Skip to main content

Advertisement

Log in

Quantum Fluctuation Relations for the Lindblad Master Equation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Accardi, L.: On the quantum Feynman-Kac formula. Milan J. Math. 48(1), 135–180 (1978)

    MathSciNet  MATH  Google Scholar 

  2. Agarwal, G.S.: Open quantum Markovian systems and the microreversibility, Z. Phys. 258, 409 (1972)

    ADS  Google Scholar 

  3. Alicki, R.: On the detailed balance condition for non-Hamiltonian systems. Rep. Math. Phys. 10, 2249 (1976)

    Article  MathSciNet  Google Scholar 

  4. Alicki, R.: The quantum open system as a model of the heat engine. J. Phys. A 12, 5 (1979)

    Article  MathSciNet  Google Scholar 

  5. Alicki, R., Lendi, K.: Quantum Dynamical Semigroups and Applications. Lecture Notes Phys., vol. 717. Springer, Berlin (2007)

    MATH  Google Scholar 

  6. Allahverdyan, A.E., Nieuwenhuizen, T.M.: Fluctuations of works from quantum sub-ensembles: the case against quantum work-fluctuation theorems. Phys. Rev. E 71, 066102 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  7. Attal, S., Gawȩdzki, K.: Private communication

  8. Attal, S., Joye, A., Pillet, C.A.: Quantum Open Systems. Vol. II: The Markovian Approach. Lecture Notes in Mathematics, vol. 1881. Springer, Berlin (2006)

    Google Scholar 

  9. Andrieux, D., Gaspard, P.: Quantum work relations and response theory. Phys. Rev. Lett. 100, 230404 (2008)

    Article  ADS  Google Scholar 

  10. Andrieux, D., Gaspard, P., Monnai, T., Tasaki, S.: Fluctuation theorem for currents in open quantum systems. New J. Phys. 11, 043014 (2009)

    Article  ADS  Google Scholar 

  11. Bauer, M., Bernard, D.: Quantum stochastic processes: a case study. J. Stat. Mech. P04016 (2011)

  12. Bochkov, G.N., Kuzovlev, Y.E.: On general theory of thermal fluctuations in non linear systems. Sov. Phys. JETP 45, 125–130 (1977)

    ADS  Google Scholar 

  13. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, London (2002)

    MATH  Google Scholar 

  14. Caldeira, A.O., Leggett, A.J.: Influence of damping on quantum interference: an exactly soluble model. Phys. Rev. A 31, 1057 (1985)

    Article  ADS  Google Scholar 

  15. Callen, H.B., Welton, T.A.: Irreversibility and generalized noise. Phys. Rev. 83, 34 (1951)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Callens, I., De Roeck, W., Jacobs, T., Maes, C., Netocny, K.: Quantum entropy production as a measure of irreversibility. Physica D 187(1–4), 11 (2002).

    Google Scholar 

  17. Campisi, M., Talkner, P., Hänggi, P.: Fluctuation theorem for arbitrary open quantum systems. Phys. Rev. Lett. 102, 210401 (2009)

    Article  ADS  Google Scholar 

  18. Campisi, M., Talkner, P., Hänggi, P.: Quantum Bochkov-Kuzovlev work fluctuation theorems. Philos. Trans. R. Soc. Lond. A 369, 291 (2011)

    Article  ADS  MATH  Google Scholar 

  19. Campisi, M., Talkner, P., Hänggi, P.: Quantum fluctuation relations: foundations and applications. Rev. Mod. Phys. 83, 771 (2011)

    Article  ADS  Google Scholar 

  20. Chernyak, V., Mukamel, S.: Effect of quantum collapse on the distribution of work in driven single molecules. Phys. Rev. Lett. 93, 048302 (2004)

    Article  ADS  Google Scholar 

  21. Chetrite, R., Gawȩdzki, K.: Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Chetrite, R., Falkovich, G., Gawȩdzki, K.: Fluctuation relations in simple examples of non-equilibrium steady states. J. Stat. Mech. P08005 (2008)

  23. Crooks, G.E.: Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 1481 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  24. Crooks, G.E.: Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721 (1999)

    Article  ADS  Google Scholar 

  25. Crooks, G.E.: On the quantum Jarzynski identity. J. Stat. Mech. P10023 (2008)

  26. Crooks, G.E.: Quantum operation time reversal. Phys. Rev. A 77, 034101 (2008)

    Article  ADS  Google Scholar 

  27. Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons and Atoms. Wiley, New York (1992)

    Google Scholar 

  28. Davies, E.B.: Quantum Theory of Open Systems. Academic Press, San Diego (1976)

    MATH  Google Scholar 

  29. Deffner, S., Lutz, E.: Nonequilibrium entropy production for open quantum systems. Phys. Rev. Lett. 107, 140404 (2011)

    Article  ADS  Google Scholar 

  30. DelMoral, P.: Feynman-Kac Formulae Genealogical and Interacting Particle Systems with Applications. Probability and Applications. Springer, New York (2004)

    Google Scholar 

  31. Derezinski, J., De Roeck, W.: Extended weak coupling limit for Pauli-Fierz operators. Commun. Math. Phys. 279, 1–30 (2008)

    Article  ADS  MATH  Google Scholar 

  32. Derezinski, J., De Roeck, W., Maes, C.: Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. 131, 341 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Derrida, B.: Non-Equilibrium steady states: fluctuations and large deviations of the density and the current. J. Stat. Mech. P07023 (2007)

  34. De Roeck, W., Maes, C.: Quantum version of free-energy—irreversible-work relations. Phys. Rev. E 69, 026115 (2004)

    Article  ADS  Google Scholar 

  35. De Roeck, W., Maes, C.: Fluctuations of the dissipated heat for a quantum stochastic model. Rev. Math. Phys. 18, 619 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. De, W.: Roeck quantum fluctuation theorem: can we go from micro to meso. C. R. Phys. 8, 674 (2007)

    Article  ADS  Google Scholar 

  37. Douarche, F., Ciliberto, S., Petrosyan, A., Rabbiosi, I.: An experimental test of the Jarzynski equality in a mechanical experiment. Europhys. Lett. 70, 593 (2005)

    Article  ADS  Google Scholar 

  38. Dumcke, R., Spohn, H.: The proper form of the generator in the weak coupling limit. Z. Phys. B 34, 419–422 (1979)

    Article  ADS  Google Scholar 

  39. Dumcke, R.: The low density limit for an N-level system interacting with a free Bose or Fermi gas. Commun. Math. Phys. 97, 331 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  40. Eisler, V.: Crossover between ballistic and diffusive transport: the quantum exclusion process. J. Stat. Mech. P06007 (2011)

  41. Engel, A., Nolte, R.: Jarzynski equation for a simple quantum system: comparing two definitions of work. Europhys. Lett. 79, 10003 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  42. Esposito, M., Mukamel, S.: Fluctuation theorems for quantum master equations. Phys. Rev. E 73, 046129 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  43. Esposito, M., Harbola, U., Mukamel, S.: Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401 (1993)

    Article  ADS  MATH  Google Scholar 

  45. Fagnola, F., Umanità, V.: Generators of detailed balance quantum Markov semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, 335 (2007). arXiv:0707.2147v2 [math-ph]

    Article  MathSciNet  MATH  Google Scholar 

  46. Fagnola, F., Umanità, V.: Detailed balance, time reversal and generators of quantum Markov semigroups. Math. Notes (Mat. Zametki) 84, 108 (2008)

    Article  MATH  Google Scholar 

  47. Fonseca Romero, K.M., Talkner, P., Hänggi, P.: Is the dynamics of open quantum systems always linear? Phys. Rev. A 69, 052109 (2004)

    Article  ADS  Google Scholar 

  48. Ford, G.W., Connell, R.F.: There is no quantum regression theorem. Phys. Rev. Lett. 77, 5 (1996)

    Google Scholar 

  49. Frigerio, A., Gorini, V., Kossakowski, A., Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys. 57(2), 97–110 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Frigerio, A., Gorini, V.: Markov dilations and quantum detailed balance. Commun. Math. Phys. 93(4), 517–532 (1984)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694 (1995)

    Article  ADS  Google Scholar 

  52. Gardiner, C.W., Zoller, P.: Quantum Noise. Springer, Berlin (2000)

    MATH  Google Scholar 

  53. Gorini, V., Kossakowski, A.: N-level system in contact with a singular reservoir. J. Math. Phys. 17, 7 (1976).

    Google Scholar 

  54. Hänggi, P., Thomas, H.: Stochastic processes: time evolution, symmetries and linear response. Phys. Rep. 88, 207 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  55. Haroche, S., Raymond, J.-M.: Exploring the Quantum. Oxford University Press, London (2006)

    Book  MATH  Google Scholar 

  56. Hatano, T., Sasa, S.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463–3466 (2001)

    Article  ADS  Google Scholar 

  57. Hepp, K., Lieb, H.: Phase transition in reservoir driven open systems with applications to lasers and superconductors. Helv. Phys. Acta 46, 573–602 (1973)

    Google Scholar 

  58. Horowitz, J.M.: Quantum trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator. Phys. Rev. E 85, 031110 (2012)

    Article  ADS  Google Scholar 

  59. Huber, G., Schmidt-Kaler, F., Deffner, S., Lutz, E.: Employing trapped cold ions to verify the quantum Jarzynski equality. Phys. Rev. Lett. 101, 070403 (2008)

    Article  ADS  Google Scholar 

  60. Hummer, G., Szabo, A.: Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl. Acad. Sci. USA 98, 3658 (2001)

    Article  ADS  Google Scholar 

  61. Komnik, A., Saleur, H.: Quantum fluctuation theorem in an interacting setup: point contacts in fractional quantum hall edge state devices. Phys. Rev. Lett. 107, 100601 (2011)

    Article  ADS  Google Scholar 

  62. Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997)

    Article  ADS  Google Scholar 

  63. Jarzynski, C.: Equilibrium free-energy differences from nonequilibrium measurements: a master-equation approach. Phys. Rev. E 56, 5018 (1997)

    Article  ADS  Google Scholar 

  64. Jarzynski, C., Wójcik, D.K.: Classical and quantum fluctuation theorems for heat exchange. Phys. Rev. Lett. 92, 230602 (2004)

    Article  ADS  Google Scholar 

  65. Klich, I.: Full counting statistics, an elementary derivation of Levitov’s formula. In: Nazarov, Yu.V., Blanter, Y.M., (eds.) Quantum Noise. Kluwer, Dordrecht (2003)

    Google Scholar 

  66. Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966)

    Article  ADS  Google Scholar 

  67. Kubo, R., Toda, M., Hashitsume, N.: Statistical Physics II: Nonequilibrium Statistical Physics. Springer, Berlin (1998)

    MATH  Google Scholar 

  68. Kurchan, J.: A quantum fluctuation theorem. arXiv:cond-mat/0007360 (2000)

  69. Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A, Math. Gen. 31, 3719 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  70. Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  71. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  72. Lindblad, G.: On the existence of quantum subdynamics. J. Phys. A 29, 4197–4207 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. Lindblad, G.: On the existence of quantum subdynamics. J. Math. Phys. 39, 5 (1998)

    Article  MathSciNet  Google Scholar 

  74. Liphardt, J., Dumont, S., Smith, S.B., Tinoco, I., Bustamante, C.: Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality. Science 7(296), 5574 (2002)

    Google Scholar 

  75. Liu, F.: A derivation of quantum Jarzynski equality using quantum Feyman-Kac formula. arXiv:1201.1557 (2012)

  76. Mallick, K.: Some exact results for the exclusion process. J. Stat. Mech. P01024 (2011)

  77. Martin, P.C., Schwinger, J.: Theory of many-particle systems. I. Phys. Rev. 115, 1342 (1959)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  78. Majewski, W.A.: The detailed balance condition in quantum statistical mechanics. J. Math. Phys. 25, 614 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  79. Monnai, T.: Unified treatment of the quantum fluctuation theorem and the Jarzynski equality in terms of microscopic reversibility. Phys. Rev. E 72, 027102 (2005)

    Article  ADS  Google Scholar 

  80. Putz, W., Woronowicz, S.L.: Passive states and KMS states for general quantum systems. Commun. Math. Phys. 58, 273–290 (1978)

    Article  ADS  Google Scholar 

  81. Mukamel, S.: Quantum extension of the Jarzynski relation: analogy with stochastic dephasing. Phys. Rev. Lett. 90, 170604 (2003)

    Article  ADS  Google Scholar 

  82. Negele, J.W., Orland, H.: Quantum Many-Particle Systems. Westview, Boulder (1988)

    MATH  Google Scholar 

  83. Pechukas, P.: Reduced dynamics need not be completely positive. Phys. Rev. Lett. 73, 8 (1994)

    MathSciNet  Google Scholar 

  84. Prost, J., Joanny, J.-F., Parrondo, J.M.R.: Generalized fluctuation-dissipation theorem for steady-state systems. Phys. Rev. Lett. 103, 090601 (2009)

    Article  ADS  Google Scholar 

  85. Saito, K., Dhar, A.: Fluctuation theorem in quantum heat conduction. Phys. Rev. Lett. 99, 180601 (2007)

    Article  ADS  Google Scholar 

  86. Saito, K., Utsumi, Y.: Symmetry in full counting statistics, fluctuation theorem, and relations among nonlinear transport coefficients in the presence of a magnetic field. Phys. Rev. B 78, 115429 (2008)

    Article  ADS  Google Scholar 

  87. Spohn, H.: Entropy production for quantum dynamical semigroups. J. Math. Phys. 19, 5 (1978)

    Article  MathSciNet  Google Scholar 

  88. Spohn, H., Lebowitz, J.L.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109 (1978)

    Article  Google Scholar 

  89. Stratonovich, R.L.: Nonlinear Nonequilibrium Thermodynamics II. Springer, Berlin (1994)

    MATH  Google Scholar 

  90. Schiff, L.I.: Quantum Mechanics. McGraw-Hill, New York (1968)

    Google Scholar 

  91. Talkner, P.: The failure of the quantum regression hypothesis. Ann. Phys. 167, 390–436 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  92. Talkner, P., Lutz, E., Hänggi, P.: Fluctuation theorem: work is not an observable. Phys. Rev. E 75, 050102(R) (2007)

    Article  ADS  Google Scholar 

  93. Talkner, P., Hänggi, P.: The Tasaki-Crooks quantum fluctuation theorem. J. Phys. A 40, F569 (2007)

    Article  ADS  MATH  Google Scholar 

  94. Talkner, P., Campisi, M., Hänggi, P.: Fluctuation theorems in driven open quantum systems. J. Stat. Mech. P02025 (2009)

  95. Tasaki, H.: Jarzynski relations for quantum systems and some applications. arXiv:cond-mat/0009244 (2000)

  96. Temme K, K., Wolf, M.M., Verstraete, F.: Stochastic exclusion processes versus coherent transport. arXiv:0912.0858

  97. Weidlich, W.: Fluctuation-dissipation theorem for a class of stationary open systems. Z. Phys. 248, 234 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  98. Yukawa, S.: A quantum analogue of the Jarzynski equality. J. Phys. Soc. Jpn. 69, 2367 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Acknowledgements

R.C. thanks K. Gawȩdzki for pointing out the fact that the Lindbladian character of Eq. (43) is non-trivial and the relation with detailed balance. R.C. acknowledges the support of the Koshland center for basic research. K.M. thanks M. Bauer and H. Orland for useful comments and S. Mallick for useful remarks on the manuscript. Results similar to those presented here were also reached independently by K. Gawȩdzki and S. Attal some time ago [7].

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Chetrite.

Appendices

Appendix A: Proof of Eq. (30)

The identity (30) is proved by using the differential equation technique. First, we note that both sides of Eq. (30) coincide at t=0. Then, we find that the time derivative of the l.h.s. is given by

$$\frac{d}{dt} \biggl\{ \overrightarrow{\exp} \biggl( -\int _{0}^{t} W_{u}(u)^{{\mathcal{H}}} \, du \biggr) \biggr\} = \overrightarrow{\exp} \biggl( -\int_{0}^{t} W_{u}(u)^{{\mathcal{H}}} \, du \biggr) \bigl( - W_{t}(t)^{{\mathcal{H}}} \bigr) . $$

This follows from the very definition of a time-ordered exponential.

The time derivative of the r.h.s. is given by

(A.1)

where we have used the fact that π t commutes with H t . The last equality follows from the definition (10) of W t . We have thus shown that the l.h.s. and the r.h.s coincide at t=0 and that they satisfy the same first order differential with respect to time: they are therefore identical for all times.

Appendix B: Proof of the Callen-Welton-Kubo Formula Using the Steady-State Quantum Fluctuation Dissipation Theorem (39)

In this appendix, we show that the steady-state quantum Fluctuation Dissipation Theorem (39) is equivalent, in the case of a closed system perturbed near equilibrium, to the Callen-Welton-Kubo relation [15, 67].

We start with a time-independent Hamiltonian H 0 with invariant density-matrix given by \(\pi_{0} = Z_{0}^{-1}\exp(-\beta H_{0})\). We then consider a perturbation of H 0 of the form

$$ H_{t}= H_0- h^{a}(t) O_{a} . $$
(B.1)

In a closed system, the Lindblad equation reduces to the Liouville equation

$$ \partial_{t}\rho_{t} = L_{t}^{\dagger} \rho_{t} = - i[H_t,\rho_{t}] . $$
(B.2)

The accompanying density is explicitly given by \(\pi_{t}=Z_{t}^{-1}\exp(-\beta H_{t})\). Comparing with Eq. (35), we find M a =i[O a ,.].

We now derive an explicit expression for the operators D a defined in Eq. (38). Starting with the exact formula

$$ \exp(-\beta H_{t})= \overrightarrow{\exp} \biggl( h_{t}^{a}\int_{0}^{\beta}d \alpha\exp (-\alpha H)O_{a}\exp(\alpha H) \biggr) \, \exp(-\beta H) $$
(B.3)

(which can be proved by differentiating both sides w.r.t. β), we find at first order [82]

(B.4)

This implies that

(B.5)
(B.6)

The first order perturbation of W t is then given by

Comparing with Eq. (38), we obtain the analytical expression for D a :

$$ D_{a}=-\beta \langle O_{a} \rangle_{ \pi_{0}} +\int _{0}^{\beta}d\alpha\exp (\alpha H)O_{a} \exp(-\alpha H) . $$
(B.7)

We now transform, using (8), the r.h.s. of the quantum Fluctuation Dissipation Theorem (39) as follows:

Note that in the first equality, we use the fact that π 0 is the invariant density of the unperturbed dynamics. The quantum Fluctuation Dissipation Theorem (39) can thus be rewritten as

$$ \frac{\delta \langle A(t) \rangle}{\delta h^{a}(u)}\bigg \vert _{h=0}= \bigl\langle E_{a}(u)A(t) \bigr\rangle_{ \pi_{0}} $$
(B.8)

where we have defined

$$ E_{a}=-\pi_{0}^{-1}L^{\dagger} ( \pi_{0}D_{a} ) . $$
(B.9)

From Eq. (B.7), we deduce the analytical expression of E a :

We remark that the terms on the r.h.s. can be interpreted as the analytic continuation in imaginary time (as allowed by the KMS condition [77]) of the Heisenberg representation with respect to the unperturbed Hamiltonian H. Thus, we have

$$ E_{a}=iO_{a}^{H}(-i\beta)-iO_{a}^{H}(0). $$
(B.10)

Finally, (B.8) becomes for (u<T)

(B.11)

The last equality follows from the fact that the correlation \(\langle X_{s}Y_{t} \rangle_{\pi_{0}}\) depends just on ts. This equation is the real space version of the Callen-Welton-Kubo equation. The more conventional form is obtained by performing a Fourier Transform with respect to time. The susceptibility is defined as (using causality, i.e. \(\frac{\delta \langle A(t) \rangle}{\delta h^{a}(u)}\big \vert _{h=0}=0\) if u>t)

(B.12)

where we have used Eq. (B.11) in the last equality. The symmetrized correlation can be written as

(B.13)

where we have used a KMS type identity to obtain the third equality: \(\operatorname{Tr}(\pi_{0}X^{H}(s)Y^{H}(t))=\) \(\operatorname{Tr}(\pi_{0}Y^{H}(t)X^{H}(s+i\beta))\). By using (B.12) and (B.13), we obtain

(B.14)

Note that in the expression for \(\chi_{AO_{a}}\), the operator A is considered to be the perturbation and O a the observable. We thus implicitly suppose that A=A . In particular, if we take A=O b

(B.15)

recalling that \(\chi_{AO_{a}}(-w)=\overline{\chi_{AO_{a}}(w)}\). For a closed system, an alternative proof showing that relation (31) implies Eq. (B.15) is given in [9].

Appendix C: Derivation of Eq. (51)

First, we establish the following relation, valid for two operators X and Y

$$ Tr \bigl( \pi_{0} Y P_{0}^{T}(\alpha) X \bigr) = \biggl\langle Y(0) \,\, \overrightarrow{\exp} \biggl( -\alpha\int _{0}^{T}du \, W_{u}(u) \biggr) X(T) \biggr\rangle. $$
(C.1)

The method to derive this formula is identical to the one used in Eqs. (16) to (20): we perform Dyson-Schwinger expansion \(P_{0}^{T}(\alpha)\) w.r.t. the deformation parameter α, rewrite the trace as a correlation function via Eq. (8) and the result is resummed as a time-ordered exponential. We recall that W t was defined in Eq. (10). The same equation is true for the reversed system, with α replaced by 1−α and where the reversed ‘injected-power’ operator is given by \(W_{t}^{R} = -(\pi_{t}^{R})^{-1} \partial_{t} \pi_{t}^{R} \):

$$ \operatorname{Tr} \bigl( \pi_{0} Y P_{0}^{R,T}(1-\alpha) X \bigr) = \biggl\langle Y(0) \, \overrightarrow{\exp} \biggl( -(1-\alpha) \int _{0}^{T}du \, W_{u}^R(u) \biggr) X(T) \biggr\rangle^{R} . $$
(C.2)

Replacing both sides of Eq. (50) by the relations (C.1) and (C.2) and inserting the duality identity (47) into Eq. (C.1), leads to the Fluctuation Relation (51) for an open quantum Markovian system.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chetrite, R., Mallick, K. Quantum Fluctuation Relations for the Lindblad Master Equation. J Stat Phys 148, 480–501 (2012). https://doi.org/10.1007/s10955-012-0557-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0557-z

Keywords

Navigation