A qubit control pulse is synthesized at low (IF) frequency by an arbitrary waveform generator as two channels, I(t) and Q(t), then upconverted to the qubit's microwave frequency by an IQ mixer driven by a local oscillator (LO):
RF(t) = I(t)·cos(ω_LO t) − Q(t)·sin(ω_LO t)
I(t) = A(t)·cos(ω_IF t)
Q(t) = (1+ε)·A(t)·sin(ω_IF t + δφ) [real hardware's gain/phase error]
With a perfect mixer (ε = 0, δφ = 0) this produces one clean tone at ω_LO + ω_IF — single-sideband upconversion. Any gain imbalance ε or phase imbalance δφ between the I and Q channels lets a second, unwanted tone leak through at the mirror frequency ω_LO − ω_IF (the "image"), with amplitude:
A_upper = ½√(1+(1+ε)² + 2(1+ε)cos δφ) (desired)
A_image = ½√(1+(1+ε)² − 2(1+ε)cos δφ) (image)
Image Rejection Ratio = 20·log₁₀(A_upper / A_image) dB
A real mixer also lets a fraction of the bare LO tone bleed straight through to the output (LO/carrier leakage), independent of any modulation. On a chip crowded with many qubits at different frequencies, the image tone is not just wasted power — if a neighboring qubit happens to sit near f_LO − f_IF, that unwanted tone can off-resonantly drive it, corrupting its state. This is one reason real control electronics are calibrated (DC-offset nulling for LO leakage, gain/phase correction for image rejection) before running any algorithm.
This 2D view derives the exact same closed-form amplitudes as the 3D version, but additionally plots the literal time-domain sum RF(t) = A(t)·[cos(ω_IF t)cos(ω_LO t) − (1+ε)sin(ω_IF t+δφ)sin(ω_LO t)] + leakage·cos(ω_LO t) — drag the waveform panel to scrub through time, or use the mouse wheel over it to zoom the time axis. (On-screen ω_LO/ω_IF are compressed to a legible display ratio; the dB readouts use the exact physics formulas above, unaffected by that display choice.)
- Gain / phase imbalance sliders — set ε and δφ; watch the spectrum bar for the image tone grow, the image path light up, and the neighbor-qubit marker pulse.
- LO leakage slider — sets the constant carrier bleed shown as the amber spectrum bar and glow at the mixer.
- Pulse width slider — widens or narrows the Gaussian envelope A(t) that gates every burst in the waveform panel.
- Drag / wheel on waveform — pan and zoom the time axis of the live RF(t) trace.
- Presets — "Ideal mixer" zeroes all three error sources; "Typical hardware" loads representative lab-grade calibration residuals.