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I am new to quantum mechanics and this is probably a dumb question. What experiments can we do to produce a quantum harmonic oscillator? For example, for a classical harmonic oscillator, we can attach a ball to a spring, make it vibrate, and observe how its position change as time goes on. For quantum harmonic oscillator, how do we set up such an experiment, and what can we measure in the experiment?

EDIT: I searched around and found another post asking about one-dimensional infinite-well model. But I didn't found one for harmonic oscillator. References on experimental realization of quantum one-dimensional infinite-well model

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    $\begingroup$ Any diatomic molecule is a quantum harmonic oscillator and we can measure its oscillations using infrared light. As a general rule we only see the quantised oscillations in systems around the size of molecules. $\endgroup$ Apr 1 '20 at 17:00
  • $\begingroup$ Any system that has a minima in its potential profile as a function of some parameter will behave as a harmonic oscillator to small perturbations in that parameter. $\endgroup$ Apr 2 '20 at 6:22
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    $\begingroup$ Superconducting circuits are a fantastic candidate because we can engineer the frequency, impedance, and even anharmonic terms in the potential. $\endgroup$
    – DanielSank
    Apr 4 '20 at 1:27
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You may be interested in this review on cavity optomechanics

https://arxiv.org/abs/1303.0733

Harmonic oscillators have energy levels spaced by $\hbar \omega$. The thermal energy in a harmonic oscillator is given by $kT$. If $kT \gg \hbar \omega$ then many energy levels of the harmonic oscillator will be occupied and the oscillator will behave like a thermal classical oscillator, not so interesting. The challenge then to creating a quantum harmonic oscillator is to get an oscillator cold enough (or high enough oscillation frequency) such that we have $\hbar \omega \gg kT$.

One way to do this is to literally make a tiny oscillator (like a micromembrane or nanowire) and cool it to cryogenic temperatures.

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Smooth binding potentials - Morse, Lennard-Jones, Poschl-Teller - that go to $0$ as $r\to \infty$ usually have a minimum. Expanding about this minimum gives a locally harmonic potential: hence the vibrational energy levels in molecules. For some molecules the anharmonic can be quite small so you can have 10s of vibrational level also exactly equidistant and behave to a good approximation like a harmonic oscillator.

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