Part 1: Experimental & Computational Introduction to Quantum Mechanics
Authors: Alona Kosobokova (Astrophysics student), Anurag Sahoo (AI Researcher). It's also on medium.
This chapter is a part of a series to investigate Quantum Mechanics and its Applications through theory, experimentation, and quantum computation.
In Part 1, we are going to build an intuition for a quantum mechanical phenomenon through a careful analysis of experimental observations.
In Part 2, the reader will see how a deterministic view of the universe does not support the weirdness we see in the quantum world.
Introduction
According to a classical understanding, light is a transverse electromagnetic wave. To explain, the transverse mode of electromagnetic radiation is a particular electromagnetic field pattern where the electric and magnetic field lines are perpendicular to each other (i.e. transverse) and to the direction of propagation. Light is called linearly polarized if its electric field oscillation is restricted to one direction: the direction of polarization. Most light sources in nature emit unpolarized light, which consists of wave trains whose directions of oscillation are completely random.
As demonstrated further in the post, the light may be polarized by passing it through a special material, which transmits only the component of light polarized along a particular direction and completely absorbs the component perpendicular to that direction. (Polarization of Light, 2017, 3)
And that would be it, but at the beginning of the 20th century, Max Planck and Albert Einstein introduced the notion of a photon: quanta of light. (Read more about the Photoelectric effect and Planck’s Law). To make sense of the previously unexplainable photoelectric effect, they suggested that light (electromagnetic wave) might be actually quantized, and each quantum particle (here: photon) carries energy E proportional to the frequency of the light:
Interesting fact, as you can notice, a photon in a spectrum of visible light (400 –700nm) carries very little energy, nevertheless, our eyes can detect as few as five to seven photons on the retina. (MIT Open Course: Quantum Mechanics, 2020)
Particles in the quantum world, unlike the “classical particles”, possess a set of properties that makes them behave in a very different manner. In Classical Mechanics (CM), a particle is some zero-size object that carries energy and at any point of time has a definite position and velocity. In Quantum Mechanics (QM) however, a particle is something like an indivisible amount of energy or momentum that propagates. It carries a set of new strange properties some of which we aim to illustrate in this article.
In the following section, we are going to demonstrate the fundamental Quantum Mechanical phenomenon at play and analyze photons passing through different configurations of linear polarized filters.
Polarizing Light
Consider the following sequences of polarizers:
As you can see when we align two polarisers along the same axis we get all the light from polariser 1 passing through polarizer 2. Aligning them perpendicularly will let no light through as the second polariser completely blocks the light from the first polarizer.
But what will happen if between those two we insert one more linear polarizer oriented towards some degree along the x-axis?
A fraction of light actually gets through all three of them now. Adding one more polarizer — get’s more light through, but why? Let’s look at what’s going on more closely.
Electromagnetic fields follow linear Maxwell’s equations and they are considered the basic classical description of propagating electromagnetic waves. It is a big topic with a lot of interesting things to learn and requires a whole other article on its own. To simplify our process, we will use the so-called Malus’ Law which is a special case description of the properties of light polarization and can be fully derived from Maxwell’s equations, and it is sufficient for our purposes. After passing through polarizer 1 the light is linearly polarized, so we can apply Malus’ Law.
If a polarized light having intensity I_o enters a polarizer with a transmission axis making an angle alpha with respect to the polarization plane, after passing through the polarizer the light will have intensity:
and its polarization plane is parallel to the transmission axes of the polarizer.
Both of our previous experiments satisfy this condition: when alpha = 90 degrees, cos(alpha) = 0, hence intensity passing through the second polarizer equals zero. When alpha = 0 degrees, cos(alpha) = 1 and we can see that all of the light from the first polarizer gets through the second.
We can also apply Malus’s Law to predict the result for our three polarizers setup:
Rearranging the equation above,
Thus, our function is expected to show maximum values at alpha = 45 and 135 degrees, and minimum values at alpha = 0 and 90 degrees.
Experiment
Our experimental set up consists of:
Light Source
Polarizer 1, and polarizer 3 oriented at a 90-degree angle to each other
Polarizer 2 between them, oriented at angle alpha along the axis-x
A high sensitivity photometer.
Rotating the second polariser we are able to achieve different alpha angles. Then using the photometer (a device that detects the relative intensity of light in LUX), we measure the intensity of light that passes through, for each angle from 45 to 90 with an increment of 4 degrees.
The below graph shows the relationship between alpha and relative luminosity measured by the photometer.
Analysis and the Quantum Part
Now let’s think about these results in a different way. If the light had strictly a wave nature, the interpretation would be pretty straightforward: A fraction of energy passes through and the rest doesn’t. But remember, as predicted at the beginning of the 20th century the light is quantized.
So what do these results mean for a single photon?
The photon unlike a sound or a water wave is a quantum object, and as such, it either passes through the polarizer completely or not at all. (“Bell’s Theorem: The Quantum Venn Diagram Paradox”, 2017.) But think about this, all photons are totally identical! How do completely identical particles behave differently in the same experiment? How does each photon ‘pick’ to go through or ‘be blocked’?
The experiments have been done, where photons were shot on the polarizers ‘one by one’. Still, we get the same result: some number of photons go through and some don’t. The question is: When shot independently, how does every single particle know about the choice of all the other particles before and after it in order to maintain the final observed distribution? How do they keep the ‘connection’ or ‘knowledge’ between them? This is where a classical deterministic picture of the Universe breaks.
The only thing we can predict for a single photon is the probability of it going or not going through. Let’s see what that probability is going to be for a photon polarized along axis-x and passing through the polarizer oriented at alpha degrees to the x-axis.s
The fraction of energy blocked by the polarizer is:
Thus, a single photon has a probability cos²(alpha) of going through the polarizer oriented at a degree alpha with respect to the preferable direction of polarization, and the probability to be blocked is sin²(alpha). In this situation, we say the photon is in a superposition state which is described by the following notation.
Or,
There have been different attempts to explain this strange behavior. Say to assume that photons are not identical after all, they carry some differences we can’t detect yet, so-called hidden variables. In the next post, using the Bell Inequality experiment we will attempt to demonstrate why this is not the case, and in fact, photons can not have any kind of undetected hidden variable.
Quantum Computer Simulation of the Experiment
Apart from physical experimentation, we built a quantum algorithm using the Qiskit package for simulating our polarizer set up. Github link.
The full circuit requires 5 qubits and 4 classical bits. It has the following sub-circuits:
The full circuit is shown below:
Fig 11 shows that the simulator behaves in a very close way to the theory and experiment dictate. Moreover, since the sub-circuits are general purpose units, it allows us to quickly simulate more complicated configurations of polarizers for our coming future experiments.
Thank you for getting through the whole post! :) Special thank you to the SFSU Physics & Astronomy department for support. Specifically Anthony Kelly, and Peter Verdone for providing the whole experimental setup and Dr. Joseph Barranco for support and encouragement.
Bibliography
“Bell’s Theorem: The Quantum Venn Diagram Paradox.” 2017. https://www.youtube.com/watch?v=zcqZHYo7ONs&feature=emb_title.
“MIT Open Course: Quantum Mechanics.” 2020. https://courses.edx.org/courses/course-v1:MITx+8.04x+3T2020/course/.
Polarization of Light. 2017. https://www8.physics.utoronto.ca/~phy224_324/experiments/polarization-of-light/polar.pdf.