Abstract
The fluctuation-dissipation theorem (FDT) is a central result in statistical physics, both for classical and quantum systems. It establishes a relationship between the linear response of a system under a time-dependent perturbation and time correlations of certain observables in equilibrium. Here we derive a generalization of the theorem which can be applied to any Markov quantum system and makes use of the symmetric logarithmic derivative (SLD). There are several important benefits from our approach. First, such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. Second, and more important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. Third, it clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state.
Highlights
The first version of the fluctuation-dissipation theorem (FDT) was derived by Callen and Welton [1] and subsequently generalized by Kubo [2, 3] in the context of linear response theory
Such a formulation clarifies the relation between classical and quantum versions of the equilibrium FDT. More important, it facilitates the extension of the FDT to arbitrary quantum Markovian evolution, as given by quantum maps. It clarifies the connection between the FDT and quantum metrology in systems with a non-equilibrium steady state
The key point in our derivation is the use of the symmetric logarithmic derivative (SLD), Λλ, of a density matrix ρλ depending on a real parameter λ, defined as:
Summary
A relevant particularization of the static fluctuation dissipation relation Eq (4) is the case of a quantum system with Hamiltonian H0 − λA at thermal equilibrium. In such a case the density matrix is ρλ = e−β(H0−λA)/Z(λ), where β = 1/(kBT ) is the inverse temperature and Z(λ) ≡ Tr e−β(H0−λA) is the partition function of the system. The equilibrium static susceptibility of an arbitrary observable B under the perturbation λA is denoted as χsB and obeys Eq (4). When the SLD is expressed in the eigenbasis of the unperturbed Hamiltonian H0 |n = En |n , one can obtain some interesting relationships for the equilibrium static susceptibility.
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