In this paper we consider further the quantum-statistical properties of radiation in nondegenerate parametric-type amplifiers. In particular we find moment-generating functions and probability distribution functions for the field variables at the output of an amplifier for various input conditions. We find that a classical description of the input fields and of the amplification process is completely valid provided we take correctly into account the response of the amplifier to the input zero-point noise fields. This result is valid for inputs of arbitrarily small power.
In this paper we consider further the quantum-statistical properties of radiation in nondegenerate parametric-type amplifiers. In particular we find moment-generating functions and probability distribution functions for the field variables at the output of an amplifier for various input conditions. We find that a classical description of the input fields and of the amplification process is completely valid provided we take correctly into account the response of the amplifier to the input zero-point noise fields. This result is valid for inputs of arbitrarily small power.
Fluctuations in the amplification and absorption of waves by quantum processes are considered. Assuming for each quantum the probability (per unit time) a of producing another quantum, probability b of being absorbed, and assuming a probability c that a new quantum is introduced, a set of differential equations is obtained. By solving these equations, a complete expression for the probability of distribution of quanta is obtained, as well as expressions for the average values and fractional fluctuation. The expressions developed are applied in particular to maser-type amplifiers, and certain fluctuations in the amplification of electromagnetic waves are pointed out which are important when their quantum character becomes significant. This condition can occur in maser-type amplifiers, where thermal and extraneous noises may be very small. For such an amplifier, a is proportional to the number of excited molecules, whereas b consists of a term proportional to the number of molecules in the ground state plus terms due to certain other losses. The noise temperature of a maser-type traveling-wave amplifier is the "effective temperature" a ( a - b ) -1 h ν/ k . In superregenerative and regenerative amplifiers using resonant cavities, the noise temperature is rather similar, if losses through the input coupling hole are excluded from b . In any case, the limiting noise for an ideal amplifier corresponds to a classical noise temperature of h ν/ k .
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 properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.
Crossover between thermal and quantum noise in the externally-pumped Josephson-junction parametric amplifiers is theoretically followed. Two different approaches are used for the analysis : i. Josephson junction is replaced with its RSJ model with the equilibrium source of fluctuations; ii. microscopic theory of the tunnel junctions is used, which takes into account not only thermal and quantum but also shot noise. Because of inconvenience of the noise temperature TNfor the amplifier sensitivity characterization in the quantum limit, we express our results in terms of the figure ΘNwhich is essentially the amplifier output noise energy, reduced to its input (in the classical limit,\Theta_{N}=k_{B}T_{N}). For the nondegenerate amplifier, the minimum value of ΘN. equals hω/2 and is due to quantum noise (zero-point oscillations) in the idle-frequency resonator of the amplifier. For the degenerate amplifier, sensitive to only one quadrature component of the input signal, ΘNcan be made much less than the above "quantum limit" even at operation temperatures much higher thanh\omega/k_{B}, if the Josephson junction characteristic frequency ωcis high enough,h\omega_{c}\gsimk_{B}T. Our analysis of the experimental situatlon shows that the values\Theta_{N}<h\omega/2can be achieved using two-dimensional arrays of a large number (N\simeq100) single-tunnel-junction interferometers with low inductances, inductively coupled to a microwave cavity.
Quantum fluctuations in optical amplification are investigated with a nondegenerate optical parametric amplifier whose internal idler mode is coupled to a squeezed vacuum. Reductions of the inherent quantum noise of the amplifier are observed with a minimum noise level 0.7 dB below the usual noise level of the amplifier with its internal idler mode in a vacuum state. With a small coherent field as the signal input, the amplified output exhibits an improvement in signal-to-noise ratio of 0.5 dB for the case of a squeezed-vacuum as compared to a vacuum state for the amplifier's internal mode.