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arXiv:1712.09561 · Physical review. B./Physical review. B

Theory of interaction-dependent instability in quantum detection by means of a Luttinger liquid tunnel junction: A rigorous theorem

Gleb A. Skorobagatko

Published Jul 10, 2018• 3 citations
arXivHTMLPDF

Abstract

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 $g<{g}_{cr}$ and the region of quantum detection breakdown for $g>{g}_{cr}$. Moreover, this abrupt decrease of QPC quantum detector efficiency $Q$ with the increase of $g$ in the close vicinity of value ${g}_{cr}$ represents evidence of interaction-dependent instability of all the quantum detection procedures for any Luttinger liquid QPC quantum detector at sufficiently low temperatures ${T}_{cr}(g)$. 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.

DOI
10.1103/physrevb.98.045409

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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 TTT 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≈gcr(T)g\ensuremath{\approx}{g}_{cr}(T)g\ensuremath≈gcr​(T) of Luttinger liquid correlation parameter ggg (0<g\ensuremath≤10<g\ensuremath{\le}10<g\ensuremath≤1) which serves as a sharp boundary between the region of good (or even perfect) quantum detection at g<gcrg<{g}_{cr}g<gcr​ and the region of quantum detection breakdown for g>gcrg>{g}_{cr}g>gcr​. Moreover, this abrupt decrease of QPC quantum detector efficiency QQQ with the increase of ggg in the close vicinity of value gcr{g}_{cr}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){T}_{cr}(g)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.
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} $.
We consider the effect of electron-electron interactions on a voltage biased quantum point contact in the tunneling regime used as a detector of a nearby qubit. We model the leads of the quantum point contact as Luttinger liquids, incorporate the effects of finite temperature and analyze the detection-induced decoherence rate and the detector efficiency, QQQ. We find that interactions generically reduce the induced decoherence along with the detector's efficiency, and strongly affect the relative strength of the decoherence induced by tunneling and that induced by interactions with the local density. With increasing interaction strength, the regime of quantum-limited detection ($Q\ensuremath{\rightarrow}1$) is shifted to increasingly lower temperatures or higher bias voltages respectively. For small to moderate interaction strengths, QQQ is a monotonously decreasing function of temperature as in the noninteracting case. Surprisingly, for sufficiently strong interactions we identify an intermediate temperature regime where the efficiency of the detector increases with rising temperature.
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.
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.
In this Ph.D. thesis I present the background and providefurther details to the work presented in the appended papers. The background consists of brief overviews of Kondo physics, Luttinger liquid theory, and conformal field theory. My own research is divided into three parts: the first deals with the interplay between Luttinger and Kondo physics (described in Papers I and II), the secondaddresses scanning tunneling microscopy (STM) response of a Luttinger liquid with an edge or an impurity (described in Paper III), and the third explores the boundary Green's function of 1D electrons in the spin-incoherent regime (described in Paper IV). In Papers I and II a setup is proposed which allows for acontrolled study of Kondo and Luttinger liquid physics. Itconsists of a quantum box, biased by a gate voltage, and side-coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using boundary conformal field theory techniques we predict that for the case of spin-polarized electrons the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage. In the limit of zero magnetic field the Luttinger liquid behavior gets masked by two-channel Kondo screening, leading to a logarithmic scaling of the differential capacitance with temperatureand gate voltage. These effects should be possible to studyexperimentally, using the recently developed single-electrontransistor (SET) electrometer technique.In Paper III the finite-temperature local spectral weight (LSW) of a Luttinger liquid with a hard wall boundary is calculated. Close to the boundary the LSW exhibits characteristic oscillations indicative of spin-charge separation. The line shape of the LSW is also found to have a Fano-like asymmetry, a feature originating from the combined effect of electron-electron interaction and scattering off the boundary. Our results can be used to predict how edges and impurities influencethe STM response of one-dimensional electron systemsat low temperatures and voltage bias. Applications to STM onsingle-walled carbon nanotubes are also discussed.Paper IV, finally, presents a study of the spin-incoherent regime of 1D strongly interacting electrons. For sufficiently low densities the potential energy dominates the kinetic energy, and one can easily reach thespin-incoherent regime where the spin exchange energy is much less than the temperature. We have generalized the description of the spin-incoherent regime to account for the presence of a boundary. By calculating the exact Green's function we find that the charge sector critical exponent is highly sensitive to the boundary, strongly modifying the tunneling of electrons close to it. Our approach also allows for a detailed description of the crossover between boundary and bulk regimes.
Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.