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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Noise in detection of qubit states using a quantum point contact

Neil P. Oxtoby, He-Bi Sun, Howard M. Wiseman

Published May 9, 2003• 0 citations
Publisher

Abstract

We present a model for detection of the states of a coupled quantum dots (qubit) by a quantum point contact. Most proposals for measurements of states of quantum systems are idealized. However in a real laboratory the measurements cannot be perfect due to practical devices and circuits. The models using ideal devices are not sufficient for describing the detection information of the states of the quantum systems. Our model therefore includes the extension to a non-ideal measurement device case using an equivalent circuit. We derive a quantum trajectory that describes the stochastic evolution of the state of the system of the qubit and the measuring device. We calculate the noise power spectrum of tunnelling events in an ideal and a non-ideal quantum point contact measurement respectively. We found that, for the strong coupling case it is difficult to obtain information of the quantum processes in the qubit by measurements using a non-ideal quantum point contact. The noise spectra can also be used to estimate the limits of applicability of the ideal model.

DOI
10.1117/12.500397

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•Jan 1, 2007•2 citations
We present a model for detection of the states of a coupled quantum dots (qubit) by a quantum point contact. Most proposals for measurements of states of quantum systems are idealized. However in a real laboratory the measurements cannot be perfect due to practical devices and circuits. The models using ideal devices are not sufficient for describing the detection information of the states of the quantum systems. Our model therefore includes the extension to a non-ideal measurement device case using an equivalent circuit. We derive a quantum trajectory that describes the stochastic evolution of the state of the system of the qubit and the measuring device. We calculate the noise power spectrum of tunnelling events in an ideal and a non-ideal quantum point contact measurement respectively. We found that, for the strong coupling case it is difficult to obtain information of the quantum processes in the qubit by measurements using a non-ideal quantum point contact. The noise spectra can also be used to estimate the limits of applicability of the ideal model.
We propose a model for non-ideal monitoring of the state of a coupled quantum dot qubit by a quantum tunnelling device. The non-ideality is modelled using an equivalent measurement circuit. This allows realistically available measurement results to be related to the state of the quantum system (qubit). We present a quantum trajectory that describes the stochastic evolution of the qubit state conditioned by tunnelling events (i.e. current) through the device. We calculate and compare the noise power spectra of the current in an ideal and a non-ideal measurement. The results show that when the two qubit dots are strongly coupled the non-ideal measurement cannot detect the qubit state precisely. The limitation of the ideal model for describing a realistic system may be estimated from the noise spectra.
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.
We propose a quantum trajectory analysis of a scheme to measure the states of a coupled-dot device (qubit) where there is a fluctuating energy gap Δ between the two states. The system consists of the qubit and a readout dot coupled to source and drain leads. The tunnel rate through the detector is conditioned by the occupation number of the nearer quantum dot (target) of the qubit and therefore probes the states of the qubit. We derive a Lindblad-form master equation to calculate the unconditional evolution of the qubit and a conditional stochastic master equation calculating the conditional evolution for different tunneling rates. The results show the effects of various device parameters and provide the optimum selection and combination of the system structure.
Full text: Recently, we presented a model for continuous DC-biased measurement of a coupled quantum dot (CQD) charge qubit by realistic measurement devices. Modelling realistic measurement devices as ideal detectors embedded in an equivalent measurement circuit (see figure), our aim was to describe the evolution of the qubit state conditioned on the macroscopic output of the external circuit, i.e. the measured current. We generalised a recently developed quantum trajectory theory for realistic photodetectors to treat solid-state detectors-primarily a single electron transistor (SET). This method yields stochastic equations whose (numerical) solutions are the 'realistic quantum trajectories' of the conditioned qubit state. Here we present numerical solutions and analysis for the SET equations. The solutions elucidate the regimes in which (Rabi) electron tunnelling between the CQDs can be detected in a realistic measurement. Further to this, the solutions reveal the conditions under which electron tunnelling events through the SET can be observed, which is related to the question of the nature of the back-action noise of the SET. Copyright (2005) Australian Institute of Physics.
We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD's) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schr\"odinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.
To obtain information about a system of interest a measurement has to be made. In experiments that probe the quantum nature of our world, the system itself is, in general, necessarily affected by the act of measurement. If the system is weakly coupled to its bath and the dynamics are such that information concerning the system is spread throughout the many degrees of freedom of the bath, and the bath is being measured then a stochastic master equation for the conditioned state of the system can be found. This is termed a quantum trajectory equation. Realistic detectors are not perfect. Information is lost in the conversion to a signal that the observer can use. This loss may occur in the detector itself, in the circuit containing the detector (described by a response time and electronic noise) or at the circuit output (electronic output noise). In order to obtain a true quantum trajectory for the experiment, the observer must condition the state of the quantum system on results that are available in the laboratory rather than on the microscopic events considered previously in quantum trajectories. A method for treating this was first proposed by Warszawski, Wiseman and Mabuchi [Phys. Rev. A 65, 023802 (2002)], in which the quantum system is embedded within a supersystem that also contains the state of the detector. They applied their theory to photodetectors of various sorts. Warszawski has also done the preliminary work on applying this theory to detecting the state of a pair of quantum dots using a SET (single-electron transistor) [MSc. Thesis, Griffith University (2001)]. The resulting theory is termed 'realistic' quantum trajectory theory. In this thesis, the approach of Warszawski, et al.is applied to various solidstate readout devices. These include the SET, the QPC (quantum point contact), and the RF-QPC (radio-frequency QPC). Numerically obtained realistic quantum trajectories for the QPC agree with heuristic results. In particular, in certain limits, the realistic quantum trajectories can take on the appearance of ideal quantum trajectories. This thesis also resolves a problem in solid-state continuous quantum measurement theory by deriving a quantum trajectory model for a SET-monitored charge qubit, that guarantees physically meaningful qubit states. The particular limit necessary to achieve this is discussed, and the SET measurement quality is analysed using techniques borrowed from quantum optics. Conditions for which the SET can approach operation at the limit allowed by quantum mechanics are given. This is also done for the QPC, for which the results agree with previous work.