# Shot noise at high frequencies (can it really be ideal white noise?)

Quantum shot noise (either optical intensity noise or electrical current noise) described by the noise spectral density of $2 e I$ (electrically) or $2 h \nu P$ (optically).

So it is white noise. I know this basically comes from the derivation, where we model electron or photon events as infinitely short (with delta impulses). See e.g. Eq 1 in http://123.physics.ucdavis.edu/shot_files/ShotNoise.pdf.

$$I(t)=\sum_j q \;\delta(t-t_j)$$

So far so good, but can this really be true in an exact sense? A consequence would be that with growing bandwidth, the fluctuation (variance) grows indefinitely. That cant be physically correct.

So things I would like to know specifically:

• Are there any experimental results where the spectral noise density of shot noise is measured? If e.g. there is a (nonzero) transit time of electrons, we should see some interesting things happen in the noise density at the inverse transit time. has something like this been observed?

• Is there a "more correct" (quantum physical) theory to derive shot noise? I.e. gives the correct spectral density? If yes, what is it and what is the idea behind the theory?

• Might there be a fundamental reason why it is impossible to measure shot noise at high frequencies? e.g. there is a fundamental limit for the gain bandwidth product of amplifiers (with which we could observe the shot noise at arbitrary frequencies in, for example, a diode)?

• Shot noise is modeled using the Poisson Distribution, because we believe that each particle's arrival is independent of all others. All measurements made that I know of, both optical and electronic, agree with this model. – Carl Witthoft Jan 25 '14 at 21:48
• @CarlWitthoft: I understand that. But besides independence of arrivals, there is also the assumption of infinitely short/instantaneous arrivals. (cf. Eq 1 in the mentioned paper). If you would replace the delta function with some finite duration impulse, shot noise is no longer white noise, but decays at certain (extremely high) frequency. I just wonder how white noise can actually be physical (the infinite variance problem). – Andreas H. Jan 25 '14 at 22:00
• I rather doubt you can perform a measurement of the temporal spread of a single photon's ejection of a band electron in any meaningful way. Either there's an electron or there isn't-- there is no continuum. – Carl Witthoft Jan 26 '14 at 12:49
• I wrote a somewhat involved answer, and I really hope it helps. In the future, try to limit your posts to one single question at a time. It helps the answers remain focused which usually means they're higher quality. – DanielSank Dec 28 '14 at 6:23

Are there any experimental results where the spectral noise density of shot noise is measured?

Absolutely, yes. Here are just a few experiments measuring co-called "quantum shot noise":

R.J. Schoelkopf, P.J. Burke, A.A.Kozhevnikov, D.E. Prober, and M.J. Rooks, Physical Review Letters, Vol 78, No. 17, p. 3370, April 1997.

A. A. Kozhevnikov, R. J. Schoelkopf, and D. E. Prober. Physical Review Letters, Vol 84, No. 15, April 2000

R. J. Schoelkopf, A. A. Kozhevnikov, D. E. Prober, M. J. Rooks. Physical Review Letters, Vol 80, No. 11, March 1998

Dissertation on shot noise thermometry [PDF]

If e.g. there is a (nonzero) transit time of electrons, we should see some interesting things happen in the noise density at the inverse transit time. Has something like this been observed?

Yes, you observe it every day. If the charge pulses induced by electrons flowing in circuits were really delta functions then as you know the noise spectral density would be constant (i.e. white noise). That would mean that there is an infinite amount of power in the noise. This is obviously nonsense. The spectral density decreases away from the white noise limit as you get to frequencies larger than $1/T$ where $T$ is the characteristic width of the electron pulse shape. In conventional consumer electronics, those pulse shapes are determined by the nonzero resistance and capacitance of the circuits which act as filters.

I looked around to see if I could find a plot of the shot noise measured in a vacuum tube circuit so you could really see how the spectral density rolls off at higher frequencies, but I didn't find quite what I wanted yet $^{[a]}$. If you look around you may find something nice.

Is there a "more correct" (quantum physical) theory to derive shot noise? E.g. gives the correct spectral density? If yes, what is it and what is the idea behind the theory?

Now you're getting into something subtle. What people call "quantum shot noise" is not the same thing as what you imagine when you draw picture of randomly timed pulses travelling down a wire. The term "shot noise" makes the most sense in a classical situation where you throw a hand full of shot (little beads) in the air and let them fall on the ground. As they land, you get a random train of sound pulses. These sounds pulses can be thought of as a noise whose spectral density can be computed from knowledge of the pulse shapes (which are not delta functions) and the statistics of the arrival times (e.g. average of 5 shots per second with Poisson distribution in time). That's the classical shot noise you already know about.

In quantum mechanics, if you sit there with a photon counter in front of a laser beam, your counter will click at random times. This has to do with the fundamental nature of quantum mechanics and what's going on when you measure the laser beam: your detector sucks a photon out of the laser beam, and this happens at a random, but statistically describable time. This "noise" is called "quantum shot noise" and is really not the same thing as the shot noise we discussed above. They're similar because they both involve random arrival times of little signals, but the origins are so different that it bugs me that these are called the same thing. A canonical paper on quantum shot noise is this review article by Aash Clerk et al.

Might there be a fundamental reason why it is impossible to measure shot noise at high frequencies? e.g. there is a fundamental limit for the gain bandwidth product of amplifiers (with which we could observe the shot noise at arbitrary frequencies in, for example, a diode)?

In any real system there are fundamental limits. For example, if I make an amplifier out of silicon then i definitely can't use it at frequencies $f$ high enough that $h f$ is bigger than the silicon bandgap. The energy in such signals is enough to cause my silicon to stop acting like a nice semiconductor, so my circuit won't work as designed. There are other problems which come into play way before that one though. Even at frequencies around $10\,$GHz you have problems with stray capacitance and inductance screwing up your electronics. The fundamental issue there is that the inductance of a physical piece of metal of linear dimension $d$ is roughly equal to $L \approx \mu_0 d$ (by dimensional analysis). The impedance of an inductor is $Z = 2\pi f L$, so as you raise the frequency your stray impedance goes up and what you hoped would act like a wire is now acting like an inductor. To fix this you can try to lower $L$ by lowering $d$, E.g. shrinking your device's physical size, but eventually you run out of room. There are lots of other fundamental physical limitations like this.

$[a]$: The problem is that in the old days when people did physics experiments with vacuum tubes they plotted things in a way which is kind of hard to grok.

Regarding your second question, shot noise for optical measurements does arise naturally when you consider the electromagnetic waves to be quantized. It is, in its most basic form, the zero-point energy of each mode of the radiation.

To quote Jeffrey Shapiro, at the risk of being sensationalist,

Local oscillator shot noise is a semiclassical fiction; the noise seen in homodyne detection (with an ideal local oscillator) is local oscillator quantum noise, plus n~l quantum noise, plus signal quantum noise.

• Thanks, do you have any reference that derives (short and explicit) shot noise from these zero-point fluctuation? – Andreas H. Jan 25 '14 at 22:14
• Nothing particularly short. Try having a look at Gerry & Knight's Quantum Optics textbook, and particularly the first two chapters, but if that's too long or complicated I don't really know what to recommend – Emilio Pisanty Jan 25 '14 at 22:32