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Conference on Optoelectronic and Microelectronic Materials and Devices (COMMAD)

Measurement of the states of a coupled dot device

Haoyu Sun, Howard M. Wiseman, Dian Wahyu Utami, Gerard J Milburn

Published Aug 25, 2005• 0 citations
Publisher

Abstract

We analyse a method to measure the states of a coupled dot device (qubit). We derive a master equation to describe the unconditional evolution of the qubit when the measured records are averaged over and a conditional stochastic master equation describing the conditional evolution of the qubit. The approach is also extended to model a measurement scheme when there is a fluctuating energy gap between two electron states. From the calculated results the relevant device parameters can be estimated and optimum combination of the parameters can be selected.

DOI
10.1109/commad.2000.1022904

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We analyse a method to measure the states of a coupled dot device (qubit). We derive a master equation to describe the unconditional evolution of the qubit when the measured records are averaged over and a conditional stochastic master equation describing the conditional evolution of the qubit. The approach is also extended to model a measurement scheme when there is a fluctuating energy gap between two electron states. From the calculated results the relevant device parameters can be estimated and optimum combination of the parameters can be selected.
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.
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.
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 provide a unified view for the master equation (ME) approach and the quantum trajectory (QT) approach to the measurement problem of an electron tunnelling between two coherently coupled quantum dots (CQD’s) using a low transparency point contact (PC) or tunnel junction as a detector. We show that the master equation for the reduced or “partially” reduced density matrix can be obtained when an average or “partial” average is taken on the conditional, stochastic density matrix over the possible outcomes of the measurement. Then we simulate ) , ( t N P , the probability of finding N electron that have tunnelled through the PC in time t , using 10000 QT’s and measurement records. The simulation results of ) , ( t N P are in very good agreement with those obtained from the Fourier analysis of the “partially” reduced density matrix. The theory of QT’s (also known as stochastic Schrodinger equation or stochastic ME) has been developed in last ten years mainly in the quantum optics community to describe open quantum system subject to continuous quantum measurements. But it was introduced to the context of solid-state mesoscopics only recently (e.g., Goan et. al. 2001 and references therein). The CQD system, considered here, forms a single charge quantum bit (qubit), and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dots. The QT or stochastic quantum-jump ME of the density matrix operator ) (t c
We provide a unified picture for the master equation approach and the quantum trajectory approach to a measurement problem of a two-state quantum system (a qubit), an electron coherently tunneling between two coupled quantum dots (CQD's) measured by a low transparency point contact (PC) detector. We show that the master equation of ``partially'' reduced density matrix can be derived from the quantum trajectory equation (stochastic master equation) by simply taking a ``partial'' average over the all possible outcomes of the measurement. If a full ensemble average is taken, the traditional (unconditional) master equation of reduced density matrix is then obtained. This unified picture, in terms of averaging over (tracing out) different amount of detection records (detector states), for these seemingly different approaches reported in the literature is particularly easy to understand using our formalism. To further demonstrate this connection, we analyze an important ensemble quantity for an initial qubit state readout experiment, P(N,t), the probability distribution of finding N electron that have tunneled through the PC barrier(s) in time t. The simulation results of P(N,t) using 10000 quantum trajectories and corresponding measurement records are, as expected, in very good agreement with those obtained from the Fourier analysis of the ``partially'' reduced density matrix. However, the quantum trajectory approach provides more information and more physical insights into the ensemble and time averaged quantity P(N,t). Each quantum trajectory resembles a single history of the qubit state in a single run of the continuous measurement experiment. We finally discuss, in this approach, the possibility of reading out the state of the qubit system in a single-shot experiment.
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.