Quantum trajectories for the realistic measurement of a solid-state charge qubit

Abstract

We present a new model for the continuous measurement of a coupled quantum dot charge qubit. We model the effects of a realistic measurement, namely adding noise to, and filtering, the current through the detector. This is achieved by embedding the detector in an equivalent circuit for measurement. Our aim is to describe the evolution of the qubit state conditioned on the macroscopic output of the external circuit. We achieve this by generalizing a recently developed quantum trajectory theory for realistic photodetectors [P. Warszawski, H. M. Wiseman and H. Mabuchi, Phys. Rev. A 65 023802 (2002)] to treat solid-state detectors. This yields stochastic equations whose (numerical) solutions are the "realistic quantum trajectories" of the conditioned qubit state. We derive our general theory in the context of a low transparency quantum point contact. Areas of application for our theory and its relation to previous work are discussed.

Model for monitoring of a charge qubit using a radio-frequency quantum point contact including experimental imperfections

“…This is achieved by expressing J cos 0 (t) and J sin ω0 (t) in terms of both the observed process, J sin (t), plus some other unobserved processes, as done in Ref. 21. Averaging over the unobserved processes results in the following linear quantum trajectory equation in the RWA:…”

“…Our derivation of the realistic quantum trajectory equation describing the conditional evolution of the combined oscillator-qubit state closely follows the standard techniques. 21,23 The details are included in Appendix B, with the final result being the following superoperator Kushner-Stratonovich (SKS) equation:…”

“…The realistic quantum trajectory derivation then proceeds in an analogous manner to Ref. 21, where the circuit state was described by one parameter -the charge on a capacitor. In general the oscillator circuit needs two parameters to describe its state, thus hinting at a complication in deriving the realistic equations for the rf configuration.…”

“…Two of us recently extended such work to condition the qubit state on a macroscopic signal that is realistically available to an observer in a dc-QPC measurement. 21 In particular, this extension considered the noisy, filtering characteristic of an external circuit, including an amplifier. The result is a corrupted version of the idealized measurement results upon which the qubit evolution can be conditioned.…”

“…We also consider conditioning the qubit state on measurement results available to a realistic observer, within the framework of realistic quantum trajectory theory. 21,22,23,29 In this approach, the bare charge-qubit detector (QPC in our case) is embedded in a realistic circuit, and an equation is derived that describes the evolution of the combined circuit-plus-qubit state conditioned on measurement results available to a realistic observer.…”