imported>WikiHarold: fix

2026-01-28T08:36:55Z

fix

← Older revision	Revision as of 08:36, 28 January 2026
(No difference)

imported>WikiHarold: fix

2026-01-28T08:36:55Z

fix

New page

{{Short description|Amplifier that uses quantum mechanical methods}}
In [[HandWiki:Physics|physics]], a '''quantum amplifier''' is an [[Engineering:Amplifier|amplifier]] that uses [[Physics:Quantum mechanics|quantum mechanical]] methods to amplify a signal; examples include the active elements of [[Physics:Laser|laser]]s and [[Physics:Optical amplifier|optical amplifier]]s.

== Introduction ==
An [[Engineering:Amplifier|amplifier]] increases the amplitude of its input. While classical amplifiers take in classical signals, quantum amplifiers take in quantum signals, such as [[Physics:Coherent state|coherent state]]s. 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. Besides amplifying the intensity of the input, quantum amplifiers can also increase the [[Physics:Quantum noise|quantum noise]] present in the signal.

==Exposition==
The physical [[Physics:Electric field|electric field]] in a paraxial single-mode pulse can be approximated with [[Physics:Quantum superposition|superposition]] of modes; the electric field <math>~E_{\rm phys}~</math> of a single mode can be described as

:<math> \vec E_{\rm phys}(\vec x)~=~ \vec e~ \hat a~ M(\vec x)~\exp(ikz-{\rm i}\omega t) ~+~ {\rm Hermitian~conjugate}~</math>

where
*<math>~\vec x =\{x_1,x_2,z \}~</math> is the spatial coordinate vector, with ''z'' being the propagation direction,
*<math>~\vec e ~</math> is the [[Physics:Polarization (waves)|polarization]] vector of the pulse,
*<math>~k~</math> is the wave number in the ''z'' direction,
*<math>~\hat a~</math> is the annihilation operator of the photon in a specific mode <math>~ M(\vec x) ~</math>{{clarify|reason=What is a definition or link for M(x)?|date=April 2015}}.

The analysis of the noise in the system is made with respect to the mean value{{clarify|reason=Definition or link for mean value of a non-Hermitian operator|date=April 2015}} 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 <math>~ M(\vec x) ~</math>. Spatial coordinates do not appear in the solution.

Denote the expectation value of the initial field as <math>~{\left\langle\hat a\right\rangle_{\rm initial}}~</math>. 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.

The action of a quantum amplifier is a unitary transform <math> \hat U </math>, acting on the initial state <math>~|{\rm initial}\rangle~</math> and producing the amplified state <math>~|{\rm final}\rangle~</math>. In the [[Schrödinger equation#Particles as waves|Schrödinger representation]], this is

:<math>~|{\rm final}\rangle = U |\rm initial \rangle </math>

The amplification depends on the expectation value <math>~\langle \hat a\rangle ~</math> of the field operator <math>~\hat a~</math> and its dispersion{{Clarify|reason=dispersion does not seem to be a standard nomenclature|date=June 2025}} <math>~\langle \hat a^\dagger \hat a\rangle - \langle \hat a^\dagger \rangle \langle \hat a\rangle~</math>. A coherent state is a state with minimal uncertainty; when the state is transformed, the uncertainty may increase. This increase can be interpreted as the added [[Physics:Quantum noise|noise]] of the amplifier.

The [[Engineering:Gain (electronics)|gain]] <math>~G~</math> is defined as follows:

:<math> G= \frac{\left\langle\hat a\right\rangle _{\rm final}}{\left\langle\hat a\right\rangle _{\rm initial}} </math>

The can be written also in the [[Physics:Matrix mechanics|Heisenberg representation]]; the changes are attributed to the amplification of the field operator. Thus, the evolution of the operator ''A'' is given by <math>~ \hat A =\hat U^\dagger \hat a \hat U~ </math>, while the [[Physics:Quantum state|state vector]] remains unchanged. The gain is given by

:<math>~ G= \frac{\left\langle\hat A\right\rangle _{\rm initial}}{\left\langle\hat a\right\rangle _{\rm initial}}~</math>

In general, the gain <math>~G~</math> may be complex, and it may depend on the initial state. For laser applications, the amplification of [[Physics:Coherent state|coherent state]]s is important. Therefore, it is usually assumed that the initial state is a coherent state characterized by a complex-valued initial parameter <math>~\alpha~</math> such that <math>~~|{\rm initial}\rangle=|\alpha\rangle~</math>. 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.

:<math>{\rm noise}= \langle \hat A^\dagger \hat A\rangle -\langle \hat A^\dagger \rangle\langle \hat A\rangle - \left(\langle \hat a^\dagger \hat a\rangle -\langle \hat a^\dagger \rangle\langle \hat a\rangle\right)</math>

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.

==Linear phase-invariant amplifiers==

Linear phase-invariant (also known as phase-insensitive or phase preserving<ref>{{cite journal | last1=Clerk | first1=A. A.| last2=Devoret | first2=M. H. | last3=Girvin | first3=S. M.| last4=Marquardt | first4=Florian | last5=Schoelkopf | first5=R. J. | title=Introduction to Quantum Noise, Measurement and Amplification | journal=Reviews of Modern Physics | volume=82 | issue=2 | pages=1155–1208 | date = 15 April 2010| doi=10.1103/RevModPhys.82.1155 | arxiv=0810.4729 | bibcode=2010RvMP...82.1155C}}</ref>) amplifiers may be described as follows. Assume that the unitary operator <math>~\hat U~</math> amplifies in such a way that the input <math>~\hat a~</math> and the output <math>~\hat A={\hat U}^\dagger \hat a \hat U~</math> are related by a linear equation

:<math>~\hat A = c \hat a + s \hat b^\dagger,</math>

where <math>~c~</math> and <math>~s~</math> are [[Physics:C-number|c-number]]s and <math>~\hat b^\dagger~</math> is a creation operator characterizing the amplifier. Without loss of generality, it may be assumed that <math>~c~</math> and <math>~s~</math> are [[Real number|real]]. The commutator of the field operators is invariant under unitary transformation <math>~\hat U~ </math>:

:<math>\hat A\hat A^\dagger -\hat A^\dagger\hat A =\hat a\hat a^\dagger -\hat a^\dagger \hat a=1</math>

From the unitarity of <math>~\hat U~ </math>, it follows that <math>~ \hat b~ </math> satisfies the canonical commutation relations for operators with Bose statistics:

:<math> ~\hat b\hat b^\dagger -\hat b^\dagger \hat b=1~</math>

The c-numbers are then<ref name="inva">{{cite journal|title=Quantum limit of noise of a phase-invariant amplifier| author=D. Kouznetsov |author2=D. Rohrlich |author3=R.Ortega|year=1995|journal=[[Physical Review A]]|volume=52|issue=2|pages=1665–1669|doi=10.1103/PhysRevA.52.1665| pmid=9912406 |bibcode=1995PhRvA..52.1665K|arxiv = cond-mat/9407011 | s2cid=19495906 }}</ref>

:<math> ~c^2 \!-\! s^2=1~</math>

Hence, the phase-invariant amplifier acts by introducing an additional [[Physics:Bosonic|bosonic]] mode. The gain and added noise of this amplifier are

:<math>~~G\!=\!c~~</math>

and

:<math>~~{\rm added~noise} =c^2\!-\!1.</math>

The coefficient <math>~~ g\!=\!|G|^2~~</math> is sometimes called the ''intensity amplification coefficient''. The noise of the linear phase-invariant amplifier is given by <math>g-1</math>. The gain can be dropped by splitting the beam; the estimate above gives the minimal possible noise of the linear phase-invariant amplifier.

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

To obtain a large amplification coefficient with minimal noise, one may use [[Homodyne detection|homodyne detection]], constructing a field state with known amplitude and phase, corresponding to the linear phase-invariant amplifier.<ref name="ande">{{cite journal|author=Vincent Josse|author2=Metin Sabuncu |author3=Nicolas J. Cerf |author4=Gerd Leuchs |author5=Ulrik L. Andersen |title=Universal Optical Amplification without Nonlinearity|journal=[[Physics:Physical Review Letters|Physical Review Letters]]|volume=96|issue=16|article-number=163602|year=2007|doi=10.1103/PhysRevLett.96.163602|bibcode=2006PhRvL..96p3602J|arxiv = quant-ph/0603119 |pmid=16712228|s2cid=7706450 }}</ref> The [[Uncertainty principle|uncertainty principle]] sets the lower bound of [[Physics:Quantum noise|quantum noise]] in an amplifier. In particular, the output of a laser system and the output of an optical generator are not coherent states.

==Nonlinear amplifiers==
Nonlinear amplifiers do not have a linear relation between their input and output. The maximum noise of a nonlinear amplifier cannot be much smaller than that of an idealized linear amplifier.<ref name="inva"/> This limit is determined by the [[Derivative|derivative]]s of the mapping function; a larger derivative implies an amplifier with greater uncertainty.<ref name="Qmap">{{cite journal|author=D. Kouznetsov|author2=D. Rohrlich|title=Quantum noise in the mapping of phase space|journal=[[Physics:Optics and Spectroscopy|Optics and Spectroscopy]]|volume=82|issue=6|pages=909–913|year=1997|url=http://www.maik.rssi.ru/abstract/optics/97/optics6_97p909abs.htm|bibcode=1997OptSp..82..909K|access-date=2007-12-28|archive-date=2016-03-03|archive-url=https://web.archive.org/web/20160303184907/http://www.maik.rssi.ru/abstract/optics/97/optics6_97p909abs.htm}}</ref> 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 oscillator|parametric oscillator]]s, which amplify while shifting the phase of the input.

==References==
{{reflist}}

==Further reading==
*{{cite journal | author = Kim M. S., Lee K. S., Bužek V. | year = 1993 | title = Amplification of superposition states in phase-sensitive amplifiers | journal = Phys. Rev. A | volume = 47 | issue = 5 | pages = 4302–4307 | bibcode = 1993PhRvA..47.4302K | doi = 10.1103/PhysRevA.47.4302 | pmid = 9909437 }}
*{{cite journal | author = Bondurant R. S. | year = 1993 | title = Quantum noise properties of a nonlinear amplifier | journal = Phys. Rev. Lett. | volume = 71 | issue = 11 | pages = 1709–1711 | bibcode = 1993PhRvL..71.1709B | doi = 10.1103/PhysRevLett.71.1709 | pmid = 10054478 }}
*{{cite journal | author = Mu Yi, Savage C. M. | year = 1994 | title = Phase-sensitive above-threshold laser amplifiers | journal = Phys. Rev. A | volume = 49 | issue = 5 | pages = 4093–4100 | bibcode = 1994PhRvA..49.4093M | doi = 10.1103/PhysRevA.49.4093 | pmid = 9910709 }}
*{{cite journal | author = Vaccaro John A., Pegg D. T. | year = 1994 | title = Phase properties of optical linear amplifiers | journal = Phys. Rev. A | volume = 49 | issue = 6 | pages = 4985–4995 | bibcode = 1994PhRvA..49.4985V | doi = 10.1103/PhysRevA.49.4985 | pmid = 9910820 }}
*{{cite journal | author = Loudon Rodney, Jedrkiewicz Ottavia, Barnett Stephen M., Jeffers John | year = 2003 | title = Quantum limits on noise in dual input-output linear optical amplifiers and attenuators | journal = Phys. Rev. A | volume = 67 | issue = 1 | page = 043803 | arxiv = quant-ph/0212012 | bibcode = 2003PhRvA..67a3803K | doi = 10.1103/PhysRevA.67.013803 | s2cid = 5334606 }}
*{{cite journal | author = Lamprecht C., Ritsch H. | year = 2003 | title = Theory of excess noise in unstable resonator lasers | journal = Phys. Rev. A | volume = 67 | issue = 1 | article-number = 013805 | arxiv = quant-ph/0203122 | bibcode = 2003PhRvA..67a3805V | doi = 10.1103/PhysRevA.67.013805 | s2cid = 119102253 }}

{{Quantum mechanics topics}}
{{emerging technologies|quantum=yes|other=yes}}

[[Category:Quantum optics]]

{{Sourceattribution|Quantum amplifier}}

Physics:Quantum amplifier - Revision history

imported>WikiHarold: fix

imported>WikiHarold: fix