Abstract
We present a theory of frequency-dependent counting statistics of electron transport through nanostructures within the framework of Markovian quantum master equations. Our method allows the calculation of finite-frequency current cumulants of arbitrary order, as we explicitly show for the second- and third-order cumulants. Our formulae generalize previous zero-frequency expressions in the literature and can be viewed as an extension of MacDonald's formula beyond shot noise. When combined with an appropriate treatment of tunneling using, e.g., the Liouvillian perturbation theory in Laplace space, our method can deal with arbitrary bias voltages and frequencies, as we illustrate with the paradigmatic example of transport through a single resonant level model. We discuss various interesting limits, including the recovery of the fluctuation-dissipation theorem near linear response, as well as some drawbacks inherent to the Markovian description arising from the neglect of quantum fluctuations.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Correlations of the current through mesoscopic conductors are of fundamental importance, as they can reveal information about the internal quantum dynamics of the conductor and electron–electron interactions. We present a theory of frequency-dependent current correlations in the context of quantum master equations. The formalism allows for the calculation of the Nth-order current cumulant at finite frequencies.
Main results. We derive equations for the frequency-dependent noise and skewness spectra of the current distribution and for the cumulant generating function. Our formalism assumes a Markovian approximation and is exemplified to second order in perturbation theory in the system–reservoir coupling. A method to calculate charge/voltage correlations is also discussed. To illustrate the method, we analyze the example of a single resonant level model. We identify the regions of applicability of the theory by comparison with the exact solution and studying the equilibrium and non-equilibrium fluctuation-dissipation theorem. In particular, we show that the Markovian description includes shot and thermal fluctuations but does not capture the physics of quantum noise, for which a non-Markovian theory is needed.
Wider implications. Markovian quantum master equations are broadly used in the literature. We present a Markovian theory of finite-frequency counting statistics of electron transport. This extends previous studies at zero frequency and we expect new investigations using our skewness formula. Also, the equation for the noise spectrum should be seen as a potential alternative to MacDonald's formula.
Figure. Noise near linear response (eV/kT = 0.0005) as a function of frequency. For comparison we also show the non-equilibrium fluctuation-dissipation theorem (NEFDT) and the exact solution. S(2)(ω) is flat for the whole range of frequencies, and coincides with the equilibrium FDT as expected. The NEFDT, however, disagrees with these two, showing also quantum fluctuations, which are absent in the Markovian noise spectrum. The quantum noise steps shown by the NEFDT are, however, at 
ω = 2
, in contrast to the exact solution, which shows steps at ω = . This is due to the fact that the NEFDT works well for tunnel junctions, but does not capture partition noise. This becomes clear also from the saturation value at large frequencies, as described in the text. Parameters: hΓL/kT = hΓR/kT = 0.05. The inset compares the exact solution with the Markovian approximation for a different regime, namely eV/kT = 25. We see that while the Markovian limit is flat for all frequencies, the exact solution presents a dip at ω = ±|ε±eV/2|. Rest of parameters: = 0, hΓL/kT = hΓR/kT = 0.25.