(p.245) C Fluctuation–dissipation theorem
(p.245) C Fluctuation–dissipation theorem
This appendix presents the derivation of the fluctuation–dissipation theorem, describing, for example, equilibrium noise in electronic circuits. However, the theorem itself is much more general, and applies to any dissipative process in a linear (or linearized) system at equilibrium. The derivation starts by defining the quantum linear response theory, the linear response coefficients (susceptibilities) and the associated noise. With these definitions the actual derivation of the theorem is rather straightforward.
C.1 Linear response theory and susceptibility
Linear response theory describes a system characterized by a time-independent Hamiltonian H 0. Observables of this system are denoted by A i and the corresponding operators by , such that
where |n S(t)〉 are the energy eigenstates of the system in the Schrödinger picture, and ρ n = exp(−ϵn/k B T) = 〈n S(t) |ρ|n S(t) 〉 are the diagonal elements of the equilibrium density matrix.
Assume that at time t = t 0 we start to apply a force f(t) that acts on the observable A j. The resultant Hamiltonian becomes
The force f(t) is assumed to act in this system as a scalar function,1 such that it commutes with the operators . The resultant change in the observable A i(t) can be conveniently characterized in the interaction picture (see Appendix A.2.1), where |n I(t)〉 = U(t,t 0)|n I(t 0)〉 = U(t,t 0)|n S(t 0)〉, because the states of the interaction and the Schrödinger pictures coincide at the initial time t 0. Here the time-evolution operator is
and the operator is the operator in the interaction picture.
(p.246) Let us consider the linear response of observable A i to force f. In this case the time-evolution operator is
As the time evolution of the states in the Schrödinger picture can be obtained from , we get for the time dependence of operator A i
Here is the (Heisenberg) time dependence in the absence of the force.
A similar procedure for higher orders in f(t) would yield commutators of the form , and so on. Now, a linear quantum system is characterized by the property that the commutators between the operators are scalars,2 and thus their further commutators vanish. For such systems we may terminate the series to first order, and eqn (C.5) is exact. Otherwise we have to require that f(t) is small such that higher orders do not contribute much.
Defining a response coefficient (or susceptibility)
allows us to represent the time dependence of observable A i by
In a stationary system χ(t, t′) = χ(t − t′). In that case eqn (C.7) is a convolution. Hence, in the Fourier transformed space it is
where A i(ω), χ ij(ω) and f(ω) are the Fourier transforms of A i(t), χ ij(t − t′) and f(t). Thus, the change in observable A i(ω) due to the applied force f(ω) is described through the susceptibility χ(ω). To be precise, only the imaginary part of χ(ω) describes a dissipative process, and the real part is the reactive part.
(p.247) C.2 Derivation of the fluctuation–dissipation theorem
Now let us consider the noise correlation function,3
This is not yet the symmetrized version introduced above, but we may symmetrize it at the end of the calculation. For simplicity, let us prove the fluctuation–dissipation relation in the case i = j, as this is what we mostly need, and then the proof is slightly simpler than in the general case. We may then drop the indices from S ij and χ ij altogether.
First, note that in the stationary case S(t) follows the symmetry . Now separate the real and imaginary parts of the correlator,
From the last equality we get . The Fourier transform of χ satisfies
We also have
The last relation follows from the fact that χ(t) is real (which in turn comes from the reality of C(t)).
This relation uses the cyclic property of the trace and a simple reordering of the exponents. Correspondingly, the Fourier transform yields the detailed balance relation
(p.248) Finally, we get for C(ω)
The proof is slightly longer if i ≠ j, but a similar theorem holds also in that case (see Exercise 6.2).
In an electrical system the admittance Y(ω) = Z −1(ω) can be defined through the response of the current to a time-dependent vector potential :4 it is the vector potential that enters the Hamiltonian rather than the electric field . Using the fact that the current density follows the vector potential via , we hence find σ(ω) = χ(ω)/(iω), or in the case of full current Y(ω) = χ(ω)/(iω). Therefore, the fluctuation–dissipation relation for current noise reads (see Fig. 6.4)
Here the prefactor 2 comes from the definition of the noise correlator. This correlator consists of two parts, symmetric and antisymmetric in the frequency. As Re[Y(ω)] = Re[Y(−ω)], these are the first and second terms in the square brackets, respectively. The symmetrized correlator of eqn (6.1) is the one usually accessed in experiments; it captures only the coth(·)-part of the above expression. However, the non-symmetrized correlator can also be viewed differently: as discussed in Sec. 6.6, the negative-frequency noise corresponds to the ‘emitted’ and the positive-frequency noise to the ‘absorbed’ fluctuations.
(1) However, it can still be an operator of some other system, and hence one should be careful when trying to commute f(t) with itself at different times.
(2) A prime example is harmonic oscillator with and .
(3) Here the operators are written in the interaction picture, but I drop the subscript I for convenience.
(4) In this case one typically chooses the gauge where the scalar potential vanishes.