Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (qubits) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase qubit (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi frequency, and two-photon Rabi frequency are compared to measurements made on a dc SQUID phase qubit with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.
We have observed Rabi-like oscillations in a current-biased dc superconducting quantum interference device presenting enhanced coherence times compared to our previous realization [J. Claudon et al., Phys. Rev. Lett. 93, 187003 (2004)]. This Josephson device behaves as an anharmonic oscillator which can be driven into a coherent superposition of quantum states by resonant microwave flux pulses. Increasing the microwave amplitude, we study the evolution of the Rabi frequency from the two-level regime to the regime of multilevel dynamics. When up to three levels are involved, the Rabi frequency is a clear signature of a quantum behavior. At higher excitation amplitude, classical and quantum predictions for the Rabi frequency converge. This result is discussed in the light of a calculation of the Wigner function. In particular, our analysis shows that pronounced quantum interferences always appear in the course of the Rabi-like oscillations.
We report spectroscopic measurements of discrete two-level systems (TLSs) coupled to a dc superconducting quantum interference device phase qubit with a $16\text{ }\ensuremath{\mu}{\text{m}}^{2}$ area
Al/AlOx/Al junction. Applying microwaves in the 10--11 GHz range, we found eight avoided level crossings with splitting sizes from 10 to 200 MHz and spectroscopic lifetimes from 4 to 160 ns. Assuming the transitions are from the ground state of the composite system to an excited state of the qubit or an excited state of one of the TLS states, we fit the location and spectral width to get the energy levels, splitting sizes, and spectroscopic coherence times of the phase qubit and TLSs. The distribution of splittings is consistent with noninteracting individual charged ions tunneling between random locations in the tunnel barrier and the distribution of lifetimes is consistent with the
AlOx in the junction barrier having a frequency-independent loss tangent. To check that the charge of each TLS couples independently to the voltage across the junction, we also measured the spectrum in the 20--22 GHz range and found tilted avoided level crossings due to the second excited state of the junction and states in which both the junction and a TLS were excited.
We report measurements of Rabi oscillations and spectroscopic coherence times in an
Al/AlOx/Al and three
Nb/AlOx/Nb dc SQUID phase qubits. One junction of the SQUID acts as a phase qubit and the other junction acts as a current-controlled nonlinear isolating inductor, allowing us to change the coupling to the current bias leads in situ by an order of magnitude. We found that for the Al qubit, a spectroscopic coherence time ${T}_{2}^{\ensuremath{\ast}}$ varied from 3 to 7 ns and the decay envelope of Rabi oscillations had a time constant ${T}^{\ensuremath{'}}=25\phantom{\rule{0.3em}{0ex}}\text{ns}$ on average at 80 mK. The three Nb devices also showed ${T}_{2}^{\ensuremath{\ast}}$ in the range of 3--8 ns, but ${T}^{\ensuremath{'}}$ was 9--15 ns, just about 1/2 the value we found in the Al device. For all the devices, the time constants were roughly independent of the isolation from the bias lines, implying that noise and dissipation from the bias leads were not the principal sources of dephasing and inhomogeneous broadening.
The properties of phase escape in a dc superconducting quantum interference device (SQUID) at 25 mK, which is well below quantum-to-classical crossover temperature
Tcr, in the presence of strong resonant ac driving have been investigated. The SQUID contains two
Nb/Al-AlOx/Nb tunnel junctions with Josephson inductance much larger than the loop inductance so it can be viewed as a single junction having adjustable critical current. We find that with increasing microwave power
W and at certain frequencies $\ensuremath{\nu}$ and $\ensuremath{\nu}/2$, the single primary peak in the switching current distribution, which is the result of macroscopic quantum tunneling of the phase across the junction, first shifts toward lower bias current
I and then a resonant peak develops. These results are explained by quantum resonant phase escape involving single and two photons with microwave-suppressed potential barrier. As
W further increases, the primary peak gradually disappears and the resonant peak grows into a single one while shifting further to lower
I. At certain
W, a second resonant peak appears, which can locate at very low
I depending on the value of $\ensuremath{\nu}$. Analysis based on the classical equation of motion shows that such resonant peak can arise from the resonant escape of the phase particle with extremely large oscillation amplitude resulting from bifurcation of the nonlinear system. Our experimental result and theoretical analysis demonstrate that at
T⪡Tcr, escape of the phase particle could be dominated by classical process, such as dynamical bifurcation of nonlinear systems under strong ac driving.
I discuss experimental and theoretical results on an LC filtered dc SQUID phase qubit. This qubit is an asymmetric aluminum dc SQUID, with junction critical currents 1.5 and 26.8 μA, on a sapphire substrate. The layout differs from earlier designs by incorporating a superconducting ground plane and weakly coupled coplanar waveguide microwave drive line to control microwave-qubit coupling. I begin with a discussion of quantizing lumped element circuit models. I use nodal analysis to construct a 2d model for the dc SQUID phase qubit that goes beyond a single junction approximation. I then discuss an extension of this ``normal modes'' SQUID model to include the on-chip LC filter with design frequency ∼ 180 MHz. I show that the filter plus SQUID model yields an effective Jaynes-Cummings Hamiltonian for the filter-SQUID system with coupling g / 2 π ∼ 32 MHz. I present the qubit design, including a noise model predicting a lifetime T1 = 1.2 μs for the qubit based on the design parameters. I characterized the qubit with measurements of the current-flux characteristic, spectroscopy, and Rabi oscillations. I measured T1 = 230 ns, close to the value 320 ns given by the noise model using the measured parameters. Rabi oscillations show a pure dephasing time Tφ = 1100 ns. The spectroscopic and Rabi data suggest two-level qubit dynamics are inadequate for describing the system. I show that the effective Jaynes-Cummings model reproduces some of the unusual features.