Superconducting Resonator — Two-Level-System Loss & Quality Factor

Material

Environment

Temperature 15 mK

Surface TLS defect density medium

Drive power (photon number n̄) 3

Readout

Loaded Q

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Linewidth κ/2π

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Ring-down time τ

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Dominant loss

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How it works

A microwave resonator stores field energy in a standing wave. Its quality factor Q sets how many oscillations that energy survives before decaying: energy falls as E(t) = E₀·exp(−ωt/Q), so a resonator rung and released rings down over a time τ = Q/ω. Three loss channels add in parallel (1/Q_total = 1/Q_TLS + 1/Q_qp + 1/Q_rad):

Q_TLS(T,n̄)  ∝ (1/δ_TLS) · √(1+n̄/n_c) · coth(ħω/2k_BT)
Q_qp(T)     ∝ exp(+Δ(Tc)/k_BT)      (quasiparticles freeze out as T→0)
Q_rad                                (fixed, geometry-set)

Temperature — cooling suppresses thermally-excited quasiparticles exponentially (Q_qp rises fast below T_c/10), but TLS defects stay lossy until ħω ≫ k_BT fully saturates their two levels.
Surface TLS defect density — amorphous oxide at interfaces hosts two-level-system defects that absorb photons resonantly; cleaner interfaces (lower density) directly raise Q_TLS.
Drive power (n̄) — pumping more photons into the resonator saturates the TLS defects (they can only absorb once before re-emitting), so Q_TLS rises with power — the classic "TLS saturation" curve measured in every real resonator test.
Material (T_c) — a higher superconducting transition temperature pushes the quasiparticle-loss onset to higher T, which is why niobium (T_c≈9.2 K) tolerates warmer stages better than aluminium (T_c≈1.2 K).

This is the same measurement — launch a pulse, watch the ring-down, fit Q — used to qualify every low-loss resonator that reads out a superconducting qubit.

Field amplitude (live) Envelope (decaying)

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