Quantum Amplifier - Introduction

Introduction

An amplifier increases the amplitude of whatever goes through it. While classical amplifiers take in classical signals, quantum amplifiers take in quantum signals like coherent states. This does not mean that the output is a coherent state; optical quantum amplifiers with feedback are examples. Besides amplifying the intensity of the input, quantum amplifiers increase the uncertainty of the signal, unlike a classical amplifier.

The physical electric field in a paraxial single-mode pulse can be approximated with superposition of modes; the partial field of such a mode can be described with

where

• is vector of spatial coordinates,
• is the vector of polarization of the pulse,
• is wavevector,
• is the operator of annihilation of photon in some specific mode .

The amplitude of field in this mode must be able to be detected. All the quantum-mechanical analysis of the noise is about the mean value of the annihilation operator and its uncertainty; the technical problems with detection of the real and imaginary parts of the projection of the field to a given mode must be solved. Therefore, spatial coordinates do not appear in the deduction, and symbols may be used for Hermitian and anti-Hermitian parts of the annihilation operator.

Assume that the mean value of the initial field . Physically, in terms of a laser, the initial state may correspond to the coherent pulse at the input of the optical amplifier, and the final state may correspond to the output pulse. The amplitude-phase behavior of the pulse must be known, and only the quantum state of the corresponding mode is important. In this case, the pulse may be treated in terms of a single-mode field.

Hence, a quantum amplifier is a unitary transform, which converts the input quantum state to the amplified state . This is the quantum amplifier in the wave function representation.

The amplification may change as the mean value of the field operator and its dispersion change. The coherent state represents the minimum uncertainty; as the state is transformed, the uncertainty may increase. This increase can be interpreted as noise in the amplifier.

The transformation of state can be written as follows:

The coefficient of amplification can be defined as follows:

This expression can be written also in the matrix representation; all the changes may be attributed to the amplification to the operator of field, and may be assumed, keeping the vector of state unchanged. Then:

In general, the coefficient may be complex, and it may depend on the initial state. For application to lasers, the amplification of coherent states are important. Therefore, it is usually assumed that the initial state is a coherent state characterized with a complex initial parameter such that . Even after such a restriction, the coefficient of amplification may depend on the amplitude or phase of the initial field. In the following consideration, only the matrix representation is used; all the brackets are assumed to be evaluated with respect to the initial coherent state.

The noise of the amplifier can be defined as follows:

This quantity characterizes the increase of uncertainty of the field due to amplification. As the uncertainty of the field operator at the coherent state does not depend on its parameter, the quantity above shows how different from the coherent state is the output field.