Session H29: Focus Session: Superconducting Qubits: Amplifiers and Read-out

Frequency tunable non-degenerate Josephson amplifier for qubit readout

We have developed a new ultra low noise microwave amplifier based on the Josephson parametric converter (JPC), which overcomes a practical weakness of devices of previous generations: having sufficient frequency tunability to easily match the qubit readout frequency. The JPC consists of two superconducting microwave resonators that are coupled to each other through a ring of four Josephson junctions, threaded by a magnetic flux and providing the non-linearity for the amplification process. The non-linearity is of the trilinear form involving the minimal number of modes, and allows ideal non-degenerate parametric amplification at the quantum limit of noise. In our new tunable version, the junctions responsible for amplification are shunted by a cross of four larger junctions, which for our purpose can be regarded as linear inductors, as in the work of Roch et al.[1]. The JPC has now a unique bias point at any applied flux and is tunable over more than half a gigahertz. We are currently using this amplifier in conjunction with a quantum non-demolition measurement of a transmon qubit and have observed quantum jumps with fidelity larger than 90{\%}. [1] N. Roch, E. Flurin, F. Nguyen, P. Morfin, P. Campagne-Ibarcq, M. H. Devoret, and B. Huard, in preparation.   

Microstrip SQUID amplifiers for quantum information science

Recent progress in SQUID amplifiers suggests that these devices might approach quantum-limited sensitivity in the microwave range, thus making them a viable option for measurement of superconducting quantum systems. In the microstrip SQUID amplifier configuration, gains of around 20dB are possible at frequencies of several hundred MHz, and the gain is limited by the maximum voltage modulation available from the SQUID. One route for increasing the voltage modulation involves using larger resistive shunts, however maintaining non-hysteretic device operation requires smaller junction capacitances than is possible with conventional photolithographically patterned junctions. Operating at higher frequencies requires a shorter input coil which reduces mutual inductance between the coil and washer and therefore gain. We have fabricated microstrip SQUID amplifiers using submicron Al-AlOx-Al junctions and large shunts. The input coil and SQUID washer are optimized for producing high gain at frequencies in the gigahertz range. Recent measurements of gain and noise temperature will be discussed as well as demonstrations of these devices as a first stage of amplification for a superconducting system   

Measurement of the phase non-reciprocity of the Josephson parametric converter

Non-reciprocal devices such as circulators and isolators play a pivotal role in many microwave experiments on quantum superconducting circuits. However, non-reciprocity in these devices is achieved using ferrites and permanent magnets which are not suitable for on chip integration. We have built a symmetric two port device by pairing two Josephson parametric converters (JPCs) working in pure conversion mode, each of which is driven with an independent pump tone. We observed a non-reciprocal phase shift between the two ports, which depends on the phase difference between the two pumps. The present noiseless system constitutes an important step towards the implementation of a noiseless gyrator, the main building block of an on-chip circulator for back-action free quantum measurement.   

Performance of the doubly pumped four-wave Josephson parametric amplifier

The degenerate Josephson Parametric Amplifier (JPA) is a promising quantum-limited amplifier for the measurement of mesoscopic systems. In the single pump scheme, the amplification process utilizes two photons at the drive frequency to produce a signal and an idler photon. However the large reflected pump tone at the drive frequency strongly acts back on the measured system. Several circulators prevent this back action, but come at a non-negligible cost in system noise temperature. Two drive tones symmetrically detuned from the original drive frequency provide the necessary pump energy but with the reflected tones now far removed from the signal frequency. We have investigated the performance of the doubly pumped JPA with pump detunings from 50 to 1000 MHz. The expected gain profile has been measured with at least 20 dB of gain and no loss of bandwidth. A gigahertz range of tunability, no degradation in noise temperature, and improvements in dynamic range are also expected.   

Analytical Calculation of Gain and Noise of DC Squid Microwave Amplifier

The dc SQUID microwave amplifier, based on Josephson junctions, is employed in a wide spectrum of applications ranging from dark matter detection to the readout of superconducting qubits. A crucial advantage offered by this device is the separation of input and output channels, unlike conventional Josephson parametric amplifiers, so that it does not require a nonreciprocal device such as a circulator for its operation. The mechanism underlying the directional gain in the SQUID microwave amplifier, however, has so far remained elusive. We present a first principles, analytical calculation, based on scattering theory, of a practical SQUID amplifier which elucidates the underlying nonlinear mode mixing responsible for the directional operation of the device. The gain and quantum noise characteristics of a SQUID operated as a microwave voltage amplifier are discussed. Work supported by IARPA and ARO (AK, MHD and JC) and NSF (AK and MHD).   

Observing quantum jumps of a transmon qubit with a Josephson parametric converter

A high fidelity linear quantum non-demolition (QND) readout of a superconducting qubit opens up the possibility of observing quantum jumps and is a prerequisite for quantum feedback and error correction. This readout is challenging since the qubit, the readout resonator and the following amplifier chain have to be simultaneously optimized to achieve the desired performance. We fabricated a superconducting transmon qubit at 5.7 GHz coupled to a compact resonator at 7.5 GHz, designed to produce a dispersive shift ($\chi )$ of 6 MHz of the resonator frequency when the qubit is excited. The resonator linewidth matches $\chi $ to produce maximum readout contrast in a transmission measurement, while maintaining a Purcell limited T$_{1}$ of about 3 $\mu $s. Using a Josephson parametric converter that is tuned to match the resonator frequency, we achieved a system noise temperature of the following amplifier chain of about 0.5 K, roughly thrice the standard quantum limit. Using these optimized parameters, we measured the qubit state with about 5 photons in the readout resonator and observed quantum jumps with fidelity above 90 {\%}. Further, by looking at the statistics of the jumps and the evolution of the qubit population in single shot traces, we find that the average qubit T$_{1}$ during the readout matches the Purcell limited T$_{1}$, as expected for a QND measurement.   

Quantum measurement in action with the transmon qubit

High fidelity, rapid quantum non-demolition readout of superconducting qubits greatly facilitates tests of single qubit measurement theory. We have realized such readout in an experiment comprised of a transmon coupled to a compact resonator, which is in turn connected via an isolator and circulator to a tunable Josephson parametric converter (JPC) operated as a phase-preserving parametric amplifier. When the qubit state is measured with an rf tone corresponding to an average cavity circulating power of 5 photons, fidelity exceeds 90{\%} for a measurement duration of 240 ns ($\sim $0.1 T1). This performance allows the observation of quantum trajectories of the qubit, showing discrete jumps and a bimodal distribution of measurement results, despite the linear character of the amplifier. This provides further support for the quantum nature of superconducting artificial atoms. We have conducted Stern-Gerlach type experiments, in which the qubit is repeatedly measured along different axes. Results are in good agreement with theoretical predictions of the effect of partial measurement on qubit state evolution.   

Monitoring the state of a superconducting quantum bit using weak measurements

We demonstrate continuous weak measurement of the state of a superconducting transmon qubit in the circuit QED architecture using a Josephson parametric amplifier. The near quantum-limited noise performance of the amplifier enables us to obtain a high-fidelity measurement record which can be used to reconstruct the qubit state evolution during measurement by employing the quantum Bayesian formalism. While simultaneously driving the qubit at its Larmor frequency and measuring it weakly, we are able to resolve the spectral signature of Rabi oscillations present in the measurement record with a high signal-to-noise ratio (SNR). We discuss current limitations in this measurement scheme and suggest ways to further optimize the SNR. Our results suggest a route to implement quantum feedback to steer the state of a superconducting qubit.   

Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback

Recent progress in the development of quantum-noise-limited superconducting parametric amplifiers has enabled high-fidelity, continuous measurements of superconducting quantum bits (qubits). We exploit this functionality while leveraging improved coherence times in transmon qubits to show that it is now possible to obtain a faithful record of real-time qubit dynamics during measurement. We weakly measure the qubit while it is continuously driven at the Larmor frequency. The phase of the resulting Rabi oscillations diffuses slowly, primarily due to the measurement. We monitor this phase diffusion and correct for it by feeding back on the qubit drive amplitude. This locks the Rabi oscillations to a classical reference signal and the oscillations persist indefinitely with a reduced visibility set by the feedback efficiency. We perform tomography on the feedback stabilized state and suggest routes to further optimize the feedback efficiency. Such capabilities suggest a measurement based route for implementing continuous quantum error correction.   

High fidelity readout of a superconducting qubit using heralded state preparation

We report measurements of a superconducting flux qubit, coupled via a shared inductance, to a quasi-lumped element 5.78-GHz readout resonator formed by the parallel combination of an interdigitated capacitor and a meander line inductor. A Josephson parametric amplifier with near-quantum-limited noise performance is used to increase the measurement sensitivity. We demonstrate a continuous, high-fidelity readout with sufficient bandwidth and signal-to-noise ratio to resolve quantum jumps in the flux qubit. We achieve a readout fidelity of 91\%, limited primarily by $T_1$ decay between state preparation and measurement. The fast, high-visibility, QND character of the readout allows for many successive readouts within a time $T_1$. We exploit this capability to herald pure ground and excited state ensemble populations by post-selecting only for certain states after an initial readout. This method enables us to eliminate errors due to imperfect state preparation, increasing the fidelity to 94\%. We also present a precise budget of fidelity loss and an analysis of the readout backaction.   

Experimental approaches to improve the single shot measurement fidelity of a superconducting charge qubit

We discuss various experimental approaches to improve the single shot measurement fidelity of a superconducting charge qubit. Dispersive readout is optimized on a transmon coupled to a superconducting coplanar waveguide resonator. Measurement parameters, such as microwave power and frequency are varied. Also control theory is adapted to construct a genetic algorithm which optimizes the shape of the drive pulse. Additionally, we attempt to reduce noise and increase SNR by employing a SLUG amplifier. Using these techniques, we discuss the feasibility of reaching the measurement fidelity needed for scalable quantum computation with superconducting circuits.   

Multiplexed dispersive readout for the superconducting phase qubit

Scaling to multiple qubit circuits requires state readout with maximum reliability and the minimum number of readout lines. Here, we introduce a multiplexed readout scheme for superconducting phase qubits. We replace our standard readout SQUIDs with inductively coupled resonators so that the measured state of the qubit (left or right side of the potential well) is read out as a shift of the resonator frequency. We connected several readout resonators to a single feedline and use a multi-tone microwave reflection measurement to simultaneously read out the states of multiple qubits using a single cable. Together with the compact lumped LC resonator design, the efficiency of chip space usage is greatly improved.   

Ground State Misidentification in Superconducting Qubits

To achieve fault tolerant quantum computation, it is necessary to maximize measurement fidelity and minimize readout-induced measurement errors. A new protocol was developed to measure $P_1(|g>)$, the probability of measuring the excited state without exciting the qubit, while not including stray tunneling present in superconducting phase qubits. We have confirmed the expected trend in $P_1(|g>)$ with device temperature. We then compared $P_1(|g>)$ for phase qubits with different readout mechanisms and found that it is $\sim$3\% for our dispersive readout scheme and $\sim$1.5\% for our prior SQUID-based readout scheme. We have further applied microwave power to the flux bias and microwave drive lines to understand the source of this difference.   

Autoresonant readout of a high Q resonator coupled to a superconducting quantum bit

The frequency of a nonlinear oscillator changes with oscillation amplitude. When a high-Q, nonlinear oscillator is excited with a frequency chirped drive, the system can respond at either low or high oscillation amplitude depending on whether the drive excitation is below or above a critical value, respectively -- a phenomenon known as autoresonance. We exploit this nonlinear phenomenon to read out the state of a superconducting transmon qubit coupled to a high-Q nonlinear resonator. Because the excitation is non-equilibium, the resonator can be read out faster than its energy decay time. The fidelity for mapping the qubit state onto the oscillator can be as high as $80\%$ and is limited by the $T_1$ lifetime of the qubit, and readily achievable pulse parameters.   

Qubit Measurement with a Nonlinear Cavity Detector Beyond Linear Response

We consider theoretically the use of a driven, nonlinear superconducting microwave cavity to measure a coupled superconducting qubit. This is similar to setups studied in recent experiments.\footnote{M. Hatridge {\it et al.} Phys.Rev.B, 83,134501 (2011)}$^,$\footnote{F.R. Ong {\it et al.} PRL 106,167002 (2011)} In a previous work, we demonstrated that for weak coupling (where linear response theory holds) one misses the quantum limit on QND detection in this system by a large factor proportional to the parametric gain.\footnote{C. Laflamme and A.A. Clerk, Phys. Rev. A 83, 033803 (2011)} Here we calculate measurement backaction beyond linear response by using an approximate mapping to a detuned degenerate parametric amplifier having both linear and dispersive couplings to the qubit. We find surprisingly that the backaction dephasing rate is far more sensitive to corrections beyond linear response than the detector response. Thus, increasing the coupling strength can significantly increase the efficiency of the measurement. We interpret this behavior in terms of the non-Gaussian photon number fluctuations of the nonlinear cavity. Our results have applications to quantum information processing and quantum amplification with superconducting microwave circuits.