Driven quantum circuits and conductors: A unifying perturbative approach

We develop and exploit an out-of-equilibrium theory, valid in arbitrary dimensions, which does not require initial thermalization. It is perturbative with respect to a weak time-dependent (TD) Hamiltonian term, but is nonperturbative with respect to strong coupling to an electromagnetic environment or to Coulomb or superconducting correlations. We derive unifying relations between the current generated by coherent radiation or statistical mixture of radiations, superimposed on a dc voltage $V_{dc}$, and the out-of-equilibrium dc current which encodes the effects of interactions. We extend fully the lateral band-transmission picture, thus quantum superposition, to coherent many-body correlated states. This provides methods for a determination of the carrier's charge $q$ free from unknown parameters through the robustness of the Josephson-like frequency. We have derived similar relations for noise (I. Safi, arXiv:1401.5950) which have been exploited, recently, to determine the fractional charge in the fractional quantum Hall effect (FQHE) within the Jain series [M. Kapfer et al., Science (to be published)]. The present theory allows for breakdown of inversion symmetry and for asymmetric rates for emission and absorption of radiations. This generates rectification exploited here to propose methods to measure the charge $q$, as well as spectroscopical analysis of the out-of-equilibrium dc current and the third cumulant of non-Gaussian source of noise. We also apply the theory to the Tomonaga-Luttinger liquid (TLL), showing a counterintuitive feature: A Lorentzian pulse superimposed on $V_{dc}$ can reduce the current compared to its dc value, at the same $V_{dc}$, questioning the terminology “photoassisted.” Beyond a charge current, the theory applies to operators such as spin current in the spin Hall effect or voltage drop across a phase-slip Josephson junction.

Figure 1

Family of quantum circuits to which the perturbative theory might apply. A tunneling junction between similar or different electrodes with strong internal and mutual Coulomb interactions, coupled strongly to an electromagnetic environment which could possibly include another quantum conductor. The tunneling amplitude $\mathrm{\Gamma }\left(t\right)$ depends on time, as well as, possibly, a local magnetic field, gate voltages, or voltages across the conductor or the environment, all being encoded into an effective complex function $f\left(t\right)$ in Eq. (2). The box can refer to a Josephson junction (JJ) with a weak Josephson energy or to a phase-slip JJ.

Figure 2

A quantum point contact (QPC) in the FQHE at $\nu =2/5$. This corresponds to the first Jain filling factor series, $\nu =p/(2p+1)$ with integer number $p$, where one expects $q=e/(2p+1)$. To experimentally determine $q$, [1] a RF excitation at frequency $f$ is added to the dc voltage $V_{dc}$ applied to contact (0). This contact injects carriers at the bulk filling factor ${\nu }_B=2/5$ into two chiral fractional edge modes (red lines). The carriers are partitioned by the QPC into transmitted and reflected current measured at contacts (1) and (2), from which the current fluctuations are recorded and their cross-correlation performed. The charge $q=e/3$ has been determined through the Josephson relation $f_J=q{V}_{dc}/\hslash$. The charge $q=e/5$ has been also measured for a weak backscattering of the inner edge. The charge determination is based on the robust relation for the photoassisted noise within our perturbative theory [2], the first to address photoassisted transport beyond the TLL description of abrupt edges and simple fractions. As it is independent on the precise model for the edges, all series of filling factors, mutual Coulomb interactions, inhomogeneities at the QPC, or edge reconstruction could in principle be also addressed.

Figure 3

Weak impurity in a TLL with $K=1/3,3/4$, subject to a single Lorentzian pulse with width $2{\tau }_1$ superimposed on a dc drive ${\omega }_J=q{V}_{dc}/\hslash$. The backscattering current is enhanced (resp. reduced) for $K>1/2$ (resp. $K<1/2$).

Figure 4

The strong backscattering regime (or a tunneling barrier) in a TLL with $K=1/3,3/4$, subject to a Lorentzian pulse with width $2{\tau }_1$ superimposed on a dc drive ${\omega }_J=q{V}_{dc}/\hslash$. The difference between the induced current $I_f\left({\omega }_J\right)$ and the dc current $I_{dc}\left({\omega }_J\right)$ at the same ${\omega }_J$, renormalized by $I_{dc}({\omega }_J=1/2{\tau }_1)$, is positive $K=1/3$, and almost vanishing for $K=3/4$.