# How can we create superposition in QM?

How can we force a particle (let's say that we know this particle has spin up) to be in a superposition of spin up and down?

Wouldn't literally any interaction of it with anything cause it to be in superposition, since we can never know the exact position/speed of the thing it is interacting with, so surely the outcome is unknown?

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For a particle with spin 1/2, creating superpositions of up and down is easy: measure the particle's spin along the $x$ direction. This will collapse the state onto either of the states $$|+\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)\textrm{ or } |-\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle-|\downarrow\rangle),$$ with the measurement outcome telling you which one was made.

Your second point is quite a bit more delicate. In an ideal case, our particle is sitting in empty space and not interacting with anything except whatever apparatus we use to manipulate it. That apparatus might indeed turn out to be a second particle, in which case, yes, the uncertainty principle forbids us from knowing its exact position+speed. This does not, however, imply that this will automatically put our state into a superposition. It means exactly that and nothing more. The outcome can still be perfectly determined, since all that really matters is that the energy (i.e. the hamiltonian) of the interaction be precisely known, as well as the time it was turned on for.

Your main misconception, however, is that unknown outcomes automatically mean superposition states, which is wrong. It is quite different to say the system is in the state $|+\rangle$ than to say it is 50% of the time in $|\uparrow\rangle$ and 50% in $|\downarrow\rangle$. With the first one, one can perform interference experiments, while the second one contains no information at all. Particularly, for evenly-weighted superposition states, the relative phase of the two components is crucial. If I have some superposition state of the form $$|\phi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+e^{i\phi}|\downarrow\rangle),$$ but I don't know what the phase $\phi$ is (which in practice means that I have a big box of states in different superpositions with $\phi$ evenly spread between $0$ and $2\pi$), then it's as good as a statistical mixture that can be prepared classically. Why is this? It's because each superposition state can be set up to make interference fringes that can be detected with enough particles. However, if I change the phase, it will offset the fringes by half a fringe spacing, so that if I take a long average I won't see any fringes at all. This means that you need a very controlled interaction to make useful superposition states.

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In case of magnetic spin, the direction of spinning (and hence the form of the superposition) is changed using magnets, as in a Stern-Gerlach experiment.

In case of optical spin (polarization), the direction of polarization (and hence the form of the superposition) is changed using a polarizer, as in a Mach-Zehnder experiment.

There are many other ways to achieve well-designed superpositions, sometines quite involved - especially when the system is complex, the accuracy requirements are high, and the number of systems available is very small. This is part of the art of experimentation....

Of course, a random interaction will always introduce superpositions. But this is useless as one doesn't know which ones. It is the reason why maintaining the purity of a state is often very hard - the main reason why it is difficult to boild quantum computers.

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This isn't for a superposition of spin up/down but superposition states have been created:

The most recent Nobel Prize winner Haroche managed to do this:

He developed a new field called cavity quantum electrodynamics (CQED), whereby the properties of an atom are controlled by placing it in an optical or microwave cavity. Haroche used CQED to control the properties of individual photons. He used CQED to develop a system which is in a superposition of two very different quantum states until a measurement is made on the system. Haroche managed to control a single atom in a cavity and then went on to use his quantum control techniques to create a superposition state of entangled microwave photons. Recently, he used a single atom to count the number of photons in a cavity, managing to show that the cavity contained one photon without actually destroying the photon, as would normally happen when a quantum state is measured.

So although this may not be exactly what you're asking for (a superposition of quantum spin states), it does show you how photons have experimentally been placed in a superposition state in the past.

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