Measuring the Decoherence of a Quantronium Qubit with the Cavity Bifurcation Amplifier

I Introduction

Quantum circuits based on Josephson junctions are candidates for the fundamental building block of a quantum computer - a quantum bit, or qubit qcircuits. Several implementations have been tested vion; Delft-echo; Semba; Clarke; Rob; Martinis; qlab-JQ, which may be distinguished by the variable controlling the state of the qubit (charge, flux or phase) and the technique used for the readout. These systems can be individually addressed, controlled and read, making them some of the most advanced solid state qubits.

An enabling characteristic of many superconducting qubits is the existence of an optimal bias point where the qubit is immune to first order to fluctuations in external control parameters. Nonetheless, even systems operated at this “sweet spot” have coherence times limited by second order fluctuations of the external control parameters schon. In this paper we describe measurements of a quantronium qubit operated at this optimal bias point vion; qlab-JQ employing the newly developed Cavity Bifurcation Amplifier (CBA) – a fast, dispersive, scalable readout system based on a driven non-linear superconducting resonator etienne; vladimir. With this architecture we can measure the fluctuations in the quantronium’s coherence on time scales as short as a second, allowing us to probe the 1/f property of charge noise on these scales. These measurements confirm previous studies of the limitations of coherence times in charge qubits schon; Nak-noise and casts additional light on the fluctuating character of $T_{2}$ itself and how its value depends on the measurement protocol.

The non-linearity of the CBA is provided by a Josephson junction placed in the center of a $\lambda$/2 on-chip coplanar waveguide resonator with both an input and output coupling capacitor playing the role of Fabry-Perot cavity mirrors (see Fig 1.) When driven with a microwave signal at frequency $\nu_{d}$ such that $\nu_{d}<\nu_{0}\left(1-\sqrt{3}/Q\right)$, where $\nu_{0}$ is the resonator small oscillation natural frequency and $\ Q$ the loaded quality factor, this system can have two dynamical metastable states which differ by their oscillation amplitude and phase. As the driving power $P$ is ramped past the bifurcation power $P_{b}$ , the CBA switches from the state of low amplitude to the state with high amplitude. We detect the state of the non-linear oscillator by monitoring the amplitude and phase change of the microwave signal transmitted by the resonator. In parallel with the CBA’s junction we place a split Cooper pair box (SCPB), a circuit known as the quantronium, giving the resonator two bifurcation powers $P_{b}^{\left|0\right\rangle}$ and $P_{b}^{\left|1\right\rangle}<P_{b}^{\left|0\right\rangle}$ depending on the state of the qubit, $\left|0\right\rangle$ or $\left|1\right\rangle$. The two qubit states are mapped into the two metastable states of the CBA by ramping quickly the power $P$ to a level intermediate between $P_{b}^{\left|0\right\rangle}$ and $P_{b}^{\left|1\right\rangle}$. If the quantronium qubit is in $\left|1\right\rangle$, the CBA will switch to the high oscillating state, whereas if it is in $\left|0\right\rangle$ the CBA will remain in the low oscillating state.

This readout has the advantage of being non-dissipative as the readout junction never switches into the normal state, unlike the original DC-biased quantronium readout vion. This dispersive readout minimally disturbs the qubit state and since after switching we do not need to wait for quasiparticles to relax, the repetition rate is only limited by the relaxation time, $T_{1}$, of our qubit and the $Q$ of our resonator. Like the DC readout, the CBA readout can latch siddiqi-qm, allowing enough time for the measurement of the complex amplitude of the transmitted wave, and therefore excellent signal to noise ratio. These characteristics were also present in the Josephson bifurcation amplifier siddiqi; siddiqi-qm; qlab-JQ, which implemented a bifurcating non-linear oscillator using a lumped element capacitor in parallel with the junction. However this capacitor was fabricated using a Cu/Si3N4/Al multilayer structure which was difficult to fabricate and integrate with more then one qubit. Also the parallel plate geometry suffered from inherent stray inductive elements. In contrast, the CBA is fabricated using a simple coplanar waveguide geometry with no stray elements. The resonance frequency $\nu_{0}$ and the quality factor $Q$ are controlled by the resonator length and output capacitor respectively. The CBA geometry thus offers the possibility of designing a multi-resonator chip with multiplexed readouts, which would accommodate up to 10 qubits at once, an important step towards scalable quantum computing. The present work, in addition to the assessment of 1/f noise in a new architecture, is a first step in this direction.

II Sample fabrication and characterization

The resonator is initially fabricated using photolithography on a bare Si wafer Luigi. A LOR5A/S1813 optical resist bilayer is used and the development is optimized to have at least 50nm of undercut beneath the S1813 to avoid wavy edges and to obtain a sloped edge on the resonator. This sloped edge is obtained by evaporating a 200nm thick Al layer onto the sample at 0.2 nm/s with an angle of $5^{\circ}$ and with a stage rotation of $10^{\circ}$/s. Next the quantronium is fabricated using electron beam lithography. We use a MMA/PMMA resist bilayer and the Dolan bridge double angle evaporation technique to fabricate our junctions Dolan. For this sample the split Cooper pair box is fabricated first inside the resonator, followed by the readout junction in a separate step using new e-beam resist. Using a hollow cathode Ar ion gun we obtain an ohmic contact between the two e-beam layers and the resonator.

III Qubit characterization

We performed gate charge and flux modulations while keeping the qubit in its ground state, to check that we have flux periodicity and 2e charge periodicity, as shown in Fig 2.

IV Qubit manipulation and coherence experiments

Once the qubit parameters are known, we can perform experiments on the qubit to determine the qubit’s quality in terms of its energy relaxation time $T_{1}$ and dephasing time $T_{2}$. An essential part of these experiments is the need to control the state of the qubit precisely. This is achieved by applying a microwave pulse to the gate with rectangular envelope, of amplitude $A$, and time length $\tau$. In a frame rotating at the gate pulse carrier frequency, $\nu_{s}$, the qubit state then precesses at a frequency, $\nu_{p}$ given by cottet:

where the Rabi frequency rabi $\nu_{Rabi}=2E_{CP}\Delta N_{g}\left\langle 0\left|N\right|1\right\rangle/h$ involves the gate modulation ampitude in units of Cooper pairs $\Delta N_{g}=C_{g}A/2e$ and charge operator matrix element between the excited and ground states $\left\langle 0\left|N\right|1\right\rangle$. For $A=0$ we have free evolution at the Ramsey frequency ramsey $\nu_{Ramsey}=|\nu_{01}-\nu_{s}|$.

The maximum constrast measured is about 60%, lower then the maximum contrast of over 99.9%, calculated for the ideal case of a non-relaxing qubit given the measured parameters of the resonator. This loss can be attributed to three main sources. First the qubit relaxes before the readout takes place, because of its finite $T_{1}$, resulting in a 10% loss in contrast. Second the qubit relaxes to the ground state as the readout voltage approaches the bifurcation voltage, resulting in a further 25% loss in contrast. Other measurements (not described here) have suggested that this loss in contrast could be from Stark shifting the qubit to lower frequencies during readout, where it can come in resonance with spurious transitions, possibly due to defects in the substrate or in the tunnel barrier. The remaining loss could be accounted for by the fact that the transition between the two oscillating states of the CBA is broadened by more than a factor of 5 from that expected.

V Conclusion

We have successfully implemented an improved version of the bifurcation amplifier based on an on-chip CPW resonator as a readout for the quantronium qubit. It offers ease of fabrication and a larger range of operating parameters ($\omega_{0}$, Q) compared to the original JBA implementation qlab-JQ. Using this readout which captures in real time the fluctuations in qubit parameters, we have demonstrated that the main source of decoherence in our sample is charge noise. By using a larger $E_{J}/E_{CP}$, we could reduce the curvature with gate charge of the levels of the Cooper pair box and we should be able to reduce the charge noise induced decoherence transmon. Furthermore, the CBA geometry is particularly well adapted to the multiplexing of the simultaneous readout of several qubits. We have started in this direction by successfully measuring the bifurcation of 5 CBAs with only one input and one output line. This configuration offers a path for scaling up of superconducting circuits up to several tens of qubits.

The authors would like to thank D. Esteve, S. Fissette, J.M. Gambetta, S. Girvin, D. Prober and D. Vion for useful discussions and assistance. This work was supported by NSA through ARO grant No. W911NF-05-01-0365, the Keck foundation, and the NSF through grant No. DMR-032-5580. L. Frunzio acknowledges partial support from CNR-Istituto di Cibernetica, Pozzuoli, Italy.