Dephasing of a Superconducting Qubit Induced by Photon Noise

We have studied the dephasing of a superconducting flux qubit coupled to a dc-SQUID based oscillator. By varying the bias conditions of both circuits we were able to tune their effective coupling strength. This allowed us to measure the effect of such a controllable and well-characterized environment on the qubit coherence. We can quantitatively account for our data with a simple model in which thermal fluctuations of the photon number in the oscillator are the limiting factor. In particular, we observe a strong reduction of the dephasing rate whenever the coupling is tuned to zero. At the optimal point we find a large spin-echo decay time of $4\mu \mathrm{s}$.

Figure 1

(a) SEM picture of the sample. The flux qubit is the small loop containing four Josephson junctions in a row; the SQUID is constituted by the outer loop containing the two large junctions. The bar equals $1\mu \mathrm{m}$. (b) Measuring circuit diagram. The SQUID, represented by its Josephson inductance $L_J$, is shunted by an on-chip capacitor $C_{\mathrm{sh}}$ through superconducting lines of inductance $L$, forming the plasma mode.

Figure 2

(a) Top: principle of the spectroscopy experiments: a bias current pulse of amplitude $I_{bpl}<I_C\sim 1\mu \mathrm{A}$ is applied while a microwave pulse (MW) probes the qubit resonance frequency. The qubit state is finally measured by a short bias current pulse as discussed in Ref. 3. Bottom: Qubit spectroscopy for $I_{bpl}$ varying between 0 and $0.6\mu \mathrm{A}$ with steps of $0.1\mu \mathrm{A}$ (bottom to top). The curves were offset by 100 MHz for clarity. The solid curves are fits to the formula for ${\nu }_q$. (b) Curve $\lambda (I_b)$ deduced from the spectroscopy curves as explained in the text. The solid line is a parabolic fit to the data. The decoupling condition is satisfied at $I_b^{*}=180\pm 20\mathrm{nA}$. (c) Calculated frequency shift $\delta {\nu }_0(I_b,ϵ)$ for the parameters of our sample. The white scale corresponds to $-20\mathrm{MHz}$, the black to $+40\mathrm{MHz}$. Along the dotted line ${ϵ}_m(I_b)$, $\delta {\nu }_0=0$.

Figure 3

(a) Qubit line shape at the optimal point. The solid line is a fit assuming a double Lorentzian shape. (b) Rabi oscillations (frequency 100 MHz) at the optimal point. The inset shows well-behaved oscillations with nearly no damping during the first 100 ns. (c) Measurement of $T_1$ at the optimal point; the dashed gray line is an exponential fit of a time constant $4\mu \mathrm{s}$. (d) Spin-echo pulse sequence and signal at the optimal point.

Figure 4

(a) Measurement of $T_1$ versus $I_b$ at the flux-noise insensitive point $ϵ=0$. (b) Measurement of $T_{\mathrm{echo}}$ (circles), $t_2$ (squares), and of the qubit frequency (triangles), as a function of $ϵ$ for $I_b=I_b^{*}$ (top) and $I_b=0\mu \mathrm{A}$ (bottom). The dotted line is a fit to the formula for ${\nu }_q$; the solid black line is the prediction of Eq. (1) for $T=70\mathrm{mK}$ and $Q=150$. (c) Value of $ϵ$ for which $t_2$ is maximum (full squares) compared to the theoretical ${ϵ}_m(I_b)$ (full line).