qdesignoptimizer.estimation.purcell_limit module
- qdesignoptimizer.estimation.purcell_limit.purcell_t1_transmon_resonator(chi, kappa, alpha, f_q, f_r)[source]
Calculates Purcell T1 limit for a transmon-resonator system in the dispersive regime.
Valid only for a two-level system approximation of the transmon (ground and first excited state), as discussed in “Controlling the Spontaneous Emission of a Superconducting Transmon Qubit”.
The transmon has negative anharmonicity in reality; here alpha is passed as its positive absolute value. Using the bosonic chi convention (H = hbar*chi*a†a*b†b) and expressing chi in terms of the physical coupling g and detuning delta = f_q - f_r:
chi = -2 * g^2 * alpha / (delta * (delta - alpha)) => (g/delta)^2 = -chi * (delta - alpha) / (2 * alpha * delta)
Then computes the Purcell decay rate and corresponding T1:
Gamma_P [rad/s] = kappa [rad/s] * (g/delta)^2 T1 = 1 / Gamma_P = 1 / (2*pi * kappa_Hz * (g/delta)^2)
NOTE: The formula diverges near the straddling regime (delta ≈ alpha, i.e. f_r between f_q and f_q + alpha) where the dispersive approximation breaks down.
All input frequencies must be in consistent units (Hz, i.e. cycles/s, not rad/s).
chi and alpha are taken as absolute values. This is consistent with pyEPR, which only returns absolute values for chi and anharmonicity.
- Parameters:
chi (float) – Absolute dispersive shift in the bosonic convention (Hz).
kappa (float) – Resonator linewidth (Hz).
alpha (float) – Qubit anharmonicity as a absolute value (Hz); the transmon anharmonicity is physically negative, but is passed here as its magnitude.
f_q (float) – Qubit frequency (Hz).
f_r (float) – Resonator frequency (Hz).
- Returns:
Purcell-limited T1 in seconds.
- Return type:
References
Koch et al., Phys. Rev. A 76, 042319 (2007) [transmon dispersive shift]
Blais et al., Rev. Mod. Phys. 93, 025005 (2021) [circuit QED review, Sec. V]
Houck et al., PRL 101, 080502 (2008) [Controlling the Spontaneous Emission of a Superconducting Transmon Qubit]