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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^^
ABERDEEN PROVING GROUND, MARYLAND
BALLISTIC RESEARCH LABORATORIES
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
BALLISTIC RESEARCH LABORATORIES
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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