Recent experiments by Yoshihara et al. [Phys. Rev. Lett. 97, 167001 (2006)] provided information on decoherence of the echo signal in Josephson-junction flux qubits at various bias conditions. These results were interpreted assuming a Gaussian model for the decoherence due to
1∕f noise. Here we revisit this problem on the basis of the exactly solvable spin-fluctuator model reproducing detailed properties of the
1∕f noise interacting with a qubit. We consider the time dependence of the echo signal and conclude that the results based on the Gaussian assumption need essential reconsideration.
We report detailed measurements of the relaxation and dephasing time in a flux-qubit measured by a switching DC SQUID. We studied their dependence on the two important circuit bias parameters: the externally applied magnetic flux and the bias current through the SQUID in two samples. We demonstrate two complementary strategies to protect the qubit from these decoherence sources. One consists in biasing the qubit so that its resonance frequency is stationary with respect to the control parameters ({\it optimal point}) ; the second consists in {\it decoupling} the qubit from current noise by chosing a proper bias current through the SQUID. At the decoupled optimal point, we measured long spin-echo decay times of up to
4μs.
We have studied decoherence in a system where two Josephson-junction flux qubits share a part of their superconducting loops and are inductively coupled. By tuning the flux-bias condition, we control the sensitivities of the energy levels to flux noises in each qubit. The dephasing rate of the first excited state is enhanced or suppressed depending on the amplitudes and the signs of the sensitivities. We have quantified the
1/f flux noises and their correlations and found that the dominant contribution is by local fluctuations.
In order to gain a better understanding of the origin of decoherence in superconducting flux qubits, we have measured the magnetic field dependence of the characteristic energy relaxation time (T(1)) and echo phase relaxation time (T(2)(echo)) near the optimal operating point of a flux qubit. We have measured T(2)(echo) by means of the phase cycling method. At the optimal point, we found the relation T(2)(echo) approximately 2T(1). This means that the echo decay time is limited by the energy relaxation (T(1) process). Moving away from the optimal point, we observe a linear increase of the phase relaxation rate (1/T(2)(echo)) with the applied external magnetic flux. This behavior can be well explained by the influence of magnetic flux noise with a 1/f spectrum on the qubit.
For many types of superconducting qubits, magnetic flux noise is a source of pure dephasing. Measurements on a representative dc superconducting quantum interference device (SQUID) over a range of temperatures show that ${S}_{\ensuremath{\Phi}}(f)={A}^{2}/{(f/1\phantom{\rule{0.28em}{0ex}}\text{Hz})}^{\ensuremath{\alpha}}$, where ${S}_{\ensuremath{\Phi}}$ is the flux noise spectral density,
A is of the order of 1 $\ensuremath{\mu}{\ensuremath{\Phi}}_{0}\phantom{\rule{0.28em}{0ex}}{\text{Hz}}^{\ensuremath{-}1/2}$, $0.61\ensuremath{\leqslant}\ensuremath{\alpha}\ensuremath{\leqslant}0.95$, and ${\ensuremath{\Phi}}_{0}$ is the flux quantum. For a qubit with an energy level splitting linearly coupled to the applied flux, calculations of the dependence of the pure dephasing time ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ of Ramsey and echo pulse sequences on $\ensuremath{\alpha}$ for fixed
A show that ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ decreases rapidly as $\ensuremath{\alpha}$ is reduced. We find that ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ is relatively insensitive to the noise bandwidth, ${f}_{1}\ensuremath{\leqslant}f\ensuremath{\leqslant}{f}_{2}$, for all $\ensuremath{\alpha}$ provided the ultraviolet cutoff frequency ${f}_{2}>1/{\ensuremath{\tau}}_{\ensuremath{\varphi}}$. We calculate the ratio ${\ensuremath{\tau}}_{\ensuremath{\varphi},E}/{\ensuremath{\tau}}_{\ensuremath{\varphi},R}$ of the echo (
E) and Ramsey (
R) sequences and the dependence of the decay function on $\ensuremath{\alpha}$ and
f2. We investigate the case in which ${S}_{\ensuremath{\Phi}}({f}_{0})$ is fixed at the ``pivot frequency'' ${f}_{0}\ensuremath{\ne}1$ Hz while $\ensuremath{\alpha}$ is varied and find that the choice of
f0 can greatly influence the sensitivity of ${\ensuremath{\tau}}_{\ensuremath{\varphi},E}$ and ${\ensuremath{\tau}}_{\ensuremath{\varphi},R}$ to the value of $\ensuremath{\alpha}$. Finally, we present calculated values of ${\ensuremath{\tau}}_{\ensuremath{\varphi}}$ in a qubit corresponding to the values of
A and $\ensuremath{\alpha}$ measured in our SQUID.
We present a measurement of the relaxation and the dephasing times in a flux qubit. In order to improve coherence of the qubit, two external parameters were optimized: the applied flux through the qubit loop and the bias current of the SQUID which serves as a readout device of the qubit state. At the optimal point the dephasing time measured with spin‐echo technique was twice longer than the energy relaxation time. By changing one of the two bias parameters while keeping the other at the optimal value, one can separate the contribution of the noise in each parameter to the decoherence of the qubit.