Phenomenological study of decoherence in solid-state spin qubits due to nuclear spin diffusion

We present a study of the prospects for coherence preservation in solid-state spin qubits using dynamical decoupling protocols. Recent experiments have provided the first demonstrations of multipulse dynamical decoupling sequences in this qubit system, but quantitative analyses of potential coherence improvements have been hampered by a lack of concrete knowledge of the relevant noise processes. We present calculations of qubit coherence under the application of arbitrary dynamical decoupling pulse sequences based on an experimentally validated semiclassical model. This phenomenological approach bundles the details of underlying noise processes into a single experimentally relevant noise power spectral density. Our results show that the dominant features of experimental measurements in a two-electron singlet-triplet spin qubit can be replicated using a $1/{\omega }^2$ noise power spectrum associated with nuclear spin flips in the host material. Beginning with this validation, we address the effects of nuclear programming, high-frequency nuclear spin dynamics, and other high-frequency classical noise sources, with conjectures supported by physical arguments and microscopic calculations where relevant. Our results provide expected performance bounds and identify diagnostic metrics that can be measured experimentally in order to better elucidate the underlying nuclear spin dynamics.

Figure 1

Qubit coherence in the presence of $1/{\omega }^2$ noise. (a) Effect of noise scaling $\alpha$ on predicted $T_2^{(\text{FID})}$. A value of $\alpha \approx 5\times {10}^{16}$ most closely reproduces the observed $T_2^{(\text{FID})}$. The dashed line represents the $1/e$ error rate. (b) Expected $1/e$ coherence time for CPMG and UDD pulse sequences as a function of pulse number for different values of $\alpha$.

Figure 2

The effects of a high-frequency cutoff on calculated spin-echo signal. (a) Schematic of $S(\omega )$ including a high-frequency cutoff to account for the dynamics of nuclear flip-flops. The high-frequency cutoff ${\omega }_C$ is variable, as is the form of the rolloff above ${\omega }_C$. (b) $\gamma$ as a function of ${\omega }_C$ for different forms of the high-frequency rolloff. (Inset) $T_2^{(\text{Echo})}$ as a function of ${\omega }_C$ for the same parameters. (c) Error probability as a function of free-precession time in a spin-echo experiment. Data approximated with a best-fit superexponential: No cutoff, $\gamma =3$; 12 dB/octave, $\gamma =3.62$; 24 dB/octave, $\gamma =3.81$; 36 dB/octave, $\gamma =3.84$; exponential, $\gamma =3.65$; Gaussian, $\gamma =3.81$. For these rolloffs, $T_2^{(\text{Echo})}\approx 30-37\hspace{0.5em}\mu$s. ${\chi }^2$ values for fits using $\gamma =4$: 12 dB/octave, ${\chi }^2=0.010$ ; 24 dB/octave, ${\chi }^2=0.002$; 36 dB/octave, ${\chi }^2=0.002$; exponential, ${\chi }^2=0.009$; Gaussian, ${\chi }^2=0.001$. $\alpha =5\times {10}^{16}$.

Figure 3

The effect of a high-frequency cutoff in the $1/{\omega }^2$ noise power spectrum on qubit coherence under the application of multipulse dynamical decoupling. Qubit error probability (coherence) is expressed as a color scale, with a value of 0.5 corresponding to complete dephasing. Left panels correspond to CPMG, right panels correspond to UDD. Individual columns for panels (c)–(n) denoted with high-frequency rolloff in use. Black dashed line corresponds to the $1/e$ decay contour. White dashed lines correspond to constant error contours beginning at 1% and diminishing by 100$\times$ per step, from right to left. Contours thus correspond to error probability ${10}^{-2}$, ${10}^{-4}$, ${10}^{-6}$, etc. Contours for values of error probability below ${10}^{-12}$ are not shown. (o), (p) Extracted $T_2$ for CPMG (o) and UDD (p) given different values of ${\omega }_C$ and the high-frequency rolloff. Lines are power-law fits to the data holding the $n=0$ intercept fixed at $T_2^{(\text{FID})}=26$ ns, and varying $\psi$. Extracted fit parameters: Gaussian rolloff, ${\psi }_{1\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.956$, ${\psi }_{10\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.817$, ${\psi }_{100\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.647$, ${\psi }_{1\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.999$, ${\psi }_{10\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.891$, ${\psi }_{100\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.692$; 12 dB/octave rolloff, ${\psi }_{1\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.769$, ${\psi }_{10\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.762$, ${\psi }_{100\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{UDD})}=0.628$, ${\psi }_{1\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.788$, ${\psi }_{10\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.748$, ${\psi }_{100\mathrm{\hspace{0.5em}}\mathrm{kHz}}^{(\text{CPMG})}=0.608$. For these calculations, $\alpha =5\times {10}^{16}$.