Energy and power fluctuations in ac-driven coherent conductors

Using a scattering matrix approach we study transport in coherent conductors driven by a time-periodic bias voltage. We investigate the role of electron-like and hole-like excitations created by the driving in the energy current noise and we reconcile previous studies on charge current noise in these kinds of systems. The energy noise reveals additional features due to electron-hole correlations. These features should be observable in power fluctuations. In particular, we show results for the case of a harmonic and biharmonic driving and of Lorentzian pulses applied to a two-terminal conductor, addressing recent experiments [Gabelli and Reulet, Phys. Rev. B 87, 075403 (2013) and Dubois et al., Nature (London) 502, 659 (2013)].

Figure 1

Energy landscape of a two-terminal conductor with a central scatterer. (a) dc-biased conductor: electrons within the bias window can be transmitted through the scatterer. (b), (c) Examples of possible processes occurring in the presence of a time-dependent driving (shaded areas indicate the amplitude of the ac oscillations). In (b) an electron with energy in the bias window gets promoted above the chemical potential of the left reservoir by interacting with the ac part of the driving. The corresponding vacancy does not contribute to transport as it cannot be filled by an electron tunneling from the right reservoir (i.e., it is reflected with probability 1). In contrast, in (c) both the electron as well as the remaining hole can be transmitted through the scatterer.

Figure 2

Sketch of a possible inelastic two-particle scattering event induced by the time-dependent driving applied to the left lead. The two reservoir states at energies $E_{-\ell }$ and $E_{-k}$ are connected to the incoming states at the scatterer at energies $E$ and $E_q$ by inelastic process via two possible pairs of energy paths.

Figure 3

Plots of the interference contributions to the charge and energy noise as a function of the dc offset of the driving. (a) Line shape of the applied voltages: harmonic driving $V_{\mathrm{L}}^{\mathrm{h}}\left(t\right)=\overline{V}+V_{0}cos\left(\Omega t\right)$ and biharmonic driving of the form $V_{\mathrm{L}}^{\mathrm{bh}}\left(t\right)=\overline{V}+V_{0}cos\left(\Omega t\right)+\frac{V_0}{2}cos\left(2\Omega t\right)$. (b) Interference contributions to the charge noise $S_{\mathrm{int}}^{\left(ii\right)}$ and $S_{\mathrm{int}}^{(i\ne j)}$ as well as their sum $S_{\mathrm{int}}$. (c) Interference contributions to the energy noise $S_{\mathrm{int}}^{\mathcal{E}\left(ii\right)}$ and $S_{\mathrm{int}}^{\mathcal{E}(i\ne j)}$, as well as the total contribution of the interference terms $S_{\mathrm{int}}^{\mathcal{E}}$. In all panels, $k_{\mathrm{B}}T=0$ and $e{V}_0=2\hslash \Omega$. Full lines correspond to the case of harmonic driving, while dashed lines represent the case of biharmonic driving.

Figure 4

Plots of the transport contributions to the charge and energy noise as a function of the dc offset of the driving. In both panels, full lines correspond to the case of harmonic driving $V_{\mathrm{L}}^{\mathrm{h}}\left(t\right)=\overline{V}+V_{0}cos\left(\Omega t\right)$ and dashed lines to biharmonic driving of the form $V_{\mathrm{L}}^{\mathrm{bh}}\left(t\right)=\overline{V}+V_{0}cos\left(\Omega t\right)+\frac{V_0}{2}cos\left(2\Omega t\right)$, with $e{V}_0=2\hslash \Omega$. (a) Transport contributions to the charge noise $S_{\mathrm{tr}}^{\left(ii\right)}$ and $S_{\mathrm{tr}}={\sum }_i{S}_{\mathrm{tr}}^{\left(ii\right)}$. (b) Transport contributions to the energy noise $S_{\mathrm{tr}}^{\mathcal{E}\left(ii\right)}$ and $S_{\mathrm{tr}}^{\mathcal{E}}={\sum }_i{S}_{\mathrm{int}}^{\mathcal{E}\left(ii\right)}$. In all panels, $k_{\mathrm{B}}T=0$.

Figure 5

(a) Shape of the ac excitation $V_{\mathrm{L}}\left(t\right)=V_{0}cos\left(\Omega t\right)+\frac{V_0}{2}cos(2\Omega t+\phi )$ for different phase shifts: $\phi =0$ (dashed line), $\phi =\pi /2$ (full line), $\phi =\pi$ (dashed-dotted line). (b), (c) Plots of the transport contributions to the charge and energy noise for the type of biharmonic driving introduced in (a), as a function of the phase $\phi$ between the two harmonic components. In all panels, $e{V}_0=2\hslash \Omega$ and $k_{\mathrm{B}}T=0$.