Generation of a SingleElectron Gaussian Wave
Packet Using Dynamical Quantum Dots
SUNGGUEN RYU AND H.S. SIM^{1}
^{1}DEPARTMENT OF PHYSICS, KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY
ABSTRACT
Generating and detecting a prescribed singleelectron state is an important step towards solidstate fermion optics and related quantum information processing. Here, we introduce our recent proposal for an experimentally feasible way of generating a single electron in a Gaussian wave packet, using a quantumdot pump operating in gigahertz frequency. With the help of a strong magnetic field, the generation is insensitive to the details of nonadiabatic pump operation and the shape of the pump potential. The spatial distribution of the wave packet can be detected by using a dynamical potential barrier.
INTRODUCTION
Ondemand singleelectron sources have been developed, opening roads toward a fermion version of quantum optics and related quantum processing [1, 2, 3]. Among them, a quantumdot (QD) pump [4] has the unique property of emitting hot electrons of ~100 meV above the Fermi energy. The pump had been originally studied for metrology, and there is growing interest on how to apply it to fermion quantum optics. Its application is plausible as the scattering of hot electrons by phonons and other electrons can be suppressed [5].
For its application, it is crucial to realize a QD pump emitting a prescribed electron wave packet with a useful form. A detector for measuring an emitted packet is also necessary for realizing the fermion optics. In this article, we introduce our recent theoretical proposal [6] for generation and detection of a singleelectron Gaussian wave packet using a QD pump.
Fig. 1: Generation of a coherent state in a typical quantumdot pump [7]. The dynamical quantum dot is formed by gates G1 and G2 (blue, inset) in GaAS 2DEG (green, inset). A strong magnetic field B=14 T is applied. Gate voltage V_{G1} adiabatically changes in the capturing process of the pump and then abruptly changes in the emission process. The other gate voltage is constant in time. The groundstate electron captured in QD evolves to a coherent state (cone) moving along the E×B drift (curve) as the barriers rise. This figure is a modified version of a figure in Ref. [6].
GENERATION OF A GAUSSIAN WAVE PACKET
A QD pump utilizes a dynamical quantum dot in a twodimensional electron gas (2DEG), which is formed by gate voltages V_{G1} and V_{G2}; See Fig. 1. A pump cycle is driven by modulating V_{G1} and consists of two processes; first, electrons are captured in the QD when the entrance barrier is lowered below the Fermi energy, and the captured electrons are emitted through the exit barrier when the entrance barrier is sufficiently raised. The capturing process has been analyzed by the semiclassical rate equation, while the emission process is little studied theoretically.
We analyzed the emission process, by solving the wave packet dynamics of a captured electron in the dynamical QD. Due to the slow capturing process, we could focus on the regime where only one electron is captured adiabatically in the QD. Time evolution of the captured electron in the emission process is not trivial, since the QD confinement potential U_{QD} nonadiabatically changes in time s. It is crucial that a strong magnetic field makes the evolution simple and insensitive to process details. At a time s=0 of the capturing process, the electron is in the ground state of the QD, which is the Gaussian state when the cyclotron frequency ω_{c} is much larger than the QD confinement frequency ω_{0}. We find that its time evolution ψ(r,s) is well described by
provided that U_{QD} changes slowly in length scale (e is the electron charge) and time scale ω_{c}^{1} [6]. ψ_{c} is a coherent state, as r_{c}(s) and p_{c}(s) follow the classical E×B drift motion. When the coherent state gains sufficient energy, it is emitted through the exit barrier and a Gaussian wave packet is generated in the quantum Hall edge (see Fig. 2 (a)) in the regime where the tunneling time through the exit barrier is small. [6].
DETECTION OF A GAUSSIAN WAVE PACKET
An important feature of a Gaussian wave packet is that it satisfies the Heisenberg minimal uncertainty
ℏ/2. We suggest using the quantum dot pump for experimentally confirming the minimal uncertainty [6].
The energy and time distribution of the emitted wave packet can be experimentally studied by measuring current I_{T} through a potential barrier induced by gate G3 (see Fig. 2). To obtain the energy distribution, one prepares the potential barrier of timeindependent height U_{3} and measures P_{E} (U_{3} )≡(ef)^{1} ∂I_{T }/∂U_{3}. f is the pumping frequency. To obtain the time distribution of arrival at the detector, one rapidly raises U_{3} from 0 to a large value (much larger than the energy of the emitted state) at time s_{3} and measures P_{T}(s_{3})≡(ef)^{1} ∂I_{T }/∂s_{3}. Fig. 2 shows a simulation of the detection. From the energy and time distributions, one can estimate the uncertainty of the measured wave packet, and study whether the uncertainty approaches the limiting value of the minimal uncertainty.
We introduced the general dynamical property of an electron wave packet in a 2D dynamic quantum dot under a strong magnetic field and demonstrated that a Gaussian wave packet can be generated and detected. This generation corresponds to that of a single photon in quantum optics. We believe that this generation will be useful for experimental studies of coherent dynamics of an electron in solid state devices.
Fig. 2: (a) Schematic view of generation and detection of a singleelectron Gaussian wave packet. The wave packet is emitted from the pump in Fig. 1. Its energy (ψ(ε)^{2}) and arrivaltime (A_{D}(s)) distributions are experimentally measured by using gate G3 of 200 nm width and voltage V_{G3}. To measure the time distribution, V_{G3} changes 1 V in 50 ps. (b), (c) Numerical results of P_{E}(E) and P_{T}(s), compared with ψ(ε)^{2} and A_{D}(s). This figure is a modified version of a figure in Ref. [6].
References
[1] J.P. Pekola, et al., Rev. Mod. Phys. 85, 1421 (2013).
[2] G. Fève, et al., Science 316, 1169 (2007).
[3] J. Dubois, et al., Nature (London) 502, 659 (2013).
[4] S.P. Giblin, et al., Nat. Commun. 3, 930 (2012).
[5] C. Emary, A. Dyson, S. Ryu, H.S. Sim, and M. Kataoka, Phys. Rev. B 93, 035436 (2016).
[6] Sungguen Ryu, M. Kataoka, and H.S. Sim, Phys. Rev. Lett. 117, 146802 (2016).
[7] J.D. Fletcher, et al., Phys. Rev. Lett. 111, 216807 (2013).

Sungguen Ryu is a postdoctoral researcher in quantum electron correlation and transport theory group at the Korea Advanced Institute of Science and Technology (KAIST). He received his PhD from KAIST for the theoretical study of quantum dynamics in singleelectron quantumdot pumps. His current research field is theoretical mesoscopic physics with a focus on nonequilibrium timedependent phenomena. 
