Single-electron source: Adiabatic versus nonadiabatic emission

We investigate adiabatic and nonadiabatic emission of single particles into an edge state using an analytically solvable dynamical scattering matrix model of an on-demand source. We compare adiabatic and nonadiabatic emissions by considering two geometries: a collider geometry where two emitters are coupled to two different edge states and a series geometry where two emitters are coupled to the same edge state. Most effects observed for adiabatic emitters also occur for nonadiabatic emitters. In particular this applies to effects arising due to the overlap of wave packets colliding at a quantum point contact. Specifically we compare the Pauli peak (the fermionic analog of the bosonic Hong-Ou-Mandel dip) for the adiabatic and nonadiabatic collider and find them to be similar. In contrast we find a striking difference between the two operating conditions in the series geometry in which particles are emitted into the same edge state. Whereas the squared average charge current can be nullified for both operating conditions, the heat current can be made to vanish only with adiabatic emitters.

Figure 1

Single-electron sources (SES) emit a train of electrons (black pulses) alternating with holes (red pulses) into an edge states (serving as an electronic waveguide). Electron and hole pulses are well separated in time and space. The shape of the single-particle wave packets depends crucially on the way the source is driven: (a) Adiabatic emission: the SES is driven by a smooth periodic potential, the pulse as a function of time has a Lorentzian shape with width $2{\Gamma }_{\tau }$. (b) Nonadiabatic emission: the SES is driven by a pulsed periodic potential, the pulse as a function of time has an exponential shape characterized by the dwell time ${\tau }_D$.

Figure 2

The two-particle emitter consists of two cavities specified by same parameters and coupled to the same chiral edge state. Electrons propagate along edge states shown as blue solid lines. Cavity $B$ is tuned to emit a hole at the time when the electron emitted by cavity $A$ reaches cavity $B$. (a) Adiabatic emission: when the driven potential is slow and smooth, an electron emitted by the cavity A is completely reabsorbed by cavity $B$. The re-absorption process is the time-reversed emission process which is possible because of the symmetric shape of the single-particle states. Both the charge current $I\left(t\right)$ and the dc heat flow $I_{dc}^Q$ nullify. (b) Nonadiabatic emission: the cavities are driven by pulsed potentials and electron-hole pairs are emitted. Since this pair is neutral, the time-dependent current is zero, $I\left(t\right)=0$. However, both the electron and the hole carry energy. Because of the asymmetric shape of the pulses reabsorption cannot be a time-reversed emission process. There is no absorption effect, the generated dc heat flow is not zero: $I_{dc}^Q>0$.

Figure 3

A mesoscopic electron collider. Two single-electron sources, $A$ and $B$, driven by pulsed potentials emit electrons into the waveguides. The emitted electrons collide at quantum point contact $C$. The metallic reservoirs are labeled by number from 1 to 4.

Figure 4

The Pauli peak: The shot noise per particle of an electron collider, $\mathcal{P}={\mathcal{P}}_{34}/\left(2N{\mathcal{P}}_0\right)$, as a function of the time delay $\delta t\equiv t_{-}^A-t_{-}^B=t_{+}^A-t_{+}^B$ normalized by ${\Gamma }_{\tau }$ for an adiabatic emission, Eq. (19) (red dashed line), and by ${\tau }_D$ for a nonadiabatic emission, Eq. (18) (black solid line).

Figure 5

Adiabatic emission: The real part of $S(t,E)$, Eq. (A5a), is shown. The time $t$ is measured in units of the period of the drive $\mathcal{T}=2\pi /\Omega$. The energy $E$ is measured from the Fermi energy $\mu$ in units of the level spacing $\Delta$. Only one period for both $t$ and $E$ is shown. The transmission probability of the quantum point contact connecting the cavity is $T=0.5$. Other parameters correspond to the optimal operating conditions.

Figure 6

Nonadiabatic emission: The real part of $\delta S_{\mathrm{in}}(t,E)$, Eq. (A16a), with $\chi =-$, is shown close to the time of emission of an electron, $t\sim t_{-}$. The time $t$ is measured in units of the dwell time ${\tau }_D=\tau /T$. The energy $E$ is measured from the Fermi energy $\mu$ in units of the level spacing $\Delta$. Only one period for $E$ is shown. The visible ripples reflect oscillations in time with the period of $\tau$. The transmission probability of the quantum point contact connecting the cavity is $T=0.5$. Other parameters correspond to the optimal operating conditions.