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Physical Review Letters

Coherent Time Evolution of a Single-Electron Wave Function

M. Kataoka, M. R. Astley, A. L. Thorn, Daniel K. L. Oi, C. H. W. Barnes, Christopher J. B. Ford, David V. Anderson, G. A. C. Jones, Ian A. Farrer, D. A. Ritchie, M. Pepper

Published Apr 13, 2009• 71 citations
Publisher

Abstract

Observation of coherent single-electron dynamics is severely limited by experimental bandwidth. We present a method to overcome this using moving quantum dots defined by surface acoustic waves. Each dot holds a single electron, and travels through a static potential landscape. When the dot passes abruptly between regions of different confinement, the electron is excited into a superposition of states, and oscillates unitarily from side to side. We detect these oscillations by using a weak, repeated measurement of the current across a tunnel barrier, and find close agreement with simulations.

DOI
10.1103/physrevlett.102.156801

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Observation of coherent single-electron dynamics is severely limited by experimental bandwidth. We present a method to overcome this using moving quantum dots defined by surface acoustic waves. Each dot holds a single electron, and travels through a static potential landscape. When the dot passes abruptly between regions of different confinement, the electron is excited into a superposition of states, and oscillates unitarily from side to side. We detect these oscillations by using a weak, repeated measurement of the current across a tunnel barrier, and find close agreement with simulations.
Motivated by recent experimental observations [M. Kataoka et al., Phys. Rev. Lett. 102, 156801 (2009)], we propose here a theoretical approach to implement quantum computation with bound states of electrons in moving quantum dots generated by the driving of surface acoustic waves. Differing from static quantum dots defined by a series of static electrodes above the two-dimensional electron gas (2DEG), here a single electron is captured from a 2DEG reservoir by a surface acoustic wave and then trapped in a moving quantum dot (MQD) transported across a quasi-one-dimensional channel, wherein all the electrons have been excluded by the actions of the surface gates. The flying qubit introduced here is encoded by the two lowest levels of the electron in the MQD, and the Rabi oscillation between these two levels could be implemented by applying finely selected microwave pulses to the surface gates. By using the Coulomb interaction between the electrons in different MQDs, we show that a desirable two-qubit operation, i.e., iswap gate, could be realized. Readouts of the present flying qubits are also feasible with the current single-electron detected technique.
Gallium arsenide is piezoelectric, so it is possible to generate coupled mechanical and electrical surface acoustic waves (SAWs) by applying a high-frequency voltage to a transducer on the surface of GaAs. By combining SAWs with existing low-dimensional nanostructures one can create a series of dynamic quantum dots corresponding to the minima of the travelling electric wave, and each dot carries a single electron at the SAW velocity (∼ 2800 m/s). These devices may be of use in developing future quantum information processors, and also offer an ideal environment for probing the quantum mechanical behaviour of single electrons. This thesis describes a numerical and theoretical study of the dynamics of an electron in a range of geometries. The numerical techniques for solving the time-dependent Schr ̈dinger equation with an arbitrary time-dependent potential will be described in Chapter 2, and then applied in Chapter 3 to calculate the transmission of an electron through an Aharonov-Bohm (AB) ring. It will be seen that an important property of the techniques used in this thesis is that they can be easily adapted to study realistic geometries, and we will see features in the AB oscillations which do not arise in simplified analytic descriptions. In Chapter 4, we will then study a device consisting of two parallel SAW channels separated by a controllable tunnelling barrier. We will use numerical simulations to investigate the effect of electric and magnetic fields upon the electron dynamics, and develop an analytic model to explain the simulation results. From the model, it will be apparent that it is possible to use this device to rotate the state of the electron to an arbitrary superposition of the first two eigenstates. We then introduce coherent and squeezed states in Chapter 5, which are ex- cited states of the quantum harmonic oscillator. Coherent and squeezed electronic states may be of use in quantum information processing, and could also arise due to unwanted perturbations in a SAW device. We will discuss how these states can be controllably generated in a SAW device, and also discuss how they could then be detected. In Chapter 6 we describe how to use the motion of a SAW to create a rapidly- changing potential in the frame of the electron, leading to a nonadiabatic excita- tion. The nonadiabatically-excited state oscillates from side to side within a 1D channel on a few-picosecond timescale, and this motion can be probed by placing a tunnelling barrier at one side of the channel. Numerical simulations will be performed to show how this motion can be controlled, and the simulation results will be seen to be in good agreement with recent experimental work performed by colleagues. Finally, we will show that this device can be used to measure the initial state of an electron which is an arbitrary superposition of the first two eigenstates.
We propose an experiment to observe interference of a single electron as it is transported along two parallel quasi-one-dimensional channels trapped in a single minimum of a traveling periodic electric field. The experimental device is a modification of the surface-acoustic-wave-based quantum processor. Interference is achieved by creating a superposition of spatial wave functions between the two channels and inducing a relative phase shift via either a transverse electric field or a magnetic field. The interference can be used to estimate the decoherence time of an electron in this type of solid-state device.
The time evolution of the quantum mechanical state of an electron is calculated in the framework of the effective-mass envelope function theory for an InAs/GaAs quantum dot. The results indicate that the superposition state electron density oscillates in the quantum dot, with a period on the order of femtoseconds. The interaction energy Eij between two electrons located in different quantum dots is calculated for one electron in the ith pure quantum state and another in the jth pure quantum state. We find that E11〉E12〉E22, and Eij decreases as the distance between the two quantum dots increases. We present a parameter-phase diagram which defines the parameter region for the use of an InAs/GaAs quantum dot as a two-level quantum system in quantum computation. A static electric field is found to efficiently prolong the decoherence time. Our results should be useful for designing the solid-state implementation of quantum computing.
In this thesis, the coherent interaction of single semiconductor quantum dots and ultrafast optical pulses is studied. Under certain conditions, localized exciton transitions in quantum dots can be seen as semi-isolated two-level systems. While this description is sufficient for the explanation of some observations in coherent experiments, it is sometimes necessary to explicitly consider coupling of the discreet quantum states confined to the dot with the environment. We start out from simple, classical examples of coherent spectroscopy and then turn towards experiments where the interaction with the vicinity of the dot becomes an important factor. First, a novel method for transient differential reflectivity spectroscopy of single quantum systems is introduced. It is a pure far-field optical technique which does not require any sophisticated sample preparation steps which makes it applicable to a broad range of structures. Pump pulses excite the sample structure and probe pulses read out the pump-induced changes in the system after a variable delay time. In the case of a single dipole, the signal is given in the form of the spectral inteferogram between the backscattered wave from the particle and the probe light which is reflected at the sample surface. This form of homodyne detection amplifies the weak scattered wave from the particle and thus makes this kind of spectroscopy for single quantum dots feasible. In the remainder of this thesis our spectroscopic method is applied to either characterize the coherent properties of single quantum dots, to prepare and read-out a desired quantum state or to deliberately manipulate them. Coherence times and oscillator strengths are determined for localized exciton transitions. Arbitrary population states can be written by driving coherent population oscillations using resonant pulses, while entangled superpositions of two exciton states in a single dot are investigated by quantum beats on transient differential spectra. We finally exploit the interaction between the dot and a nearby absorbing layer to switch the dot's absorption spectrum on ultrafast timescales via light-induced transient electric fields.
We have investigated the dynamics of bound particles in multilevel current-carrying quantum dots. We look specifically in the regime of resonant tunnelling transport, where several channels are available for transport. Through a non-Markovian formalism under the Born approximation, we investigate the real-time evolution of the confined particles including transport-induced decoherence and relaxation. In the case of a coherent superposition between states with different particle number, we find that a Fock-space coherence may be preserved even in the presence of tunneling into and out of the dot. Real-time results are presented for various asymmetries of tunneling rates into different orbitals.