Journal of Statistical Physics

, Volume 131, Issue 5, pp 969–987 | Cite as

Master Equation in Phase Space for a Uniaxial Spin System

  • Yuri P. Kalmykov
  • William T. Coffey
  • Serguey V. Titov


A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.


Spins Uniaxial spin systems Quasi-probability distributions Wigner distributions Master equation Fokker-Planck equation 


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  1. 1.
    de Groot, S.R., Suttorp, L.G.: Foundations of Electrodynamics. North-Holland, Amsterdam (1972), Chaps. VI and VII Google Scholar
  2. 2.
    Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Berlin (2001) zbMATHGoogle Scholar
  3. 3.
    Haken, H.: Laser Theory. Springer, Berlin (1984) Google Scholar
  4. 4.
    Narducci, L.M., Bowden, C.M., Bluemel, V., Carrazana, G.P.: Phase-space description of the thermal relaxation of a (2J+1)-level system. Phys. Rev. A 11(1), 280–287 (1975) CrossRefADSGoogle Scholar
  5. 5.
    Takahashi, Y., Shibata, F.: Spin coherent state representation in non-equilibrium statistical mechanics. J. Phys. Soc. Jpn. 38(3), 656–668 (1975) CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Takahashi, Y., Shibata, F.: Generalized phase space method in spin systems—spin coherent state representation. J. Stat. Phys. 14(1), 49–65 (1976) CrossRefADSGoogle Scholar
  7. 7.
    Perelomov, A.M.: Generalized Coherent States and Their Applications. Springer, Berlin (1986) zbMATHGoogle Scholar
  8. 8.
    Nashitsume, N., Shibata, F., Shingu, M.: Quantal master equation valid for any time scale. J. Stat. Phys. 17(4), 155–169 (1977) CrossRefADSGoogle Scholar
  9. 9.
    Shibata, F., Takahashi, Y., Nashitsume, N.: A generalized stochastic Liouville equation. Non-Markovian versus memoryless master equations. J. Stat. Phys. 17(4), 171–187 (1977) CrossRefADSGoogle Scholar
  10. 10.
    Shibata, F.: Theory of nonlinear spin relaxation. J. Phys. Soc. Jpn. 49(1), 15–24 (1980) CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Shibata, F., Asou, M.: Theory of nonlinear spin relaxation. II. J. Phys. Soc. Jpn. 49(4), 1234–1241 (1980) CrossRefADSGoogle Scholar
  12. 12.
    Shibata, F., Ushiyama, C.: Rigorous solution to nonlinear spin relaxation process. J. Phys. Soc. Jpn. 62(2), 381–384 (1993) CrossRefADSGoogle Scholar
  13. 13.
    Kalmykov, Y.P., Coffey, W.T., Titov, S.V.: Phase-space formulation of the nonlinear longitudinal relaxation of the magnetization in quantum spin systems. Phys. Rev. E 76(5), 051104 (2007) CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Stratonovich, R.L.: On distributions in representation space. Zh. Eksp. Teor. Fiz. 31(6), 1012–1020 (1956) [Sov. Phys. JETP 4(6), 891–898 (1957)] Google Scholar
  15. 15.
    Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40(2), 153–174 (1975) CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Scully, M.O., Wodkiewicz, K.: Spin quasi-distribution functions. Found. Phys. 24(1), 85–107 (1994) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Radcliffe, J.M.: Some properties of coherent spin states. J. Phys. A: Math. Gen. 4(3), 313–323 (1971) CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Atomic coherent states in quantum optics. Phys. Rev. A 6(6), 2211–2237 (1972) CrossRefADSGoogle Scholar
  19. 19.
    Agarwal, G.S.: Relaxation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions. Phys. Rev. A 24(6), 2889–2896 (1981) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Agarwal, G.S.: State reconstruction for a collection of two-level systems. Phys. Rev. A 57(1), 671–673 (1998) CrossRefADSGoogle Scholar
  21. 21.
    Dowling, J.P., Agarwal, G.S., Schleich, W.P.: Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms. Phys. Rev. A 49(5), 4101–4109 (1994) CrossRefADSGoogle Scholar
  22. 22.
    Brif, C., Mann, A.: Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries. Phys. Rev. A 59(2), 971–987 (1999) CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Várilly, J.C., Gracia-Bondía, J.M.: The Moyal representation for spins. Ann. Phys. (NY) 190(1), 107–148 (1989) CrossRefADSzbMATHGoogle Scholar
  24. 24.
    Klimov, A.B.: Exact evolution equations for SU(2) quasidistribution functions. J. Math. Phys. 43(5), 2202–2213 (2002) CrossRefADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932) CrossRefADSzbMATHGoogle Scholar
  26. 26.
    Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer, Berlin (1989) zbMATHGoogle Scholar
  27. 27.
    Coffey, W.T., Kalmykov, Y.P., Waldron, J.T.: The Langevin Equation, 2nd edn. World Scientific, Singapore (2004) zbMATHGoogle Scholar
  28. 28.
    García-Palacios, J.L.: Solving quantum master equations in phase space by continued-fraction methods. Europhys. Lett. 65(6), 735–741 (2004) CrossRefADSGoogle Scholar
  29. 29.
    García-Palacios, J.L., Zueco, D.: The Caldeira–Leggett quantum master equation in Wigner phase space: continued-fraction solution and application to Brownian motion in periodic potentials. J. Phys. A: Math. Gen. 37, 10735–10770 (2004) CrossRefADSzbMATHGoogle Scholar
  30. 30.
    Coffey, W.T., Kalmykov, Yu.P., Titov, S.V., Mulligan, B.P.: Semiclassical master equation in Wigner’s phase space applied to Brownian motion in a periodic potential. Phys. Rev. E 75(4), 041117 (2007) CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Coffey, W.T., Kalmykov, Yu.P., Titov, S.V.: Solution of the master equation for Wigner’s quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential. J. Chem. Phys. 127(7), 074502 (2007) CrossRefADSGoogle Scholar
  32. 32.
    Slichter, C.P.: Principles of Magnetic Resonance. Springer, Berlin (1990) and references therein Google Scholar
  33. 33.
    Blum, K.: Density Matrix: Theory and Applications. Plenum, New York (1996) Google Scholar
  34. 34.
    Zubarev, D., Morozov, V., Röpke, G.: Statistical Mechanics of Nonequilibrium Processes, vol. 1. Akademie, Berlin (2001) Google Scholar
  35. 35.
    Garanin, D.A.: Quantum thermoactivation of nanoscale magnets. Phys. Rev. E 55(3), 2569–2572 (1997) CrossRefADSGoogle Scholar
  36. 36.
    García-Palacios, J.L., Zueco, D.: Solving spin quantum master equations with matrix continued-fraction methods: application to superparamagnets. J. Phys. A: Math. Gen. 39, 13243–13284 (2006) CrossRefADSzbMATHGoogle Scholar
  37. 37.
    Zueco, D., García-Palacios, J.L.: Longitudinal relaxation and thermoactivation of quantum superparamagnets. Phys. Rev. B 73, 104448 (2006) CrossRefADSGoogle Scholar
  38. 38.
    Wangsness, R.K., Bloch, F.: The dynamical theory of nuclear induction. Phys. Rev. 89, 728–739 (1953) CrossRefADSzbMATHGoogle Scholar
  39. 39.
    Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1998) Google Scholar
  40. 40.
    Agarwal, G.S.: Brownian motion of a quantum oscillator. Phys. Rev. A 4(2), 739–747 (1971) CrossRefADSGoogle Scholar
  41. 41.
    Kalmykov, Y.P., Coffey, W.T., Titov, S.V.: Phase-space equilibrium distribution function for spins. J. Phys. A: Math. Theor. 41(10), 105302 (2008) CrossRefADSGoogle Scholar
  42. 42.
    Weiss, U.: Quantum Dissipative Systems, 2nd edn. World Scientific, Singapore (1999) zbMATHGoogle Scholar
  43. 43.
    Geva, E., Rosenman, E., Tannor, D.: On the second-order corrections to the quantum canonical equilibrium density matrix. J. Chem. Phys. 113(4), 1380–1390 (2000) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Yuri P. Kalmykov
    • 1
  • William T. Coffey
    • 2
  • Serguey V. Titov
    • 2
    • 3
  1. 1.Laboratoire de Mathématiques, Physique et SystèmesUniversité de PerpignanPerpignan CedexFrance
  2. 2.Department of Electronic and Electrical EngineeringTrinity CollegeDublinIreland
  3. 3.Institute of Radio Engineering and ElectronicsRussian Acad. Sci.FryazinoRussia

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