Quantum noise
Encyclopedia
Quantum noise is uncertainty of some physical quantity due to its quantum origin.

In the case of number of particles (photons or electrons), the quantum noise is also called shot noise
Shot noise
Shot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...

. Most optical communications use amplitude modulation. In this case, the quantum noise appears as shot noise
Shot noise
Shot noise is a type of electronic noise that may be dominant when the finite number of particles that carry energy is sufficiently small so that uncertainties due to the Poisson distribution, which describes the occurrence of independent random events, are of significance...

 only.

In the case of uncertainty of electric field in some laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...

s, the quantum noise is not just shot noise; uncertainties of both amplitude and phase contribute to the quantum noise. This issue becomes important in the case of noise of a quantum amplifier
Quantum amplifier
A quantum amplifer is an amplifier that uses quantum mechanical methods to amplify a quantum signal; examples include the active elements of optical lasers and optical re-transmitters....

, which preserves the phase. The phase noise becomes important at the frequency modulation
Frequency modulation
In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...

 or phase modulation
Phase modulation
Phase modulation is a form of modulation that represents information as variations in the instantaneous phase of a carrier wave.Unlike its more popular counterpart, frequency modulation , PM is not very widely used for radio transmissions...

 of waves with energy of quantum comparable to the energy of a signal (which is believed to be more robust with respect to additive noise than an amplitude modulation).

Origin of quantum noise

Quantum noise may appear in any system while conventional sources of noise (industrial noise
Industrial noise
Industrial noise is usually considered mainly from the point of view of environmental health and safety, rather than nuisance, as sustained exposure can cause permanent hearing damage. Traditionally, occupational noise has been a hazard linked to heavy industries such as ship-building and...

, vibration, fluctuations of voltage in the electric power supply, thermal noise due to Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, etc.) are somehow suppressed. Generally, quantum noise can be considered as error of description of any physical system within classical (not quantum) theory. In an electric circuit, the random fluctuations of a signal due to discrete character of electrons can be called quantum noise.
The random error of interferometric measurement of position due to discrete character of photons registered can be attributed to quantum noise. Even uncertainty of position of a probe in the probe microscopy may cause the quantum noise although this is not dominant mechanism that determines the resolution of such a device. In most cases, quantum noise refers to the fluctuations of signal in extremely accurate optical systems with stabilized lasers and efficient detectors.

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 the most of 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
Coherent state
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behaviour of a classical harmonic oscillator...

. 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
Quantum amplifier
A quantum amplifer is an amplifier that uses quantum mechanical methods to amplify a quantum signal; examples include the active elements of optical lasers and optical re-transmitters....

.

In general, amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...

 is a device which may increase the mean value of some parameter at its output dependently 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
Reverberation
Reverberation is the persistence of sound in a particular space after the original sound is removed. A reverberation, or reverb, is created when a sound is produced in an enclosed space causing a large number of echoes to build up and then slowly decay as the sound is absorbed by the walls and air...

 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
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

, 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
Coherent state
In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resembles the oscillating behaviour of a classical harmonic oscillator...

 with the same amplitude and phase. The lower bound for this noise follows from the fundamental properties of the operator of creation and annihilation.
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