Really doing mechanics at the quantum level

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Since before the development of micro- and nanoelectromechanical techniques, there has been an interest in making actual mechanical widgets that show quantum behavior.  There is no reason that we should not be able to make a mechanical resonator, like a guitar string or a cantilevered beam, with a high enough resonance frequency so that when it is placed at low temperatures ( \(\hbar \omega \gg k_{\mathrm{B}}T\)), the resonator can sit in its quantum mechanical ground state.  Indeed, achieving this was Science's breakthrough of the year in 2010.  

This past week, a paper was published from ETH Zurich in which an aluminum nitride mechanical resonator was actually used as a qubit, where the ground and first excited states of this quantum (an)harmonic oscillator represented \(|0 \rangle\) and \(|1 \rangle\).  They demonstrate actual quantum gate operations on this mechanical system (which is coupled to a more traditional transmon qubit - the setup is explained in this earlier paper).  

One key trick to being able to make a qubit out of a mechanical oscillator is to have sufficiently large anharmonicity.  An ideal, perfectly harmonic quantum oscillator has an energy spectrum given by \((n + 1/2)\hbar \omega\), where \(n\) is the number of quanta of excitations in the resonator.  In that situation, the energy difference between adjacent levels is always \(\hbar \omega\).  The problem with this from the qubit perspective is, you want to have a quantum two-level system, and how can you controllably drive transitions just between a particular pair of levels when all of the adjacent level transitions cost the same energy?  The authors of this recent paper have achieved a strong anharmonicity, basically making the "spring" of the mechanical resonator softer in one displacement direction than the other.  The result is that the energy difference between levels \(|0\rangle\) and \(|1\rangle\) is very different than the energy difference between levels \(|1\rangle\) and \(|2\rangle\), etc.  (In typical superconducting qubits, the resonance is not mechanical but an electrical \(LC\)-type, and a Josephson junction acts like a non-linear inductor, giving the desired anharmonic properties.)  This kind of mechanical anharmonicity means that you can effectively have interactions between vibrational excitations ("phonon-phonon"), analogous to what the circuit QED folks can do.  Neat stuff.