Consider a quantum LC oscillator with capacitor energy EC/h=100{“version”:”1.1″,”math”:”\(E_C/h = 100\)”} MHz and inductor energyEL/h=5{“version”:”1.1″,”math”:” \(E_L/h = 5\)”} GHz in its ground state. What is the fluctuation in charge Qzpf{“version”:”1.1″,”math”:”\(Q_\text{zpf}\)”} expressed in terms of the electronic charge e{“version”:”1.1″,”math”:”\(e\)”}?

To measure the state of a transmon qubit, we make use of the capacitive coupling between the qubit and a resonator captured by the Hamiltonian H=g[σ+a+σ−a†]{“version”:”1.1″,”math”:”\(H = g [\sigma_+ a + \sigma_- a^\dagger]\)”}. Given that the qubit frequency ωq=(2π)5{“version”:”1.1″,”math”:”\(\omega_q = (2\pi)\, 5 \)”} GHz, the resonator frequency ωr=(2π)4{“version”:”1.1″,”math”:”\(\omega_r = (2\pi)\,4 \)”} GHz, and the capacitive coupling g=(2π)10{“version”:”1.1″,”math”:”\(g = (2\pi)\, 10\)”} MHz: Considering the qubit to be in the state |0⟩{“version”:”1.1″,”math”:”\(\vert 0 \rangle\)”}, what is the magnitude of the shift in the resonator’s frequency (assuming ℏ=1{“version”:”1.1″,”math”:”\(\hbar = 1\)”})?

To measure the state of a transmon qubit, we make use of the capacitive coupling between the qubit and a resonator captured by the Hamiltonian H=g[σ+a+σ−a†]{“version”:”1.1″,”math”:”\(H = g [\sigma_+ a + \sigma_- a^\dagger]\)”}. Given that the qubit frequency ωq=(2π)5{“version”:”1.1″,”math”:”\(\omega_q = (2\pi)\, 5 \)”} GHz, the resonator frequency ωr=(2π)4{“version”:”1.1″,”math”:”\(\omega_r = (2\pi)\,4 \)”} GHz, and the capacitive coupling g=(2π)10{“version”:”1.1″,”math”:”\(g = (2\pi)\, 10\)”} MHz, are we in a dispersive regime (assuming ℏ=1{“version”:”1.1″,”math”:”\(\hbar =1\)”})?

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