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.
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.
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.
Johnson noise is a small random voltage that appears between terminals of any resistor interacting with its thermal bath at temperature T. It looks like continuous, but the discreteness of the electrical charge suggests its discrete origin coming from the charge noise due to random translocations of individual electrons between terminals. The capacitance allowing these translocations would quantize the energy entering the resistor in this way, thus acting as the antenna of the resistor to pick up thermal energy in the form of charge unbalances (fluctuations of energy) between its terminals. The subsequent relaxations of these fluctuations by the conductance G=1/R of the resistor (the collective reaction of all its carriers) would give rise to its Johnson noise. This collective reaction to dissipate fluctuations of energy caused by individual electrons, agrees with the Fluctuation-Dissipation framework that Callen and Welton proposed in 1951 for noisy processes.
In this talk, we are briefly surveying our recent results [1-3] about two very popular yet often misunderstood concepts in physical informatics: (i) The existence of Johnson noise at near to absolute zero temperature has been debated many times yet it is generally accepted [1]. We point out the fundamental problems [2] with the particular approach the related quantum theories [4] of Fluctuation-Dissipation Theorem are using. Then we prove that the existence of zero-point noise would allow the construction of a perpetual motion machine [1]. Finally, we cite early works pointing out that the observed “zeropoint” noise in experiments [7] with phase-sensitive linear amplifiers is an amplifier-noise [5,6] due to the uncertainty principle, and it does not exist in the resistor in an objective way, independently from the measurement [1,2]. Thus a correct derivation of the Fluctuation-Dissipation Theorem must include [1] the type of experimental setup used for the measurement. (ii) The general opinion is that information entropy and thermal entropy are interchangeable. This belief triggered Brillouin's negentropy principle of information [8], and Landauer's claimed principle [9] about energy dissipation during information erasure that has been debated on many occasions [10-19]. Here we show the newest and perhaps the simplest arguments [3] proving that the two types of entropies are apples and oranges and are not interchangeable.
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.