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arXiv:0803.0958 · Physical Review B

Quantum nondemolition-like fast measurement scheme for a superconducting qubit

I. Serban, B. L. T. Plourde, Frank K. Wilhelm

Published Aug 12, 2008• 19 citations
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Abstract

We present a measurement protocol for a flux qubit coupled to a dc-superconducting quantum interference device (SQUID), representative of any two-state system with a controllable coupling to a harmonic-oscillator quadrature, which consists of two steps. First, the qubit state is imprinted onto the SQUID via a very short and strong interaction. We show that at the end of this step the qubit dephases completely, although the perturbation of the measured qubit observable during this step is weak. In the second step, information about the qubit is extracted by measuring the SQUID. This step can have arbitrarily long duration, since it no longer induces qubit errors.

DOI
10.1103/physrevb.78.054507

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We present a measurement protocol for a flux qubit coupled to a dc-superconducting quantum interference device (SQUID), representative of any two-state system with a controllable coupling to a harmonic-oscillator quadrature, which consists of two steps. First, the qubit state is imprinted onto the SQUID via a very short and strong interaction. We show that at the end of this step the qubit dephases completely, although the perturbation of the measured qubit observable during this step is weak. In the second step, information about the qubit is extracted by measuring the SQUID. This step can have arbitrarily long duration, since it no longer induces qubit errors.
We theoretically study the quantum nondemolition measurements of a flux qubit coupled to a noisy superconducting quantum interference device (SQUID). The obtained analytical results indicate that the measurement probability is frequency-dependent in a short time scale and has a close relationship with the measurement-induced dephasing. Furthermore, when the detuning between the driven and bare resonator equals the coupling strength, we can obtain the maximum measurement rate that is determined by the character of the noise in the SQUID. Finally, we analysed the mixed effect caused by coupling between the non-diagonal term and the external variable. It is found that the initial information of the qubit is destroyed due to quantum tunneling between the qubit states.
We propose a generalization of dispersive qubit readout that provides the time evolution of a flux qubit observable.Our proposal relies on the nonlinear coupling of the qubit to a harmonic oscillator with high frequency, representing a dc superconducting quantum interference device.Information about the qubit dynamics is obtained by recording the oscillator response to resonant driving and subsequent lock-in detection.The measurement process is simulated for the example of coherent qubit oscillations.This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity.
We theoretically describe the weak measurement of a two-level system (qubit) and quantify the degree to which such a qubit measurement has a quantum nondemolition (QND) character. The qubit is coupled to a harmonic oscillator, which undergoes a projective measurement. Information on the qubit state is extracted from the oscillator measurement outcomes, and the QND character of the measurement is inferred from the result of subsequent measurements of the oscillator. We use the positive operator valued measure (POVM) formalism to describe the qubit measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the oscillator, and (ii) quantum tunneling between the qubit states ∣0⟩|0⟩∣0⟩ and ∣1⟩|1⟩∣1⟩ during measurements. Our theory can be applied to QND measurements performed on superconducting qubits coupled to a circuit oscillator.
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.
We present a protocol for measuring the quadrature of a harmonic oscillator (HO). The HO is coupled to a qubit, with an interaction modulated by the qubit control and effectively proportional to the HO quadrature III. Repeated measurement of the qubit leads to gradually increasing information on the quadrature III, leading to squeezing. We derive an analytical formula for the quadrature variance, (ΔI)2=1/(1+4ϕ2s)(ΔI)^2 = 1/(1+4ϕ^2 s)(ΔI)2=1/(1+4ϕ2s), with ϕϕϕ the product of interaction strength and interaction time and sss the number of repetitions of the measurement. We discuss the robustness of this scheme against decoherence. We find that this protocol could lead to significant squeezing in a realistic setup formed of a superconducting flux qubit used to measure an electrical or mechanical resonator.
We propose a generalization of dispersive qubit readout that provides the time evolution of a flux qubit observable. Our proposal relies on the nonlinear coupling of the qubit to a harmonic oscillator with high frequency, representing a dc superconducting quantum interference device. Information about the qubit dynamics is obtained by recording the oscillator response to resonant driving and subsequent lock-in detection. The measurement process is simulated for the example of coherent qubit oscillations. This corroborates the underlying measurement relation and also reveals that the measurement scheme possesses low backaction and high fidelity.