Dynamical-decoupling noise spectroscopy at an optimal working point of a qubit

I present a theory of environmental noise spectroscopy via dynamical decoupling of a qubit at an optimal working point. Considering a sequence of $n$ pulses and pure dephasing due to quadratic coupling to Gaussian distributed noise $\xi \left(t\right)$, I use the linked-cluster (cumulant) expansion to calculate the coherence decay. Solutions allowing for reconstruction of spectral density of noise are given. For noise with correlation time shorter than the time scale on which coherence decays, the noise filtered by the dynamical decoupling procedure can be treated as effectively Gaussian at large $n$, and well-established methods of noise spectroscopy can be used to reconstruct the spectrum of ${\xi }^2\left(t\right)$ noise. On the other hand, for noise of dominant low-frequency character $(1/f^{\beta }$ noise with $\beta >1)$, an infinite-order resummation of the cumulant expansion is necessary, and it leads to an analytical formula for coherence decay having a power-law tail at long times. In this case, the coherence at time $t$ depends both on spectral density of $\xi \left(t\right)$ noise at $\omega =n\pi /t$, and on the effective low-frequency cutoff of the noise spectrum, which is typically given by the inverse of the data acquisition time. Simulations of decoherence due to purely transverse noise show that the analytical formulas derived in this paper apply in this often encountered case of an optimal working point, provided that the number of pulses is not very large, and that the longitudinal qubit splitting is much larger than the transverse noise amplitude.

Figure 1

(a) Frequency-domain filter function for CP sequence with $n=2$, 4, and 10 pulses. The plotted function $F\left(z\right)={\pi }^2{\left|{\tilde{f}}_t\left(z\right)\right|}^2/4t^2$ with $z=\omega t/\pi$ has a dominant peak at $z_1\approx n$, and the next peak at $z_2\approx 3n$ is $\approx 9$ times smaller. (b) Calculation of $W_l\left(t\right)$ for linear coupling $v_1\xi \left(t^{\prime }\right){\widehat{\sigma }}_z$ to noise with spectral density $v_{1}S\left(\tilde{\omega }\right)=A/{\tilde{\omega }}^{3/2}+B/[{\gamma }^2+{(\tilde{\omega }-{\omega }_p)}^2]$ where $\tilde{\omega }=\omega /v_1$, and $A={\pi }^{7/2}/4,B=10,{\omega }_p=10$ are dimensionless constants. (c) Demonstration of spectroscopy in this case: the results of exact calculation of $W_l\left(t\right)$ from (b) are used to reconstruct $v_{1}S\left(\omega \right)$ using Eq. (8).

Figure 2

Numerical calculation of $|{\chi }_4\left(t\right)|$ (circles) for Ornstein-Uhlenbeck noise with $\sigma =1$ and $v_2{t}_c=1$, done for $n=11$ pulses of the CP sequence. The dashed line is the “single-peak” (SP) approximation for $|{\chi }_4|=\frac{1}{2}{R}_4$ from Eq. (25), while the solid line is the SP approximation for $t/n\ll t_c$ from Eq. (26). The latter formula agrees quite well with the numerical calculation for $t/n<t_c<t$ (i.e., for $1<t/t_c<n)$. ${\chi }_2^2\left(t\right)$ is shown as crosses, and one can see that for $t/t_c\ll 1$ we have $|{\chi }_4\left(t\right)|\approx {\chi }_2^2\left(t\right)$, in agreement with Eq. (35) which holds in this regime.

Figure 3

Decoherence function at an OWP for CP sequences with $n=1$, 11, and 21. Symbols are the results of numerical simulation with OU noise (with $\sigma =1)$ obtained using standard algorithms [58]. The coupling is $v_2=1/t_c$. The solid lines are the Gaussian approximation $W\left(t\right)=exp[-{\chi }_2\left(t\right)]$, while the dashed lines are $W\left(t\right)=exp[-{\chi }_2\left(t\right)-{\chi }_4\left(t\right)]$ with ${\chi }_2\left(t\right)$ and ${\chi }_4\left(t\right)$ given in Eqs (24) and (26), respectively. The divergence of the latter signifies the failure of the approach in which only a few cumulants are taken into account. Already for $n=11$ the characteristic decay time scale $T_2$ is well described by the Gaussian approximation, while for $n=21$ the two lowest cumulants are enough to describe the coherence decay by an order of magnitude from the initial value.

Figure 4

Decoherence due to OU noise (with $\sigma =1)$ at an OWP for CP sequence with $n=1$, 5, and 11. Symbols are the results of numerical simulation. For each $t$ the averaging time was $T_M=Mt$ with $M={10}^6$, so that the resulting ${\sigma }_0^2$ was well approximated by the total power of the OU noise, ${\sigma }^2$. With coupling $v_2={10}^5/t_c$, the coherence decay in the presented time range is due to the $1/{\omega }^2$ tail of $S\left(\omega \right)$. The solid lines are obtained using Eq. (36). For $n=5$ the dotted line is the Gaussian approximation, and the dashed line is $W\left(t\right)\sim t^{-3/2}$ asymptotics from Eq. (37).

Figure 5

Decoherence due to transverse OU noise with $\sigma =1$ for CP sequence with $n=1$ (spin echo). The transverse coupling constant is $v_t=\sqrt{2\Omega v_2}$ and $v_2={10}^5/t_c$. The notation $(i,{\pi }_j)$ specifies the initial direction of the qubit $(i=x,y)$ and the axis about which the $\pi$ pulse is applied $(j=x,y)$. The solid lines are obtained using Eq. (36).

Figure 6

Decoherence due to transverse OU noise with $\sigma =1$ for CP sequence with $n=4$. Parameters and symbols are the same as in Fig. 5.

Figure 7

Decoherence due to transverse OU noise (with $\sigma =1)$ at an OWP for CPMG $(y,{\pi }_y)$ sequence with $n=1$, 11, and 21. Symbols are the results of numerical simulation for transverse noise with $v_t$ couplings chosen such that for qubit splitting $\Omega =10v_2{\sigma }^2$ (upper panel) and $\Omega =100v_2{\sigma }^2$ (lower panel) the results are expected to match the effective Hamiltonian results from Fig. 3, where $v_2=1/t_c$ was used. Solid lines are the Gaussian approximation $W\left(t\right)=exp[-{\chi }_2\left(t\right)]$, while the dashed line for $n=21$ is $W\left(t\right)=exp[-{\chi }_2\left(t\right)-{\chi }_4\left(t\right)]$.

Figure 8

Decoherence due to transverse OU noise for CPMG $(y,{\pi }_y)$ sequence with $n=1$, 5, and 11. Symbols are the results of numerical simulation for transverse noise with $v_t$ coupling chosen such that the results are expected to match the effective Hamiltonian calculations from Fig. 4 (where $v_2={10}^5/t_c$ has been used). The circle and cross symbols correspond to two values of $\Omega$. The solid lines are obtained using Eq. (36).