Strongly correlated transport of interacting electrons through the one-dimensional tunnel contact is considered within the Luttinger liquid model of one-dimensional electrodes on arbitrary time scales t>ħ/Λg. Using previous results, namely, the Self-equilibration theorem proven in Ref. [20] the exact distribution functions P(N,t) for the probabilities of N electrons to tunnel through the contact by the time t are derived and plotted as 3D intensity graphs called real-time tunneling maps in a whole available region of Luttinger liquid correlation parameter values, bias voltages, and temperatures. The tunneling maps being obtained reveal tunneling anomalies - unusually high tunneling probabilities for the small numbers of electrons on the short-time scales beyond the steady state regime of electron tunnel transport. Especially, this concerns the case of strong short-range electron-electron interaction in the leads at g≲1/2. The effect being found can be treated as the direct consequence of the self-equilibration phenomenon. It is shown that features of such an anomalous electron tunneling beyond the steady state regime can be used for precise experimental measurements of small differences between Luttinger liquid correlation parameter values in different one-dimensional quantum-point contacts.
Equilibration to the state for a wide class of Luttinger liquid ballistic weakly linked tunnel contacts is extensively studied. Quantum fluctuations of tunnel current are considered in all orders in tunnel coupling and out of the equilibrium in the time domain. Especially, two important mathematical statements: Self-equilibration (SE-)theorem and Self-equilibration (SE-)lemma on the exact re-exponentiation of thermal average from the Keldysh-contour-ordered evolution operator for arbitrary weakly linked Luttinger liquid tunnel contact are proven. Demonstrated proof of SE-theorem and SE-lemma represents first evidence of a novel emergent phenomenon of in the dynamics of quantum fluctuations of electron transport through ballistic tunnel junctions. This phenomenon and all related real-time full counting statistics are shown to be much more general as compared to the usual Levitov-Lesovik scattering approach for the non-interacting electrons, though SE-theorem also contains known results obtained within the Levitov-Lesovik scattering approach as corresponding limiting case for lowest order cumulants at $ g=1 $. As the result, corresponding differential equation of for time-dependent Keldysh partition function of tunnel contact is derived and explained. As the consequence of obtained results, a universal character of self-equilibration of tunnel current in one-dimensional weakly linked tunnel junctions is revealed and studied on the level of non-equilibrium Fano factor. As well, a new measure of disequilibrium in such systems - the steady flow rate - is also introduced and discussed.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
The low-temperature regime of charge-qubit decoherence due to its Coulomb interaction with electrons tunneling through Luttinger liquid quantum-point contact (QPC) is investigated. The study is focused on quantum detector properties of Luttinger liquid QPC. Earlier results on related problems were approximate, up to the second order in small electrostatic coupling between charge qubit and QPC. However, here it is shown that in low- (and zero-) temperature limits the respective perturbative decoherence and acquisition of information timescales both tend to diverge, thus shadowing a true picture of low-temperature quantum detection for such quantum systems. Here it is shown that one can successfully circumvent these difficulties in order to restore a complete and exact picture of low-temperature decoherence and quantum detection for charge qubits measured by arbitrary Luttinger liquid QPC. To do this, here I prove two general mathematical statements [summation (S) theorem and (S) lemma] about exact re-exponentiation of Keldysh-contour ordered
T exponent for an arbitrary Luttinger liquid tunnel Hamiltonian. The resulting exact formulas are believed to be important in a wide range of those Luttinger liquid problems, where real-time quantum field dynamic is crucial. As the result, decoherence and acquisition of information timescales as well as QPC quantum detector efficiency rate are calculated exactly and are shown to have a dramatic dependence on repulsive interaction between electrons in one-dimensional (1D) leads of QPC. In particular, it is found that at temperatures close to zero there exists a certain well-defined threshold value $g\ensuremath{\approx}{g}_{cr}(T)$ of Luttinger liquid correlation parameter
g ($0<g\ensuremath{\le}1$) which serves as a sharp boundary between the region of good (or even perfect) quantum detection at
g<gcr and the region of quantum detection breakdown for
g>gcr. Moreover, this abrupt decrease of QPC quantum detector efficiency
Q with the increase of
g in the close vicinity of value
gcr represents evidence of interaction-dependent instability of all the quantum detection procedures for any Luttinger liquid QPC quantum detector at sufficiently low temperatures
Tcr(g). The reasons behind these effects are discussed. Also, it is shown that the low-temperature detection instability effect is able to explain the large mismatch between expected and observed decoherence timescales in two recent experiments [Gorman et al., Phys. Rev. Lett. 95, 090502 (2005); Petersson et al., Phys. Rev. Lett. 105, 246804 (2010)] on charge-qubit quantum dynamics.
We study how electron-electron interactions renormalize tunneling into a Luttinger liquid beyond the lowest order of perturbation in the tunneling amplitude. We find that the conventional fixed point has a finite basin of attraction only in the point contact model, but a finite size of the contact makes it generically unstable to the tunneling-induced breakup of the liquid into two independent parts. In the course of renormalization to the nonperturbative-in-tunneling fixed point, the tunneling conductance may show a nonmonotonic behavior with temperature or bias voltage.
Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.
The low-temperature limit of charge qubit decoherence due to its Coulomb interaction with electrons tunneling through Luttinger liquid quantum-point contact (QPC) is investigated. The study is focused on quantum detector properties of Luttinger liquid QPC. Efficiency of Luttinger liquid qurrent-carrying quantum point-contact (QPC) as quantum detector for the coupled charge qubit quantum state is characterized by the ratio of decoherence time and the acquisition of information time and how both depend on electron-electron interaction has remained an interesting open question. Here this question is answered in its most challenging zero- (and near zero-) temperature realization where the latter two characteristic time-scales, both tends to zero shadowing true picture of quantum detection in the zero-temperature limit. In order to resolve this difficulty two important mathematical statements (S-theorem and S-lemma) have been proven here for the first time. As the result, it was established here that low-temperature properties of QPC quantum detector with 1D electrodes are defined explicitly by the competition between two different time-scales those reflecting two different types of physical processes in the system. These processes are: i) the orthogonality catastrophe-type decoherence of charge-qubit, one is analogous to the decay of Kondo-resonance narrowed by QPC plasmons; ii) Kane-Fisher suppression of the electron tunneling through Luttinger liquid QPC resulting in the divergence of electron tunneling time. In particular, it has been found that at temperatures close to zero there exists a certain well-defined threshold value $ g=g_{cr} $ of Luttinger liquid correlation parameter which serves as a sharp boundary between region of good (or even perfect) quantum detection at $ g<g_{cr} $ and the region of quantum detector break down where $ g>g_{cr} $.