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What is the average flux carried by the first excited state of the quantum LC oscillator with resonant frequency ω0=(2π)×2GHz{“version”:”1.1″,”math”:”\(\omega_0 = (2\pi) \times 2 \, \text{GHz}\)”} ? Note that Φ0{“version”:”1.1″,”math”:”\(\Phi_0\) “} denotes the flux quantum.

Consider a quantum LC oscillator with capacitor energy EC/h=100{“version”:”1.1″,”math”:”\(E_C/h = 100\)”} MHz and inductor energy EL/h=5{“version”:”1.1″,”math”:” \(E_L/h = 5\)”} GHz in its ground state. What is the fluctuation in flux Φzpf{“version”:”1.1″,”math”:”\(\Phi_\text{zpf}\)”} (expressed in terms of the magnetic flux quantum Φ0{“version”:”1.1″,”math”:”\(\Phi_0\)”})?

We have seen that to design a two-qubit gate, two transmon qubits are capacitively coupled giving rise to a transverse coupling between the two transmon qubits. When on-resonance, the Hamiltonian of this interaction in the rotating wave approximation is given by H=g[σ−σ++σ+σ−]{“version”:”1.1″,”math”:”\(H = g [\sigma_- \sigma_+ + \sigma_+ \sigma_-]\)”}. What is the final state of the two qubits [up to an overall phase] starting from the initial state is |10⟩{“version”:”1.1″,”math”:”\(\vert 1 0 \rangle\)”} when the time duration for which the transmons are in-resonance is t=π/(4g){“version”:”1.1″,”math”:”\(t=\pi/(4g)\)”}?

An important characteristic of a quantum LC oscillator is that its energy eigenstates are equally spaced.  In contrast, a Transmon (i.e., quantum anharmonic oscillator with capacitive energy EC{“version”:”1.1″,”math”:”\(E_C\)”} and inductive energy EJ{“version”:”1.1″,”math”:”\(E_J\)”}) due to Josephson junction introduces shifts in the energy levels of the oscillator.  To first order in perturbation theory, what is the energy spacing between the ground (i.e., |0⟩{“version”:”1.1″,”math”:”\(\vert 0 \rangle\)”}) and the first (i.e., |1⟩{“version”:”1.1″,”math”:”\(\vert 1 \rangle\)”}) excited state?

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