Wednesday, April 04, 2007

Quantum Dot

Here is where the mystery begins. Imagine the smooth flat surface of a black granite slab. Just in front of you there is a circular pit, about 20 cm wide, maybe twice as deep. It could be used for storing a stack of small dinner plates.

This imagined reality has been magnified one hundred million times. My circular depression is a quantum dot, two nanometres in diameter. And in the centre, I am going to trap an electron.

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There he is again. Second day on the job and he still hasn’t a clue. ‘Spread some of that hardcore’ he was told. Gave us a blank look like he’d never heard of hardcore before. Next thing I knew, he was on his knees, pushing half bricks around with his hands. What an idiot.

The lads call him Prof. He’s some kind of student. Imperial College. No idea what he’s studying. He says he can’t explain it, we wouldn’t be interested. Well most of them wouldn’t be, but at least he could make the attempt, couldn’t he?

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Even magnified one hundred million times, an electron has no measurable size at all. Analogies with things like marbles just don’t work. If I want to start with a motionless electron placed at the centre of my circular depression, I will have to heap up the electron’s ‘quantum amplitude’ at that point. I have to make a wave which is high in the centre, and pretty much zero everywhere else.

I think back to dropping a sugar cube into a cup of coffee. The cube hits the surface and makes a hole. The coffee rushes into the gap from all sides, collides in the middle and surges up in a tall splashy column. That’s what I want in the centre of my quantum dot.

I remind myself that whereas the coffee column immediately falls back and is lost in concentric ripples, I would like my electron wave, as far as possible, to stay put in the centre of its quantum dot home. The symbol for the electron’s wave is y, a function of position and time - y(x, y, z, t). The value of y at a place and time is a complex number, which can’t be easily visualised. The textbooks tell me to square the modulus of y, and even that isn’t meant to directly represent a part of physical reality. The value of ‘y-squared’ at that place and that time is just the probability of observing my electron there. But we might equally find it somewhere else instead if we cared to look.

In my imagination, the y wave is like a crystalline needle in the centre of the pit. It seems to shimmer through different colours as it begins to evolve. But look, the peak is dropping, the base of the needle filling out. And now it has completely collapsed, the y wave just slopping around at the bottom of the depression, like water in a child’s inflatable swimming pool on a sunny afternoon. The slopping y wave means my electron has become delocalised. It could be anywhere in the quantum dot.

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Today it was concrete spreading. We put the steel grid down and then the mixer backed up and the wet concrete came down the chute. The lads had to work fast before it set. It has to be spread around with the shovels, making sure it fills all the cavities. Don’t want any bubbles or spaces to cause weakness later. Prof was put at the back where he just had to level the stuff out. The lads are experienced, they know where to poke, pat and direct it. But even that he messed up, breaking the rhythm of the work. He’s a liability. It’s only we’re desperate for labour during the summer.

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Why did my well-placed electron spread out? To make that initial needle-like y wave at the centre of the depression, I had to add together waves of many different wavelengths. They then constructively interfered to heap up in the centre, and cancel everywhere else. But the shorter, more jagged waves correspond to higher momentum ‘options’ for the electron (that’s just the way the maths works).

Only if my electron had had zero momentum would it have stayed in the centre, but unfortunately that’s impossible to realise, because the wave which delivers zero momentum is hardly a wave at all: it has infinite wavelength, so it potentially exists, probability spread thinly, pretty much everywhere.

No, to get my electron tightly localised, I had to use some very high frequency, short wavelength y waves and these gave the electron significant momentum. Once the clock started ticking, the wave started to spread and my sharp needle collapsed. Since my electron doesn’t have enough energy to classically escape the quantum dot, its wave function is just smeared around: small-scale choppy, large-scale pretty much flat - in configuration space, that is.

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The lads joss him rotten. Big busty page three girls are thrust at him. ‘Like a bit of that, eh?’ they roar, rejoicing in their wit. He seems to smile sheepishly, trying for the words to deflect them until they tire of the effort. In truth he’s little enough game, more an object of contempt, as is anyone who can‘t hack the job. I don’t know why he puts up with it. He answers evasively, needs the money, he says.

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So here is the mystery. In that real quantum dot, two nanometres wide, what exactly is happening? It’s far too small to take a look with a microscope. And shooting energetic photons at the dot would smash up the delicate thing we’re trying to understand.

It’s tempting to say this weird y wave really is sloshing around, but the maths makes that difficult. To start with, the wave is almost always just doing its slopping thing, it very rarely goes back to being a needle. But if we do any experiment, that’s what we seem to see: an electron at a definite place. There’s a name for this - the collapse of the wave function - but the maths gives no reason for such a collapse back to a needle to occur.

And then there is the problem of entanglement. If two electrons interact, the y function going forwards is a function of both their attributes - y(x1, y1, z1, x2, y2, z2, t). This can’t in general be reduced to separate y waves for each electron, each propagating cosily in our comfortable three dimensional space. In the maths, the y wave exists in something called Hilbert space which is not our normal universe at all - in fact it has an infinite number of dimensions. In Hilbert space, the wave function y is called the ‘state vector’.

Each point coordinate (x, y, or z) in our real universe becomes a dimension, or axis, in Hilbert space. For example, there is an axis corresponding to exactly x = 27.3 light years from earth in the direction of Andromeda. The y state vector has a projection of its amplitude onto this axis, as to any other, a projection which might be zero, or might not, depending on the energy I gave my electron and its ability to tunnel out of my quantum dot.

But the y state vector can equally be projected onto other axes, representing specific values of observable quantities like momentum or energy, rather than location.

The state vector turns effortlessly as time advances, projecting different amplitudes onto its infinity of axes. And these amplitudes give the probabilities (via squaring) of the particle being at that location, having that momentum, being in that orientation, possessing that energy.

In fact the y function doesn’t even know it’s an electron! It’s gloriously agnostic as to what kind of entity it is, because that is set by the parameters of its initial conditions and its time-evolution equation.

So here is how I see it. The y wave (or function, or state vector) for my electron is like a bookkeeping spreadsheet file. It describes all the options my electron might have: position options, momentum options, energy options, orientation options; all of which are time-indexed, and might also depend on what other ‘particle spreadsheets’ contain. And for each option, it tells me the probability that the electron might exhibit that value if it was measured (via the complex amplitude). But I shouldn’t mistake the spreadsheet for the real electron itself. What that might be is a complete mystery.

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I don’t see many students. I thought they spent all their time drinking and partying. I can’t imagine Prof partying, he’d be stuck against a wall with his goofy smile thinking ‘deep thoughts’. I’m not, what is it, ‘anti-intellectual’. If they’ve got brains, why not use them? But if his kind of brains makes him the sort of misfit he is, better of without them I say. The lads reckon he’s bloody useless, more trouble than he’s worth. Even after a few pints at lunchtime you still can’t get inside him, can’t figure him out.

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I think the problem is that we’re trying to understand the universe with caveman perceptions and concepts. It’s like if we tried to understand Bach in terms of bashing different sizes of dried hollow fruit gourd with old bones. We could get a rough theory by asking what kinds of gourds and bones explain the Brandenburg Concertos, but it’s pushing the envelope a bit.

I wonder if down at the Planck Length (1.6 × 10−35 metres) there isn’t some kind of hazy, shimmering reality, continually throwing up resonances and ephemeral stabilities. Quantum foam in the multiverse. On a scale billions and billions of times greater, we macroscopic creatures see a world of apparent particles, the elements of cabbages and kings. But it’s all illusion, just gourds and bones. But contemplating the ultimate reality, I risk straying into mysticism, and that’s an indulgence best taken in small measure.

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Prof left yesterday. We were glad to see the back of him, truth to tell. We know what we need here and he’s just not up to it. In a crueller world, he wouldn’t survive, and we’d sure as hell have to pay a lot less taxes to support him.

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Note: an illustration of this story can be seen here.