Microscopic models for charge-noise-induced dephasing of solid-state qubits

Several experiments have shown qubit coherence decay of the form $exp[-{(t/T_2)}^{\alpha }]$ due to environmental charge-noise fluctuations. We present a microscopic description for temperature dependencies of the parameters $T_2$ and $\alpha$. Our description is appropriate to qubits in semiconductors interacting with spurious two-level charge fluctuators coupled to a thermal bath. We find distinct power-law dependencies of $T_2$ and $\alpha$ on temperature depending on the nature of the interaction of the fluctuators with the associated bath. We consider fluctuator dynamics induced by first- and second-order tunneling with a continuum of delocalized electron states. We also study one- and two-phonon processes for fluctuators in either GaAs or Si. These results can be used to identify dominant charge-dephasing mechanisms and suppress them.

Figure 1

(a) A qubit (Q) is coupled to an ensemble of independent fluctuators. Each fluctuator $\left(F_n\right)$ is itself coupled to an independent bath $\left(B_n\right)$. (b) A two-level fluctuator. Because of the interaction with its bath, the two-level fluctuator is excited at the rate ${\gamma }_{\uparrow }$ and relaxes at the rate ${\gamma }_{\downarrow }$. (c) Two-level fluctuators can be, e.g., two localized states (represented by the green wave functions) between which a charge can tunnel. (d) The qubit evolves under the influence of sharp control $\pi$ pulses.

Figure 2

Approximate compressed-exponential form of $C^e\left(t_s\right)$ when ${\tau }_n\equiv \tau \forall n$. (a) The slope of $f^e\left(t_s\right)$ in log-log scale gives ${\alpha }^e\left(t_s\right)$ [see Eq. (27)]. (b) We approximate $C^e\left(t_s\right)\simeq exp[-{(t_s/T_2^e)}^{{\alpha }^e}]$ by taking ${\alpha }^e\equiv {\alpha }^e\left(T_2^e\right)$. When ${\alpha }^e\left(T_2^e\right)\simeq 3$ (in the slow-noise limit, where $T_2^e/\tau \ll 1$ is in the blue area), decay is faster than exponential and $T_2^e$ is given by Eq. (30). When ${\alpha }^e\left(T_2^e\right)\simeq 1$ (in the fast-noise limit, where $T_2^e/\tau \gg 1$ is in the red area), the decay is purely exponential and $T_2^e\simeq T_2\mathrm{M}$ is given by Eq. (23). (c)–(e) Comparison of the exact (solid black line) and compressed-exponential (dashed red line) forms of $C^e\left(t_s\right)$. (c) $\eta =10$, (d) $\eta =0.1$, (e) $\eta =0.01$. (f) Maximum error made by taking $C^e\left(t_s\right)\simeq exp[-{(t_s/T_2^e)}^{{\alpha }^e}]$ with ${\alpha }^e\equiv {\alpha }^e\left(T_2^s\right)$, Eq. (38). Dots correspond to (c)–(e).

Figure 3

Tunneling processes between localized electron states and a continuum of delocalized states. (a) Direct tunneling. This first-order process is only allowed if $|{\epsilon }_n-\mu |\lesssim k_{\mathrm{B}}T$, where ${\epsilon }_n$ is the energy of the localized state for fluctuator $n$ and $\mu$ is the chemical potential of the electron reservoir. (b) Cotunneling between pairs of localized states forming a fluctuator $n$. This second-order process occurs if $|{\epsilon }_{\alpha n}-{\epsilon }_{\beta n}|\lesssim k_{\mathrm{B}}T$.

Figure 4

Coupling of a fluctuator consisting of two localized electron states interacting with a phonon bath. We consider transitions up to second order in the electron-phonon interaction. (a) Direct phonon absorption. (b) Excitation due to the two-phonon sum process. (c) Raman excitation. (a)–(c) We also include the corresponding relaxation processes (not shown). (d) Equilibration rate for a fluctuator coupled to phonons through the deformation and piezoelectric mechanisms, calculated with Eqs. (56) to (58). Solid red line: GaAs lattice. Dashed blue line: silicon lattice. In GaAs [74, 76, 77], $\Xi =a\left({\Gamma }_{1\mathrm{c}}\right)=-8.6$ eV, $e_{14}=-0.16\mathrm{C}/{\mathrm{m}}^2, v_{\mathrm{LA}}=5210\mathrm{m}/\mathrm{s}, v_{\mathrm{TA}}=3070\mathrm{m}/\mathrm{s}, \hslash {\omega }_{\mathrm{D}}/k_{\mathrm{B}}=360$ K, and $\varepsilon =12.9{\varepsilon }_0$. In silicon, $\Xi ={\Xi }_d+\frac{1}{3}{\Xi }_u, {\Xi }_d=5$ eV, ${\Xi }_u=8.77$ eV, $e_{14}=0, v_{\mathrm{LA}}=9040\mathrm{m}/\mathrm{s}, v_{\mathrm{TA}}=5400\mathrm{m}/\mathrm{s}$, and $\hslash {\omega }_{\mathrm{D}}/k_{\mathrm{B}}=640$ K. For both GaAs and silicon, we take $|{\wp }_{\alpha \beta }^n|=1\mathrm{nm}, {\wp }_0^n=10\mathrm{nm}, \hslash {\omega }_n=10\mathrm{neV}$, and $\hslash {\omega }_{\gamma }=1\mathrm{meV}$.

Figure 5

Hahn-echo coherence time $T_2^e$ and corresponding stretching parameter ${\alpha }^e$. (a) Coherence time from the Raman phonon process in GaAs (solid red line) and silicon (dashed blue line). (b) Corresponding stretching parameter. (c) Coherence time for the direct tunneling process. (d) Corresponding stretching parameter. (a), (b) Material parameters take the same values as given in the caption of Fig. 4 for GaAs and silicon. For the Raman process, we choose ${\wp }_{\alpha \beta }^n=0, {\wp }_0^n=100\mathrm{nm}, \hslash {\omega }_{\gamma }^n=200\mu \text{eV}$, and $D\left(0\right){\Omega }_n^2=1\times {10}^3 {\mathrm{s}}^{-1} \forall n$. These numbers are chosen to give $T_2^e\sim 1\mu \text{s}$ and ${\alpha }^e\simeq 1$ for $T\sim 100\text{mK}$, consistent with Ref. [3]. The decay is given by a compressed exponential with ${\alpha }^e>1$ at low temperature, but becomes exponential (${\alpha }^e=1$) at higher temperature. (c), (d) For the direct tunneling process, we choose $D\left(0\right){\Omega }_n^2=1\times {10}^2 {\mathrm{s}}^{-1}$ and $\frac{2\pi }{\hslash }{D}_{\mathrm{el}}\left({\epsilon }_{\alpha n}\right){\left|t_n\left({\epsilon }_{\alpha n}\right)\right|}^2={10}^6 {\mathrm{s}}^{-1} \forall n$. The behaviors of $T_2^e\left(T\right)$ and ${\alpha }^e\left(T\right)$ are radically different for the Raman mechanism and tunneling mechanism, which makes them easily distinguishable. As explained in the main text, various other qubit-dephasing channels can become relevant at higher temperatures, possibly obscuring the crossovers seen here in any given experiment.

Figure 6

Parameter ${\beta }^e={\alpha }^e-1$. (a) Raman process. Black circles (black triangles): ${\beta }^e$ for GaAs (silicon), from the numerical method of Eqs. (26), (27) and (62), (63). Solid red line (dashed blue line): analytical temperature dependence for ${\tau }_c\lesssim T_2^s$ from Table 3 for GaAs (silicon). (b) Direct tunneling. Red circles: numerical method. Solid black line: analytical prediction. Assumptions and microscopic parameters are the same as in Fig. 5.