Decoherence of Flux Qubits due to $1/f$ Flux Noise

We have investigated decoherence in Josephson-junction flux qubits. Based on the measurements of decoherence at various bias conditions, we discriminate contributions of different noise sources. We present a Gaussian decay function extracted from the echo signal as evidence of dephasing due to $1/f$ flux noise whose spectral density is evaluated to be about $({10}^{-6}{\Phi }_0{)}^2/\mathrm{Hz}$ at 1 Hz. We also demonstrate that, at an optimal bias condition where the noise sources are well decoupled, the coherence observed in the echo measurement is limited mainly by energy relaxation of the qubit.

Figure 1

Scanning electron micrograph of a sample and a sketch of the measurement setup. The qubit shares a part of the loop with a SQUID shunted by an on-chip capacitor. A current bias line ($I_b$) and a voltage measurement line ($V_m$) consist of lossy coaxial cables and are connected to the SQUID via on-chip resistors. Microwave current pulses are fed through an on-chip control line ($I_{\mathrm{MW}}$) inductively coupled to the qubit. The sample chip is enclosed in a copper shield box, and all of the wires are electrically shielded as well. CPF stands for a copper-powder filter. Magnetic flux bias is applied with an external superconducting coil connected to a battery-powered current source. The sample is cooled to 20 mK in a dilution refrigerator magnetically shielded with three $\mu$-metal layers at room temperature.

Figure 2

Measurements of (a) energy relaxation, (b) free-induction decay, and (c) echo decay at the optimal bias condition $n_{\varphi }=n_{\varphi }^{*}$ and ${\mathrm{I}}_b={\mathrm{I}}_b^{*}$. In each panel, the plot with a longer decay time is for sample A, and the other is for sample B. The insets depict the applied pulse sequence for each measurement. In the measurement of free-induction decay, the microwave carrier frequency was detuned from the qubit frequency $E_{01}/h$ by 10 MHz. Solid curves in (a) and (c) are exponential fits.

Figure 3

Decoherence at various SQUID bias currents $I_b$ (sample A). Qubit flux bias is kept at $n_{\varphi }=n_{\varphi }^{*}$. (a) Spectroscopically determined $n_{\varphi }^{*}$ vs $I_b$. The solid curve is a polynomial fit. (b) Energy-relaxation rate ${\Gamma }_1$ vs $I_b$. (c) Dephasing component $⟨e^{i\phi (t)}{⟩}_E$ in the echo measurements for different values of $I_b$. The solid curve is an exponential decay, $\exp (-{\Gamma }_{\phi E}t)$. (d) FID dephasing rate ${\Gamma }_{\phi F}$ and echo dephasing rate ${\Gamma }_{\phi E}$ as a function of $I_b$. See text for the explanations of the fitting curves in (b) and (d).

Figure 4

Decoherence at various flux biases $n_{\varphi }$. SQUID bias current is kept at $I_b=I_b^{*}$. (a) Dephasing component $⟨e^{i\phi (t)}{⟩}_E$ in the echo measurements for different values of $n_{\varphi }$ (sample A). The solid curve is a Gaussian fit, $\exp [-({\Gamma }_{\phi E}^{g}t{)}^2]$. (b) Energy-relaxation rate ${\Gamma }_1$ and echo dephasing rate ${\Gamma }_{\phi E}^g$ vs $n_{\varphi }$. The solid line is a $|\partial E_{01}/\partial n_{\varphi }|$ fit to ${\Gamma }_{\phi E}^g$. (c) FID dephasing rate ${\Gamma }_{\phi F}^g$ vs $n_{\varphi }$. The solid line is a $|\partial E_{01}/\partial n_{\varphi }|$ fit. (d)–(f) Data for sample B.