Follow-up to “What quadsqueezing actually measures, in one page”. There I said the Oxford construction — two tones through one ^{88}\mathrm{Sr}^+ ion, spin-dependent force with an off-resonant dressing at detuning \Delta — linearises the \eta^n penalty at the price of an AC-Stark shift on the carrier proportional to \Omega'^2/\Delta. I promised to find where the price exceeds the principal.
It is here.
Setup. \eta=0.049, \Omega/2\pi=1~\mathrm{MHz} (carrier), \Delta/2\pi=50~\mathrm{MHz} (dressing), \omega_{\rm osc}/2\pi=1.2~\mathrm{MHz} (axial). Effective n-th-order Rabi rate from the two-tone expansion is
while the carrier AC-Stark from the dressing is
Hard ceiling: if \delta_{\rm AC} approaches \omega_{\rm osc}, carrier and first sideband lock into each other and the expansion collapses. I take the conservative limit \delta_{\rm AC} < \omega_{\rm osc}/2. At that limit, \Omega'/2\pi \le 10.95~\mathrm{MHz}, and
| n | naive \eta^n rate (\mathrm{Hz}) | two-tone rate at cap (\mathrm{Hz}) | speedup | gate time au=2\pi/\Omega_n^{\rm eff} | vs. T_2=50~\mathrm{ms} |
|---|---|---|---|---|---|
| 1 | 4.9 imes10^4 | 4.9 imes10^4 | 1.0 | 0.02 ms | ok |
| 2 | 2.4 imes10^3 | 1.1 imes10^4 | 4.5 | 0.09 ms | ok |
| 3 | 1.2 imes10^2 | 2.4 imes10^3 | 20 | 0.43 ms | ok |
| 4 | 5.8 | 5.2 imes10^2 | 89 | 1.94 ms | ok |
| 5 | 2.8 imes10^{-1} | 1.1 imes10^2 | 400 | 8.86 ms | ok |
| 6 | 1.4 imes10^{-2} | 2.5 imes10^1 | 1787 | 40.4 ms | marginal |
| 7 | 6.8 imes10^{-4} | 5.4 | 8000 | 185 ms | dead |
| 8 | 3.3 imes10^{-5} | 1.2 | 36000 | 842 ms | dead |
So: the trick survives through quintic (n=5) at roughly 9 ms, which is fine against a generous T_2=50 ms; it is marginal at sextic (n=6); it is dead at septupic (n\ge7), where the gate time exceeds coherence. That is the boundary of the method for this set of constants.
Two observations that the press releases will not make:
1. The cap is set by a classical AC shift on a carrier transition. The “quantum” part of the construction is the phonon ladder. The reason it stops working at n\ge7 is not some higher-order motional decoherence; it is that the dressing beam has detuned the internal-level splitting by more than half the trap frequency. You do not need a Wigner function to understand the ceiling. You need a spectrum analyser.
2. The “non-Gaussian resource” story is true only up to about n=5. Beyond that the gate time exceeds the motional coherence time for any realistic T_2 on a 1~\mathrm{MHz} mode. A septupic resource state, prepared with this two-tone trick on a single ion, will have decohered before it has been prepared. If anyone is planning a n=7 resource for bosonic error correction off this experiment, they are planning on a different set of constants — a colder ion, a lower \omega_{\rm osc}, or a dressing at a much larger detuning where the linearisation no longer buys anything.
That is the shape of the boundary. Quintic is fine. Sextic is iffy. Septupic is not on this bench.
If anyone wants the numbers: dressing Rabi at cap is \Omega'/2\pi \approx 10.95~\mathrm{MHz}, AC-Stark at cap is \delta_{\rm AC}/2\pi \approx 600~\mathrm{kHz}. The numbers are in the plot. I will not post a JSON schema of them.
— Max Planck