Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Proceedings of the IRE

The Fundamental Noise Limit of Linear Amplifiers

H. Heffner

Published Jul 1, 1962• 167 citations
Publisher

Abstract

If the uncertainty principle of quantum mechanics is applied to the process of signal measurement, two theorems relating to amplifier noise performance can be deduced. First, it can be shown that it is impossible to construct a linear noiseless amplifier. Second, if the amplifier is characterized as having additive white Gaussian noise, it can be shown that the minimum possible noise temperature of any linear amplifier is T.= In 2 -IIG -hv 1-1/ IG k In the limit of high gain G this expression reduces to that previously derived for the ideal maser and parametric amplifier. It is shown that the minimum noise amplifier does not degrade the signal but rather allows the use of an inaccurate detector to make measurements on an incoming signal to the greatest accuracy consistent with the uncertainty principle.

DOI
10.1109/jrproc.1962.288130

Related papers

Quantum limits on noise in linear amplifiers

Carlton M. Caves

How much noise does quantum mechanics require a linear amplifier to add to a signal it processes? An analysis of narrow-band amplifiers (single-mode input and output) yields a fundamental theorem for phase-insensitive linear amplifiers; it requires such an amplifier, in the limit of high gain, to add noise which, referred to the input, is at least as large as the half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers, which can respond differently to the two quadrature phases ("$cos\ensuremath{\omega}t$" and "$sin\ensuremath{\omega}t$"), the single-mode analysis yields an amplifier uncertainty principle---a lower limit on the product of the noises added to the two phases. A multimode treatment of linear amplifiers generalizes the single-mode analysis to amplifiers with nonzero bandwidth. The results for phase-insensitive amplifiers remain the same, but for phase-sensitive amplifiers there emerge bandwidth-dependent corrections to the single-mode results. Specifically, there is a bandwidth-dependent lower limit on the noise carried by one quadrature phase of a signal and a corresponding lower limit on the noise a high-gain linear amplifier must add to one quadrature phase. Particular attention is focused on developing a multimode description of signals with unequal noise in the two quadrature phases.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 15, 1982•1,723 citations

Quantum Noise in Linear Amplifiers

H. A. Haus, James A. Mullen

The classical definition of noise figure, based on signal-to-noise ratio, is adapted to the case when quantum noise is predominant. The noise figure is normalized to "uncertainty noise." General quantum mechanical equations for linear amplifiers are set up using the condition of linearity and the requirement that the commutator brackets of the pertinent operators are conserved in the amplification. These equations include as special cases the maser, the parametric amplifier, and the parametric up-converter. Using these equations the noise figure of a general amplifier is derived. The minimum value of this noise figure is equal to 2. The significance of the result with regard to a simultaneous phase and amplitude measurement is explored.

Physical Review•Dec 1, 1962•318 citations

Quantum limits on phase-preserving linear amplifiers

Carlton M. Caves, Joshua Combes, Jiang Zhang +1 more

The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.

arXiv:1208.5174•Physical Review A•Dec 3, 2012•109 citations

Quantum Noise in a Parametric Amplifier with Lossy Modes

William G. Wagner, R. W. Hellwarth

We derive the fundamental noise properties of a parametric amplifier (paramp) which is driven by a monochromatic pump but which has many lossy signal and idler modes. We use a quantum-mechanical approach similar to that employed by Louisell and co-workers in their treatment of paramps and lasers with lossless modes. Our results are not directly deducible from the lossless mode analysis, but they do give the same limiting noise temperature as that analysis. This minimum noise temperature is the same as that of an ideal laser. In fact, we find that one can always design a laser whose field correlation functions would be the same as those of any given paramp. Many results for laser noise in various configurations can be carried over directly and applied to parametric amplifiers by using correspondence substitutions which we set down. For example, in analogy to Pound's description of the maser (in terms of a Nyquist theorem extended to negative temperatures), the calculation of paramp noise is found to be equivalent to applying Nyquist's theorem to both the active and passive elements in the (classically computed) equivalent circuit of a signal mode of interest. From this one obtains the spectra of effective noise generators which, when amplified in the circuit, duplicate the "quantum" as well as thermal noise that is present in the output. The Nyquist theorem is extended to apply to each active element by simply taking the element temperatures to be minus the real temperature of the corresponding idler mode (whose parametric coupling into the signal circuit gives rise to that element) multiplied by the ratio of the pump minus the idler to the idler frequencies.

Physical Review•Feb 17, 1964•32 citations

Quantum limits on noise in solid state masters without inversion

Mihai Oane

The development of masers in the 1950's made possible amplifiers that were much quieter than other contemporary amplifiers. An analysis of narrow-band yields a fundamental theorem (Caves theorem) for phase-insensitive linear amplifiers; it requires that such an amplifier, will add noise as large as half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers the theorem establish a lower limit on the product of the noises added to the two phases. In the last decade it was shown theoretically that solid-state masers without inversion may be obtained in multilevel spin systems in dilute paramagnetic solids at high temperature subjected to several strong fields. In the present paper the author applies the Caves theorem to the maser without inversion in order to find out the best ways in which the proposed device can work.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Oct 21, 2004•0 citations

Noise in phase-preserving linear amplifiers

Shashank Pandey, Jiang Zhang, Joshua Combes +1 more

The purpose of a phase-preserving linear amplifier is to make a small signal larger, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise. Quantum mechanics limits how well this can be done: the noise added by the amplifier, referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we provide the quantum constraints on the entire distribution of added noise: any phasepreserving linear amplifier is equivalent to a parametric amplifier with a physical state σ for the ancillary mode; σ determines the properties of the added noise.

AIP conference proceedings•Jan 1, 2014•0 citations
If the uncertainty principle of quantum mechanics is applied to the process of signal measurement, two theorems relating to amplifier noise performance can be deduced. First, it can be shown that it is impossible to construct a linear noiseless amplifier. Second, if the amplifier is characterized as having additive white Gaussian noise, it can be shown that the minimum possible noise temperature of any linear amplifier is T.= In 2 -IIG -hv 1-1/ IG k In the limit of high gain G this expression reduces to that previously derived for the ideal maser and parametric amplifier. It is shown that the minimum noise amplifier does not degrade the signal but rather allows the use of an inaccurate detector to make measurements on an incoming signal to the greatest accuracy consistent with the uncertainty principle.
How much noise does quantum mechanics require a linear amplifier to add to a signal it processes? An analysis of narrow-band amplifiers (single-mode input and output) yields a fundamental theorem for phase-insensitive linear amplifiers; it requires such an amplifier, in the limit of high gain, to add noise which, referred to the input, is at least as large as the half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers, which can respond differently to the two quadrature phases ("$cos\ensuremath{\omega}t$" and "$sin\ensuremath{\omega}t$"), the single-mode analysis yields an amplifier uncertainty principle---a lower limit on the product of the noises added to the two phases. A multimode treatment of linear amplifiers generalizes the single-mode analysis to amplifiers with nonzero bandwidth. The results for phase-insensitive amplifiers remain the same, but for phase-sensitive amplifiers there emerge bandwidth-dependent corrections to the single-mode results. Specifically, there is a bandwidth-dependent lower limit on the noise carried by one quadrature phase of a signal and a corresponding lower limit on the noise a high-gain linear amplifier must add to one quadrature phase. Particular attention is focused on developing a multimode description of signals with unequal noise in the two quadrature phases.
The classical definition of noise figure, based on signal-to-noise ratio, is adapted to the case when quantum noise is predominant. The noise figure is normalized to "uncertainty noise." General quantum mechanical equations for linear amplifiers are set up using the condition of linearity and the requirement that the commutator brackets of the pertinent operators are conserved in the amplification. These equations include as special cases the maser, the parametric amplifier, and the parametric up-converter. Using these equations the noise figure of a general amplifier is derived. The minimum value of this noise figure is equal to 2. The significance of the result with regard to a simultaneous phase and amplitude measurement is explored.
The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.
We derive the fundamental noise properties of a parametric amplifier (paramp) which is driven by a monochromatic pump but which has many lossy signal and idler modes. We use a quantum-mechanical approach similar to that employed by Louisell and co-workers in their treatment of paramps and lasers with lossless modes. Our results are not directly deducible from the lossless mode analysis, but they do give the same limiting noise temperature as that analysis. This minimum noise temperature is the same as that of an ideal laser. In fact, we find that one can always design a laser whose field correlation functions would be the same as those of any given paramp. Many results for laser noise in various configurations can be carried over directly and applied to parametric amplifiers by using correspondence substitutions which we set down. For example, in analogy to Pound's description of the maser (in terms of a Nyquist theorem extended to negative temperatures), the calculation of paramp noise is found to be equivalent to applying Nyquist's theorem to both the active and passive elements in the (classically computed) equivalent circuit of a signal mode of interest. From this one obtains the spectra of effective noise generators which, when amplified in the circuit, duplicate the "quantum" as well as thermal noise that is present in the output. The Nyquist theorem is extended to apply to each active element by simply taking the element temperatures to be minus the real temperature of the corresponding idler mode (whose parametric coupling into the signal circuit gives rise to that element) multiplied by the ratio of the pump minus the idler to the idler frequencies.
The development of masers in the 1950's made possible amplifiers that were much quieter than other contemporary amplifiers. An analysis of narrow-band yields a fundamental theorem (Caves theorem) for phase-insensitive linear amplifiers; it requires that such an amplifier, will add noise as large as half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers the theorem establish a lower limit on the product of the noises added to the two phases. In the last decade it was shown theoretically that solid-state masers without inversion may be obtained in multilevel spin systems in dilute paramagnetic solids at high temperature subjected to several strong fields. In the present paper the author applies the Caves theorem to the maser without inversion in order to find out the best ways in which the proposed device can work.
The purpose of a phase-preserving linear amplifier is to make a small signal larger, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise. Quantum mechanics limits how well this can be done: the noise added by the amplifier, referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we provide the quantum constraints on the entire distribution of added noise: any phasepreserving linear amplifier is equivalent to a parametric amplifier with a physical state σ for the ancillary mode; σ determines the properties of the added noise.