Floquet Theory of Electron Waiting Times in Quantum-Coherent Conductors

Abstract

We present a Floquet scattering theory of electron waiting time distributions in periodically driven quantum conductors. We employ a second-quantized formulation that allows us to relate the waiting time distribution to the Floquet scattering matrix of the system. As an application we evaluate the electron waiting times for a quantum point contact, modulating either the applied voltage (external driving) or the transmission probability (internal driving) periodically in time. Lorentzianshaped voltage pulses are of particular interest as they lead to the emission of clean single-particle excitations as recently demonstrated experimentally. The distributions of waiting times provide us with a detailed characterization of the dynamical properties of the quantum-coherent conductor in addition to what can be obtained from the shot noise or the full counting statistics.Introduction.-A surge of interest in dynamic quantum conductors has recently led to a number of groundbreaking experiments [1][2][3][4][5][6]. An on-demand coherent single-electron source based on a submicron capacitor [1,7] has been experimentally realized and successfully operated in the gigahertz regime [2]. Recently, the fermionic analogue of an optical Hong-Ou-Mandel experiment was performed to demonstrate that two such on-demand sources produce indistinguishable electronic quantum states [3]. Additionally, clean single-particle excitations have been created on top of a Fermi sea by applying a periodic sequence of Lorentzian-shaped voltage pulses to an electrical contact [4,5] following a pioneering theoretical proposal by Levitov and co-workers [8][9][10].These experimental breakthroughs hold promises for future gigahertz quantum electronics with precisely synchronized single-particle operations. One may envision circuit architectures with driven single-electron emitters coupled to the edge states of a quantum Hall conductor (or to the helical edge states in a topological insulator [11,12]) serving as rail tracks for charge and information carriers by guiding them to beam splitters (quantum point contacts) and particle interferometers for further processing. To facilitate progress towards this goal, a detailed understanding of the single-particle emitters and their statistical properties is required.In one approach, the full counting statistics of emitted charge is analyzed [13][14][15][16][17]. The charge fluctuations are typically integrated over many periods of the driving and important short-time physics may be lost. In a complementary approach, one considers the distribution of waiting times between charge carriers [18][19][20][21][22][23][24][25]. This view on quantum transport seems promising as picosecond single-electron detection is now becoming feasible [6]. A quantum theory of electron waiting times has recently been developed for voltage-biased mesoscopic conductors [20], however, so far without an explicit driving. To describe the statistical properties of coherent single-electron emitters, a theory of waiting time distributions (WTD)...

Waiting time distribution of quantum electronic transport in the transient regime

“…In the case of ac bias, averaging over a period was carried out so that WTD depends only on t again. 21 For the transient process, time translational symmetry is broken and there is no time periodicity either in the transport. Hence WTD depends on two time indices.…”

“…Recently a scattering quantum theory for WTD was formulated by Albert et al that can be used to study WTD of dc electronic quantum transport 19,20 and has been extended to steady state ac regime. 21 Despite of the success of this quantum theory, there are many open questions remained to be answered. For instance, since this quantum theory is applicable only at zero temperature and can not be used for transient dynamics, it is clearly desirable to develop a new theory at finite temperatures and in the transient regime so that the fluctuation theory 22 can be discussed and switching dynamics can be studied.…”

“…[2][3][4] The theory of GF for current was generalized to a general quantum mechanical variable by Nazarov and Kindermann, 23 and was extended to study short time behavior of dc and ac current using the wavepacket formalism. 19,21,24 In this paper, we develop a theoretical formalism for WTD for coherent conductors. Using the non-equilibrium Green's function 25,26 and path integral method in the two-time quantum measurement scheme 27 , we obtain GF for the electron transport system which allows us to study FCS and WTD in dc, ac, and transient regimes.…”

Full-counting statistics of charge and spin transport in the transient regime: A nonequilibrium Green's function approach

“…In the presence of ac bias, averaging over a time period is needed so that WTD depends only on t as well. 16 However, in the transient transport regime, time translational invariance does not exist and there is also no time periodicity like the ac case. As discussed in details in Ref.…”

Electron waiting times of a periodically driven single-electron turnstile

“…For Coulomb blockade structures such as quantum dots and metallic islands coupled to normal-state or superconducting leads, methods based on Markovian [25,26] (and nonMarkovian [27]) master equations have been developed. For coherent conductors, the distribution of electron waiting times can be obtained from a compact determinant formula containing the scattering matrix of the system [28,29]. Moreover, transient behaviors have been described using nonequilibrium Green's functions [30][31][32].…”