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arXiv:cond-mat/0401204 · Physical Review B

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

Neil P. Oxtoby, Prahlad Warszawski, Howard M. Wiseman, He-Bi Sun, Rodney Polkinghorne

Published Apr 19, 2005• 24 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.71.165317

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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.

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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.

Journal of Physics Condensed Matter•Nov 7, 2003•14 citations
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.
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.
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.
Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the observer implies that the evolution of the conditional state of the quantum system must be non-Markovian. We present a general method of describing this evolution and apply it to avalanche photodiodes and to photoreceivers. We include the effects of efficiency, dead time, bandwidth, electronic noise and dark counts. The essential idea is to treat the quantum system and classical detector jointly, and to average over the latter to obtain the conditional quantum state. The significance of our theory is that quantum trajectories for realistic detection are necessary for sophisticated approaches to quantum feedback, and our approach could be applied in many areas of physics.
Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as, by an ideal photodetector. Here we derive (non-Markovian) quantum trajectories for realistic photodetection, including the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. We apply our theory to a realistic cavity QED scenario and investigate the impact of such detector imperfections on the conditional evolution of the system state. A practical theory of quantum trajectories with realistic detection will be essential for experimental and technological applications of quantum feedback in many areas.
The extension of quantum trajectory theory to incorporate realistic imperfections in the measurement of solid-state qubits is important for quantum computation, particularly for the purposes of state preparation and error correction as well as for readout of computations. Previously this has been achieved for low-frequency (dc) weak measurements. In this paper we extend realistic quantum trajectory theory to include radio frequency (rf) weak measurements where a low-transparency quantum point contact (QPC), coupled to a charge qubit, is used to damp a classical oscillator circuit. The resulting realistic quantum trajectory equation must be solved numerically. We present an analytical result for the limit of large dissipation within the oscillator (relative to the QPC), where the oscillator slaves to the qubit. The rf+dc\mathrm{rf}+\mathrm{dc}rf+dc mode of operation is considered. Here the QPC is biased (dc) as well as subjected to a small-amplitude sinusoidal carrier signal (rf). The rf+dc\mathrm{rf}+\mathrm{dc}rf+dc QPC is shown to be a low-efficiency charge-qubit detector, which may nevertheless be higher than the dc-QPC (which is subject to 1∕f1∕f1∕f noise).
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.