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arXiv:cond-mat/0306004 · Physical review. B, Condensed matter

Theory of weak continuous measurements in a strongly driven quantum bit

Anatoly Yu. Smirnov

Published Oct 17, 2003• 38 citations
arXivHTMLPDF

Abstract

Continuous spectroscopic measurements of a strongly driven superconducting qubit by means of a high-quality tank circuit (a linear detector) are under study. Output functions of the detector---namely, a spectrum of voltage fluctuations and an impedance---are expressed in terms of the qubit spectrum and magnetic susceptibility. The nonequilibrium spectrum of the current fluctuations in the qubit loop and the linear response function of the driven qubit coupled to a heat bath are calculated with Bloch-Redfield and rotating-wave approximations. Back-action effects of the qubit on the tank and the tank on the qubit are analyzed quantitatively. We show that the voltage spectrum of the tank provides detailed information about the frequency and decay rate of Rabi oscillations in the qubit. It is found that both an efficiency of spectroscopic measurement and measurement-induced decoherence of the qubit demonstrate a resonant behavior as the Rabi frequency approaches the resonant frequency of the tank. We determine conditions when the spectroscopic observation of the Rabi oscillations in the flux qubit with the tank circuit can be considered as a weak continuous quantum measurement.

DOI
10.1103/physrevb.68.134514

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Continuous spectroscopic measurements of a strongly driven superconducting qubit by means of a high-quality tank circuit (a linear detector) are under study. Output functions of the detector---namely, a spectrum of voltage fluctuations and an impedance---are expressed in terms of the qubit spectrum and magnetic susceptibility. The nonequilibrium spectrum of the current fluctuations in the qubit loop and the linear response function of the driven qubit coupled to a heat bath are calculated with Bloch-Redfield and rotating-wave approximations. Back-action effects of the qubit on the tank and the tank on the qubit are analyzed quantitatively. We show that the voltage spectrum of the tank provides detailed information about the frequency and decay rate of Rabi oscillations in the qubit. It is found that both an efficiency of spectroscopic measurement and measurement-induced decoherence of the qubit demonstrate a resonant behavior as the Rabi frequency approaches the resonant frequency of the tank. We determine conditions when the spectroscopic observation of the Rabi oscillations in the flux qubit with the tank circuit can be considered as a weak continuous quantum measurement.
Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi oscillations in a phase qubit. The external source, typically in GHz range, induces transitions between the qubit levels. The resulting Rabi oscillations of supercurrent in the qubit loop induce the voltage oscillations across the coil of a high quality resonant tank circuit, inductively coupled to the phase qubit. It is the presence of these voltage oscillations in the detected signal which reveals the existence of Rabi oscillations in the qubit. A detailed calculation for zero and nonzero temperatures are made for the case of persistent current qubit. According to the estimates for decoherence and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi frequency in the MHz range.
Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi oscillations in a phase qubit. The external source, typically in the GHz range, induces transitions between the qubit levels. The resulting Rabi oscillations of supercurrent in the qubit loop are detected by a high-quality resonant tank circuit, inductively coupled to the phase qubit. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase qubit), which explicitly account for the back action of a tank circuit. According to the results of our simulations the Rabi oscillations in the MHz range can be detected using the conventional NMR pulse Fourier technique.
We study theoretically dynamic properties of an interferometer-type Josephson charge qubit consisting of a single-Cooper-pair transistor (with capacitive gate) closed by a superconducting inductive loop. Analyzing the radio frequency response of the device to microwave fields within a two-band model, we present basic features of a method for observation of various Rabi oscillation modes. For this purpose, we apply the dressed-state approach in conjunction with the rotating-wave approximation to the strongly driven quantum bit. We demonstrate the existence of multiphoton excitations and Rabi frequency transitions between quantum levels connected with them. The resulting oscillations of the supercurrent in the interferometer modify the spectral loop susceptibility. This effect may be detected via the coil of a high-quality tank circuit inductively coupled to the qubit by measuring the tank-qubit impedance variations belonging to it.
We infer the high-frequency flux noise spectrum in a superconducting flux qubit by studying the decay of Rabi oscillations under strong driving conditions. The large anharmonicity of the qubit and its strong inductive coupling to a microwave line enabled high-amplitude driving without causing significant additional decoherence. Rabi frequencies up to 1.7 GHz were achieved, approaching the qubit's level splitting of 4.8 GHz, a regime where the rotating-wave approximation breaks down as a model for the driven dynamics. The spectral density of flux noise observed in the wide frequency range decreases with increasing frequency up to 300 MHz, where the spectral density is not very far from the extrapolation of the 1/f1/f1/f spectrum obtained from the free-induction-decay measurements. We discuss a possible origin of the flux noise due to surface electron spins.
A superconducting charge-flux qubit coupled to a high-Q tank circuit was studied in a low-frequency electric field. A fine structure of the multiphoton resonance lines and quantum interference effects associated with the excitation of a quasi-two-level system due to the Landau–Zener–Stückelberg tunneling was observed. The results obtained for multiphoton resonant excitations and low-frequency oscillations of the average occupation of quantum levels were compared using different parameters of the measuring circuit. The mechanism responsible for the fine structure of resonance lines was considered. The method to measure the impedance arising in the tank circuit due to the oscillations of the superconducting current in the qubit and the main sources of decoherence were discussed.