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.
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.
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]
Noise that can be attributed to vacuum fluctuations, usually referred to as quantum noise, is examined. It is shown that vacuum fluctuations of a single uncoupled mode do not constitute noise, in the sense of a random process, but the superposition of the fluctuations of a large number of modes does constitute, formally, noise. The effect of vacuum fluctuations of the free-space radiation field on a harmonic oscillator, a nondegenerate parametric amplifier, and a degenerate parametric amplifier, all driven by a prescribed sinusoidal field, is compared with the effect of classical noise. It is found that the coordinates of all systems respond in a formally similar manner to both vacuum fluctuations and classical noise. However, the resonance fluorescence spectrum---the evidence of ``detection''---is completely different for the two kinds of noise. The spectrum of the harmonic oscillator does not exhibit noise in response to vacuum fluctuations, but does so in response to classical noise. The spectra of the two types of parametric amplifiers do exhibit noise in response to vacuum fluctuations, but this noise differs from that in the classical case. An explanation for the difference is offered, based on the fact that the quantum fluctuations cannot do work, but can noise-modulate power from an outside source, which, for the parametric amplifiers, is the pump. In the analysis of noise from the degenerate parametric amplifier, it is shown that squeezed noise, viewed as an oscillation of the dispersion with a sufficiently low minimum, is generated in the same manner in the case of classical noise as in the case of quantum noise, and is due to phase conjugation.
The interaction between a coherently oscillating radiation field, such as is generally encountered in micro-wave experiments, and a number of similar atomic systems coupled to the field through their electric dipole moments is analyzed for the case of resonance between atomic system and field, with both the field and molecules treated quantum-mechanically. Expressions are derived for the expectation value of field strength and the energy, and comparison between the two makes it possible to distinguish between coherent and incoherent parts of the energy, the former having its counterpart in the expectation value of the field strength, while the latter has no such counterpart. Terms corresponding to induced and spontaneous emission are identified, and it is shown that the latter includes both coherent and incoherent components. Special situations related to masers and to the coherence of spontaneous radiation are discussed. The distinction between the behavior of correlated and uncorrelated states is examined, and it is shown that both the coherent emission from uncorrelated states and the incoherent emission from correlated states may be proportional to the square of the number of molecules. The phase of the field is predictable for the coherent emission but unpredictable for the incoherent emission.
The fundamental source of noise in laser and optical amplifier is spontaneous emission. However, it is possible to reduce this noise by preparing the gain medium (atoms) in coherent superposition of states. There are essentially two schemes for generating atomic coherence in optical amplifiers. \n \nFirst scheme involves the preparation of atoms in a coherent superposition of states before injecting them inside the cavity. In the other scheme atoms (initially pumped incoherently) are continuously driven by an external pump field inside the cavity. Almost all the studies concerning the quenching of spontaneous emission noise in optical amplifiers, based on the notion of atomic coherent superposition, ignore the finite bandwidth of the field that is responsible for generating coherence. In this dissertation, we have examined the effect of the finite bandwidth associated with the phase fluctuations of the driving field on a number of noise quenching schemes. \n \nIt is known that coherent preparation of atoms can, under certain conditions, lead to noise-free amplification and almost 50 percent phase squeezing in a two-photon correlated emission laser. We study the effect of phase fluctuations in the injected atoms on the phase noise of the field in these systems. We discuss the conditions under which squeezing is preserved in the two-photon linear amplifiers. We also show that, even for very large phase fluctuations in the injected atoms, almost 50 percent phase squeezing can be achieved. The corresponding gain is however significantly reduced. \n \nWe also study a system of phase-sensitive amplifier where atoms are prepared in a coherent superposition of states by a strong classical field. We study the role of pump phase fluctuations associated with the finite linewidth on the quantum statistical properties of the radiation field. A study of output field characteristics reveals that the presence of a weak pump field gives rise to equal variances in both the field quadratures which are independent of linewidth. On the other hand, the presence of strong pump field alters the output field characteristics with the generation of squeezing which is later on destroyed due to the presence of phase fluctuations
In the previous articles of this series, dealing with the interaction between a number of molecules and the electromagnetic field in a resonant cavity, both the molecules and the field were treated by perturbation theory. The perturbation restriction on the field is removed in the present article, allowing large changes in the field, but the molecules are still assumed to undergo a small change during the time under consideration. The justification for this type of analysis, involving the generalization of the conventional concepts of induced and spontaneous emission, the applicability to a molecular amplifier during the buildup period, and the re-examination of a calculation by Serber and Townes concerning the fundamental limits of molecular amplification, is discussed.Two different molecular distributions are considered. In one (the resonant case) all molecules have the same frequency as the cavity, and in the other (the nonresonant case) there is a uniform frequency distribution. The molecules are assumed to be initially in an emissive state. Several types of driving fields are considered. Expressions are obtained for the field operators by the solution of a Volterra integral equation, and expectation values are obtained for the field strength and field energy.In the resonant case, both the coherent and incoherent fields increase exponentially after a sufficiently long time, no matter how small the initial gain is. Their ratio becomes constant and is equal to the number of photons in the driving field only in the absence of dissipation. An interesting related result is the fact that the signal-to-noise ratio for constant signal input power increases as the cavity dissipation increases. An estimate of the total time for which the theory is valid is obtained from a consideration of the energy emitted by the molecules. Contact is made with perturbation theory for sufficiently small gain and short time.In the nonresonant case the effect of the molecules is shown to be that of a negative dissipation. In contrast to the resonant case, the gain becomes exponential only if the negative dissipation exceeds, in absolute value, the true dissipation. The ratio of induced to spontaneous emission is, in this case also, equal to the number of photons in the driving field only in the absence of disipation. However, the signal-to-noise ratio for constant input power drops with increasing cavity dissipation.