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arXiv:cond-mat/0303049 · Physical review. B, Condensed matter

Method for direct observation of coherent quantum oscillations in a superconducting phase qubit: Computer simulations

Ya. S. Greenberg

Published Dec 30, 2003• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.68.224517

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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.
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.
We analyze the dynamics of a qubit-resonator system coupled with a thermal bath and external electromagnetic fields. Using the evolution equations for the set of Heisenberg operators that describe the whole system, we derive an expression for the resonator field, that includes the resonator-drive, the resonator-bath, and resonator-qubit interactions. The renormalization of the resonator frequency, caused by the qubit-resonator interaction, is accounted for. Using the solutions for the resonator field, we derive the equation that describes the qubit dynamics. The dependence of the qubit evolution during the measurement time on the fidelity of a single-shot measurement is studied. The relation between the fidelity and measurement time is shown explicitly. We proposed a novel adiabatic method for the phase qubit measurement. The method utilizes a low-frequency, quasi-classical resonator inductively coupled to the qubit. The resonator modulates the qubit energy, and the back reaction of the qubit causes a shift in the phase of the resonator. The resonator phase shift can be used to determine the qubit state. We have simulated this measurement taking into the account the energy levels outside the phase qubit manifold. We have shown that, for qubit frequencies in the range of 8-12GHZ, a resonator frequency of 500 MHz and a measurement time of 100 ns, the phase difference between the two qubit states is greater than 0.2 rad. This phase difference exceeds the measurement uncertainty, and can be detected using a classical phase-meter. A fidelity of 0.9999 can be achieved for a relaxation time of 0.5 ms. We also model and simulate a microstrip-SQUID amplifier of frequency about 500 MHz, which could be used to amplify the resonator oscillations in the phase qubit adiabatic measurement. The voltage gain and the amplifier noise temperature are calculated. We simulate the preparation of a generalized Bell state and compute the relaxation times required for achieving high fidelities. We consider two capacitively coupled phase qubits similar to that used in the experimental work of R. Bialczak et al. (Nature Physics, 6, 409, 2010). The Bell state was created using a NOT gate and a universal entangling gate. We solved the equation of motion for the density matrix with the Bloch relaxation times, T{sub 1} and T{sub 2}. For every qubit we took into consideration all the states in the shallow well including the states outside the qubit manifold. Also we took into consideration unwanted interaction between the qubits during the application of the NOT gate. The NOT gate was implemented using a Gaussian pulse assuming the frequency difference between the two qubits is 200 MHz. An entangling gate was implemented reducing this frequency difference to zero. We have found the optimal parameters for the Gaussian pulse and the optimal duration for the entangling gate. We have shown that the maximum fidelity of 0.99 can be achieved for the minimum relaxation time, T{sub 1} = 1.2 {micro}s, if T{sub 2} = 2T{sub 1}. By slightly increasing the required value of T{sub 1}, one can sharply reduce the required value for T{sub 2}.
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.
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.