Finite sizes and smooth cutoffs in superconducting circuits

We investigate the validity of two common assumptions in the modeling of superconducting circuits: first, that the superconducting qubits are pointlike and, second, that the UV behavior of the transmission line is not relevant to the qubit dynamics. We show that in the experimentally accessible ultrastrong-coupling regime and for short (but attainable) times, the use of an inaccurate cutoff model (such as a sharp cutoff or none at all) could introduce very significant inaccuracies in the model's predictions.

Figure 1

The switching function $\chi \left(t\right)$.

Figure 2

Relative difference of spontaneous emission probabilities when comparing (a) shape functions over a range of sizes $\sigma$, (b) shape functions over a range of cutoff scales $\varepsilon$, (c) cutoff functions over a range of cutoff scales $\varepsilon$, and (d) cutoff functions over a range of sizes $\sigma$. Markers are included to distinguish lines. Markers indicate which pair of shape functions or cutoff functions is being compared and are listed in the legends to the right of (b) and (d). The interaction times are fixed at $r=r_0$ and $T=T_0$. Shapes are compared using exponential cutoff and cutoffs using Gaussian shape. Sizes are compared while the cutoff is kept constant at ${\varepsilon }_0$, and cutoff scales are compared while the size is kept constant at ${\sigma }_0$. The behavior of the vacuum excitation probabilities is similar.

Figure 3

Relative difference of vacuum excitation probability comparing (a) exponential and Gaussian, (b) exponential and Lorentzian, (c) exponential and sharp, (d) Gaussian and Lorentzian, (e) Gaussian and sharp, and (f) sharp and Lorentzian cutoff functions across a range of ramp-up and constant interaction times $r$ and $T$.

Figure 4

Relative difference of probability of spontaneous emission comparing (a) exponential and Gaussian, (b) exponential and Lorentzian, (c) exponential and sharp, (d) Gaussian and Lorentzian, (e) Gaussian and sharp, and (f) sharp and Lorentzian cutoff functions across a range of ramp-up and constant interaction times $r$ and $T$.

Figure 5

Relative difference of vacuum excitation probability between pairs of cutoff functions across a range of interaction time scales $T$ for fixed size ${\sigma }_0$, shape (Gaussian), cutoff scale ${\varepsilon }_0$, and ramp-up time $r=0.1r_0$. Markers are included to distinguish lines.