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arXiv:cond-mat/0408412 · Physical Review B

Electron spin tomography through counting statistics: A quantum trajectory approach

Holger Schaefers, Walter T. Strunz

Published Feb 23, 2005• 6 citations
arXivHTMLPDF

Abstract

We investigate the dynamics of electron spin qubits in quantum dots. Measurement of the qubit state is realized by a charge current through the dot. The dynamics is described in the framework of the quantum trajectory approach, widely used in quantum optics. The relevant master equation dynamics is unraveled to simulate stochastic tunneling events of the current through the dot. Quantum trajectories are then used to extract the counting statistics of the current. We show how, in combination with an electron spin resonance field, counting statistics can be employed for quantum-state tomography of the qubit state. Further, it is shown how decoherence and relaxation time scales can be estimated with the help of counting statistics, in the time domain.

DOI
10.1103/physrevb.71.075321

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We investigate the dynamics of electron spin qubits in quantum dots. Measurement of the qubit state is realized by a charge current through the dot. The dynamics is described in the framework of the quantum trajectory approach, widely used in quantum optics. The relevant master equation dynamics is unraveled to simulate stochastic tunneling events of the current through the dot. Quantum trajectories are then used to extract the counting statistics of the current. We show how, in combination with an electron spin resonance field, counting statistics can be employed for quantum-state tomography of the qubit state. Further, it is shown how decoherence and relaxation time scales can be estimated with the help of counting statistics, in the time domain.
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
A Monte Carlo simulation on the basis of quantum trajectory approach is carried out for the measurement dynamics of a single-electron spin resonance. The measured electron, which is confined in either a quantum dot or a defect trap, is tunnel coupled to a side reservoir and continuously monitored by a mesoscopic detector. The simulation not only recovers the observed telegraphic signal of detector current, but also predicts unique features in the output power spectrum which are associated with electron dynamics in different regimes.
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 describe time-dependent single-electron transport through quantum dots in the Coulomb blockade regime. Coherent dynamics of a single charge qubit in a double quantum dot is discussed with full one-qubit manipulation. Strength of decoherence is controlled with the applied voltage, but uncontrolled decoherence arises from electron–phonon coupling and background fluctuations. Then energy-relaxation dynamics is discussed for orbital and spin degree of freedom in a quantum dot. The electron–phonon interaction and spin–orbit coupling can be investigated as the dissipation problem. Finally, charge detection measurement is presented for statistical analysis of single-electron tunnelling transitions and for a sensitive qubit read-out device.
Stochastic systems feature, in general, both coherent dynamics and incoherent transitions between different states. We propose a method to identify the coherent part in the full counting statistics for the transitions. The proposal is illustrated for electron transfer through a quantum-dot spin valve, which combines quantum-coherent spin precession with electron tunneling. We show that by counting the number of transferred electrons as a function of time, it is possible to distill out the coherent dynamics from the counting statistics even in transport regimes, in which other tools such as the frequency-dependent current noise and the waiting-time distribution fail.
We propose a theoretical protocol for reconstructing the density matrix of a single-electron spin qubit using spin-polarized transport. The system consists of a quantum dot coupled to ferromagnetic reservoirs and subject to a magnetic field lying in the xyxyxy plane of the Bloch sphere. Spin-dependent tunneling events measured along the x±x\pmx±, y±y\pmy±, and z±z\pmz± quantization axes give rise to probability distributions that encode the quantum state of the qubit. The open-system dynamics are described using a Lindblad master equation, which captures the time evolution of the spin under continuous coupling to the reservoirs. By counting tunneling events for four different magnetic alignments, we formulate a scheme for reconstructing the full density matrix of the qubit. The resulting simulation data are analyzed using machine-learning techniques to process the measured probability distributions and infer the corresponding density matrix elements. The proposed model enables complete access to the open-system density matrix, including both population probabilities and relative phase information. Successful state reconstruction demonstrates the validity and robustness of the approach, highlighting its applicability to experimentally accessible spin-transport platforms.