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Soviet Physics Uspekhi

Quantum Nyquist formula and the applicability ranges of the Callen–Welton formula

Виталий Л. Гинзбург, Лев П. Питаевский

Published Feb 28, 1987• 22 citations
Publisher

Abstract

We discuss Yu. L. Klimontovich's objections to the generally accepted derivations of the fluctuation-dissipation theorem and his proposed additional restrictions on the applicability of this theorem. We demonstrate that Yu. L. Klimontovich's arguments contradict the basic principles of statistical physics and hence cannot be correct.

DOI
10.1070/pu1987v030n02abeh002813

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We discuss Yu. L. Klimontovich's objections to the generally accepted derivations of the fluctuation-dissipation theorem and his proposed additional restrictions on the applicability of this theorem. We demonstrate that Yu. L. Klimontovich's arguments contradict the basic principles of statistical physics and hence cannot be correct.
The quantum fluctuation dissipation theorem (QFDT) in the Callen–Welton [ Phys. Rev.83 (1951) 34] form is critically revisited. We show that the role of the system eigenvalues is in general not correctly accounted for by the accepted form of the QFDT. As a consequence, a series of quantum results claimed in the literature, like the presence of zero point fluctuations, the violation of the quantum regression hypothesis, the non-white spectrum of the Langevin force, etc. emerge as a consequence of an incorrect application of the theorem. In this context the case of the single harmonic oscillator is illustrated as a typical example where the accepted form of the QFDT is proven to fail.
The fluctuation-dissipation relations (FDR) in physical systems are studied at all levels of the statistical description. The most general FDR are the relations for the fluctuations of many-body distribution functions. It is pointed out the problem of formulation of FDR is related to the problem of deriving irreversible equations based on the reversible equations of classical and quantum mechanics. The FDR are divided into two classes: 1) FDR for fluctuations with infinite correlation times (collisionless approximation), which correspond to infinitely narrow resonances, and 2) FDR for fluctuations with finite correlation times (collisional approximation). The corresponding spectral densities have finite widths, determined by the collision integrals. The fundamental questions about which different viewpoints have been published in the literature are critically analyzed: 1) the limits of applicability of the Callen-Welton formula and 2) the quantum generalization of Nyquist's formula for the intensity of a Langevin source of oscillatory systems. It is shown that the traditional form of the quantum Nyquist formula, is not well-founded and leads to unphysical consequences. A different expression, used in the literature, for the quantum Nyquist formula is examined. It is not universal, but holds in many important cases. Its region of applicability is determined by the corresponding quantum kinetic equations. The consequences of the two forms of the quantum Nyquist formula, which can be checked experimentally, are studied. The question of the formulation of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source and the corresponding diffusion coefficient in quantum systems. It arises, in particular, also in quantum electrodynamics in the calculation of the Lamb shift (Sec. 12). In this connection two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the subtraction formalism of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula. The exposition is illustrated with many specific examples.
The Callen-Welton formula (fluctuation-dissipation theorem) of voltage and current noise of a resistance are the sum of Nyquist's classical Johnson noise equations and a (quantum) zero-point term with power density spectrum proportional to frequency and independent of temperature. At zero temperature, the classical Nyquist term vanishes however the zero-point term produces non-zero noise voltage and current. We show that the claim of zero-point noise directly contradicts to the Fermi-Dirac distribution, which defines the thermodynamics of electrons according to quantum-statistical physics. As a consequence, the Johnson noise must be zero at zero temperature, which is in accordance with Nyquist's original formula. Further investigation shows that the Callen-Welton derivation has conceptual errors such as neglecting phonon scattering, disregarding the Pauli principle during calculating the transition probabilities and using bosonic (linear oscillator) energies leading to the zero-point noise artifact. Following Kleen's proposal, the possible origin of the heterodyne (Koch - van Harlingen - Clark) experimental results are also discussed in terms of Heffner theory of quantum noise of frequency/phase-sensitive linear amplifiers. Experiments that failed to see the zero-point noise term are also mentioned.
We suggest taking the fluctuation-dissipation theorem of Callen and Welton [Phys. Rev. 83, 34 (1951)] as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown [Phys. Rev. 122, 345 (1961); Nuovo Cimento 19, 154 (1961)]. It is shown that the essential physical contents of the Caldeira-Leggett model [Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983)] such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and Anderson's orthogonality theorem is also noted.
This chapter provides a summary of all results concerning the fluctuation-dissipation theorem needed to understand many aspects of noisy systems. The proof is a succinct one in density matrix language, with a review of the latter provided for those who wish to follow the proof. Callen and Welton (1951) were the first to emphasise the universal nature of the relation between fluctuations and dissipation. They provided a quantum-mechanical derivation that involved detailed sums over the eigenstates. In this chapter, density operators are introduced to deal with an ensemble of systems rather than a single system. The response function, equilibrium theorems, hermiticity and time reversal, and application of the fluctuation-dissipation theorem to a harmonic oscillator are discussed. To make some of the abstract concepts of dissipative motion become meaningful, they are illustrated by using a reservoir consisting of a continuum of harmonic oscillators.