Theory of qubit noise characterization using the long-time cavity transmission

Noise-induced decoherence is one of the main threats to large-scale quantum computation. In an attempt to assess the noise affecting a qubit, we go beyond the standard steady-state solution of the transmission through a qubit-coupled cavity in input-output theory by including dynamical noise in the description of the system. We solve the quantum Langevin equations exactly for a noise-free system and treat the noise as a perturbation. In the long-time limit the corrections may be written as a sum of convolutions of the noise power spectral density with an integration kernel that depends on external control parameters. Using the convolution theorem, we invert the corrections and obtain relations for the noise spectral density as an integral over measurable quantities. Additionally, we treat the noise exactly in the dispersive regime and again find that noise characteristics are imprinted in the long-time transmission in convolutions containing the power spectral density.

Figure 1

The system under consideration consists of a cavity-coupled qubit (green circle) with fluctuating energy separation ${\omega }_q$ due to noise $\delta X\left(t\right)$ affecting its control parameters. The cavity mirrors (gray) allow for an input field $b_{\mathrm{in}}$ at the left port and an output field $b_{\mathrm{out}}$ at the right port, thereby creating a measurable transmission signal $A\left(t\right)$. We find that the averaged transmission probability ${\langle \langle \left|A\right|}^2\rangle \rangle$ receives noise corrections in the form of sums over convolutions $\mathcal{C}=S★K$ of the power spectral density $S\left(\omega \right)$ of $\delta X$ with an integration kernel $K$. As a result, the spectral density can be extracted from the averaged transmission probability by using the convolution theorem.

Figure 2

Normalized averaged long-time transmission probability. Noise corrections are visible whenever ${\langle \langle \left|A\right|}^2\rangle \rangle /|A_0{|}^2\ne 1$. (a) Plot of ${\langle \langle \left|A\right|}^2\rangle \rangle /|A_0{|}^2$ as a function of the qubit frequency ${\omega }_q$ and the probe frequency ${\omega }_p$. The effect of the noise is most pronounced at the qubit-probe resonance ${\omega }_q={\omega }_p$. (b) Line cuts at different detunings as indicated by the dashed lines in (a). Analytical results (solid lines) are drawn according to Eq. (7), while the numerical results are obtained by averaging over ${10}^3$ solutions of the Lindblad master equation that corresponds to the exact Langevin equations. We include normally distributed quasistatic noise with zero mean and standard deviation $\sigma =\delta X_{\text{rms}}$ and allow for up to 12 photons in the cavity. (c) Temperature dependence of the averaged transmission probability. At high temperatures the noise corrections are washed out. (d) Plot of ${\langle \langle \left|A\right|}^2\rangle \rangle /|A_0{|}^2$ at the critical parameter settings $\mathrm{\Lambda }=0$, i.e., ${\kappa }_c=\gamma \pm 4g\sqrt{-\langle {\sigma }_z\rangle }$ for $\kappa \gtrless \gamma$. Analytical results (solid lines) are drawn according to Eq. (16), while the numerical results (squares) are obtained as in (b). (e) Comparison between the cases of a diagonalizable and nondiagonalizable system matrix $L$. We find that the transmissions for $\kappa \ne {\kappa }_c$ smoothly transition to the transmission at $\kappa ={\kappa }_c$ (dashed line). The parameters are $\gamma t={10}^3$, $\lambda \delta X_{\text{rms}}={10}^{-3}{\omega }_c$, $g=0.01{\omega }_c$, ${\gamma }_1=0.01{\omega }_c$, ${\gamma }_{\phi }=0.005{\omega }_c$, and ${\kappa }_{\text{int}}=0$ and (a), (b), (d), and (e) ${\omega }_q/T=10$, (a)–(c) $\kappa =0.01{\omega }_c$, and (c)–(e) ${\delta }_0=0$.

Figure 3

Schematic of the procedure to determine the power spectral density $S$. In the first phase, the averaged squared transmission is recorded for fixed $\mathrm{\Delta }$ as a function of the detuning ${\delta }_0$ to extract the convolutions ${\mathcal{C}}_j$. After repeating this for many values of $\mathrm{\Delta }$, the second phase is initiated. The Fourier transform of any of the convolutions is computed and finally the spectral density by applying the convolution theorem.

Figure 4

Normalized deviation from the empty cavity transmission probability ${\delta \langle \langle \left|A\right|}^2\rangle \rangle /|A_{\infty }{|}^2$. Here ${\delta \langle \langle \left|A\right|}^2\rangle \rangle /|A_{\infty }{|}^2$ is plotted as a function of the cavity-probe detuning ${\mathrm{\Delta }}_c$ and temperature $T$ (a) in the noise-free case and (b) in the presence of white noise of amplitude $S_0=0.1{\delta }_0$. The noise affects the transmission features at the qubit frequency ${\mathrm{\Delta }}_c={\delta }_0$, but not at the cavity frequency ${\mathrm{\Delta }}_c=0$. At high temperatures $T\gtrsim {\omega }_q$, the noise signatures are washed out. (c) Line cuts at $T=0.5{\delta }_0$ as indicated by the horizontal lines in (a) and (b). The parameters are $g=\kappa =0.05{\delta }_0$, ${\kappa }_{\text{int}}=0$, ${\gamma }_1=0.05{\delta }_0$, ${\gamma }_{\phi }=0.025{\delta }_0$, and $\lambda =0.9$.