Quantum Noise - Coherent States and Noise of A Quantum Amplifier

Coherent States and Noise of A Quantum Amplifier

Although the coherent states can be realized in wide variety of physical systems, they refer mainly to the state of optical light. In most cases, the light of a laser can be interpreted as a classical wave. Within quantum mechanics, such a wave can be approximated with a coherent state. At total amount of photons of order of 108 (which still corresponds to very moderate energy), the relative error of measurement of intensity due the quantum noise is only of order of 10−5 which is good precision for most of applications. Quantum noise becomes important at the amplification of a small signal. Roughly, the quantum uncertainty of the quadrature components of the field is amplified as well as the signal; and the resulting uncertainty appears as noise. This determines the lower limit of noise of a quantum amplifier.

In general, an amplifier is a device which may increase the mean value of some parameter at its output dependent on its input. Amplifiers reproduce some parameters of the input signal. For example, the electric circuit which replaces any set of input with a predefined output (such as a system which plays back a prefabricated audio recording given any audible input) is not an amplifier. A powerful laser in the way of an optical beam which replaces the beam with stronger one also is not an amplifier. But a method for replicating an input signal at a higher output volume (such as an acoustical system which makes the song of a singer hearable over a large stadium, or an active medium which produces photons in amounts roughly proportional to its input) should be considered an amplifier.

A quantum amplifier is an amplifier which operates close to the quantum limit of its performance. In this sense, the acousto-electrical circuit which converts the small electric signal from a microphone to the powerful sound wave (which may cause reverberation of resonant materials) is not a quantum amplifier. The minimal noise of a quantum amplifier depends on the property of the input signal, which is reproduced at the output. In a narrow sense, the optical quantum amplifier reproduces both amplitude and phase of the input wave. Usually, the amplifier amplifies many modes of the optical field (and special efforts are required to reduce the number of these modes). In the idealized case, one may consider just one mode of the electromagnetic field, which corresponds to a pulse with definite polarization, definite transversal structure and definite arrival time, duration and frequency, with uncertainties limited with the Heisenberg uncertainty principle. The input mode may carry some information in its amplitude and phase; the output signal carries the same phase but larger amplitude, roughly proportional to the amplitude of the input pulse. Such an amplifier is called phase-invariant amplifier. Mathematically, quantum amplification can be represented with an unitary operator, which entangles the state of the optical field with internal degrees of freedom of the amplifier. This entanglement appears as quantum noise; the uncertainty of the field at the output is larger than that of the coherent state with the same amplitude and phase. The lower bound for this noise follows from the fundamental properties of the operator of creation and annihilation.