Conclusion

In this paper we have derived several expressions for the Keldysh action A for a general multi-terminal, time-dependent scatterer. This object is defined as the (logarithm of the) trace of the density matrix of the scatterer after evolution forwards and backwards in time with different Hamiltonians: eA = TrhT+exp −i Z t1 t0 dtH+(t) ρ0T−exp i Z t1 t0 dtH−(t) i. (2.90) Our main result is a compact formula for the action in terms of reservoir Green functions and the scattering matrix of the scatterer (Eq. 2.1). We have shown how to perform the trace over Keldysh indices explicitly when reservoirs are characterized by filling factors. Thus we obtained a formula (Eq. 2.2) generalizing the Levitov counting statistics formula. We have also explicitly performed the trace over channel indices for a two termi- nal scatterer (Eq. 2.4). In this case we demonstrated that the Keldysh action only depends on the scattering matrix through the eigenvalues of the transmission matrix. To illustrate the utility of the Keldysh action, and confirm the correctness of our results, we considered Full Counting statistics and the Fermi Edge singularity. We found that our results agree with the existing literature.

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Quantum tunneling detection

of two-photon and

two-electron processes

3.1

Introduction

The quantum nature of electron transfer in coherent conductors is seldom explicitly manifested in averaged current-voltage curves. To reveal it one should measure current noise and/or the higher-order correlations of cur- rent comprising the Full Counting Statistics that arise from the transfer [1]. Such measurements not only reveal the discrete nature of the charges trans- ferred, they also quantify quantum many-body effects in electron transport and may be used for the detection of pairwise entanglement of transferred particles [2, 3, 4, 5]. If the noise is measured at frequencies in the quantum range, ~_{ω ≫kBT}, the measurement amounts to the detection of photons

produced by the current fluctuations. This aspect is important in view of attempts to transfer quantum information from electrons to photons and back [6].

It was demonstrated in Ref. [7] that one needs a quantum detector to measure quantum noise. Indeed, any classical measurement of a fluc- tuating quantity would give a noise spectrum symmetric in frequency,

S(ω) = S(−ω). A quantum tunneling detector is generally a quantum

two-level system with a level separation ε >0. The result of detection are

two transition rates: Γup from the lower to the higher level and Γdown for

either absorption or emission of a photon of matching energy~_ω=ε. One

can define the noise spectrum in such a way that it is proportional to the transition ratesS(_±ε/~_{)∝Γup,down(ε)}. Differences between_Γup,down thus

manifest the quantum nature of noise. If the source of noise is a coherent conductor biased by a voltage V, detector signals in the range ε < eV

are readily interpreted in terms of single electron transfers through the conductor. The maximum energy gain available for electrons in the course of such transfer is eV. Consequently this value also limits the energy of

the emitted photon. The previous research has not addressed the energy rangeε > eV. It remained unclear if the detector detects anything and — if it does — what it detects.

A first proposal for the experimental realization of a quantum tunneling detector included transitions between two localized electron states in a double quantum dot [8]. However it does not matter much if the tunneling occurs between localized or delocalized electron states and if all tunnel events are accompanied by the same energy transfer ε. In most practical cases the energy dependence of the ratesΓup,down can be readily extracted

from the measurement results. This is why quantum tunneling detection has been experimentally realized in a superconducting tunnel junction [9] and in a single quantum dot [10].

In this Chapter we study quantum tunneling detection in the range

eV < ε <2eV assumingε, eV _≫kBT. The motivation is that for these en-

ergies the detector is not sensitive to single-electron one-photon processes described above and its output — the transition rateΓup— is determined

by much more interesting two-particle processes. It is clear from plain energy considerations that transitions may originate fromtwo-photonpro-

cesses. Such two-photon absorption can occur given any non-equilibrium photon distribution bounded by eV, not necessarily produced by a co- herent conductor. Less obvious and specific for a coherent conductor is a cooperative two-electron process. Indeed, if two electrons team up in

crossing the conductor they can emit a single photon with an energy up to

2eV. Essential for this cooperation are electron-electron interactions. It is known [11] that the most important electron-electron interaction in this energy range is due to the electromagnetic environment of the conductor, the same environment in which the non-equilibrium photons dwell.

We quantify the signals due to two-photon and two-electron events and find them to be of the same order of magnitude. We also show that part of the signal is due to quantum interference of these two processes: We

e

A

T

Z

α

V

ε

Figure 3.1. Model circuit: A coherent conductor with transmission eigenvalues

Tn is connected in series with an external impedanceZ. This structure is biased

with a voltage V. The shared node A is capacitively coupled to a quantum tunneling detector. The coupling constant is α. The tunneling detector consists of an electron that can tunnel between the lowest two states of a double quantum dot. The level splittingεof this two level system can be tuned by a gate voltage.

demonstrate how different contributions can be separated in experiments thereby facilitating the direct observation of two-particle processes in the context of quantum transport.