Quantum Amplifier - Linear Phase-invariant Amplifiers

Linear Phase-invariant Amplifiers

Linear phase-invariant amplifiers are mathematical abstractions: assume that the unitary operator amplifies in such a way that the input and the output are related by a linear equation

where and are c-numbers and is an operator in the amplifier. Without loss of generality, it may be assumed that and are real. The unitary transformation preserves the commutator of the field operators, so:

From the unitarity of, it follows that satisfies the standard commutation relation for the Bose operators

and

Hence, the amplifier is equivalent to an additional mode of the field with a large amount of energy stored, and behaves as a boson. Calculating the gain and the noise of this amplifier, we get

and

The coefficient should be interpreted as the intensity amplification coefficient, and the noise of the linear amplifier as . The gain can be dropped out by the splitting of the beam; therefore, the estimate above is the minimal noise of the linear quantum amplifier.

The linear amplifier has an advantage over the multi-mode amplifier because if several modes of a linear amplifier are amplified by the same factor, the noise in each mode is determined independently; modes in a linear quantum amplifier are independent.

For a large amplification coefficient, the minimal noise can be realized with homodyne detection and construction of a field state with known amplitude and phase, which corresponds to the linear phase-invariant amplifier. The uncertainty principle sets the lower bound of quantum noise in an amplifier. In particular, the output of a laser system and the output of an optical generator are not coherent states.