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arXiv:0906.0906 · Physical Review B

Quantum nondemolition measurements of a qubit coupled to a harmonic oscillator

Luca Chirolli, Guido Burkard

Published Nov 18, 2009• 7 citations
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Abstract

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 $|0⟩$ and $|1⟩$ during measurements. Our theory can be applied to QND measurements performed on superconducting qubits coupled to a circuit oscillator.

DOI
10.1103/physrevb.80.184509

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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 derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
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.
The present work makes a brief introduction to QND (Quantum non demolition measurement) measurements and its characteristics. In addition, a qubit coupled to a forced quantum harmonic oscillator which is described theoretically as an example of a QND measurement in a qubit. The formalism developed for this type of coupled quantum systems is developed within the quantum theory of computation. As part of the study of QND measurements, the flow qubits making use of quantum superconducting interferometers (SQUIDs) are introduced. The analysis of this computer schema attempts to introduce the reader to the concepts of quantum computing such as qubit, which is the basic component that allows information to be processed quantumly. The objective of this work is to characterize whether the elaborated measures on the coupled qubit are QND or not. In this sense, the application of the exposed formalism will allow us to glimpse the scope and limitations of coupled qubits in the development and application of quantum computing systems to this day. Additionally, the application of this theory can be applied to QND measurements on superconducting qubits coupled to a quantum harmonic oscillator. All this process is subject to the analysis and methodology provided by the history of science and technology.