A transmon is to be designed with transition frequency and anharmonicity

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Quantum Information and Technology, Ch Superconducting qubits and circuit QED

Problem

A transmon is to be designed with transition frequency f01=5.500 GHzf_{01} = 5.500\,\mathrm{GHz} and anharmonicity α/2π=−300 MHz\alpha/2\pi = -300\,\mathrm{MHz}, using ℏω01≈8EJEC−EC\hbar\omega_{01} \approx \sqrt{8E_JE_C} - E_C and α≈−EC/ℏ\alpha \approx -E_C/\hbar. ((a)) Find EJ/hE_J/h and EC/hE_C/h in GHz. ((b)) Compute EJ/ECE_J/E_C and comment on where this places the design relative to the usual transmon operating point EJ/EC≈50E_J/E_C \approx 50. ((c)) A reference transmon with EJ/h=12.5 GHzE_J/h = 12.5\,\mathrm{GHz}, EC/h=0.25 GHzE_C/h = 0.25\,\mathrm{GHz} (EJ/EC=50E_J/E_C = 50) has a first-level charge dispersion of about 10 kHz10\,\mathrm{kHz}. Taking the dispersion to scale as EC e−8EJ/ECE_C\,e^{-\sqrt{8E_J/E_C}}, estimate the charge dispersion of this design. ((d)) This transmon is coupled to a readout resonator with g/2π=80.0 MHzg/2\pi = 80.0\,\mathrm{MHz}. Using the transmon dispersive shift χ=g2ΔαΔ+α\chi = \dfrac{g^2}{\Delta}\dfrac{\alpha}{\Delta+\alpha}, find the (positive) detuning Δ/2π\Delta/2\pi needed to give χ/2π=−4.00 MHz\chi/2\pi = -4.00\,\mathrm{MHz}. ((e)) Choosing κ=2∣χ∣\kappa = 2|\chi|, state κ/2π\kappa/2\pi and compute the Purcell-limited T1T_1, γP≈κ(g/Δ)2\gamma_P \approx \kappa(g/\Delta)^2.

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