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Blog for September 2024

22nd September 2024 - Bell's Inequality

Bell’s inequality is both incredibly simple and incredibly powerful, and speaks to the fundamental nature of reality; it will be the focus of this blog.

So what is Bell’s inequality? It was derived by John Stewart Bell in the 1960s, and has now been tested by experiments. The following derivation is based on work by Eugene Wigner, Bernard d’Espagnat and Anton Zeilinger, who helped in providing a more digestible understanding of Bell’s original work.

To understand Bell’s inequality, we start by thinking about identical twins. We choose 100 twins, where the twins are selected with the following requirement: a twin must have either blue or brown eyes, brunette or blond hair, and is either tall or short; twins with hair colour different from brunette or blond, or eye colour different from blue or brown, or heights different from tall or short, are not selected.

We define, F_tall as the fraction of tall twins and Fshort as the fraction of short twins, and therefore F_tall + F_short = 1; similarly, F_blue + F_brown = 1 and F_brunette + F_blond = 1.

We define, F_tall&blue as the fraction of twins that are tall and have blue eyes, and F_{tall&blue&brunette} as the fraction of twins that are tall and have blue eyes and have brunette hair.

Using this terminology, F_tall&blue = F_{tall&blue&brunette} + F_{tall&blue&blond} because F_brunette + F_blond = 1.

The two terms ( F_{tall&blue&brunette} and F_{tall&blue&blond}) on the right hand side of the above equation can be expressed as follows:

F_{tall&blue&brunette} + F_{tall&brown&brunette} = F_{tall&brunette}

and therefore: F_{tall&blue&brunette} = F_{tall&brunette} – F_{tall&brown&brunette}

Similarly,

F_{tall&blue&blond} + F_{short&blue&blond} = F_blue&blond

and therefore, F_{tall&blue&blond} = F_blue&blond – F_{short&blue&blond}

Using the above, it is clear that the following is true:

F_tall&blue = F_{tall&brunette} − F_{tall&brown&brunette} + F_blue&blond – F_{short&blue&blond}

Removing the negative terms, leaves the following inequality:

F_tall&blue ≤ F_{tall&brunette} + F_blue&blond

This inequality says that the number of twins that are tall and have blue eyes will always be less than or equal to the number of twins that are tall and have brunette hair + the number of twins that have blue eyes and blond hair.

Now, if we associate the properties of twins with the properties of photons (the quantum particles of light), we arrive at Bell’s inequality.

The property of a photon we will use is its polarisation, which is the property that Polaroid sunglasses use to reduce the glare of the sun. The polarisation of a photon can be thought of as an arrow attached to the photon that represents the direction of the oscillating electromagnetic field; we assume that this arrow always point in the same direction for a specific photon prior to impacting the sunglasses, and that a group of typical un-polarised photons have their arrows randomly pointing in every different direction. Polaroid sunglasses work by allowing a fraction of photons to pass through, with the fraction dependent upon the direction of the polarisation arrow. If the polarisation arrow is aligned with the direction of the molecules in the sunglasses, then the photon passes through. If the polarisation arrow is perpendicular to the molecules in the sunglasses, then the photon does not pass through. If the polarisation arrow is somewhere between aligned with and perpendicular with the molecules in the sunglasses, then a fraction of the photons pass through; the fraction, from Malus’ Law, is found to be proportional to the angle θ between the polarisation arrow and the direction of the molecules in the sunglasses: ∝ cos²θ.

In the experiment that will be performed, two entangled photons (the twins) will each be passed through a polariser (sunglasses) which can be set to transmit a photon with a polarisation arrow set to one of three angles (0^∘,30^∘,60^∘); equivalent to height, eye colour and hair colour. The resulting photon will then be passed through a beam splitter, which according to Malus’ Law, will result in either a horizontally polarised photon (H) or a vertically polarised photon (V); equivalent to getting either tall or short if height is selected, blue or brown if eye colour is selected or brunette or blond if hair colour is selected.

Here is the required association, where θ represents the angle of the polariser (sunglasses), and H or V represents the resulting photon from the beam splitter:

tall→H_0^∘ , short→V_0^∘ , blue→H_30^∘ , brown→V_30^∘ , brunette→H_60^∘ , blond→V_60^∘

Which gives Bell’s Inequality for these photons:

F_{H_0^∘ & H_30^∘} ≤ F_{H_0^∘ & H_60^∘} + F_{H_30^∘ & V_60^∘}

The experiment, when performed, gives the following impossible result:

0.75 ≤ 0.25 + 0.25

i.e. 0.75 ≤ 0.50

Which is truly shocking! What does this mean?

Well, when an inequality is violated, it means an assumption(s) leading to the derivation of the inequality must be wrong.

Looking back at the relatively simple derivation of the inequality above, the main assumption is local realism: i.e. the twins and photons are assumed to have definite properties of height, hair colour and eye colour that persist whether they are being observed or not, and that these properties can only be influenced by local surroundings.

For the twins, this is true – local realism is true, and the properties of the twins exist whether being observed or not.

But for entangled photons, as we have seen in previous blogs, local realism is not true, and we must give up the cherished belief that quantum particles have definite properties before observation – they do not. Bell’s inequality and the experiments performed with polarised photons show that local realism is not true in the quantum realm!

Interestingly, with a knowledge of entanglement and use of Malus’ Law, this violation of the inequality was expected: cos²(30^∘) = 0.75 and cos²(60^∘) = 0.25, which are the values observed from the experiment discussed above.

Future blogs will continue to explore what it means for quantum particles (of which we are all made) not to obey local realism; local realism is something that we as a collection of non local realistic quantum particles, take for granted!

This blog is dedicated to an amazing mum, who though no longer with us, is entangled in all our thoughts.

Extra Blog for July 2024

21st July 2024 - quantum reality

These two pictures hold an incredible mystery for reality.

They demonstrate clearly that quantum particles (we are all made from atoms, and therefore quantum particles), have a very different reality to our classical intuition.

The picture below shows the arrangement of a relatively simple experiment: a source of light from (S), impacts a beam splitter (BS₁). A half of the light is transmitted along the path (T) and a half of the light is reflected along (R). Both paths of light hit a mirror (M) and then make their way to detectors (D₁) and (D₂). The detectors each record that half of the light comes from path (T) and half from path (R, exactly as expected. This is a simple experiment that has been repeated many times. It has also been undertaken using a statistically significant number of single photons of light, where again the detectors record on average, one half of the photons at each detector.

A beam splitter comprises two triangular glass prisms, glued together with an epoxy, with the epoxy set at a thickness to ensure that light of a particular frequency transmits a half and reflects a half. It is a standard piece of optical equipment.

The following picture shows the same arrangement as above, but with a further beam splitter (BS₂) located at the intersection of the transmitted and reflected light paths. Intuition would suggest that as for the first beam splitter (BS₁), one half of the photons along the transmitted path (T) would be transmitted and one half reflected, and one half of the photons along the reflected path (R) would be transmitted and one half reflected, resulting in each detector again recording 50% of the light reaching each of them. Indeed, if the reflected path is blocked from reaching the second beam splitter, then the light from the transmitted path (T) is indeed split into transmitted and reflected halves that lead to the detectors recording 50% of the light reaching each of them. Exactly as expected.

But, the picture below shows that if the reflected path (R) is allowed to interact with the transmitted path (T), a very unexpected thing happens – ALL of the light ends up at (D₁) and NONE of the light ends up at (D₂). This is true, even if single photons are passed through the beam experiment, one at a time.

What is going on?

The pictures above are from quantum Tutorials by Frank Rioux, Emeritus Professor of Chemistry

In terms of quantum mechanics, which describes the behaviour of quantum particles such as photons, there is no mystery, and the expected result is obtained. This is because, in reality, the single light photon interacting with the first beam splitter (BS₁) results in a new quantum state that is a superposition of both transmitted and reflected photon states; the superposition expresses the 50% probability of the photon being transmitted or reflected, and like Shrodinger’s cat, the photon exists in a quantum state of being both reflected and transmitted, until “detected”. When this quantum superposition interacts with the second beam splitter (BS₂), the transmitted and reflected parts of the superposition interfere, just as in the experiment discussed in the first Blog July 2024 Blog, such that constructive interference occurs at (D₁) and destructive interference occurs at (D₂): all the photons are detected at (D₁) when the second beam splitter (BS₂) is in position, demonstrating that the classical interpretation that a photon exists and is real and passes through the experiment in either (T) or (R) is incorrect, and conflicts with this simple experiment which has been repeated many time. The quantum interpretation is that the photon passes through the experiment as a superposition of both (T) and (R), with a probability of being reflected of 50% and a probability of being transmitted of 50%. Mathematically, the photon is actually represented as 1(√2)(T+iR), very different to a classical representation which would be either T or R.

So in reality, the quantum photon particle actually goes through all paths to the detector in a state that represents the probability of being a particular state when “detected”, rather than as a specific state – reality, really is strange!

Recently, I was in a shop with four others, in which one of us asked the others, how much do we think this dress costs – four very different values were assigned to this garment. It was as if, at this point, until one of us took the dress to the front of the shop to ask the price thereby “detecting” the real value of this dress, the value for this dress for the four of us was a quantum superposition of all these guessed values; the analogy to the experiment above breaking down at the point we attempted to pay for it, as the shop had a value in mind much greater than any of the guessed values!

Next time we will look at Bell’s Inequality, which conclusively shows that quantum particles have no defined values for their parameters until “detected” – future Blogs will look at the measurement problem of quantum mechanics (i.e. what does “detected” actually mean), and how is it that since we and pretty much everything we interact with are comprised of quantum particles, that the world we live in appears so different to this strange quantum world of probabilities and quantum reality.

Blog for July 2024

8th July 2024 - double slit experiment

This is the first post from this new website.

The topic will be Reality, and what it really is.

So Young’s double slit experiment with monochromatic light, results in the characteristic bright and dark bands of light on the screen, which form as a consequence of the interference of the diffracted light from each slit.

This is as expected if light is treated as a wave, and resembles the same pattern that one sees if water is observed passing through two slits, or small gaps.

Repeating this same experiment with low intensity light, such that the energy represents a single photon passing through the slits at any one time, and waiting until a large number of these photons have sequentially passed through the slits, amazingly produces the same pattern of light and dark bands; the individual photons produce random points on the screen, which when added together produce this interference pattern. The same has been done with single electrons passing through two slits, and again the light and dark bands are produced by the sum of these single electrons. Closing one of the slits, and a standard diffraction pattern results. So these single particles passing through the slits one at a time, produce a pattern dependent on whether a single or double slit is present.

In quantum mechanics, these single particles are not point like particles, but instead are represented as spread out waves of probability: the particle is associated with a wave of probability, that represents the likelihood of being found with a specific momenta at a specific location, in accordance with Heisenberg’s uncertainty principle; one cannot represent these particles as familiar particles that exist in a particular location with a particular momenta, but instead the particle is a superposition of all these states, and these superposition of states can interfere.

So, what is real? What is happening in this double slit experiment with photons and electrons? The impact on the screen is definitely observable to us, as is the interference pattern or diffraction pattern. The probability wave is not real to us, we are unable to observe this, but we can calculate it, and we are able to measure the associated probabilities.

The question of this quantum measurement problem (i.e. how to go from probability wave to a definitive point on the screen), as well as what reality means for a quantum particle will be explored through these blogs. We are all made of atoms, which comprise these same quantum particles, so this question of what is real, really matters!