Quantum amplifier

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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.

Contents

[edit] Introduction

The analysis of quantum amplifiers implies application of quantum mechanics. There is important question about the noise of the quantum amplifier. For the basic properties of quantum amplifiers, the abstract mathematical terminology is used. In the most of cases, quantum amplifiers deal with a coherent (quasi-classical) signal. The weak light from a coherent optical communication system needs to be amplified before application. In this sense, all optical re-transmitters can be considered as quantum amplifiers. The optical quantum amplifier with feedback may generate coherent light. In this sense, the active element of any laser is also quantum amplifier. For the analysis of noise properties of quantum amplifiers, the mathematical idealization below is used.

[edit] Single-mode amplifier

Mathematically, a quantum amplifier is a unitary transform \hat U, which converts the initial quantum state of a system into an amplified state; this is concept of Quantum amplifier in the Schroedinger representation. In particular, the initial state can be a coherent state \langle \hat A^\dagger \hat A\rangle - \langle \hat A^\dagger \rangle \langle \hat A\rangle. It can be written as follows: ~|{\rm final}\rangle = U |\rm initial \rangle.

The quantum amplifier is characterized with the coefficient of amplification ~G~. Usually, one uses the coherent state for definition of G:

G= \frac {\left\langle\hat a\right\rangle _{\rm final}} {\left\langle\hat a\right\rangle _{\rm initial}}

Here, \hat a is the field operator, or the annihilation operator. 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 {\left\langle\hat A\right\rangle _{\rm initial}} {\left\langle\hat a\right\rangle _{\rm initial}}~

We assume that the mean value of the initial field ~{\left\langle\hat a\right\rangle_{\rm initial}} \ne 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, 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.

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, ~~|{\rm initial}\rangle=|\alpha\rangle~. Even after such a restriction, the coefficient of amplification may depend on the amplitude 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.

Important question is about the noise of a quantum amplifier,

{\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)

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 the characterization shows how different from the coherent state is the output field.

[edit] Linear phase-invariant amplifier

is of course, a mathematical abstraction. Assume, that the initial field ~\hat a~ and the final field ~\hat A~ 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 A\hat A^\dagger -\hat A^\dagger\hat A = \hat a\hat a^\dagger -\hat a^\dagger \hat a=1

Form Unitarity of ~\hat U~, it follows [[1]], that the operator of the amplifier ~ \hat b~ also satisfies the standard commutation relation for the Bose operators; ~\hat b\hat 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 ~~{\rm 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.

[edit] Nonlinear amplifier

In more general case, the mean value of the output field is some function of the input filed; so, we can speak about nonlinear mapping of phase space. Last century, there were some speculations about the possibility of improving the sensitivity of detectors amplifying the real input fields with a nonlinear quantum amplifier. It might be not so prudent that the original papers by V.Dubrovich were not published, but the criticisms in ref.[2] got wide circulation; that paper is shows that similar lower bounds for the noise of quantum amplifier hold also for nonlinear amplifiers.

[2] D. Kouznetsov, D. Rohrlich. Quantum noise in the mapping of phase space. English version: Optics and Spectroscopy, July 1997, v.83, N6, p.909-913.

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