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Quantum phase uncertainties in the classical limit

J. D. Franson

Published May 1, 1994• 0 citations
OpenAlex

Abstract

Several sources of phase noise, including spontaneous emission noise and the loss of coherence due to which-path information, are examined in the classical limit of high field intensities. Although the origin of these effects may appear to be quantum-mechanical in nature, it is found that classical analogies for these effects exist in the form of chaos.

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Quantum noise in the classical limit

J. D. Franson

Spontaneous emission noise can be thought of as being due to vacuum fluctuations and may appear to be inherently quantum-mechanical in nature. The same can be said of the loss of coherence in which-path experiments, which is due to the entanglement of a photon with some other system in one path through an interferometer. Both of these processes have been considered in the limit of high field intensities in order to determine whether or not they have a classical analogy. It was found that the same phenomena occur classically as well due to chaos. The extent to which these phenomena can or cannot be considered to be inherently quantum-mechanical will be discussed.

Annual Meeting Optical Society of America•Jan 1, 1993•0 citations

Uncertainty in determining the phase for an optical field due to the particle nature of light

Z. Y. Ou, Qichang C. Su

When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.

Laser Physics•Jan 1, 2003•3 citations

Quantum noise in photonics

CHARLES H. HENRY, Rudolf F. Kazarinov

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]

Reviews of Modern Physics•Jul 1, 1996•220 citations

Phase noise in quantum physics

Barry C. Sanders

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.

•Nov 1, 1987•0 citations

Incoherence, Quantum Fluctuations, and Noise

I. R. Senitzky

An examination is made of the relationship between the uncertainty principle and minimum amplifier noise. First, the concept of coherence is discussed, and an incoherence parameter is defined in terms of the uncertainty that enters into the uncertainty principle. Harmonic oscillator states are examined for coherence. The concept of noise is then discussed and contrasted with incoherence, noise referring to behavior in time of a single system while incoherence involving comparison among members of an ensemble. It is shown, with illustrations, that the two concepts are different, and that an incoherent field of a cavity mode need not exhibit noise. In particular, the zero-point field in a lossless cavity is not noise. The superposition of many incoherent effects, however, usually leads to noise. Spontaneous emission is examined both for coherence and noise. It is shown that the spontaneous emission field of a single molecule is incoherent but does not exhibit noise; the (low order) spontaneous emission from a molecular beam, however, does constitute noise. Spontaneous emission from complex systems is also discussed. The origin of fundamental noise in an amplifier is investigated and is shown to come from spontaneous emission by the amplification mechanism. It is concluded that fundamental noise cannot be determined by a consideration of quantum fluctuations of---or by the application of the uncertainty principle to---the electromagnetic field only, as has been done in several recent articles. The physical significance of the zero-point field is analyzed, and is shown to lie in a formal contribution to spontaneous emission by the mechanism coupled to the field, provided this mechanism is treated quantum mechanically.

Physical Review•Dec 15, 1962•21 citations

Quantum Noise Reduction In Optical Fields

Daniel F. Walls, Matthew J. Collett, Alistair S. Lane +1 more

Current laser systems have phase and amplitude fluctuations which are partially quantum in origin. The amplitude fluctuations of a highly stabilized laser are shot noise limited, that is they have quantum fluctuations due to the Poissonian fluctuations of photons in a coherent state. These photon number fluctuations have been shown to have an origin in the incoherent pumping mechanism used for standard lasers. It has recently been demonstrated experimentally that by suppressing the amplitude fluctuations of the pump a laser output with reduced amplitude fluctuations is possible. The phase fluctuations in a laser are due to phase diffusion and are substantially above the quantum limit. Techniques to reduce the laser phase fluctuations have been proposed making use of correlated spontaneous emission.

Digest on Nonlinear Optics: Materials, Phenomena and Devices•Jan 1, 1990•0 citations
Several sources of phase noise, including spontaneous emission noise and the loss of coherence due to which-path information, are examined in the classical limit of high field intensities. Although the origin of these effects may appear to be quantum-mechanical in nature, it is found that classical analogies for these effects exist in the form of chaos.
Spontaneous emission noise can be thought of as being due to vacuum fluctuations and may appear to be inherently quantum-mechanical in nature. The same can be said of the loss of coherence in which-path experiments, which is due to the entanglement of a photon with some other system in one path through an interferometer. Both of these processes have been considered in the limit of high field intensities in order to determine whether or not they have a classical analogy. It was found that the same phenomena occur classically as well due to chaos. The extent to which these phenomena can or cannot be considered to be inherently quantum-mechanical will be discussed.
When any two optical fields are superposed, a fringe pattern will be formed if the observation interval is short enough (within the coherent time) and a phase value may be extracted from the fringe and assigned as the phase difference between the fields. When measured against a common reference field, phase can be defined for an optical field. However, when the intensities of the optical fields are low or the observation time is so short that only a few photons are detected, the fringe pattern is unrecognizable due to the particle nature of light. By analyzing the fringe patterns simulated by a quantum Monte Carlo method, we find that there exist uncertainties in determining the phase value. This uncertainty is universal for any optical field with a finite photon number and decreases to zero as the photon number of the field increases to infinity or the classical limit is approached. The existence of this uncertainty in determining the phase value makes it problematic to define a quantum mechanical phase. An interference experiment with coherent state is performed to confirm the theoretical predictions. The implication of the above is that phase is a classical quantity that is associated with a large number of particles and may not have a quantum mechanical analogy.
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]
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.
An examination is made of the relationship between the uncertainty principle and minimum amplifier noise. First, the concept of coherence is discussed, and an incoherence parameter is defined in terms of the uncertainty that enters into the uncertainty principle. Harmonic oscillator states are examined for coherence. The concept of noise is then discussed and contrasted with incoherence, noise referring to behavior in time of a single system while incoherence involving comparison among members of an ensemble. It is shown, with illustrations, that the two concepts are different, and that an incoherent field of a cavity mode need not exhibit noise. In particular, the zero-point field in a lossless cavity is not noise. The superposition of many incoherent effects, however, usually leads to noise. Spontaneous emission is examined both for coherence and noise. It is shown that the spontaneous emission field of a single molecule is incoherent but does not exhibit noise; the (low order) spontaneous emission from a molecular beam, however, does constitute noise. Spontaneous emission from complex systems is also discussed. The origin of fundamental noise in an amplifier is investigated and is shown to come from spontaneous emission by the amplification mechanism. It is concluded that fundamental noise cannot be determined by a consideration of quantum fluctuations of---or by the application of the uncertainty principle to---the electromagnetic field only, as has been done in several recent articles. The physical significance of the zero-point field is analyzed, and is shown to lie in a formal contribution to spontaneous emission by the mechanism coupled to the field, provided this mechanism is treated quantum mechanically.
Current laser systems have phase and amplitude fluctuations which are partially quantum in origin. The amplitude fluctuations of a highly stabilized laser are shot noise limited, that is they have quantum fluctuations due to the Poissonian fluctuations of photons in a coherent state. These photon number fluctuations have been shown to have an origin in the incoherent pumping mechanism used for standard lasers. It has recently been demonstrated experimentally that by suppressing the amplitude fluctuations of the pump a laser output with reduced amplitude fluctuations is possible. The phase fluctuations in a laser are due to phase diffusion and are substantially above the quantum limit. Techniques to reduce the laser phase fluctuations have been proposed making use of correlated spontaneous emission.