Quantum decoherence of a charge qubit in a spin-fermion model

We consider quantum decoherence in solid-state systems by studying the transverse dynamics of a single qubit interacting with a fermionic bath and driven by external pulses. Our interest is in investigating the extent to which the lost coherence can be restored by the application of external pulses to the qubit. We show that the qubit evolution under various pulse sequences can be mapped onto Keldysh path integrals. This approach allows a simple diagrammatic treatment of different bath excitation processes contributing to qubit decoherence. We apply this theory to the evolution of the qubit coupled to the Andreev-fluctuator bath in the context of widely studied superconducting qubits. We show that charge fluctuations within the Andreev-fluctuator model lead to a $1/f$ noise spectrum with a characteristic temperature depedence. We discuss the strategy for suppression of decoherence by the application of higher-order (beyond spin echo) pulse sequences.

Figure 1

(a) Dependence of ${\widehat{V}}_C\left(t\right)$ on time along the Keldysh contour. (b) The plot of the function $f_n\left(t^{\prime }\right)$ for the SE sequence $\left(n=1\right)$.

Figure 2

(Color online) Correlated tunneling of two electrons with opposite spins from the impurity sites in the insulator into the superconductor. An electron from the $i$th impurity with energy below the gap $\Delta$ tunnels into the superconductor, propagates over distances of the order of coherence length $\xi$, and recombines with another electron with opposite spin from the $j\text{th}$ site into a Cooper pair. The amplitude for such Andreev processes decays exponentially with distance between the impurity sites $A_{lj}\propto \text{exp}\left(-\left|{\mathbf{r}}_l-{\mathbf{r}}_j\right|/\pi \xi \right)$, see Eq. (14).

Figure 3

(Color online) Dyson’s equation for the retarded Green’s function ${\mathbf{G}}^R$ in the Born approximation. Here we have adopted the convention of Ref. 38. The advanced and Keldysh Green’s functions are obtained analogously resulting in Eqs. (18, 19).

Figure 4

(Color online) Log-log plot of the noise spectral density $S_Q\left(\omega \right)$ for Andreev-fluctuator model. The plot is obtained by randomly generating the positions ${\mathbf{r}}_l∊\left[-5,5\right]\xi$ and energies ${\epsilon }_l∊\left[-1,1\right]\text{K}$ of 50 impurity sites and then numerically integrating Eq. (23). Here we assumed that $v_i=v$, on-site repulsion $U\to \infty$, and the sites are occupied with equal number of electrons with spins up and down. We used $p_F^{-1}={10}^{-2}\xi$, $A_0=0.1\text{K}$, and $T=0.1\text{K}$. The dashed (blue) and dot-dashed (black) lines, shown for comparison, correspond to $1/f$ and $1/f^{1.3}$ noise spectra.