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arXiv:cond-mat/0412270 · Physical Review Letters

Continuous Measurement of a Microwave-Driven Solid State Qubit

S. D. Barrett, Thomas M. Stace

Published Jan 13, 2006• 30 citations
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Abstract

We analyze the dynamics of a continuously observed, damped, microwave-driven solid state charge qubit, consisting of a single electron in a double well potential. The microwave field induces transitions between the qubit eigenstates, which have a profound effect on the detector output current. Useful information about the qubit dynamics, such as dephasing and relaxation rates, and the Rabi frequency, can be extracted from the detector conductance and output noise power spectrum. We also propose a technique for single-shot electron spin readout, for spin based quantum information processing, which has a number of practical advantages over existing schemes.

DOI
10.1103/physrevlett.96.017405

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We analyze the dynamics of a continuously observed, damped, microwave-driven solid state charge qubit, consisting of a single electron in a double well potential. The microwave field induces transitions between the qubit eigenstates, which have a profound effect on the detector output current. Useful information about the qubit dynamics, such as dephasing and relaxation rates, and the Rabi frequency, can be extracted from the detector conductance and output noise power spectrum. We also propose a technique for single-shot electron spin readout, for spin based quantum information processing, which has a number of practical advantages over existing schemes.
Developing fast and accurate control and readout techniques is an important challenge in quantum information processing with semiconductor qubits. Here, we study the dynamics and the coherence properties of a GaAs/AlGaAs double quantum dot charge qubit strongly coupled to a frequency-tunable high-impedance resonator. We drive qubit transitions with synthesized microwave pulses and perform qubit readout through the state-dependent frequency shift imparted by the qubit on the dispersively coupled resonator. We perform Rabi oscillation, Ramsey fringe, energy relaxation, and Hahn-echo measurements and find significantly reduced decoherence rates down to γ_{2}/2π∼3 MHz corresponding to coherence times of up to T_{2}∼50 ns for charge states in gate-defined quantum dot qubits. We realize Rabi π pulses of width down to σ∼0.25 ns.
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 have studied interactions between two capacitively coupled GaAs/AlGaAs few-electron double quantum dots. Each double quantum dot defines a tunable two-level system, or qubit, in which a single excess electron occupies the ground state of either one dot or the other. Applying microwave radiation, we resonantly drive transitions between states and noninvasively measure occupancy changes using proximal quantum point contact charge detectors. The level structure of the interacting two-qubit system is probed by driving it at a fixed microwave frequency while varying the energy detuning of both double dots. We observe additional resonant transitions consistent with a simple coupled two-qubit Hamiltonian model.
Strong coupling of semiconductor spin qubits to superconducting microwave resonators was recently demonstrated [X. Mi et al., Nature 555, 599 (2018); A. J. Landig et al., Nature 560, 179 (2018); N. Samkharadze et al., Science 359, 1123 (2018); T. Cubaynes et al., NPJ Quant. Inf. 5, 47 (2019)]. These breakthroughs pave the way for quantum information processing that combines the long coherence times of solid-state spin qubits with the long-distance connectivity, fast control, and fast high-fidelity quantum-non-demolition readout of existing superconducting qubit implementations. Here we theoretically analyze and optimize the dispersive readout of a single spin in a semiconductor double quantum dot (DQD) coupled to a microwave resonator via its electric dipole moment. The strong spin-photon coupling arises from the motion of the electron spin in a local magnetic field gradient. We calculate the signal-to-noise ratio (SNR) of the readout accounting for both Purcell spin relaxation and spin relaxation arising from intrinsic electric noise within the semiconductor. We express the maximum achievable SNR in terms of the cooperativity associated with these two dissipation processes. We find that while the cooperativity increases with the strength of the dipole coupling between the DQD and the resonator, it does not depend on the strength of the magnetic field gradient. We then optimize the SNR as a function of experimentally tunable DQD parameters. We identify wide regions of parameter space where the unwanted backaction of the resonator photons on the qubit is small. Moreover, we find that the coupling of the resonator to other DQD transitions can enhance the SNR by at least a factor of two, a ``straddling'' effect [J. Koch et al., Phys. Rev. A 76, 042319 (2007)] that occurs only at nonzero energy detuning of the DQD double-well potential. We estimate that with current technology, single-shot readout fidelities in the range 82--95% can be achieved within a few $\ensuremath{\mu}\mathrm{s}$ of readout time without requiring the use of Purcell filters.
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 study the dynamics of a charge qubit, consisting of a single electron in a double well potential, coupled to a point-contact (PC) electrometer using the quantum trajectories formalism. In contrast with previous work, our analysis is valid for arbitrary source-drain bias across the PC, but is restricted to the sub-Zeno limit. We find that the dynamics is strongly affected by inelastic tunnelling processes in the PC. These processes reduce the efficiency of the PC as a qubit readout device, and induce relaxation even when the source-drain bias is zero. We show that the sub-Zeno dynamics are divided into two regimes: low- and high-bias in which the PC current and current power spectra show markedly different behaviour. To further illustrate the division between the regimes and the inefficiency of the detector, we present simulated quantum trajectories of the conditional qubit and detector dynamics. We also describe how single shot measurements in an arbitrary basis may be achieved in the sub-Zeno regime.