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IEEE Transactions on Applied Superconductivity

Josephson Junction Comparator as a Quantum-Limited Detector for Flux Qubit Readout

Thomas J. Walls, D. V. Averin, Konstantin K. Likharev

Published Jun 1, 2007• 6 citations
Publisher

Abstract

We have applied the Langevin-Heisenberg-Lax formalism for dissipative quantum systems to a balanced Josephson junction comparator operating as a fast, single-shot detector of state of a flux (rf-SQUID) qubit. Earlier experiments and theoretical analysis have shown that such a detector may reach quantum- limited resolution when it measures signal from a high-impedance (current) source. In this work, we have calculated the flux resolution of a detector for a signal source with finite inductive impedance (e.g., a flux qubit) for both an instant switching of the comparator, and for its switching with a realistic SFQ driver. The results show that the flux resolution may be quite high: for typical experimental parameters, the signal-to-noise ratio may be on the order of 10, at 10-ps-scale measurement time. Moreover, the natural energy measure of the resolution, Eout, may be made (within a narrow range of parameters) smaller than planck/2. This situation is similar to the case of continuous flux measurements using SQUIDs. In both cases, a more adequate sensitivity measure requires an account of the fluctuation back-action of the detector on the signal source. In this work we have shown that in the case of single-shot measurements, the back-action effects as characterized by the "information/dephasing" measure, may be brought down to their fundamental limit by at least two methods: coupling quenching and feedback compensation.

DOI
10.1109/tasc.2007.898632

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We have applied the Langevin-Heisenberg-Lax formalism for dissipative quantum systems to a balanced Josephson junction comparator operating as a fast, single-shot detector of state of a flux (rf-SQUID) qubit. Earlier experiments and theoretical analysis have shown that such a detector may reach quantum- limited resolution when it measures signal from a high-impedance (current) source. In this work, we have calculated the flux resolution of a detector for a signal source with finite inductive impedance (e.g., a flux qubit) for both an instant switching of the comparator, and for its switching with a realistic SFQ driver. The results show that the flux resolution may be quite high: for typical experimental parameters, the signal-to-noise ratio may be on the order of 10, at 10-ps-scale measurement time. Moreover, the natural energy measure of the resolution, Eout, may be made (within a narrow range of parameters) smaller than planck/2. This situation is similar to the case of continuous flux measurements using SQUIDs. In both cases, a more adequate sensitivity measure requires an account of the fluctuation back-action of the detector on the signal source. In this work we have shown that in the case of single-shot measurements, the back-action effects as characterized by the "information/dephasing" measure, may be brought down to their fundamental limit by at least two methods: coupling quenching and feedback compensation.
We have designed and tested a Rapid Single-Flux-Quantum (RSFQ) circuit for experimental measurement of the fluctuation-induced threshold uncertainty of switching of SFQ comparators, which are essential components of RSFQ logic devices, analog-to-digital converters, and digital SQUIDs. Statistical density of the switching events in comparators using externally-overdamped Nb-trilayer Josephson junctions has been found to be in agreement with the distribution which follows from a simple theory based on a time-dependent harmonic-oscillator model of the device. Width of the distribution (i.e, the single-shot current resolution of the comparator) measured as a function of temperature in the range 1.7-4.2 K corresponds to fundamental (thermal/quantum) fluctuations, with no evidence of excess noise sources.>
Superconducting qubits demonstrate good perspectives as scalable blocks for quantum information processing. These systems can be individually readout, addressed, and controlled, making them some of the most attractive thin-film qubits. Quantum measurements should be performed in order to readout the final state of flux qubits. To observe the dynamics of such systems, one requires a fast magnetometer with exceptional sensitivity and low back-action. We address the problem of reducing the back-action of an RF SQUID readout circuit, which may significantly destroy a Josephson flux qubit in the case of the circuits integration. The main sources of such back-action in conventional RF SQUID and DC SQUID circuits are low-ohmic shunting resistors ensuring high damping. The advantage of flux qubits is the presence of an optimal bias point at which, in the first-order approximation, the device is immune to fluctuations in readout and control lines. Nonetheless, even such systems have characteristics limited by a second-order dissipation. The dissipation can be eliminated by using an “ideal parametric readout” based on an RF SQUID acting in non-hysteretic adiabatic mode. In this paper, we propose an RF SQUID in a non-hysteretic regime as a fast, sensitive, and scalable readout system with low back-action for measurements of a single MW photon counter based on a flux qubit.
We present a readout method for superconducting flux qubits. The qubit quantum flux state can be measured by determining the Josephson inductance of an inductively coupled dc superconducting quantum interference device (dc-SQUID). We determine the response function of the dc-SQUID and its back-action on the qubit during measurement. Due to driving, the qubit energy relaxation rate depends on the spectral density of the measurement circuit noise at sum and difference frequencies of the qubit Larmor frequency and SQUID driving frequency. The qubit dephasing rate is proportional to the spectral density of circuit noise at the SQUID driving frequency. These features of the back-action are qualitatively different from the case when the SQUID is used in the usual switching mode. For a particular type of readout circuit with feasible parameters we find that single shot readout of a superconducting flux qubit is possible.
Recent experiments on superconducting flux qubits, consisting of a superconducting loop interrupted by Josephson junctions, have demonstrated quantum coherence between two different quantum states. The state of the qubit is measured with a superconducting quantum interference device (SQUID). Such measurements require the SQUID to have high resolution while exerting minimal backaction on the qubit. By designing shunts across the SQUID junctions appropriately, one can improve the measurement resolution without increasing the backaction significantly. Using a path-integral approach to analyze the Caldeira-Leggett model, we calculate the narrowing of the distribution of the switching events from the zero-voltage state of the SQUID for arbitrary shunt admittances, focusing on shunts consisting of a capacitance Cs{C}_{s}Cs​ and resistance Rs{R}_{s}Rs​ in series. To test this model, we fabricated a dc SQUID in which each junction is shunted with a thin-film interdigitated capacitor in series with a resistor, and measured the switching distribution as a function of temperature and applied magnetic flux. After accounting for the damping due to the SQUID leads, we found good agreement between the measured escape rates and the predictions of our model. We analyze the backaction of a shunted symmetric SQUID on a flux qubit. For the given parameters of our SQUID and realistic parameters for a flux qubit, at the degeneracy point we find a relaxation time of $113\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$, which limits the decoherence time to $226\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$. Based on our analysis of the escape process, we determine that a SQUID with purely capacitive shunts should have narrow switching distributions and no dissipation.
We study a single-shot single-flux-quantum (SFQ) readout circuit based on current-to-time conversion for superconducting flux qubits. Circulating currents representing states of a qubit were converted to propagation delay time of SFQ pulses, and then detected by a time discriminator. The key parameter which determines the availability of single-shot readout is current resolution. We evaluated the current resolution by measuring the gray zone which was defined as the transition width in the output probability of the current-to-time converter with 1000 trials for each value of an input current. For reducing unwanted effects on the qubit, shunt resistors are required to be removed for Josephson junctions in the readout circuit. We designed two readout circuits for comparison, the circuit composed of shunted junctions and that of unshunted junctions. Gray zones were measured to be 4.8μA\mu{\rm A}μAfor unshunted junctions and 2.2μA\mu{\rm A}μAfor the shunted ones at 4.2 K. These gray zones may originate from Johnson noise. Operation at a reduced temperature and employment of a likelihood decision circuit will lead to reduced gray zone, i.e., improved current resolution sufficient for single-shot readout.
Single-shot readout experiments were performed on the two lowest-energy states of a superconducting qubit with three Josephson junctions embedded in a superconducting loop. We measured the qubit state via switching current Isw of a current-biased dc-SQUID, a quantum detector surrounding the qubit loop. The qubit signals were measured in a small Isw regime of the SQUID, typically less than 100 nA, where the Isw distribution is particularly narrow. The obtained single-shot data indicate that the qubit state is readout, through the flux generated by the qubit persistent-current, as energy eigenstates rather than current eigenstates.