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.
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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].
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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
This follows from the very definition of a time-ordered exponential.
The time derivative of the r.h.s. is given by
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
In a closed system, the Lindblad equation reduces to the Liouville equation
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
(which can be proved by differentiating both sides w.r.t. β), we find at first order [82]
This implies that
The first order perturbation of W t is then given by
Comparing with Eq. (38), we obtain the analytical expression for D a :
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
where we have defined
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
Finally, (B.8) becomes for (u<T)
The last equality follows from the fact that the correlation \(\langle X_{s}Y_{t} \rangle_{\pi_{0}}\) depends just on t−s. 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)
where we have used Eq. (B.11) in the last equality. The symmetrized correlation can be written as
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
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
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
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 \(W_{t}^{R} = -(\pi_{t}^{R})^{-1} \partial_{t} \pi_{t}^{R} \):
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.
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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
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DOI: https://doi.org/10.1007/s10955-012-0557-z