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

TL;DR

This paper rigorously analyzes the low-temperature behavior of charge-qubit decoherence in Luttinger liquid QPCs, revealing an interaction-dependent instability that affects quantum detection efficiency and explains experimental discrepancies.

Contribution

It proves exact mathematical statements for Luttinger liquid Hamiltonians and demonstrates how electron interactions cause instability in quantum detection at low temperatures.

Findings

Decoherence and information timescales diverge at low temperatures.

Quantum detector efficiency sharply decreases near critical interaction strength.

Interaction-dependent instability explains experimental decoherence time mismatches.

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. It is shown, that in low-temperature limit 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 I prove two general mathematical statements (S-theorem and S-lemma) about exact re-exponentiation of Keldysh-contour ordered T-exponent for arbitrary Luttinger liquid tunnel Hamiltonian. As the result, decoherence- and acquisition of information time-scales 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 1D leads of QPC. Discovered abrupt decrease of QPC quantum detector efficiency $ Q $ with the increase of $ g $ in the close vicinity of value $ g_{cr}(T) $ represents a fingerprint of interaction-dependent instability of all the quantum detection procedure for any Luttinger liquid QPC quantum detector at definite low enough temperatures $ T_{cr}(g) $. The reasons behind these effects are discussed. Also, it is shown that such the low-temperature detection instability effect is able to explain a large unclear mismatch between expected and observed decoherence timescales in two well-known experiments (J.Gorman, D.G.Hasko, D.A.Williams, Phys.Rev.Lett., 95, 090502 (2005); K.D.Petersson, J.R.Petta, H.Lu, A.C.Gossard, Phys.Rev.Lett., 105, 246804 (2010);) on charge-qubit quantum dynamics.