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arXiv:1101.0354 · Chinese Physics B

Quantum nondemolition measurements of a flux qubit coupled to a noisy detector

Wei Jiang, Yang Yu, Lian-Fu Wei

Published Aug 1, 2011• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1674-1056/20/8/080307

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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 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.
Superconducting flux qubits have many advantages as a storage of quantum information, such as broad range tunability of frequency, small-size fabricability, and high controllability. In the flux qubit-oscillator, qubits are connected to SQUID resonators for the purpose of performing dispersive non-destructive readouts of qubit signals with high fidelity. In this work, we propose a theoretical model for analyzing quantum characteristics of a flux qubit-oscillator on the basis of quantum solutions obtained using a unitary transformation approach. The energy levels of the combined system (qubit + resonator) are analyzed in detail. Equally spaced each energy level of the resonator splits into two parts depending on qubit states. Besides, coupling of the qubit to the resonator brings about an additional modification in the split energy levels. So long as the coupling strength and the tunnel splitting are not zero but finite values, the energy-level splitting of the resonator does not disappear. We conclude that quantum nondemolition dispersive measurements of the qubit states are possible by inducing bifurcation of the resonator states through the coupling.
We observe measurement-induced qubit state mixing in a transmon qubit dispersively coupled to a planar readout cavity. Our results indicate that dephasing noise at the qubit-readout detuning frequency is up-converted by readout photons to cause spurious qubit state transitions, thus limiting the nondemolition character of the readout. Furthermore, we use the qubit transition rate as a tool to extract an equivalent flux noise spectral density at f~1 GHz and find agreement with values extrapolated from a 1/f(α) fit to the measured flux noise spectral density below 1 Hz.
We have measured flux-noise correlations in a tunable superconducting flux qubit. The device consists of two loops that independently control the qubit's energy splitting and tunnel coupling. Low-frequency flux noise in the loops causes fluctuations of the qubit frequency and leads to dephasing. Since the noises in the two loops couple to different terms of the qubit Hamiltonian, a measurement of the dephasing rate at different bias points provides a way to extract both the amplitude and the sign of the noise correlations. We find that the flux fluctuations in the two loops are anticorrelated, consistent with a model where the flux noise is generated by randomly oriented unpaired spins on the metal surface.
Decoherence in superconducting qubits is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the qubit. Because of the coupling, the qubit frequency is shifted by an amount nδν0n δν_0nδν0​ if the resonator contains nnn energy quanta. Thermal fluctuations induce temporal variations n(t)n(t)n(t) and thus dephasing. We give an approximate formula for the qubit dephasing time as a function of δν0δν_0δν0​. We discuss the specific case of a flux-qubit coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-qubit interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the oscillator variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that δν0δν_0δν0​ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed.
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.