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Master Equation in Phase Space for a Uniaxial Spin System

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Abstract

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.

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References

  1. de Groot, S.R., Suttorp, L.G.: Foundations of Electrodynamics. North-Holland, Amsterdam (1972), Chaps. VI and VII

    Google Scholar 

  2. Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Berlin (2001)

    MATH  Google Scholar 

  3. Haken, H.: Laser Theory. Springer, Berlin (1984)

    Google Scholar 

  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)

    Article  ADS  Google Scholar 

  5. Takahashi, Y., Shibata, F.: Spin coherent state representation in non-equilibrium statistical mechanics. J. Phys. Soc. Jpn. 38(3), 656–668 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  6. Takahashi, Y., Shibata, F.: Generalized phase space method in spin systems—spin coherent state representation. J. Stat. Phys. 14(1), 49–65 (1976)

    Article  ADS  Google Scholar 

  7. Perelomov, A.M.: Generalized Coherent States and Their Applications. Springer, Berlin (1986)

    MATH  Google Scholar 

  8. Nashitsume, N., Shibata, F., Shingu, M.: Quantal master equation valid for any time scale. J. Stat. Phys. 17(4), 155–169 (1977)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  10. Shibata, F.: Theory of nonlinear spin relaxation. J. Phys. Soc. Jpn. 49(1), 15–24 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  11. Shibata, F., Asou, M.: Theory of nonlinear spin relaxation. II. J. Phys. Soc. Jpn. 49(4), 1234–1241 (1980)

    Article  ADS  Google Scholar 

  12. Shibata, F., Ushiyama, C.: Rigorous solution to nonlinear spin relaxation process. J. Phys. Soc. Jpn. 62(2), 381–384 (1993)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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. Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40(2), 153–174 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  16. Scully, M.O., Wodkiewicz, K.: Spin quasi-distribution functions. Found. Phys. 24(1), 85–107 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  17. Radcliffe, J.M.: Some properties of coherent spin states. J. Phys. A: Math. Gen. 4(3), 313–323 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  18. Arecchi, F.T., Courtens, E., Gilmore, R., Thomas, H.: Atomic coherent states in quantum optics. Phys. Rev. A 6(6), 2211–2237 (1972)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  20. Agarwal, G.S.: State reconstruction for a collection of two-level systems. Phys. Rev. A 57(1), 671–673 (1998)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  23. Várilly, J.C., Gracia-Bondía, J.M.: The Moyal representation for spins. Ann. Phys. (NY) 190(1), 107–148 (1989)

    Article  ADS  MATH  Google Scholar 

  24. Klimov, A.B.: Exact evolution equations for SU(2) quasidistribution functions. J. Math. Phys. 43(5), 2202–2213 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Article  ADS  MATH  Google Scholar 

  26. Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer, Berlin (1989)

    MATH  Google Scholar 

  27. Coffey, W.T., Kalmykov, Y.P., Waldron, J.T.: The Langevin Equation, 2nd edn. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  28. García-Palacios, J.L.: Solving quantum master equations in phase space by continued-fraction methods. Europhys. Lett. 65(6), 735–741 (2004)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  32. Slichter, C.P.: Principles of Magnetic Resonance. Springer, Berlin (1990) and references therein

    Google Scholar 

  33. Blum, K.: Density Matrix: Theory and Applications. Plenum, New York (1996)

    Google Scholar 

  34. Zubarev, D., Morozov, V., Röpke, G.: Statistical Mechanics of Nonequilibrium Processes, vol. 1. Akademie, Berlin (2001)

    Google Scholar 

  35. Garanin, D.A.: Quantum thermoactivation of nanoscale magnets. Phys. Rev. E 55(3), 2569–2572 (1997)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  37. Zueco, D., García-Palacios, J.L.: Longitudinal relaxation and thermoactivation of quantum superparamagnets. Phys. Rev. B 73, 104448 (2006)

    Article  ADS  Google Scholar 

  38. Wangsness, R.K., Bloch, F.: The dynamical theory of nuclear induction. Phys. Rev. 89, 728–739 (1953)

    Article  ADS  MATH  Google Scholar 

  39. Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1998)

    Google Scholar 

  40. Agarwal, G.S.: Brownian motion of a quantum oscillator. Phys. Rev. A 4(2), 739–747 (1971)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  42. Weiss, U.: Quantum Dissipative Systems, 2nd edn. World Scientific, Singapore (1999)

    MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52, Avenue de Paul Alduy, 66860, Perpignan Cedex, France

    Yuri P. Kalmykov

  2. Department of Electronic and Electrical Engineering, Trinity College, Dublin, 2, Ireland

    William T. Coffey & Serguey V. Titov

  3. Institute of Radio Engineering and Electronics, Russian Acad. Sci., Vvedenskii Square 1, Fryazino, 141190, Russia

    Serguey V. Titov

Authors
  1. Yuri P. Kalmykov
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  2. William T. Coffey
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  3. Serguey V. Titov
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Correspondence to Yuri P. Kalmykov.

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Kalmykov, Y.P., Coffey, W.T. & Titov, S.V. Master Equation in Phase Space for a Uniaxial Spin System. J Stat Phys 131, 969–987 (2008). https://doi.org/10.1007/s10955-008-9522-2

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  • DOI: https://doi.org/10.1007/s10955-008-9522-2

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