The main properties of the quantum amplifier are its amplification coefficient and uncertainty.
In the case of lasers, the uncertainty corresponds to the amplified spontaneous emission of the active medium.
The unavoidable noise of quantum amplifiers is one of the reasons for the use of digital signals in optical communications and can be deduced from the fundamentals of quantum mechanics.
While classical amplifiers take in classical signals, quantum amplifiers take in quantum signals, such as coherent states.
This does not necessarily mean that the output is a coherent state; indeed, typically it is not.
The form of the output depends on the specific amplifier design.
The physical electric field in a paraxial single-mode pulse can be approximated with superposition of modes; the electric field
of a single mode can be described as where The analysis of the noise in the system is made with respect to the mean value[clarification needed] of the annihilation operator.
To obtain the noise, one solves for the real and imaginary parts of the projection of the field to a given mode
Physically, the initial state corresponds to the coherent pulse at the input of the optical amplifier; the final state corresponds to the output pulse.
The amplitude-phase behavior of the pulse must be known, although only the quantum state of the corresponding mode is important.
The pulse may be treated in terms of a single-mode field.
, as follows: This equation describes the quantum amplifier in the Schrödinger representation.
can be defined as follows: The can be written also in the Heisenberg representation; the changes are attributed to the amplification of the field operator.
For laser applications, the amplification of coherent states is important.
Even with such a restriction, the gain may depend on the amplitude or phase of the initial field.
In the following, the Heisenberg representation is used; all brackets are assumed to be evaluated with respect to the initial coherent state.
The expectation values are assumed to be evaluated with respect to the initial coherent state.
This quantity characterizes the increase of the uncertainty of the field due to amplification.
As the uncertainty of the field operator does not depend on its parameter, the quantity above shows how much output field differs from a coherent state.
The commutator of the field operators is invariant under unitary transformation
satisfies the canonical commutation relations for operators with Bose statistics: The c-numbers are then Hence, the phase-invariant amplifier acts by introducing an additional mode to the field, with a large amount of stored energy, behaving as a boson.
Calculating the gain and the noise of this amplifier, one finds and The coefficient
The gain can be dropped by splitting the beam; the estimate above gives the minimal possible noise of the linear phase-invariant amplifier.
To obtain a large amplification coefficient with minimal noise, one may use homodyne detection, constructing a field state with known amplitude and phase, corresponding to the linear phase-invariant amplifier.
[2] The uncertainty principle sets the lower bound of quantum noise in an amplifier.
Nonlinear amplifiers do not have a linear relation between their input and output.
[3] Examples include most lasers, which include near-linear amplifiers, operating close to their threshold and thus exhibiting large uncertainty and nonlinear operation.
As with the linear amplifiers, they may preserve the phase and keep the uncertainty low, but there are exceptions.
These include parametric oscillators, which amplify while shifting the phase of the input.