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arXiv:cond-mat/0510246 · Physical Review B

Phase space topology of a switching current detector

Joachim Sjöstrand, T. H. Hansson, A. Karlhede, Jochen Walter, Erik A. Tholén, David B. Haviland

Published Apr 27, 2006• 3 citations
arXivHTMLPDF

Abstract

We examine in theory and by numerical simulation, the dynamic process of switching from a zero voltage to a finite voltage state in a Josephson junction circuit. The theoretical model describes small capacitance Josephson junctions which are overdamped at high frequencies, and can be applied to detection of the quantum state of a qubit circuit. We show that the speed and fidelity of the readout are strongly influenced by the topology of the phase space attractors. The readout will be close to optimal when choosing the circuit parameters so as to avoid having an unstable limiting cycle which separates the two basins of attraction.

DOI
10.1103/physrevb.73.132511

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We examine in theory and by numerical simulation, the dynamic process of switching from a zero voltage to a finite voltage state in a Josephson junction circuit. The theoretical model describes small capacitance Josephson junctions which are overdamped at high frequencies, and can be applied to detection of the quantum state of a qubit circuit. We show that the speed and fidelity of the readout are strongly influenced by the topology of the phase space attractors. The readout will be close to optimal when choosing the circuit parameters so as to avoid having an unstable limiting cycle which separates the two basins of attraction.
Josephson junction qubits are promising candidates for a scalable quantum processor. Such qubits are commonly manipulated by means of sequences of rf-pulses and different methods are used to determine their quantum state. The readout should be able to distinguish the two qubit states with high accuracy and be faster than the relaxation time of the qubit. We discuss and experiment with a readout method based on the switching of a Josephson junction from the zero voltage state to a finite voltage state. The Josephson junction circuit has a non-linear dynamics and when it is brought to a bifurcation point, it can be made arbitrarily sensitive to small perturbations. This extreme sensitivity at a bifurcation point can be used to distinguish the two quantum states if the topology of the phase space of the circuit leads to a quick separation into the final states where re-crossings of the bifurcation point are negligible. We optimize a switching current detector by analyzing the phase space of a Josephson junction circuit with frequency dependent damping. A pulse and hold technique is used where an initial current pulse brings the junction close to its bifurcation point and the subsequent hold level is used to give the circuit enough time to evolve until the two states can be distinguished by the measuring instrument. We generate the pulse and hold waveform by a new technique where a voltage step with following linear voltage rise is applied to a bias capacitor. The frequency dependent damping is realized by an on-chip RC-environment fabricated with optical lithography. Josephson junction circuits are added on by means of e-beam lithography. Measurements show that switching currents can be detected with pulses as short as 5 ns and a resolution of 2.5% for a sample directly connected to the measurement leads of the cryostat. Detailed analysis of the switching currents in the RC-environment show that pulses with a duration of 20 us can be explained by a generalization of Kramers' escape theory, whereas switching the same sample with 25 ns pulses occurs out of thermal equilibrium, with sensitivity and speed adequate for qubit readout.
We investigate by theory and experiment, the Josephson junction switching current detector in an environment with frequency-dependent damping. Analysis of the circuit's phase space shows that a favorable topology for switching can be obtained with overdamped dynamics at high frequencies. A pulse-and-hold method is described, where a fast switch pulse brings the circuit close to an unstable point in the phase space when biased at the hold level. Experiments are performed on Cooper pair transistors and quantronium circuits, which are overdamped at high frequencies with an on-chip RC shunt. For $20\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$ switch pulses the switching process is well described by thermal equilibrium escape, based on a generalization of the Kramers formula to the case of frequency-dependent damping. A capacitor bias method is used to create very rapid, 25ns25\phantom{\rule{0.3em}{0ex}}\mathrm{ns}25ns switch pulses, where it is observed that the switching process is not governed by thermal equilibrium noise.
We present a systematic study of the phase coherent dynamics of a superconducting three-Josephson-junction flux qubit. The qubit state is detected with the integrated-pulse method, which is a variant of the pulsed switching-dc-superconducting quantum interference device (SQUID) method. In this scheme, the dc SQUID bias current pulse is applied via a capacitor instead of a resistor, giving rise to a narrow bandpass instead of a pure low-pass filter configuration of the electromagnetic environment. Measuring one and the same qubit with both setups allows a direct comparison. With the capacitive method about four times faster switching pulses and an increased visibility are achieved. Furthermore, the deliberate engineering of the electromagnetic environment, which minimizes the noise due to the bias circuit, is facilitated. Right at the degeneracy point, the qubit coherence is limited by energy relaxation. We find two main noise contributions. White noise limits the energy relaxation and contributes to the dephasing far from the degeneracy point. 1∕f1∕f1∕f noise is the dominant source of dephasing in the direct vicinity of the optimal point. The influence of 1∕f1∕f1∕f noise is also supported by nonrandom beatings in the Ramsey and spin echo decay traces. Numeric simulations of a coupled qubit-oscillator system indicate that these beatings are due to the resonant interaction of the qubit with at least one pointlike fluctuator, coupled especially strongly to the qubit.
We demonstrate the narrow switching distribution of an underdamped Josephson junction from the zero to the finite voltage state at millikelvin temperatures. We argue that such junctions can be used as ultrasensitive detectors of the single photons in the GHz range, operating close to the quantum limit: a given initial (zero voltage) state can be driven by an incoming signal to the finite voltage state. The width of the switching distribution at a nominal temperature of about T = 10 mK was 4.5 nA, which corresponds to an effective noise temperature of the device below 60 mK.
A new phase diffusive regime in a current biased Josephson junction is theoretically explored which originates from embedding the junction in a circuit environment with anomalous dissipation. This is realized by placing parallel to the junction a resistor in series with a capacitor such that electromagnetic fluctuations effectively couple also to the charge of the junction. This leads to rich Josephson dynamics, in particular for the switching of the junction out of a zero voltage state. Modelled as the escape process of a fictitious phase-particle out of a metastable well, a detailed study reveals that anomalous dissipation has a strong impact at low temperatures when quantum tunneling dominates against thermal activation. As a manifestation, a regime is found, where for realistic circuit parameters the quantum escape process is substantially enhanced, followed by a short voltage pulse and re-trapping with high probability. This class of circuits may be leveraged for detecting microwave photons or dissipative quantum annealing processes. In addition, the analysis provides a general framework for engineering dissipative dynamics in nonlinear systems using anomalous environments.
We analyze properties of bifurcation quantum detectors based on weakly nonlinear superconducting resonance circuits, in particular, with application to quantum readout. The developed quantitative description demonstrates strong influence of higher harmonics on their characteristics. While this effect is relevant for various circuits, including the conventional Josephson bifurcation amplifier and the parametrically driven circuit, we first focus on the period-doubling bifurcation under a force driving. This kind of bifurcation is due to nominally quadratic nonlinearity, which enables parametric down-conversion of the driving signal at nearly double resonance frequency to the basic mode. We analyze the effect of higher harmonics on the dynamics of the basic mode, inherent in a nonlinear circuit, which in our case is based on a Josephson junction with a sinusoidal current-phase relation as the origin of nonlinearity. We demonstrate that effects beyond the monochromatic approximation significantly modify the bare characteristics and evaluate their contribution. Due to high sensitivity of this circuit to small variations of parameters, it can serve as an efficient detector of the quantum state of superconducting qubits.