The bandwidth of transistors in logic devices approaches the quantum limit, where Johnson noise and associated error rates are supposed to be strongly enhanced. However, the related theory — asserting a temperature-independent quantum zero-point (ZP) contribution to Johnson noise, which dominates the quantum regime — is controversial and resolution of the controversy is essential to determine the real error rate and fundamental energy dissipation limits of logic gates in the quantum limit. The Callen–Welton formula (fluctuation–dissipation theorem) of voltage and current noise for a resistance is the sum of Nyquist’s classical Johnson noise equation and a quantum ZP term with a power density spectrum proportional to frequency and independent of temperature. The classical Johnson–Nyquist formula vanishes at the approach of zero temperature, but the quantum ZP term still predicts non-zero noise voltage and current. Here, we show that this noise cannot be reconciled with the Fermi–Dirac distribution, which defines the thermodynamics of electrons according to quantum-statistical physics. Consequently, Johnson noise must be nil at zero temperature, and non-zero noise found for certain experimental arrangements may be a measurement artifact, such as the one mentioned in Kleen’s uncertainty relation argument.
Abstract There is a longstanding debate about the zero-point term in the Johnson noise voltage of a resistor. This term originates from a quantum-theoretical treatment of the fluctuation-dissipation theorem (FDT). Is the zero-point term really there, or is it only an experimental artifact, due to the uncertainty principle, for phase-sensitive amplifiers? Could it be removed by renormalization of theories? We discuss some historical measurement schemes that do not lead to the effect predicted by the FDT, and we analyse new features that emerge when the consequences of the zero-point term are measured via the mean energy and force in a capacitor shunting the resistor. If these measurements verify the existence of a zero-point term in the noise, then two types of perpetual motion machines can be constructed. Further investigation with the same approach shows that, in the quantum limit, the Johnson–Nyquist formula is also invalid under general conditions even though it is valid for a resistor-antenna system. Therefore we conclude that in a satisfactory quantum theory of the Johnson noise, the FDT must, as a minimum, include also the measurement system used to evaluate the observed quantities. Issues concerning the zero-point term may also have implications for phenomena in advanced nanotechnology.
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.
The effect of a constant electric field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}$ on the low-temperature Johnson noise of a metallic resistor is calculated by nonequilibrium diagrammatic perturbation theory. The expansion, in powers of
E2, is shown to diverge, and the divergence is associated with accumulating Joule heat. An insight into how to treat a steady state properly when there is heating is provided by the Boltzmann equation. A vertex function that corresponds to a steady-state solution of this equation is identified. With this vertex function it is possible to evaluate not only one-particle operators, for which the Boltzmann equation is sufficient, but also troublesome contributions to the current fluctuations and other two-body observables for which a many-body approach is required. The calculated corrections to the current correlations agree qualitatively, but not quantitatively, with simple phenomenological agruments. In particular, it is not possible to describe the leading effects of the electric field solely in terms of a variable local temperature. The model calculation illustrates and sheds some light on several problems that only appear when nonlinear deviations from thermal equilibrium must be considered: the complicated effects to which heat generation leads, the identification and treatment of these effects in perturbation theory, and the difficulties these effects pose for hypothetical generalizations of the fluctuation dissipation theorem. Some possible connections between the calculation and
f1 noise are also briefly explored. In particular, recent experiments that may confirm the conclusion of this calculation, that systems in which there is only impurity scattering do not exhibit
f1 noise, are noted.
We compute the evanescent-wave Johnson noise (EWJN) in the vacuum half-space above a superconductor, and the resulting relaxation time (
T1) of spin and charge qubits placed at nanometer distances from the surface. The electromagnetic response is described by a single microscopic transverse current-response kernel
Q(q,ω) for a BCS superconductor. This is computed for varying densities of both non-magnetic impurities and magnetic impurities, for arbitrary frequency and temperature and for wave vectors
q≪kF (the Fermi wavevector). When combined with the fluctuation-dissipation theorem and the nonlocal surface impedances of the half-space, this yields the magnetic and electric field noise at any distance
z≫kF−1 from the surface, from which we obtain
T1. Just below
Tc the magnetic noise is enhanced relative to the normal state by the coherence (Hebel-Slichter-type) peak of the dissipative conductivity and drops exponentially at lower temperatures; the electric noise shows no coherence peak. The theory predicts that there is a zero-temperature noise floor induced by magnetic impurities. In the gapless regime produced by pair breaking, the finite subgap density of states
ν(0) yields a temperature-independent noise spectral density and a relaxation rate bounded by
T1−1(T)≤[ν(0)/νF]2T1,N−1(T) for
T≪Tc, with equality in the extreme nonlocal regime. Here
ν(0) and
νF are the superconducting and normal-state densities of states at the Fermi energy, and
T1,N(T) is the relaxation time the same electrode would produce in its normal state at the same temperature.
Abstract In many quantum computer architectures, the qubits are in close proximity to metallic device elements. The fluctuating currents in the metal give rise to noisy electromagnetic fields that leak out into the surrounding region. These fields are known as evanescent-wave Johnson noise. The noise can decohere the qubits. We review and update the general theory of this effect for charge qubits subject to electric noise and for spin and magnetic qubits subject to magnetic noise. A mapping of the quantum-mechanical problem onto a problem in classical electrodynamics simplifies the calculations. The focus is on relatively simple geometries in which analytical calculations can be done. Results are presented for the local noise spectral density in the vicinity of cylindrical conductors such as small antennae, noise from objects that can be treated as dipoles, and noise correlation functions for several geometries. We summarize the current state of the comparison of theory with experimental results on decoherence times of qubits. Emphasis is placed on qualitative understanding of the basic concepts and phenomena.