Advertisement

Finding Stationary Solutions of the Lindblad Equation by Analyzing the Entropy Production Functional

  • A. S. Trushechkin
Article
  • 11 Downloads

Abstract

A necessary and sufficient condition is derived for a density operator to be a stationary solution for a certain class of Lindblad equations in the theory of open quantum systems. This condition is based on the properties of a functional that in some cases corresponds to entropy production. Examples are given where this condition is used to find stationary solutions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Accardi and S. Kozyrev, “Lectures on quantum interacting particle systems,” in Quantum Interacting Particle Systems: Lecture Notes of the Volterra–CIRM Int. Sch., Trento, 2000 (World Sci., Singapore, 2002), QP–PQ: Quantum Probab. White Noise Anal. 14, pp. 1–195.CrossRefGoogle Scholar
  2. 2.
    L. Accardi and S. V. Kozyrev, “Coherent population trapping and partial decoherence in the stochastic limit,” Int. J. Theor. Phys. 45 (4), 661–678 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    L. Accardi, Y. G. Lu, and I. Volovich, Quantum Theory and Its Stochastic Limit (Springer, Berlin, 2002).CrossRefzbMATHGoogle Scholar
  4. 4.
    I. Aref’eva, “Multiplicity and thermalization time in heavy-ions collisions,” EPJ Web Conf. 125, 01007 (2016).CrossRefGoogle Scholar
  5. 5.
    I. Ya. Aref’eva and M. A. Khramtsov, “AdS/CFT prescription for angle-deficit space and winding geodesics,” J. High Energy Phys. 2016 (4), 121 (2016).MathSciNetGoogle Scholar
  6. 6.
    I. Ya. Aref’eva, M. A. Khramtsov, and M. D. Tikhanovskaya, “Thermalization after holographic bilocal quench,” J. High Energy Phys. 2017 (9), 115 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    I. Ya. Aref’eva, I. V. Volovich, and S. V. Kozyrev, “Stochastic limit method and interference in quantum manyparticle systems,” Teor. Mat. Fiz. 183 (3), 388–408 (2015) [Theor. Math. Phys. 183, 782–799 (2015)].CrossRefGoogle Scholar
  8. 8.
    F. Barra, “The thermodynamic cost of driving quantum systems by their boundaries,” Sci. Rep. 5, 14873 (2015).CrossRefGoogle Scholar
  9. 9.
    H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford Univ. Press, Oxford, 2002; Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2010).zbMATHGoogle Scholar
  10. 10.
    E. B. Davies, “Markovian master equations,” Commun. Math. Phys. 39, 91–110 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Yu. N. Drozhzhinov, “Multidimensional Tauberian theorems for generalized functions,” Usp. Mat. Nauk 71 (6), 99–154 (2016) [Russ. Math. Surv. 71, 1081–1134 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” J. Math. Phys. 17 (5), 821–825 (1976).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation,” Mat. Sb. 206 (10), 71–102 (2015) [Sb. Math. 206, 1410–1439 (2015)].CrossRefGoogle Scholar
  14. 14.
    A. K. Gushchin, “Lp-estimates for the nontangential maximal function of the solution to a second-order elliptic equation,” Mat. Sb. 207 (10), 28–55 (2016) [Sb. Math. 207, 1384–1409 (2016)].CrossRefGoogle Scholar
  15. 15.
    M. O. Katanaev, “Lorentz invariant vacuum solutions in general relativity,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 149–153 (2015) [Proc. Steklov Inst. Math. 290, 138–142 (2015)].MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe,” Usp. Fiz. Nauk 186 (7), 763–775 (2016) [Phys. Usp. 59 (7), 689–700 (2016)].CrossRefGoogle Scholar
  17. 17.
    M. O. Katanaev, “Cosmological models with homogeneous and isotropic spatial sections,” Teor. Mat. Fiz. 191 (2), 219–227 (2017) [Theor. Math. Phys. 191, 661–668 (2017)].MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys. 48, 119–130 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    I. A. Luchnikov and S. N. Filippov, “Quantum evolution in the stroboscopic limit of repeated measurements,” Phys. Rev. A 95 (2), 022113 (2017).CrossRefGoogle Scholar
  20. 20.
    N. G. Marchuk, “Demonstration representation and tensor products of Clifford algebras,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 154–165 (2015) [Proc. Steklov Inst. Math. 290, 143–154 (2015)].MathSciNetzbMATHGoogle Scholar
  21. 21.
    N. G. Marchuk and D. S. Shirokov, “General solutions of one class of field equations,” Rep. Math. Phys. 78 (3), 305–326 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. N. Pechen and N. B. Il’in, “On critical points of the objective functional for maximization of qubit observables,” Usp. Mat. Nauk 70 (4), 211–212 (2015) [Russ. Math. Surv. 70, 782–784 (2015)].CrossRefzbMATHGoogle Scholar
  23. 23.
    A. N. Pechen and N. B. Il’in, “Existence of traps in the problem of maximizing quantum observable averages for a qubit at short times,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 289, 227–234 (2015) [Proc. Steklov Inst. Math. 289, 213–220 (2015)].MathSciNetzbMATHGoogle Scholar
  24. 24.
    A. N. Pechen and N. B. Il’in, “On the problem of maximizing the transition probability in an n-level quantum system using nonselective measurements,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 248–255 (2016) [Proc. Steklov Inst. Math. 294, 233–240 (2016)].MathSciNetzbMATHGoogle Scholar
  25. 25.
    H. Spohn, “Entropy production for quantum dynamical semigroups,” J. Math. Phys. 19 (5), 1227–1230 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    H. Spohn and J. L. Lebowitz, “Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs,” Adv. Chem. Phys. 38, 109–142 (1978).Google Scholar
  27. 27.
    A. Trushechkin, “Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions,” J. Math. Phys. 58 (6), 062102 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    A. S. Trushechkin, “On general definition of entropy production in Markovian open quantum systems,” in Quantum Computations (VINITI, Moscow, 2017), Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh., Temat. Obz. 138, pp. 82–98.Google Scholar
  29. 29.
    A. S. Trushechkin and I. V. Volovich, “Perturbative treatment of inter-site couplings in the local description of open quantum networks,” Europhys. Lett. 113 (3), 30005 (2016).CrossRefGoogle Scholar
  30. 30.
    B. O. Volkov, “Lévy Laplacians and instantons,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 226–238 (2015) [Proc. Steklov Inst. Math. 290, 210–222 (2015)].zbMATHGoogle Scholar
  31. 31.
    B. O. Volkov, “Stochastic Lévy differential operators and Yang–Mills equations,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20 (2), 1750008 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    I. V. Volovich, “Cauchy–Schwarz inequality-based criteria for the non-classicality of sub-Poisson and antibunched light,” Phys. Lett. A 380 (1–2), 56–58 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    I. V. Volovich and S. V. Kozyrev, “Manipulation of states of a degenerate quantum system,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 256–267 (2016) [Proc. Steklov Inst. Math. 294, 241–251 (2016)].MathSciNetzbMATHGoogle Scholar
  34. 34.
    V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations,” Teor. Mat. Fiz. 185 (2), 227–251 (2015) [Theor. Math. Phys. 185, 1557–1581 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    V. V. Zharinov, “Bäcklund transformations,” Teor. Mat. Fiz. 189 (3), 323–334 (2016) [Theor. Math. Phys. 189, 1681–1692 (2016)].CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia
  3. 3.National University of Science and Technology MISISMoscowRussia

Personalised recommendations