Quantum amplifier

In physics, the term quantum amplifier may refer to any device, which use some non-classical (quantum) effects for amplification of signal. In this sense, the active element of an optical laser can be considered as quantum amplifier.The weak light from a coherent optical communication system needs to be amplified before application. Therefore, the optical re-transmitters usually include quantum amplifiers.

Main properties of the quantum amplifier are its amplification coefficient and noise.These parameters are not independent; the higher is the amplification coefficient, thehigher is the noise. In the case of lasers, the noise corresponds to the ASE of the active medium.The unavoidable noise of quantum amplifiers is one of reasons for use of digital signals in the optical communications. The minimum of the noise can be deduced from the fundamentals of quantum mechanics. This analysis uses formalism of quantum mechanics.

Introduction

Amplifiers increase the amplitude of the field; the amplitude of field at the output is supposed to be larger than that at the input. In the most of cases, quantum amplifiers deal with quasi-classical signal. In particular, the input of the amplifier can be coherent state. Although an optical quantum amplifier with feedback (optical generator) may generate highly coherent light, formally, the output state of the amplifier is not a coherent state, (We assume, the amplification coefficient is larger than unity.)The amplifier increases the uncertanty of the field, it becomes larger than that of the coherent state.

For the analysis of noise properties of quantum amplifiers, the mathematical idealization below is used.

Single-mode amplifier

The physical electric field in a paraxial single-mode pulse can be approximated with superposition of modes;the parcial field $~E\_\{\; m\; phys\}~$ of one such a mode can be described with $vec\; E\_\{\; m\; phys\}(vec\; x)~=~\; vec\; e~\; a~\; M(vec\; x)~exp(ikz-\{\; m\; i\}omega\; t)\; ~+~\; \{\; m\; Hermitian~Conjugate\}~$where $~vec\; x\; =\{x\_1,x\_2,z\; \}~$ is vector of spatial coordinates,$~vec\; e\; ~$ is vector of polarization,$~k~$ is wavevector,$~a~$ is the operator of annihilation of photon in some specific mode$~\; M(vec\; x)\; ~$. We assume that the amplitude of field in this mode can be detected, for example, withsome kind of homodyne detection. Practically, all the quantum-mechanical analysis of the noise is about the mean value of the annihilation operator and its uncertainty; all the technical problems with detection of the real and imaginary parts of the projection of the field to a given mode $~\; M(vec\; x)\; ~$ are supposed to be solved. Therefore, the spatial coordinates do not appear in the deduction, and even symbols $~x~$ may be used for Hermitian and anti-Hermitian parts of the annihilation operator,

Assume that the mean value of the initial field $~\{leftlanglehat\; a\; ight\; angle\_\{\; m\; initial\; e\; 0~$. Physically, the initial state may correspond to the coherent pulse at the input of the optical amplifier, and the final state may correspond to the output pulse. We assume that the amplitude-phase behavior of the pulse are known(id est, we know the mode $~M~$), and the only quantum state of the corresponding mode is important. Only in this case we may treat such a pulse in terms of a single-mode field.

Mathematically, a quantum amplifier is a unitary transform $hat\; U$, which converts the input quantum state $~|\{\; m\; initial\}\; angle~$ to the amplified state$~|\{\; m\; final\}\; angle~$; this is concept of quantum amplifier in the Schroedinger representation.In particular, the initial state can be a coherent state.

The amplification may change as the mean value $~langle\; a\; angle\; ~$ of the field operator $~a~$ and its dispersion $~langle\; hat\; a^dagger\; hat\; a\; angle\; -langle\; hat\; a^dagger\; anglelangle\; hat\; a\; angle~$. The coherent state realizes the minimum of uncertainty. As the state is transformed, the uncertainty may increase. This increase can be interpreted as noise to the amplifier.

The transformation of state can be written as follows: $~|\{\; m\; final\}\; angle\; =\; U\; |\; m\; initial\; angle$

The coefficient $~G~$ of amplification can be defined as follows: $G=\; frac\{leftlanglehat\; a\; ight\; angle\; \_\{\; m\; final\{leftlanglehat\; a\; ight\; angle\; \_\{\; m\; initial$ This expression can be written also in the Heisenberg representation; we may attribute all the changes due to the amplification to the operator of field, and define $~\; hat\; A\; =hat\; U^dagger\; hat\; a\; hat\; U~$, keeping the vector of state unchanged. Then, $~\; G=\; frac\{leftlanglehat\; A\; ight\; angle\; \_\{\; m\; initial\{leftlanglehat\; a\; ight\; angle\; \_\{\; m\; initial~$

In general case, the coefficient $~G~$ may be complex, and if may depend on the initial state. For application to lasers, the amplification of coherent state is important. Therefore, it is usually assumed, that the initial state is coherent state. characterized with a complex parameter $~alpha~$, that is, $~~|\{\; m\; initial\}\; angle=|alpha\; angle~$. Even after such a restriction, the coefficient of amplification may depend on the amplitude (or even the phase) of the initial field. In the following consideration, the only Heisenberg representation is used; all the brackets are assumed to be evaluated with respect to initial coherent state.

The noise of the amplifier can be defined as follows: $\{\; m\; noise\}=\; langle\; hat\; A^dagger\; hat\; A\; angle\; -langle\; hat\; A^dagger\; anglelangle\; hat\; A\; angle\; -\; left(langle\; hat\; a^dagger\; hat\; a\; angle\; -langle\; hat\; a^dagger\; anglelangle\; hat\; a\; angle\; ight)$

This quantity characterizes the increase of uncertainty of the field due to the amplification. As the uncertainty of the field operator at the coherent state does not depend on its parameter, the quantity above shows, how different from the coherent state is the output field.

Linear phase-invariant amplifier

is of course, a mathematical abstraction. Assume, that the unitary operator $~hat\; U~$ realizes the amplifivation is such a way that the initial field $~hat\; a~$ and the final field $~hat\; A=\{hat\; U\}^dagger\; hat\; a\; hat\; U~$ are related with a linear equation

$~hat\; A\; =\; c\; hat\; a\; +\; s\; hat\; b^dagger\; ~~~,$

where $~c~$ and $~s~$ are c-numbers and $~hat\; b^dagger~$is some operator of internal degrees of freedom of the amplifier (device which provides the amplification).Without loss of generality, we may assume that $~c~$ and $~s~$ are real.The unitary transformation $~hat\; U~$ preserves the commutator of the filed operators, so, $hat\; Ahat\; A^dagger\; -hat\; A^daggerhat\; A\; =hat\; ahat\; a^dagger\; -hat\; a^dagger\; hat\; a=1$

Form unitarity of $~hat\; U~$, it follows cite journal
title= Quantum limit of noise of a phase-invariant amplifier
author=D. Kouznetsov
coauthors=D. Rohrlich, R.Ortega
journal=PRA
volume=52
issue=2
pages=1665–1569
year=1995
url=http://prola.aps.org/abstract/PRA/v52/i2/p1665_1
doi= 10.1103/PhysRevA.52.1665] ,that the operator of the amplifier $~\; hat\; b~$also satisfies the standard commutation relation for the Bose operators; $~hat\; bhat\; b^dagger\; -hat\; b^dagger\; hat\; b=1~$

and $~c^2\; !-!\; s^2=1~.$

In this sense, the amplifier is equivalent to some additional mode of the field with enormous energy stored, and the lack of excitation of this mode behave as a boson.

Calculating the gain and the noise of the linear quantum amplifier, we get$~~G!=!c~~$and $~~\{\; m\; noise\}\; =c^2!-!1~~~.~$

The coefficient $~~\; g!=!|G|^2~~$ should be interpreted as the intensity amplification coefficient,and the noise of the linear amplifier is just intensity amplification coefficient minus unity.

However, the gain can be dropped out by the splitting of the beam;therefore, the estimate above is minimal noise of the linear quantum amplifier.

The linear amplifier have advantage (at least for consideration) than the multi-mode case is straight-forward. In particular, if several modes (or even continuum) of modes of a field are amplified with the same amplification coefficient, the noise in each mode is still determined with the estimate above. In this sense, modes in a linear quantum amplifier are independent.

For large amplification coefficient, the minimal noise can be realized with homodin detection and analogous construction of a field state with measured amplitude and phase; formally this also corresponds to the linear phase-invariant amplifiercite journal
author=Vincent Josse
coauthor=Metin Sabuncu, Nicolas J. Cerf, Gerd Leuchs, and Ulrik L. Andersen
title=Universal Optical Amplification without Nonlinearity
journal=PRL
issue=75
pages=022002
year=2007
doi=10.1103/PhysRevLett.96.163602
url= http://link.aps.org/abstract/PRL/v96/e163602
volume=96] .

The fundamentals of quantum mechanics set the lower bound of quantum noise in any kind of amplifiers. In particular, the output field of laset system withmaster oscillator-power amplifier is not a coherent state. In similar way, the quantum fluctuations in the output of the optical generator with unstable cavity are stronger those for a coherent state.

Nonlinear amplifier

The amplifier can be used as a main element of the optical quantum generator. The amplifier with feedback becomes a master oscillator.If the amplifier is almost linear, then, the laser works close to the threshold; quantum fluctuations in such a laser are strong; and the most of lasers(especially stabilised lasers) work in deeply nonlinear regime. The nonlinear amplifier may still preserve the phase of the field and stabilize the amplitude.However, the noise of nonlinear amplifier cannot be much smaller than that of an idealized linear amplifier cite journal
title= Quantum limit of noise of a phase-invariant amplifier
author=D. Kouznetsov
coauthors=D. Rohrlich, R.Ortega
journal=PRA
volume=52
issue=2
pages=1665–1569
year=1995
url=http://prola.aps.org/abstract/PRA/v52/i2/p1665_1
doi= 10.1103/PhysRevA.52.1665] .

In more general case, the phase of output field has no need to follow the phase of the input field. For example, the parametric amplifier amplifies only one of quadrature components of the filed, destroying the phase relation. Generally, the mean value of the output field is some function of the input filed; so, we can speak about nonlinear mapping of phase space. However, even in this case, there are definite restrictions on the reduction of noise; the lower limit of noise follows from the derrivativeds of the mapping function. The larger are the derivatives, the larger the minimal noise of the quantum amplifier.cite journal
author = D. Kouznetsov
coauthors= D. Rohrlich
title = Quantum noise in the mapping of phase space.
journal =Optics and Spectroscopy
volume = 82
issue= 6
pages= 909–913
year=1997
url=http://www.maik.rssi.ru/abstract/optics/97/optics6_97p909abs.htm ]

References

See also
*M. S. Kim, K. S. Lee, and V. Bužek. Amplification of superposition states in phase-sensitive amplifiers. Phys. Rev. A 47, 4302 (1993)
*R. S. Bondurant. Quantum noise properties of a nonlinear amplifier. Phys. Rev. Lett. 71, 1709 (1993)
*Yi Mu and C. M. Savage. Phase-sensitive above-threshold laser amplifiers. Phys. Rev. A 49, 4093 (1994)
*Phase properties of optical linear amplifiers. John A. Vaccaro and D. T. Pegg. Phys. Rev. A 49, 4985 (1994)
* Rodney Loudon, Ottavia Jedrkiewicz, Stephen M. Barnett, and John Jeffers. Quantum limits on noise in dual input-output linear optical amplifiers and attenuators. Phys. Rev. A 67, 043803 (2003)
*C. Lamprecht and H. Ritsch. Theory of excess noise in unstable resonator lasers. Phys. Rev. A 67, 013805(2003)