Time to get serious. How does the Holometer, which is a pair of [interferometers](https://pedagoguepadawan.net/68/holometerinterferometer/) detect [holographic noise](https://pedagoguepadawan.net/66/holographicnoise/)? Fortunately, I met with Craig Hogan who is the Director of the Fermilab Center for Particle Astrophysics and a professor at University of Chicago since I wasn’t sure.

I’m going to attempt to explain why the holographic noise is measurable with the Holometer by way of analogy. In fact, there are several analogies that are applicable (Brownian motion, diffraction limits, black hole information theory, Heisenberg uncertainty principle). However, I want to be clear that we are talking about New Physics. You can’t explain this phenomenon using classical or quantum physics; it is not an application of any of the above phenomena; they serve only as analogies. That’s why it is New Physics. I’ve heard this New Physics referred in a couple of ways, but I like Planckian Physics best.

I know that we want to get to the Holometer, but first we need to introduce the Heisenberg uncertainty principle. Due to wave-particle duality, the Heisenberg uncertainty principles states that the more precisely we measure the position of a particle, the less precisely the momentum of that particle can be measured. (The uncertainty principle can also be presented in terms of energy and time, but we’ll stick with position and momentum.) While this inherent uncertainty is small, it is observable at the macroscopic level in some situations. For example, radioactive decay of a nucleus that ejects an alpha particle can only occur due to the uncertainty principle (from an energy and time perspective) via a process called tunneling. Classically, alpha decay violates conservation of energy.

Planckian Physics states that, in the same way that measuring position and momentum result in uncertainty, measuring position in perpendicular directions result in uncertainty. That is, the more precisely we measure the position of a particle in one direction, the less precisely we can measure the position of the particle in a perpendicular direction. Stated another way, precisely measuring the position of a particle in one directions results in transverse jitter in the position of the particle. This jitter is the holographic noise.

Hopefully, this explains why it is not possible to measure the holographic noise in a single dimension. For example, bouncing a photon off of a mirror and measuring when it returns cannot measure the holographic noise because this measurement only results in a transverse jitter (i.e., motion perpendicular to the motion of the photon) which would not affect the measurement.

Therefore, we need to employ an interferometer. When the position of a photon along one arm of the interferometer is precisely measured, the transverse jitter of the beam splitter results in noise in the interference measured by the photodiode.

(diagram from Professor Craig Hogan)

The beam splitter executes a random walk of one Planck length per Planck time as the photon travels the length of the arm. This answers two of the questions from the [previous post](https://pedagoguepadawan.net/81/holometerspectralanalysis/): why are the Holometer arms 40-m long and why do we look for the holographic noise at 4 MHz? It takes a photon 267 ns to travel 80 m (40 m down and 40 m back). While 267 ns may not seem like long, it is 5 x 10³⁶ Planck times. That means that, if the beam splitter may move a Planck length (1.6 x 10^-35 m) for each Planck time, the beam splitter’s transverse jitter (due to the random walk) is on the order of 10^-17 m which can be measured by an interferometer with clever spectral analysis. The holographic noise is present at all frequencies, but we are most sensitive to it at 4 MHz because that is the . and nearly constant from 0 Hz to near 3.75 MHz, where it drops off to zero (3.75 MHz is the frequency at which photos return to the beam splitter (the speed of light divided by 80 m)). (Revised Mon Jul 18 23:01:57 CDT 2011)

I haven’t yet explained why the Holometer consists of a pair of interferometers and why the theory predicts that the holographic noise of the two interferometers will be correlated if their space-time overlap and the holographic noise will not be correlated if their space-time does not overlap. Honestly, I don’t understand this aspect of the experiment yet. That’s okay because I get to meet with Professor Hogan again next week. I’m hoping to clarify my understanding by proposing several thought experiments and seeing if I can connect the resulting explanations. I think that the fact that the photons traveling down the arms are entangled is key. I proposed an experiment where two coherent photons are shot down each arm at exactly the same time and the resulting interference is measured. I would expect that the holographic noise would not be present since the photons are not entangled. However, I’m not sure if this thought experiment is possible to execute without entangling the photons in some manner. I’m going to study up on collapsing wave functions before our next meeting. If you have any questions, please comment!

This post is one in a series about The Holometer experiment and my work at Fermilab in the Summer of 2011:

* [Holometer: Holographic Noise](https://pedagoguepadawan.net/66/holographicnoise/)
* [Holometer: Interferometer](https://pedagoguepadawan.net/68/holometerinterferometer/)
* [Holometer: Spectral Analysis](https://pedagoguepadawan.net/81/holometerspectralanalysis/)
* Holometer: Transverse Jitter
* [Holometer: Correlated Interferometers](https://pedagoguepadawan.net/94/holometercorrelatedinterferometers/)
* [Holometer: Computer-Based Measurements](https://pedagoguepadawan.net/111/holometercomputerbasedmeasurements/)

This post introduces some of the concepts that are important when analyzing the data measured from the Holometer. The [last post](https://pedagoguepadawan.net/68/holometerinterferometer/) introduced the concept of an interferometer. In the Holometer, the photodiodes measure the intensity of the laser at the dark port. As described in the previous post, if everything was perfectly aligned and nothing moved, the dark port would be completely dark. However, in reality, everything is moving a bit and the intensity of laser at the dark port varies. The photodiode produces a voltage corresponding to the intensity of the laser which is sampled by a computer-based measurement system. (I’ll write much more about the computer-based measurement system later since it is the one part of this experiment with which I have expertise.) If this voltage is plotted over time, it looks like noise. That is, there is no discernible pattern to the voltages. Here is an example (not from a photodiode, but a simulated signal):

It is very difficult to see repeating patterns in a signal when there are either multiple patterns present or there is significant random noise. In order to see these repeating patterns we travel to a parallel universe of sorts. In this parallel universe we observe the signal in the frequency domain (amplitude as a function of frequency) instead of in the time domain (amplitude as a function of time). This is done with a Fourier transform. I’m not going to focus on how to calculate Fourier transforms of signals, but I do want to describe Fourier transforms conceptually.

Let’s start with a sine wave with a frequency of 4 Hz, an amplitude of 1 V, and a phase of 0 degrees. Frequency is how many cycles of the sine wave are in every second; amplitude is the difference in volts from the equilibrium position (0 V in this case) to the maximum voltage; and phase is where the sine wave starts at time 0 s. I’m not going to worry about the phase at all. It looks like this in the time-domain (graph of amplitude in Volts vs. time in seconds):

It could also be described in the frequency domain (amplitude in Vrms vs. time in seconds):

The amplitude of the signal in the frequency domain is zero everywhere except at 4 Hz, where it is 0.7 V (Vrms value of an 1-V amplitude sine wave). (For those of you wondering, I’m not going to discuss the phase information in the frequency domain.)

What if our signal is the combination of two sine waves? Let’s add a second sine wave with a frequency of 10 Hz, an amplitude of 1.5 V, and a phase of 0 degrees. The signal still looks like a repeating pattern, but is more complicated:

However, the signal is easily described in the frequency domain:

The amplitude of the signal in the frequency domain is zero everywhere except at 4 Hz and 10 Hz.

The idea behind a frequency-domain representation of a signal is that **any signal can be represented as a combination of sine waves of various frequencies, amplitudes, and phases**. Here’s an example of a signal that is a combination of five different sine waves of various frequency and amplitudes:

and here are the five frequencies clearly shown in the frequency domain:

Good luck determining that the time-domain graph was comprised of five sine waves of frequency 4 Hz, 10 Hz, 13 Hz, 15 Hz, and 18 Hz.

Let’s go back and look at the original signal in the frequency domain:

It is now evident that what appeared to be random noise is composed of a 60 Hz signal. The signal to noise ratio for this example was worse than 1/50. That is, the amplitude of the noise was more than 50 times greater than that of the 60 Hz signal. This is the power of analyzing signals in the frequency domain. When performing this analysis, the Fourier transform is calculated multiple times and the results are averaged together. This reduces the influence of the random noise and makes the repeating signal more apparent. There are other techniques besides averaging to improve the measurement such as windowing, but we aren’t going to worry about that for now.

There are many things that contribute to the noise measured by the photodiode in the interferometer. A truck driving by, seismic activity, vibrations due to a fan, 60 Hz AC power, and the digitizer itself are all possibilities. In addition, the holographic noise presented in [the first post](https://pedagoguepadawan.net/66/holographicnoise/) in this series is also present. If the two interferometers which make up the Holometer are isolated from each other, than much of their noise will be uncorrelated, that is, not related to each other. Some sources of noise would be correlated, such as seismic activity and a truck driving near the experiment. However, these sources of noise occur at much lower frequencies that the holographic noise. If the two interferometers overlap in their space-time (I’ll explain this more later), than the holographic noise should be correlated. We can remove uncorrelated signals between two signals by calculating the cross power spectrum. As an example, suppose I have two signals:

We can filter out uncorrelated signals by calculating the cross-power spectrum for the two signals (again, just focus on the concept and not how the calculation is done). The graph of the resulting cross-power spectrum for these two signals is:

Whoa! There is a 60 Hz correlated signal between the two signals. This signal was lost in the noise when looking at the individual signals in the frequency domain because it was so much weaker than the other components (unless you look really carefully). However, by minimizing the uncorrelated frequencies between the two signals, the 60 Hz correlated signal becomes more apparent. (The uncorrelated 80-Hz and 110-Hz signals are not completely eliminated, but they are minimized.)

The Holometer experiment uses these techniques to search for the correlated holographic noise signal between two adjacent interferometers.

I have the opportunity to meet with Craig Hogan this week and ask him some of my questions which should help me more clearly explain the ideas of the holographic principle, in general, and the Holometer experiment, in particular. I hope to have clear answers for questions such as:

* Why is the interferometer 40-m long?
* Given the incredible small scale of the Planck length, how to the two adjacent interferometers “share” the same space-time?
* What is the connection between the holographic information principle and the Planck length?
* Why is the holographic noise expected to be around 4 MHz?
* What’s up with [this](http://www.wired.com/wiredscience/2011/07/hologram-universe/)?

What questions do you have?

This post is one in a series about The Holometer experiment and my work at Fermilab in the Summer of 2011:

* [Holometer: Holographic Noise](https://pedagoguepadawan.net/66/holographicnoise/)
* [Holometer: Interferometer](https://pedagoguepadawan.net/68/holometerinterferometer/)
* Holometer: Spectral Analysis
* [Holometer: Transverse Jitter](https://pedagoguepadawan.net/83/holometertransversejitter/)
* [Holometer: Correlated Interferometers](https://pedagoguepadawan.net/94/holometercorrelatedinterferometers/)
* [Holometer: Computer-Based Measurements](https://pedagoguepadawan.net/111/holometercomputerbasedmeasurements/)

This is my second post in my Holometer series. The [first one](https://pedagoguepadawan.net/66/holographicnoise/) provided some background information on the Holographic Principle.

The Holometer experiment consists of a pair of 40-m-long interferometers. I don’t yet fully understand why there are two or why they are 40-m long; so, I’ll address the design of the experiment later. For now, I’ll provide some background on interferometers in general.

When discussing interferometers, I have to start with the one of the most important failed experiments: the [Michelson–Morley experiment](http://en.wikipedia.org/wiki/Michelson–Morley_experiment). This is especially the case since it was conducted at my alma matter, Case Western Reserve University. In his efforts to detect the ether, Michelson developed the interferometer:

(Michelson interferometer; source: [wikipedia](http://en.wikipedia.org/wiki/File:Interferometer.svg))

An interferometer has two perpendicular arms. Light is directed at a beam splitter (half-silvered mirror in the diagram) and is split down each arm. While the diagram refers to coherent light, and the Holometer does use a laser, coherent light is not required (and certainly not used by Michelson). The light travels the length of each arm, is reflected off mirrors at the end of each arm, and passes back through/off the beam splitter to the detector.

I’m assuming that you are familiar with the concept of [interference](http://hyperphysics.phy-astr.gsu.edu/hbase/sound/interf.html). Interference of waves is evident when playing with slinkies, dropping a pair of pebbles in a pond, or listening to two tones close in frequency (i.e., beats). Perhaps the most famous example of interference of light is [Young’s Double Slit experiment](http://en.wikipedia.org/wiki/Double-slit_experiment). In essence, when two waves of light meet, they interfere. If two peaks or two troughs of these waves align (i.e., are in phase), they constructively interfere and the detector measures a brighter light. If a peak aligns with a trough (i.e., out of phase), they destructively interfere and, if aligned exactly, the detector measures no light.

(constructive and destructive interference; source: [wikipedia](http://upload.wikimedia.org/wikipedia/commons/0/0f/Interference_of_two_waves.svg))

Michelson and Morely measured the effect of the ether on the speed of light by observing the interference patterns as their interferometer was rotated. While they didn’t know the direction of the ether, by rotating their interferometer, they were assured that at times one beam would be in the same direction as the ether while the other was perpendicular to it. If the light traveled different speeds based on its direction relative to that of the ether, the interference patterns produced by the interferometer would change as it is rotated. The interference pattern didn’t change since there was no ether and light travels at a constant speed in a given medium regardless of direction.

Given that light does travel at a constant speed, an interferometer is useful for measuring very small differences in length. If the two arms of the interferometer are exactly the same length, the two waves will constructively interfere. However, if one arm is a quarter-of-a-wavelength longer than the other, one wave will travel a total of an extra half-a-wavelength compared to the other, and the two waves will destructively interfere. Given the very small wavelength of light, this makes an interferometer a very sensitive instrument. The Holometer is designed such that one arm is a quarter-of-a-wavelength longer than the other. If this distance was exact, and nothing moved, the detector would measure no light (in fact, it is referred to it as the dark port).

However, in order for the Holometer experiment to eventually make sense, we need to look at the light in an interferometer from a different perspective: light as a particle. When a photon (i.e., particle of light) hits the beam splitter, quantum mechanics tells us that the photon travels down **both arms at the same time**. Don’t think too hard about that; just drink the kool-aid and move on. This particle perspective is key because it illustrates that, since the same photon travels down each arm and reflects back, the photon interferes with itself. This is a very important characteristic to which we will return later.

So, at this point we’ve described a very sensitive instrument – the interferometer. It can measure differences in length of an order smaller than the wavelength of light (400 nm – 750 nm). The realization that you may have at this point is that there is a huge difference in magnitude between the wavelength of light (7.5 x 10^-7 m) and Planck’s length (1.6 x 10^-35 m). How can the Holometer measure anything on the order of the holographic noise? We’ll tackle that next as we explore spectral analysis and correlation.

This post is one in a series about The Holometer experiment and my work at Fermilab in the Summer of 2011:

* [Holometer: Holographic Noise](https://pedagoguepadawan.net/66/holographicnoise/)
* Holometer: Interferometer
* [Holometer: Spectral Analysis](https://pedagoguepadawan.net/81/holometerspectralanalysis/)
* [Holometer: Transverse Jitter](https://pedagoguepadawan.net/83/holometertransversejitter/)
* [Holometer: Correlated Interferometers](https://pedagoguepadawan.net/94/holometercorrelatedinterferometers/)
* [Holometer: Computer-Based Measurements](https://pedagoguepadawan.net/111/holometercomputerbasedmeasurements/)

This summer I am working at [Fermi National Accelerator Laboratory](http://fnal.gov/) as a Teacher Research Associate as part of the [TRAC program](http://ed.fnal.gov/interns/programs/trac/). I plan on writing a series of posts about my experiences and, specifically, about the experiment with which I’m involved: [the holometer](http://holometer.fnal.gov/). Before describing my specific contributions to the experiment, I think I should start with the theory that led to holometer experiment. One of my goals this summer is to be able to explain this experiment and the theory that it is testing in a way that can be understood by my students. This will take several revisions and this post is my first draft.

The theorist involved with this experiment is [Craig Hogan](http://astro.fnal.gov/people/Hogan/) and the holometer is designed to test the the holographic principle. What is the holographic principle? Some describe this theory as claiming that the reality that we perceive is actually a three-dimensional projection from the two-dimensional reality at the edge of the universe. While that sounds cool and sci-fi, I have no idea what it means.

An analogy that I think helps explain the holographic principle is that of graphics on a computer screen. When you play Angry Birds, the bird flies across the screen in an apparently smooth path:

However, if you zoom in and look more closely, you’ll see that the bird cannot follow an arbitrarily smooth path since the screen is made of pixels. As the bird flies across the screen, it must move in discrete intervals horizontally and vertically. That is, its location on the screen is quantized. What appears to be smooth movement, is actually the bird jumping from one pixel to the next. The minimum distance the bird can move is the width of one pixel. A pixel’s width is sufficiently small that we don’t notice these jumps as we play the game.

How does this analogy apply to the holographic principle? Space-time is the screen; we are the Angry Birds; and the Planck length is the width of a pixel. To elaborate, as we move through space-time, our movement is not perfectly smooth but, rather, jumpy because the smallest distance we can move is the Planck length (1.6 x 10^-35 m)¹. Similarly to how, if we zoom in on the computer screen, we can observe the jumpiness of the Angry Bird, through an analogous magnification we should be able to observe the jumpiness, or jitter, of our movement through space-time. This jitter is the focus of Hogan’s research and is called holographic noise. The holometer experiment is designed to measure this phenomenon. How can we build an apparatus that can measure this holographic noise when the Planck length is so incredibly small? Stay tuned for the next post in this series!

Does this make any sense? Feedback is most welcome!

This post is one in a series about The Holometer experiment and my work at Fermilab in the Summer of 2011:

* Holometer: Holographic Noise
* [Holometer: Interferometer](https://pedagoguepadawan.net/68/holometerinterferometer/)
* [Holometer: Spectral Analysis](https://pedagoguepadawan.net/81/holometerspectralanalysis/)
* [Holometer: Transverse Jitter](https://pedagoguepadawan.net/83/holometertransversejitter/)
* [Holometer: Correlated Interferometers](https://pedagoguepadawan.net/94/holometercorrelatedinterferometers/)
* [Holometer: Computer-Based Measurements](https://pedagoguepadawan.net/111/holometercomputerbasedmeasurements/)

¹ The [Planck length](http://en.wikipedia.org/wiki/Planck_length) was derived from fundamental physical constants (speed of light, gravitational constant, and Planck constant) by Max Planck.