How to know if your vampire friend without murdering them accidentally is a video project made for PC3130: Quantum Mechanics II. Its aim was to teach a chosen topic on quantum mechanics to fellow students in the same class; as such, certain basic concepts are glossed over or unexplained in the video, although some are expounded in the write-up below.
Usually, to know if an object is present somewhere, one needs to interact with it in some way. Even if it were merely looked at, at least one particle of light – a “photon” – has to bounce off the object to reach the eye – a “detector”. The “measurement” here would be the answer to the question, “Is the object here?”
The referenced paper[1] takes advantage of the quantised nature of light to get around this necessity. Essentially, if a single photon goes through a path containing the object, it either interacts with the object, or not interact with the object. If it does, and the process absorbs the photon (say, the object is a vampire and the light is absorbed by them, which kills them), no light will be detected at the end of the path.
Yet, even if the photon takes the path where it doesn't interact with the object, the outcome might be different depending on the object's presence! This difference in outcome allows one to know an object's presence without actually interacting with it, thereby performing an interaction-free measurement.
The symbols used in the video, and their corresponding bra–ket notation counterparts, will be laid out in this section. Some prior familiarity with the bra–ket notation is assumed.
FADE IN:
INT. 4-DIMENSIONAL HILBERT SPACE
Space is well-lit with a background that is subtly off-white. The time dimension not immediately visible but its presence definitely felt.
Photons – individual particles of light – are the primary agent of this phenomena. Of its many properties, only two are needed here: its polarisation and its direction of travel. The polarisation is an angle between 0° (vertically polarised) and 90° (horizontally polarised), while the direction of travel is one of two: rightwards or downwards. The restriction on the direction of travel is because of how the equipment is set up, as will be clearer later.
Vertically polarised photons are denoted
A photon with an arbitrary polarisation can be written as
|00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°, ⋅ ⟩ = cos(00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°)|0°, ⋅ ⟩ + sin(00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°)|90°, ⋅ ⟩
Meanwhile, a photon travelling rightwards is
As the above photons have a polarisation of 45°, they are denoted
The Born rule states that the probability of finding a photon of state
where
Essentially, the above means
More generally, using Equation
and
Finally, the probability is given by the square of the projected length:
If the photon is polarised by 45°, there is a 50% chance of finding it to be vertically polarised, and a 50% chance of finding it to be horizontally polarised. Visually, the square of the length of the final photon indicates how likely it is to find it in that state.
Dude These are polarising beam splitters.
A singular beam splitter slides in from below to the centre of the frame.
Dude They let vertically polarised light through, and reflects horizontally polarised light.
Beam splitter duplicates itself, positions themselves one on top of the other. Two photon appears from frame left, pauses, top one passes through, and the bottom reflects downwards, in tandem with narration. Top (Bottom) photon has graphic with arrow pointing up-down (left-right).
The beam splitter, here represented by
In other words, it is only the horizontally-polarised component of the photon that gets reflected. Generally,
With a photon of polarisation 45°,
and hence the probabilities are
It is important to realise that the visuals represent the state of the photon rather than the photon itself – the photon cannot split into half! The above situation is therefore intepreted as follows:
“If a single photon with polarisation 45° moving rightwards passes through a beam splitter, there will be
Mirrors, represented with
Dude If we pass vertically polarised light through a Faraday rotator - rotating the polarisation by some angle theta - (MORE)
Both beam splitter disappears upwards as the camera follows the top photon to right frame, where a Faraday rotator is present. The arrow rotates clockwise a little as it passes through the rotator.
When light of a certain polarisation passes through a uniform magnetic field, its plane of polarisation shifts due to the Faraday effect. By controlling the strength of the magnetic field, the polarisation of a photon can be rotated to a certain angle: here depicted as
Dude (cont'd) - then through this inteferometer, the photon ends up at the bottom with polarisation theta.
Without first including Tommy, the experimental setup acts on the incoming vertically-polarised photon via the following steps:
The final state of the photon
where we have defined
Hence,
Meanwhile, the presence of Tommy means that
which is representative of the fact that the rightwards-moving photon is “absorbed” by Tommy, while the downwards-moving photon continues in its path. Therefore, with this setup,
So there is a 50% chance of detecting the photon at the bottom, and a 50% chance of Tommy absorbing the photon (and dying).
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