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Journal of the Optical Society of America B

From classical to quantum noise

H. A. Haus

Published Nov 1, 1995• 38 citations
Publisher

Abstract

An account is given of noise in electrical circuits and its relation to quantum noise connected with Heisenberg’s uncertainty principle. A phase-insensitive amplifier must introduce noise (most commonly the spontaneous emission noise) in order to satisfy the constraint imposed by quantum mechanics on the simultaneous measurement of two observables represented by noncommuting operators. Nonstationary, phase-sensitive amplifiers can employ squeezed radiation and reduce the noise below that of phase-insensitive amplifiers. Experiments on squeezing in optical fibers and the attendant noise reduction are presented. The squeezing apparatus can also be employed as a quantum nondemolition measurement of photon number. A thought experiment of a two-slit interferometer is described, in which the fringes are washed out progressively with the accuracy of the photon number determination in one of its arms. Implications of measurements of this kind for the resolution of the Einstein–Podolsky–Rosen paradox are discussed.

DOI
10.1364/josab.12.002019

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The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]

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Physical Review Letters•Sep 13, 1993•4 citations
An account is given of noise in electrical circuits and its relation to quantum noise connected with Heisenberg’s uncertainty principle. A phase-insensitive amplifier must introduce noise (most commonly the spontaneous emission noise) in order to satisfy the constraint imposed by quantum mechanics on the simultaneous measurement of two observables represented by noncommuting operators. Nonstationary, phase-sensitive amplifiers can employ squeezed radiation and reduce the noise below that of phase-insensitive amplifiers. Experiments on squeezing in optical fibers and the attendant noise reduction are presented. The squeezing apparatus can also be employed as a quantum nondemolition measurement of photon number. A thought experiment of a two-slit interferometer is described, in which the fringes are washed out progressively with the accuracy of the photon number determination in one of its arms. Implications of measurements of this kind for the resolution of the Einstein–Podolsky–Rosen paradox are discussed.
We showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise . Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states , provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.
We study the possibility of employing phase-sensitive parametric amplifiers to compensate for losses incurred by solitons propagating in optical fibers. As phase-sensitive amplification is free from the excess spontaneous noise associated with laser amplifiers, the only quantum noise source in such a system arises from the fiber losses. In addition, under phase-sensitive amplification, fluctuations in the soliton frequency are squeezed to one half of the vacuum level, thus producing a negligible contribution to the fluctuations in the soliton arrival time. This greatly reduces the Gordon–Haus jitter of the solitons. The performance of the phase-sensitive amplifier is compared with that of alternative noise-reduction schemes employing phase-insensitive amplifiers and spectral filters as well as with the limits on long-distance data transmission imposed by quantum mechanics.
While it is true that quantum mechanics imposes strict limits on the precision of measurements, we can sometimes improve measurement precision without violating the uncertainty principle. The rub is that this improvement may be formidable to achieve in practice—and always comes at the expense of a loss in the precision of a complementary measurement. Last October a team at AT&T Bell Labs succeeded in reducing the noise observed from an optical cavity below the “shot noise”—the level of fluctuations associated with variations in the vacuum fields, which is the ultimate limit to accuracy that would be predicted by semiclassical models. Although the Bell Labs experiment reduced the noise below this level by only 7 to 10%, it bolstered hopes that continued progress might eventually yield reductions in noise by factors of ten. Manipulation of the quantum noise limitations might be absolutely critical to such precision measurements as detection of gravity waves, whose faint signals are likely to be near the measurement limit of laser interferometers or mechanical oscillators. The phenomenon might possibly find applications in optical communication or optical memory systems.
The nature of phase noise in quantum optics is analyzed. In an experiment involving the measurement of the electromagnetic field the two quantities of interest are the energy and phase of the field. However, measurements of the quantities produce quantum fluctuations. The quantum fluctuations are regarded as noise in the treatment presented here. The quantum system is represented by a probability distribution, the Wigner function, and the quantum fluctuations are treated as stochastic noise associated with the quantity being measured. The difficulties of associating a quantum operator with the phase of the system are reviewed and the related energy-phase uncertainty relation is discussed. The alternate interpretation of the phase noise of a quantum system as being the classical phase noise of the Wigner function is presented. In particular the energy and phase noise of the vacuum state, the coherent state, the squeezed state and the squeezed vacuum are discussed in this way. The squeezed states of light are minimum uncertainty states with respect to the quadrature operators and exhibit noise of one quadrature below the noise level associated with the vacuum. The reduced noise level in one quadrature of the field underlies the importance of squeezed states in many practical applications where there is a need to reduce the quantum noise of one quadrature of coherent light. The periodic phase operator eliminates the difficulties associated with the multivalued nature of phase. The analysis of the vacuum and intense coherent state of Carruthers and Nieto by employing periodic phase operators is reviewed, particularly with respect to the energy-phase uncertainty relations and we generalize the approach to develop a phase operator analysis of the squeezed state in the intense field and vacuum limits. We demonstrate here for the first time that the phase operator is simply related to the phase of the squeezed state in the intense field limit and that the squeezed state is approximately an energy-phase minimum uncertainty state in the low-squeezing limit. Also we enlarge on previous work to demonstrate that the phase operator corresponds simply and unambiguously to the phase of the squeeze parameter for the strongly squeezed vacuum and the intensely squeezed vacuum is an energy-phase minimum uncertainty state for some values of phase. The occurrence of squeezing for the case of two coupled quantum oscillators is presented. The system consisting of one mode of the electromagnetic field coupled to a spinless nonrelativistic electron subjected to an harmonic potential is represented by two coupled harmonic oscillators. The dynamics are compared for the case that the rotating wave approximation is employed and for the case that the counter-rotating terms are included. These calculations have not been performed before. The parametric amplifier Hamiltonian with a nonresonant coupling is also studied in order to provide insight into the effects of the counter-rotating terms. Squeezing of the field produced by the electron is a consequence of the inclusion of the counter-rotating terms. The case of a spinless nonrelativistic electron subject to an harmonic potential and coupled to a continuum of electromagnetic field modes is also considered. The case of two coupled oscillators discussed above is generalized by replacing the oscillator which represents the single-mode field by a bath of oscillators. The effects of including counter-rotating terms and of ignoring the counter - rotating terms in the Hamiltonian are compared. The interaction is assumed to produce a frequency shift and an exponential damping term for the oscillating electron. The frequency shift is assumed to be small in either case and so the Wigner-Weisskopff approximation is employed to solve the equations of motion. We demonstrate the new results that dissipation-induced phase-dependent noise is a consequence of including the counter-rotating terms and that the noise is phase-independent for the case that the counterrotating terms are excluded. The relation between these results and recent work on quantum tunnelling in superconducting quantum interference devices is discussed. We conclude by suggesting further research related to the work in this thesis.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]