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Proof of no Johnson noise at zero temperature

László B. Kish, Gunnar A. Niklasson, Claes‐Göran Granqvist

Published Jun 1, 2015• 0 citations
OpenAlex

Abstract

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.

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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.

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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.

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Spin noise at an arbitrary spin temperature

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An ensemble of spins oriented along the $z$ direction exhibits nonzero fluctuation in the transverse ($x$- and $y$-) components of the spin angular momentum in accordance with the uncertainty principle. When the spins obey a spin temperature distribution, the mean square fluctuation in $S_{x}$ can be calculated by ensemble average of the expectation value of $S_{x}^{2}$ with respect to an equilibrium density matrix $ρ=e^{βS_{z}}/Z$. The fluctuation can also be calculated from the fluctuation-dissipation theorem as has been done in literature in the context of NMR spin noise. For spin 1/2 particles in the high temperature limit, appropriate for many NMR experiments, the two methods are known to produce the same, temperature-independent spin noise. We show that inclusion of the zero-point fluctuation term in the original Nyquist relation extends this correspondence to an arbitrary spin temperature for any spin $S$. This indicates that the uncertainty principle-limited spin projection noise can be viewed as a result of the zero-point fluctuation in the thermal bath coupled to the spins.

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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 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.
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.
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.
An ensemble of spins oriented along the zzz direction exhibits nonzero fluctuation in the transverse (xxx- and yyy-) components of the spin angular momentum in accordance with the uncertainty principle. When the spins obey a spin temperature distribution, the mean square fluctuation in SxS_{x}Sx​ can be calculated by ensemble average of the expectation value of Sx2S_{x}^{2}Sx2​ with respect to an equilibrium density matrix ρ=eβSz/Zρ=e^{βS_{z}}/Zρ=eβSz​/Z. The fluctuation can also be calculated from the fluctuation-dissipation theorem as has been done in literature in the context of NMR spin noise. For spin 1/2 particles in the high temperature limit, appropriate for many NMR experiments, the two methods are known to produce the same, temperature-independent spin noise. We show that inclusion of the zero-point fluctuation term in the original Nyquist relation extends this correspondence to an arbitrary spin temperature for any spin SSS. This indicates that the uncertainty principle-limited spin projection noise can be viewed as a result of the zero-point fluctuation in the thermal bath coupled to the spins.