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liiU TECHNICAL NOTE NO. 1^24 SEPTEIV1BER 1961

RELATIONSHIP BETWEEN ADDITIVE AND

NON-ADDITIVE QUANTUM FLUCTUATIONS

L. P. Bolyjano, Jr.

/:•-.

ASIIA

•■"'./: ()X

Department of the Army Project No. 303-05-023 Ordnance Management Structure Code No. 5210.11.167

BALLISTIC RESEARCH LABORATORIES iMift'i'

mi •" i ''■^n'm''Viin'iiäilJiiiiiil*l£&iiii£i^^

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TECHNICAL NOTE NO. Ik2k

SEPTEMBER I96I

RELATIONSHIP BEWEEN ADDITIVE AND NON-ADDITIVE QUANTUM FLUCTUATIONS

L. P. Bolgiano, Jr.

Ballistic Measurements Laboratory

Department of the Army Project No. 505-05-025 Ordnance Management Structure Code No. 5210.11.16?

ABERDEEN PROVING GROUND, MARYLAND

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TECHNICAL NOTE NO. ll*21*

LPBolgiano,Jr/bj Aberdeen Proving Ground, Maryland September 1961

RELATIONSHIP BETWEEN ADDITIVE AND NON-ADDITIVE QUANTUM FLUCTUATIONS

ABSTRACT

The limitation imposed by quantum mechanics on the measurement of

electromagnetic signals may be viewed as a non-classical noise. Theory

leads both to a minimum noise, which is additive in the sense of being

independent of signal strength, and to a non-additive noise dependent

on signal strength. The relationship between these two types of noise

is discussed. Also a short derivation is given which exhibits their

common relationship to the zero-point fluctuation attributed by

quantum theory to the electromagnetic field.

In discussions of quantum noise limitations in masers ' ' ' , there

appear at first sight to be two separate types of quantum noise predicted.

Spontaneous emission noise, or equivalently zero-point fluctuation of the

field, leads to a minimum noise when no signal is present. On the other

hand, the statistical nature of the amplification process which does not

correlate specific output photons to specific input photons gives rise

to an uncertainty in the precision with which the mean number of

incident quanta, K, can be measured equal to the [/N (for large N).

That some relationship should exist between these fluctuations is

suggested by Heisenberg's derivation of the Einstein formula ' for

fluctuations in black-body radiation. In this calculation the mean

square "particle'1 fluctuation, equal to the- number of quanta involved,

results directly from the zero-point energy of the field oscillators.

Serber and Townes ' , in fact, point out that a [/N~fluctuation in

signal energy can be obtained by assuming that the signal amplitude

is augmented by a random fluctuation of amount corresponding to the

in-phase component of a noise signal whose energy corresponds to the

zero-point fluctuation.

We wish here to point out that, using a similar argument, it is

possible to obtain a single f-rmula which gives both the I/N" fluctuation

expected for large N and also the minimum fluctuation l/2 hf per degree

of freedom expected for N = 0.

We begin by considering the description given by classical wave

theory for a sine wave plus thermal noise. That is we assume a waveform

representable by an equation of the form

I = As cos cut + A^ (t) cos (oat + 0 (t)) (l)

where the signal amplitude A is constant and the random amplitude

Aj. (t) and phase 0 (t) of the noise vary slowly relative to ü). If this

expression is squared to compute power, there result three terms

corresponding to signal power, noise power, and beats between the two.

It may seem strange to speak of beats when only one frequency appears

explicitly In equation (l). However, the slow variation of /L. (t) and 0 (t)

could be equivalently represented by writing the noise as a Fourier

expansion over a frequency band of width equal to the highest frequency

contained in A,, (t) and 0 (L).

If we conaider only time variations slow with respect to cu, the

signal power terra is steady and the noise power term fluctuates with a

mean square fluctuation equal to the mean square noise power itself.

The interaction term, which averages zero, alternately adds and

subtracts from the total power contributing a mean square fluctuation

proportional to the product of signal and noise power. The total (8)

mean square fluctuation in power which results may be expressed by

A?" = ?,, + 2P P., (2) ii s rj

where P denotes signal power and ?N mean noise power.

In order t: obtain an estimate of quantum fluctuations, we shall

interpret this power fluctuation as an energy fluctuation per unit time

at frequency i . We express the signal energy per unit time as Nhf,

and take for mean noise power and mean square noise power l/2 hf and p

(1/2 hf) respeitively, to correspond to the zero-point fluctuation.

This gives for Lne mean square energy fluctuation

AVf - {l/2 hf) r 2(Nhf) (l/2 hf)

= (N - I/.) (hf)2, (3)

and for the r^ot mean square fluctuation in energy

y^2 = hf /N f l/k . {h)

For large N this formula gives the non-additive /N fluctuation

normally associated with Poisson's "law of rare events". For N = 0,

it also gives the minimum zero-point fluctuation l/2 hf.

The significance of the non-additive I/N fluctuation for microwave (9) radiometry was dlscusaed in a previous paper . Since only the

non-additive quantum fluctuation was considered, it was necessary to

qualify the resultü as applying only if N is not too small. The

derivation given here suggests that this restriction might be

removed by replacing N by N + l/k in the formula found for the mean

square energy fluctuation.

L. P. BOLGIANO, JE.

REFERENCES

i. Slegman, A. E. Zero-point Energy as the Source of Amplifier Noise. Proc. IRE 1*9, 635 (1961).

2. Siegjnan, A. E. Thermal Noise in Microwave Systems, Part III. Microwave Journal h, 95-iOU (1961).

;. Shimoda, K., Takahasi, H. and Townes, C. H. Fluctuations in Amplification of Quanta with Applications to Maser Amplifiers. J. Phys, Soc. Japan 12, 660 (1957).

•.. Jtern , T. I, Information Rates in Photon Channels and Photon Amplifiers. IRE Int'l Conv. Rec. pt, it, 102-186 (i960) - especially formula (11) page loh.

'■-, Keisenhtrrg, W. The Physical Principles of the Quantum Theory. Dover Publu-ations, New York (1950), page 100.

0. Einstein, A. Zum gegenwärtigen Stand des Strahlungsprobieras. Phys. Zeitschrift 10, 1C5-195 (1909).

7. derber, R. und Townes, C. H. Limits on Electromagnetic Amplification due to Complementarity. Quantum Electronics, ed. by C. H. Townes, Columbia University Press, New York (i960), page P.y:.

3rillouiri, ,',. oc'.ence and [nformation Theory. Academic Press, New York djyj) pp. 2p7-&.

9. Bolgiano. L, ':'., Jr. Quantum Fjuctuations in Microwave Radiometry. IRE Trans, on M.T. and T. MTT - 9, 515-521 (l96l).

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