Minimal excitation states for heat transport in driven quantum Hall systems

We investigate minimal excitation states for heat transport into a fractional quantum Hall system driven out of equilibrium by means of time-periodic voltage pulses. A quantum point contact allows for tunneling of fractional quasiparticles between opposite edge states, thus acting as a beam splitter in the framework of the electron quantum optics. Excitations are then studied through heat and mixed noise generated by the random partitioning at the barrier. It is shown that levitons, the single-particle excitations of a filled Fermi sea recently observed in experiments, represent the cleanest states for heat transport since excess heat and mixed shot noise both vanish only when Lorentzian voltage pulses carrying integer electric charge are applied to the conductor. This happens in the integer quantum Hall regime and for Laughlin fractional states as well, with no influence of fractional physics on the conditions for clean energy pulses. In addition, we demonstrate the robustness of such excitations to the overlap of Lorentzian wave packets. Even though mixed and heat noise have nonlinear dependence on the voltage bias, and despite the noninteger power-law behavior arising from the fractional quantum Hall physics, an arbitrary superposition of levitons always generates minimal excitation states.

Figure 1

Fractional quantum Hall liquid in a four-terminal setup. Two gate voltages create a quantum point contact at $x=0$, allowing for tunneling between opposite edges. A periodic bias $V\left(t\right)$ is applied to contact 1, while contacts 3 and 4 are grounded. Backscattered currents and their fluctuations are detected in contact 2.

Figure 2

Excess mixed noise $-\mathrm{\Delta }{\mathcal{S}}_X$ as a function of the charge per period $q$ at zero temperature. The high-energy cutoff is set to ${\omega }_{\mathrm{c}}=10\omega$. Behavior for Lorentzian pulses (full black line) and sinusoidal voltage drive (dashed red line) is reported.

Figure 3

Excess heat noise $\mathrm{\Delta }{\mathcal{S}}_Q$ as a function of the charge per period $q$. Full black and dashed red lines represent Lorentzian and sinusoidal drives, respectively. The temperature is $\theta =0$ and the cutoff is ${\omega }_{\mathrm{c}}=10\omega$.

Figure 4

Time-periodic voltage drive given by Eq. (43) in the case of $N=2$ Lorentzian-shaped pulses per period at total charge $q=1$ (i.e., $\frac{1}{2}$ for each pulse). The top panel represents two completely overlapping pulses ($\alpha =0$), for which we simply have $V_2\left(t\right)=2\tilde{V}\left(t\right)$. The central and bottom panels correspond to nontrivial cases $\alpha =0.45$ and 0.9 with finite overlap between pulses. In all cases, the behavior of individual Lorentzian pulses $\tilde{V}\left(t\right)$ and $\tilde{V}\left(t-\frac{\alpha }{N}T\right)$ are depicted with dashed, thin lines.

Figure 5

Excess signals $-\mathrm{\Delta }{\mathcal{S}}_X$ (top panel) and $\mathrm{\Delta }{\mathcal{S}}_Q$ (bottom panel) as a function of $q$ for a cluster of two identical Lorentzian pulses separated by a time delay $\alpha T/2$. All curves refer to the case of $\nu =1$ and zero temperature. The cutoff is set to ${\omega }_{\mathrm{c}}=10\omega$.

Figure 6

Excess signals $-\mathrm{\Delta }{\mathcal{S}}_X$ and $\mathrm{\Delta }{\mathcal{S}}_Q$ as a function of $q$ for two identical Lorentzian pulses with time delay $\alpha T/2$ at fractional filling $\nu =\frac{1}{3}$ and zero temperature. The cutoff is set to ${\omega }_{\mathrm{c}}=10\omega$.

Figure 7

Excess mixed noise $-\mathrm{\Delta }{\mathcal{S}}_X$ as a function of $q$ for a cluster of two Lorentzian pulses. Here, $q$ is partitioned asymmetrically, with the two pulses carrying, respectively, $\frac{1}{3}$ and $\frac{2}{3}$ of the total charge per period. One should compare this figure with Figs. 5 and 6, where $-\mathrm{\Delta }{\mathcal{S}}_X$ for identical pulses is plotted. The time delay between pulses is $\alpha T/2$, with different values of $\alpha$ according to the legend. The cutoff is set to ${\omega }_{\mathrm{c}}=10\omega$ and the temperature is $\theta =0$.