Noise of a single-electron emitter

I analyze the correlation function of currents generated by a periodically driven quantum capacitor emitting single electrons and holes into a chiral waveguide. I compare adiabatic and nonadiabatic transient working regimes of a single-electron emitter and find a striking difference between the correlation functions in the two regimes. Quite generally, for a system driven with frequency $\Omega$ the correlation function depends on two frequencies $\omega$ and $\ell \Omega -\omega$, where $\ell$ is an integer. For an emitter driven nonadiabatically the correlation functions for different $\ell$ are similar and almost symmetric in $\omega$, while in the case of adiabatic driving the correlation functions for $\ell \ne 0$ are highly asymmetric in $\omega$ and exceed significantly the one corresponding to $\ell =0$. Under optimal operating conditions the correlation function for odd $\ell$ is zero.

Figure 1

Single-electron source (SES) emits a regular train of electrons or alternating electrons and holes (shown as dark areas) well separated in distance. The blue strip is an electron waveguide, e.g., the edge state of a two-dimensional electron gas in the integer quantum Hall effect regime. The arrow indicates the direction of movement of emitted particles.

Figure 2

Single-electron source (SES), a circular edge state driven by the periodic potential of the top gate (not shown), emits electrons and holes (shown as dark areas) into one of the edge states of a Hall bar. The arrows indicate the direction of movement of emitted particles as well as of electrons of the underlying Fermi sea in the edge states (shown as blue strips). 1 and 2 are metallic contacts. The contact 1 is grounded. The current and its fluctuations are measured at the contact 2 where the emitted particles arrive.

Figure 3

Time-dependent potential $U$ (black dashed line) driving the capacitor and the corresponding generated time-dependent current $I$ (blue solid line). The upper indices stand for adiabatic “$ad$” and nonadiabatic “$na$” regimes. The red thin line indicates the position of the Fermi level. When the cavity's level rises above the Fermi level an electron is emitted at time $t_{-}=\pi /2$ (upward current pulse). When the cavity's level sinks below the Fermi level, a hole is emitted at time $t_{+}=3\pi /2$ (downward current pulse). The potential $U$ and current $I$ are given in arbitrary units. Time $t$ is given in units of $1/\Omega$.

Figure 4

Excess noise power in the adiabatic regime, Eq. (47), in units of $2(e^2/\mathcal{T}){\left[{\tau }_D/\left(2{\Gamma }_{\tau }\right)\right]}^2$. The frequency $\omega$ is in units of $1/\left(2{\Gamma }_{\tau }\right)$. ${\tau }_D$ is the dwell time. ${\Gamma }_{\tau }$ is the half-width of an emitted current pulse. The mean energy of emitted particles $\mathcal{E}=\hslash /\left(2{\Gamma }_{\tau }\right)$ defines the frequency cutoff over which the excess noise power drops quickly. The temperature is zero.

Figure 5

Excess noise power in the nonadiabatic regime, Eq. (54), in units of $2(e^2/\mathcal{T})$. The frequency $\omega$ is in units of $\Delta /\hslash$. The noise power is given for different temperatures: $k_{B}T/\Delta =0.05$ (black solid line), $0.1$ (green dashed line), and $0.15$ (red dotted line). The thin black solid line is the noise power calculated with a zero-temperature efficiency factor, Eq. (57). The cutoff frequency at $\omega \sim 1/2$ is due to the energy $\sim$$\Delta /2$ of emitted particles. The transmission of the cavity QPC $\overline{T}=0.1$ and, correspondingly, the level width is $\overline{\delta }=0.008\Delta$. The dwell time ${\tau }_D\Delta /\hslash \approx 62.8$.

Figure 6

Photon-induced noise power in the adiabatic regime. The absolute value of the PIN power, Eqs. (62) and (63), is shown in units of $2(e^2/\mathcal{T}){\Omega }^2{\tau }_D{\Gamma }_{\tau }$ for different numbers of involved photons: $\ell =$ 2 (black solid line), 4 (red dotted line), and 6 (blue dashed line). The frequency $\omega$ is in units of the frequency of the drive, $\Omega$. ${\tau }_D$ is the dwell time. ${\Gamma }_{\tau }$ is the half-width of an emitted current pulse, which we choose here to be small such that $\Omega {\Gamma }_{\tau }\ll 1$. The temperature is zero.

Figure 7

Asymmetric part of the photon-induced noise power in the nonadiabatic regime, $\delta {\mathcal{P}}_{2\lambda }$, Eq. (72), is plotted in units of $2(e^2/\mathcal{T})$ for different numbers of involved photons: $2\lambda =$ 2 (black solid line), 4 (red dotted line), and 6 (blue dashed line). The frequency $\omega$ is in units of $\Delta /\hslash$. The driving frequency $\Omega =0.001\Delta /\hslash$. The temperature is $k_{B}T/\Delta =0.01$. Other parameters are the same as in Fig. 5.