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Physica Scripta

The Theory of Quantum Amplifiers

Stig Stenholm

Published Jan 1, 1986• 66 citations
Publisher

Abstract

The theory of linear quantum amplifiers is reviewed. A simplified method using a "faked vacuum" is introduced to derive amplifier master equations. Properties of various amplifiers are summarized. The solution is derived and used as the amplifier transfer function. The Caves' theory for amplifier added noise is discussed. The properties of squeezed states are reviewed, the use of the Wigner function to describe them is advocated. Finally the amplification of squeezed states is discussed in the Wigner representation.

DOI
10.1088/0031-8949/1986/t12/008

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The theory of linear quantum amplifiers is reviewed. A simplified method using a "faked vacuum" is introduced to derive amplifier master equations. Properties of various amplifiers are summarized. The solution is derived and used as the amplifier transfer function. The Caves' theory for amplifier added noise is discussed. The properties of squeezed states are reviewed, the use of the Wigner function to describe them is advocated. Finally the amplification of squeezed states is discussed in the Wigner representation.
The analysis of the quantum-mechanical parametric amplifier begun in the preceding paper is extended to describe the joint quantum state of the two interacting modes of oscillation of the system. Equations are derived for the time evolution of the Schr\"odinger density operator. Comparison with classical theory is made by describing the state of the quantum-mechanical system by means of the PPP representation. The time evolution of the weight function $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ for this representation, and of the corresponding Wigner distribution function $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$, is found. A decoupled form for the equations of motion for the field variables is exhibited which simplifies the solutions for these functions. It is shown that the amplification process gives rise to a correlation between the two mode amplitudes which becomes exceedingly strong with increasing time. The relation of this correlation to the time development of the functions $P(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ and $W(\ensuremath{\alpha}, \ensuremath{\beta}, t)$ is discussed.
This study summarizes the theory of laser-amplifier noise based on the master equation, the quantum-mechanical Langevin equation, and the rate equation. The three approaches are used differently depending on the application. Laser-amplifier noise in the direct-detection receiver is described by the master equation. Meanwhile, the quantum-mechanical Langevin equation is most effective for analyzing noise appearing in coherent optical communication systems. The rate equation approach provides a clear understanding of the intensity-noise characteristics of laser oscillators, including the amplitude squeezing.
This study summarizes the theory of laser-amplifier noise based on the master equation, the quantum-mechanical Langevin equation, and the rate equation. The three approaches are used differently depending on the application. Laser-amplifier noise in the direct-detection receiver is described by the master equation. Meanwhile, the quantum-mechanical Langevin equation is most effective for analyzing noise appearing in coherent optical communication systems. The rate equation approach provides a clear understanding of the intensity-noise characteristics of laser oscillators, including the amplitude squeezing.
This chapter reviews and extends the work on the particular class of linear amplifiers known as inverted population amplifiers. It discusses first the fundamental requirements demanded by quantum mechanics from any theory of linear amplifiers, and then describes the basic model for the atomic amplifier in a closed cavity. The chapter considers the extension of this model to allow explicit incoherent coupling of the internal cavity field to external fields, and treats finally an open system with coherent field coupling. It investigates in particular the conditions under which the amplified field retains quantum properties such as squeezing. The chapter examines the ability of the phase-insensitive systems to enhance the signal-to-noise ratio of an injected signal, and finds that in general any signal is degraded by the amplification process. Phase-insensitive atomic amplifiers destroy all quantum statistical properties of the field if the gain exceeds the low value of two.
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.