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arXiv:cond-mat/0601035 · Physical Review B

Sensitivity and back action in charge qubit measurements by a strongly coupled single-electron transistor

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

Published Jul 27, 2006• 34 citations

Abstract

We consider charge qubit monitoring (continuous-in-time weak measurement) by a single-electron transistor (SET) operating in the sequential-tunneling regime. We show that commonly used master equations for this regime are not of the Lindblad form that is necessary and sufficient for guaranteeing valid physical states. In this paper we derive a Lindblad-form master equation and a corresponding quantum trajectory model for continuous measurement of the charge qubit by a SET. Our approach requires that the SET-qubit coupling be strong compared to the SET tunneling rates. We present an analysis of the quality of the qubit measurement in this model (sensitivity versus back action). Typically, the strong coupling when the SET island is occupied causes back action on the qubit beyond the quantum back action necessary for its sensitivity, and hence the conditioned qubit state is mixed. However, in one strongly coupled, asymmetric regime, the SET can approach the limit of an ideal detector with an almost pure conditioned state. We also quantify the quality of the SET using more traditional concepts such as the measurement time and decoherence time, which we have generalized so as to treat the strongly responding regime.

DOI: 10.1103/physrevb.74.045328

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We consider charge qubit monitoring (continuous-in-time weak measurement) by a single-electron transistor (SET) operating in the sequential-tunneling regime. We show that commonly used master equations for this regime are not of the Lindblad form that is necessary and sufficient for guaranteeing valid physical states. In this paper we derive a Lindblad-form master equation and a corresponding quantum trajectory model for continuous measurement of the charge qubit by a SET. Our approach requires that the SET-qubit coupling be strong compared to the SET tunneling rates. We present an analysis of the quality of the qubit measurement in this model (sensitivity versus back action). Typically, the strong coupling when the SET island is occupied causes back action on the qubit beyond the quantum back action necessary for its sensitivity, and hence the conditioned qubit state is mixed. However, in one strongly coupled, asymmetric regime, the SET can approach the limit of an ideal detector with an almost pure conditioned state. We also quantify the quality of the SET using more traditional concepts such as the measurement time and decoherence time, which we have generalized so as to treat the strongly responding regime.

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 report an experimental observation of the backaction of a single-electron transistor (SET) measuring the Coulomb staircase of a single-electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunneling model accurately recreates these effects, confirming this mechanism of the backaction of a SET. This is a first step toward understanding the effects of quantum measurement on solid-state qubits.

We derive the quantum trajectory or stochastic (conditional) master equation for a single superconducting Cooper-pair box (SCB) charge qubit measured by a single-electron transistor (SET) detector. This stochastic master equation describes the random evolution of the measured SCB qubit density matrix which both conditions and is conditioned on a particular realization of the measured electron tunneling events through the SET junctions. Hence it can be regarded as a Monte Carlo method that allows us to simulate the continuous quantum measurement process. We show that the master equation for the ``partially'' reduced density matrix [Y. Makhlin et al., Phys. Rev. Lett. 85, 4578 (2000)] can be obtained when a ``partial'' average is taken on the stochastic master equation over the fine grained measurement records of the tunneling events in the SET. Finally, we present some Monte Carlo simulation results for the SCB/SET measurement process. We also analyze the probability distribution

P(m,t)

of finding

m

electrons that have tunneled into the drain of the SET in time

t

to demonstrate the connection between the quantum trajectory approach and the ``partially'' reduced density matrix approach.

We report an experimental observation of the backaction of a Single Electron Transistor (SET) measuring the Coulomb staircase of a single electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunnelling model accurately recreates these effects, confirming this mechanism of the backaction of an SET. This is a first step towards understanding the effects of quantum measurement on solid state qubits.