How to enhance dephasing time in superconducting qubits

We theoretically investigate the influence of designed pulse sequences in restoring quantum coherence lost due to background noise in superconducting qubits. We consider both $1/f$ noise and random telegraph noise and show that the qubit coherence time can be substantially enhanced by carefully engineered pulse sequences. Conversely, the time dependence of qubit coherence under external pulse sequences could be used as a spectroscopic tool for extracting the noise mechanisms in superconducting qubits, i.e., by using Uhrig’s pulse sequence [Phys. Rev. Lett. 98, 100504 (2007)], one can obtain information about moments of the spectral density of noise. We also study the effect of pulse sequences on the evolution of the qubit affected by a strongly coupled fluctuator and show that the non-Gaussian features in decoherence are suppressed by the application of pulses.

Figure 1

(a) The illustration of various pulse sequences with application times of $\pi$ pulses marked. SE is shown along with the PDD, CPMG, CDD, and UDD sequences with $n=5$ pulses (for CDD, this corresponds to the third order of concatenation). Ten-pulse UDD and CDD (fourth order of concatenation) are also shown. (b) The function $f\left(t;t^{\prime }\right)$ defined in Eq. (14) for two-pulse CPMG sequence.

Figure 2

(Color online) Filter functions $F\left(\omega t\right)$ for SE and various pulse sequences with $n=10$ pulses (for CDD, it corresponds to the fourth level of concatenation). The lower panel shows $F\left(\omega t\right)$ in the logarithmic scale.

Figure 3

(Color online) The dependence of (a) $W\left(t\right)$ and (b) $\chi \left(t\right)=-\text{ln}W\left(t\right)$ for SE and higher-order $\left(n=5\right)$ sequences for $1/f$ noise with $A_0/{\omega }_c=1$. UDD and CPMG give $W\left(t\right)\approx 1$ for $t<\left(2n+2\right)/{\omega }_c$. The SE signal (equivalent to PDD or CPMG with $n=1$) is also shown for comparison.

Figure 4

(Color online) The decay of qubit coherence $W\left(t\right)$ for a single TLF coupled to the qubit for (a) strong coupling $g=10$ and (b) weak coupling $g=0.2$. The results of the simulation of the RTN are shown as symbols, and the calculations in Gaussian $\left(G\right)$ approximation are shown as lines. ${\text{CDD}}_4$ (with $n=10$) gives practically the same result as UDD, and thus, it is not shown. For $g=0.2$, the Gaussian approximation agrees very well with the exact results. For clarity, only SE and CPMG with $n=10$ are shown in (b). The FID signal for strong coupling (not shown) is an oscillating function for which the SE signal is an envelope.

Figure 5

(Color online) The ratio $R\left(t\right)={\chi }^{\left(4\right)}/{\chi }^{\left(2\right)}$ plotted for sequences having $n=0\dots 5$ pulses with RTN characterized by $\gamma =1$ and energy of interaction with the qubit $v=10$ (so that $g=10$). The solid lines correspond (from left to right) to FID, SE, and CPMG with $n=3,4,5$. The dashed lines correspond to the UDD sequence with $n=3,4,5$. With increasing number of pulses, the time at which the non-Gaussian effects start to be quantitatively important [when $R\left(t\right)<1$] becomes larger, eventually surpassing the $T_2$ time of the qubit.