Spectroscopy of cross correlations of environmental noises with two qubits

A single qubit driven by an appropriate sequence of control pulses can serve as a spectrometer of local noise affecting its energy splitting. We show that by driving and observing two spatially separated qubits, one can reconstruct the spectrum of cross correlations of noises acting at various locations. When the qubits are driven by the same sequence of pulses, the real part of a cross-correlation spectrum can be reconstructed, while applying two distinct sequences to the two qubits allows for reconstruction of the imaginary part of this spectrum. The latter quantity contains information on either causal correlations between environmental dynamics at distinct locations or the occurrence of propagation of noisy signals through the environment. We illustrate the former case by modeling the noise spectroscopy protocol for qubits coupled to correlated two-level systems. While entanglement between the qubits is not necessary, its presence enhances the signal from which the spectroscopic information is reconstructed.

Figure 1

Illustration of the filtering term $|{\tilde{f}}_{\text{PDD}}^{n,T}{\left(\omega \right)|}^2$ (thick black curve). The function $\frac{T^2}{{(n+1)}^2}{sin}^2\left(\frac{\omega T}{2}\right)/{cos}^2\left(\frac{\omega T}{2(n+1)}\right)$ (blue) has the form of a series of very narrow peaks of height $T^2$ located at frequencies $m(n+1)\pi /T$ ($m=\pm 1,\pm 3,...$). The filtering term is a product of this function and ${\text{sinc}}^2\left(\frac{\omega T}{2(n+1)}\right)$ (dashed red curve), which dampens the amplitude of the $m$-th peak by a factor of $4/{\pi }^2{m}^2$.

Figure 2

Reconstruction of imaginary and real (inset) parts of the cross power in a numerical experiment. The values of the cross power have been extracted from data obtained from simulation of ${\rho }_{11,-1-1}\left(T\right)$ with (blue closed circles) and without (red open circles) corrections from higher-frequency peaks in the filtering term (see Appendix pp3). The results are compared with the exact function (solid black line). The parameters of the coupled telegraph noises were set to ${\gamma }_{0\uparrow }^B={\gamma }_{1\downarrow }^B=0, {\gamma }_{0\downarrow }^B={\gamma }_{1\uparrow }^B=4{\gamma }_A$, and $v_1=v_2=0.1{\gamma }_A$. The number of pulses $n=37$ was kept fixed and the total time $T$ was manipulated in order to sweep the wide range of frequencies.