH, Berlin, 1998).
  79. P. Hänggi, Generalized Langevin equations: A useful tool for the perplexed modeller of nonequilibrium fluctuations? in Stochastic Dynamics, edited by L. Schimansky-Geier and T. Pöschel (Springer, Berlin, 1997), pp. 15–22.
  80. U. Weiss, Quantum Dissipative Systems, 5th ed. (World Scientific, Singapore, 2021).
  81. S. N. Burmistrov, Physical Kinetics (Springer, Singapore, 2022).
  82. Solving the equation E[xξ]+ξ(t)=0 while retaining the nonlinearities would amount to computing loops with Γ, thereby overcounting the fluctuations already resummed into E.
  83. With respect to the approximation λ±≃±iω0, which would correspond to the strictly undamped case and is discussed in Appendix pp1, here we are retaining the exact imaginary part of λ±, which corresponds to keeping the exact real part of the zeros of the fluctuation operator M(ω), while discarding the linewidth. Consistently, the factor ω0=λ+λ− in Eq. (31) is also replaced by ωγ.
  84. C. M. Fraser, Calculation of higher derivative terms in the one-loop effective Lagrangian, Z. Phys. C 28, 101 (1985).
  85. F. T. Brandt, J. Frenkel, and J. C. Taylor, Noise in resistively shunted Josephson junctions, Phys. Rev. B 82, 014515 (2010).
  86. R. Gati, B. Hemmerling, J. Fölling, M. Albiez, and M. K. Oberthaler, Noise thermometry with two weakly coupled Bose-Einstein condensates, Phys. Rev. Lett. 96, 130404 (2006).
  87. S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping, Nature (London) 443, 430 (2006).
  88. A. J. E. Kreil, H. Y. Musiienko-Shmarova, P. Frey, A. Pomyalov, V. S. L'vov, G. A. Melkov, A. A. Serga, and B. Hillebrands, Experimental observation of Josephson oscillations in a room-temperature Bose-Einstein magnon condensate, Phys. Rev. B 104, 144414 (2021).
  89. K. Nakata, K. A. van Hoogdalem, P. Simon, and D. Loss, Josephson and persistent spin currents in Bose-Einstein condensates of magnons, Phys. Rev. B 90, 144419 (2014).
  90. K. Nakata, J. Zou, J. Klinovaja, and D. Loss, Magnonic φ Josephson junctions and synchronized precession, Phys. Rev. Res. 6, 033207 (2024).
  91. H. Grabert and U. Weiss, Crossover from thermal hopping to quantum tunneling, Phys. Rev. Lett. 53, 1787 (1984).
  92. P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62, 251 (1990).
  93. Only n=1 is physically significant, marking the bifurcation of the bounce orbit from the barrier-top solution; the higher poles correspond to its n-fold repetitions, which are exponentially subdominant, and their presence signals that the Gaussian treatment of the unstable direction is controlled only for T>T0.
  94. A. O. Caldeira and A. J. Leggett, Influence of dissipation on quantum tunneling in macroscopic systems, Phys. Rev. Lett. 46, 211 (1981).
  95. A. O. Caldeira and A. J. Leggett, Quantum tunnelling in a dissipative system, Ann. Phys. 149, 374 (1983).
  96. S. E. Korshunov, Quantum-mechanical tunneling with dissipation in a sloping sinusoidal potential, Sov. Phys. JETP 65, 1025 (1987).
  97. M. Hongo, T. Fujimori, T. Misumi, M. Nitta, and N. Sakai, Effective field theory of magnons: Chiral magnets and the Schwinger mechanism, Phys. Rev. B 104, 134403 (2021).
  98. S. Yuan, C. Sui, Z. Fan, J. Berakdar, D. Xue, and C. Jia, Magnonic Klein and acausal tunneling enabled by breaking the anti parity-time symmetry in antiferromagnets, Commun. Phys. 6, 95 (2023).
  99. T. C. Adorno, S. P. Gavrilov, and D. M. Gitman, Schwinger mechanism of magnon-antimagnon pair production on magnetic field inhomogeneities and the bosonic Klein effect, Phys. Rev. B 110, 014410 (2024).
  100. E. M. Lifshitz, The theory of molecular attractive forces between solids, Sov. Phys. JETP 2, 73 (1956).
  101. H. Neuberger and T. Ziman, Finite-size effects in Heisenberg antiferromagnets, Phys. Rev. B 39, 2608 (1989).
  102. P. Hasenfratz and F. Niedermayer, Finite size and temperature effects in the AF Heisenberg model, Z. Phys. B 92, 91 (1993).
  103. K. Nakata and K. Suzuki, Magnonic Casimir effect in ferrimagnets, Phys. Rev. Lett. 130, 096702 (2023).
  104. K. Nakata and K. Suzuki, Non-Hermitian Casimir effect of magnons, npj Spintron. 2, 11 (2024).
  105. S. N. Burmistrov and L. B. Dubovskii, Cavitation on single electron bubbles in liquid helium at small negative pressures, Phys. Rev. B 105, 064511 (2022).
  106. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (Dover Publications, Mineola, 2010).
  107. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge, 2012).

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Phys. Rev. B 114, 144510– Published 22 September, 2026
  • Received 19 May 2026
  • Revised 25 August 2026
  • Accepted 14 September 2026
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