Four-Wave Mixing SMPD Efficiency

Based on Balembois et al., “Cyclically operated Single Microwave Photon Counter with 10⁻²² W/√Hz sensitivity”, arXiv:2307.03614. Implements Eq. (3) (four-wave mixing transfer efficiency), Appendix G (optimal detection time), and the total-efficiency product η_theory = η_4wm · η_RO · η_D · η_qubit.

Glossary — what each variable means

χ_b, χ_w	Dispersive shifts of the transmon qubit with respect to the buffer (b) and waste (w) resonators. Quoted as χ/2π in MHz.
κ_b, κ_w	Total linewidths (energy decay rates) of the buffer and waste resonators. Quoted as κ/2π in MHz. κ_w provides the irreversible dissipation that makes the click event irreversible.
\|ξ\|	Pump amplitude (dimensionless, in units of √photons) of the off-resonant pump tone that activates the four-wave-mixing process. Controls the parametric coupling g₃ = 2\|ξ\|√(χ_bχ_w).
C	Cooperativity of the four-wave-mixing process, C = 4\|ξ\|²χ_bχ_w/(κ_bκ_w). Unit transfer efficiency is reached at C = 1.
η_4wm	Transfer efficiency of a buffer photon into a qubit excitation via the four-wave-mixing process (Eq. 3). Peaks at 1 when C = 1.
T₁	Qubit energy-relaxation time (\|e⟩ → \|g⟩). Sets how long an absorbed excitation survives in the qubit.
T_d	Detection window: duration during which the pump is on and an incoming photon can trigger a qubit excitation.
T_m	Measurement window: duration of the dispersive readout pulse on the waste resonator that checks the qubit state.
T_r	Reset window: conditional reset back to \|g⟩ (π-pulse + re-measurement if the qubit was found excited). Use the average reset time.
T_cycle	T_d + T_m + T_r, the total duration of one detect–measure–reset cycle.
η_D	Duty-cycle efficiency = fraction of time the detector is actually listening, T_d/T_cycle.
η_qubit	Probability that a qubit excitation created during detection survives until measurement, averaged over the detection window: (T₁/T_d)(1 − e^−T_d/T₁).
η_RO	Single-shot readout fidelity for the excited state — probability of reading “click” given the qubit is in \|e⟩.
η_theory	Total predicted operational efficiency: η_4wm · η_RO · η_D · η_qubit.

1. Four-wave mixing efficiency η_4wm

η_4wm = 4C/(1+C)², C = 4|ξ|² χ_bχ_w/(κ_bκ_w). The 2π factors cancel in the ratio, so χ/2π and κ/2π in MHz may be entered directly.

χ_b/2π [MHz]:

χ_w/2π [MHz]:

κ_b/2π [MHz]:

κ_w/2π [MHz]:

Pump amplitude |ξ| (dimensionless, √photons) — spot evaluation:

2. Optimal detection time (Appendix G)

η_D = T_d/(T_m+T_r+T_d), η_qubit = (T₁/T_d)(1 − e^−T_d/T₁). Optimum: T_d^opt ≈ √(2(T_m+T_r)T₁) (Eq. G3).

T₁ [μs]:

T_m [μs]:

T_r [μs]:

T_d [μs] (user value):

3. Total detection efficiency

η_theory = η_4wm · η_RO · η_D · η_qubit. Uses η_4wm(|ξ|) from section 1 and η_D, η_qubit from section 2.

Readout efficiency η_RO:

use user T_d use optimal T_d

Formulas

Quantity	Expression
Cooperativity C	4\|ξ\|² χ_bχ_w/(κ_bκ_w)
4WM efficiency η_4wm	4C/(1+C)² (Eq. 3)
Optimal pump \|ξ\|^opt	√(κ_bκ_w/(4χ_bχ_w)) (C=1)
Duty-cycle efficiency η_D	T_d/(T_m+T_r+T_d)
Qubit efficiency η_qubit	(T₁/T_d)(1 − e^−T_d/T₁)
Optimal detection time	T_d^opt ≈ √(2(T_m+T_r)T₁) (Eq. G3)
Total efficiency	η_theory = η_4wm · η_RO · η_D · η_qubit

How to compare to an experiment

Two kinds of measurement map onto the efficiencies on this page. Which one you look at determines which factors appear.

(a) End-of-detection qubit population p_e (Fig. 2a-style). Read the qubit immediately after the detection window. In the weak-signal regime (⟨n_ph⟩ ≪ 1 per cycle):

p_e(|ξ|) ≈ p_e^bg + η_4wm(|ξ|) · η_qubit · ⟨n_ph⟩

with ⟨n_ph⟩ = Φ_in · T_d the mean number of signal photons during the detection window. Only η_4wm and η_qubit enter — η_RO and η_D do not, because you are reading the population directly rather than converting to a per-second rate. Normalizing the curve to its peak gives the shape η_4wm(|ξ|) independent of the absolute calibration of Φ_in.

(b) Click rate vs incoming photon rate (Fig. 3b-style — recommended). Record the detector click rate R for several input photon rates Φ_in and fit a line:

R = η · Φ_in + α

Slope η = η_4wm · η_RO · η_D · η_qubit — the full operational efficiency from section 3 of this calculator.
Intercept α — the dark count rate (incl. thermal photons in the input line, pump-induced qubit heating, and thermal qubit population).

The linear fit extracts η and α simultaneously from one dataset, so dark counts are absorbed into the intercept and you do not need a separately calibrated “signal off” baseline (nor an accurate subtraction, which can drift between runs). For reliable numbers, sweep Φ_in over at least a decade and include a point or two deep enough that η · Φ_in ≫ α as well as a point near Φ_in = 0 to anchor the intercept.

The power sensitivity then follows from Eq. 1 of the paper: 𝒮 = ℏω√α / η.