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

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 $g\approx g_{cr}\left(T\right)$ of Luttinger liquid correlation parameter $g$ ($0<g\le 1$) 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 $T_{cr}\left(g\right)$. 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.

Figure 1

Schematic picture of the model setup similar to ones used in the experiments of Refs. [24, 25]: charge -qubit (double quantum dot or DQD with one excess electron on it) electrostatically interacting with biased QPC (which serves as the current-carrying quantum detector and a source of decoherence for the former) and three controlling gates (which independently modulate charge-qubit evolution). Panel (a) depicts the most common case of 3D quantum wires with noninteracting electrons at $g=1$ (Fermi liquid leads) in the role of QPC electrodes, while in panel (b) one can see Luttinger liquid realization of the same setup with 1D quantum wires in the role of QPC leads of interacting electrons with $g=1/3$. It will be shown in the text that low-temperature decoherence and quantum detection in systems of such type are governed by quantum states of DQD excess electrons “dressed” into Kondo-like polaronic “clouds” of virtual plasmons from QPC. Translucent red dashed ellipses around the DQD and the QPC tunnel contact in panels (a) and (b) correspond to such “dressing”: These ellipses depict different numbers of polaronic Kondo-like clouds (or “decoherence clouds”) of virtual plasmons in different cases of charge fractionalization in QPC electrodes: $g=1$ [one Kondo-like cloud in panel (a)] and $g=1/3$ [three identical Kondo-like clouds in panel (b)].

Figure 2

Low-temperature quantum detector efficiency $Q$ as the function of Luttinger liquid correlation parameter $g$ according to exact Eq. (17) is plotted for different bias voltages in two temperature regimes. In Fig. 2 for all curves the temperature is equal to $T={10}^{-6}\mathrm{\Lambda }$ (here $\mathrm{\Lambda }$ is the high-energy cutoff of the order of the Fermi energy in the leads of QPC) while the difference $eV$between chemical potentials of QPC leads ($V$ is bias voltage) varies from 2 times smaller (the lowest curve) to 8 times bigger (the top curve) than temperature $T$ (i.e., $eV$ are equal to $0.5T;T;2T;4T;6T;8T$ from the bottom to the top curve). In Fig. 2, all curves are for the case where $T={10}^{-12}\mathrm{\Lambda }$ (i.e., corresponding plots, in fact, model a zero-temperature limit of the theory) while voltages in Fig. 2 relate to the temperature in the same way as ones in Fig. 2. For both cases [Figs. 2 and 2], I put $\mathrm{\Delta }\lambda =0.2\mathrm{\Lambda }$ and $\mathrm{\Delta }\tilde{\lambda }=0.02\mathrm{\Lambda }$. Remarkably, in Fig. 2 at $T\to 0$ one can observe a well-defined threshold value $g_{cr}$ of Luttinger liquid correlation parameter, which is approximately equal to $0.57$ (for chosen parameters of the model). The abruptness of a step on the plot $Q\left(g\right)$ around the critical value $g_{cr}$ signals that a quality of quantum detection procedure is very unstable against small variations of local electron-electron interaction in QPC electrodes around the respective critical magnitude $U_{cr}$ of local electron-electron interactions in QPC leads, corresponding to situation $g\approx g_{cr}$.