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arXiv:0709.2459 · Physical Review B

Quantum theory of the low-frequency linear susceptibility of interferometer-type superconducting qubits

Ya. S. Greenberg, E. V. Il’ichev

Published Mar 14, 2008• 11 citations
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Abstract

We use the density matrix formalism to analyze the interaction of interferometer-type superconducting qubits with a high quality tank circuit, which frequency is well below the gap frequency of a qubit. We start with the ground state characterization of the superconducting flux and charge qubits. Then, by making use of a dressed state approach, we describe the qubits' spectroscopy when the qubit is irradiated by a microwave field which is tuned to the gap frequency. The last section of the paper is devoted to continuous monitoring of qubit states by using a dc superconducting quantum interference device in the inductive mode.

DOI
10.1103/physrevb.77.094513

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We use the density matrix formalism to analyze the interaction of interferometer-type superconducting qubits with a high quality tank circuit, which frequency is well below the gap frequency of a qubit. We start with the ground state characterization of the superconducting flux and charge qubits. Then, by making use of a dressed state approach, we describe the qubits' spectroscopy when the qubit is irradiated by a microwave field which is tuned to the gap frequency. The last section of the paper is devoted to continuous monitoring of qubit states by using a dc superconducting quantum interference device in the inductive mode.
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 implemented experimentally an interferometer-type charge qubit consisting of a single-Cooper-pair transistor closed by a superconducting loop that is in flip-chip configuration inductively coupled to a radio-frequency tank circuit. The tank permits us to readout the qubit state, acting as an inductance measuring apparatus. By applying continuous microwave power to the quantum device, we observed inductance alterations caused by redistributions of the energy-level populations. From the measured data we extracted the energy gap between ground and upper levels as a function of the transistor quasicharge as well as the Josephson phase across both junctions.
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.
We have analyzed a dissipative two-level quantum system (TLS) which is continuously and simultaneously irradiated by high- and low-frequency excitation. The interaction of the TLS with a high-frequency excitation is considered in the frame of the dressed-state approach. A linear response of the coupled TLS and corresponding photon field system to a signal whose frequency is of the order of the Rabi frequency is found. The response exhibits undamped low-frequency oscillations whose amplitude has a clear resonance at the Rabi frequency with the width being dependent on the damping rates of the TLS. The method can be useful for low-frequency Rabi spectroscopy in various physical systems described by a two-level Hamiltonian, such as nuclei spins in NMR, double well quantum dots, superconducting flux, charge qubits, etc. The application of the method to a superconducting flux qubit and to the detection of NMR is considered in detail.
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.