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arXiv:cond-mat/0202102 · Physical Review Letters

Quantum Entanglement in Carbon Nanotubes

Cristina Bena B, Smitha Vishveshwara, Leon Balents, Matthew P. A. Fisher

Published Jun 26, 2002• 164 citations
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Abstract

With the surge of research in quantum information, the issue of producing entangled states has gained prominence. Here, we show that judiciously bringing together two systems of strongly interacting electrons with vastly differing ground states-the gapped BCS superconductor and the Luttinger liquid-can result in quantum entanglement. We propose three sets of measurements involving single-walled metallic carbon nanotubes which have been shown to exhibit Luttinger liquid physics, to test our claim and as nanoscience experiments of interest in and of themselves.

DOI
10.1103/physrevlett.89.037901

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With the surge of research in quantum information, the issue of producing entangled states has gained prominence. Here, we show that judiciously bringing together two systems of strongly interacting electrons with vastly differing ground states-the gapped BCS superconductor and the Luttinger liquid-can result in quantum entanglement. We propose three sets of measurements involving single-walled metallic carbon nanotubes which have been shown to exhibit Luttinger liquid physics, to test our claim and as nanoscience experiments of interest in and of themselves.
The Luttinger model of the one-dimensional Fermi gas is the cornerstone of modern understanding of interacting electrons in one dimension. In fact, the enormous class of systems whose universal behavior is adiabatically connected to it are now deemed Luttinger liquids. Recently, it has been shown that metallic single-walled carbon nanotubes are almost perfectly described by the Luttinger Hamiltonian. Indeed, strongly non-Fermi liquid behavior has been observed in a variety of DC transport experiments, in very good agreement with theoretical predictions. Here, we describe how fractional quasiparticle charge, a fundamental property of Luttinger liquids, can be observed in impurity-induced shot noise.
We propose a device which implements a solid-state nanostructured electron entangler. It consists of a single-walled carbon nanotube connected at both ends to normal state electrodes and coupled in its middle part to a superconducting nanowire. Such a device acts as an electronic beam splitter for correlated electrons originating from the superconductor. We first show that it can be used to detect positive (bosonic-like) noise correlations in a fermionic system. Furthermore, it provides a source for entangled electrons in the two arms of the splitter. To generate entangled electron states, we propose two kinds of set-up based either on spin or energy filters. They respectively consist of ferromagnetic pads and of a system of electrostatic gates which define quantum dots. The fabrication of this device would require state-of-the-art nanofabrication techniques, carbon nanotube synthesis and integration, as well as atomic force microscopy imaging and manipulation.
We calculate the correlation functions and the dc conductivity of Luttinger liquid superlattices, modelled by a repeated pattern of interacting and free Luttinger liquids. In a specific realization, where the interacting subsystem is a Hubbard chain, the system exhibits a rich phase diagram with four different phases: two metals and two compressible insulators. In general, we find that the effective low-energy description amalgamates features of both types of liquids in proportion to their spatial extent, suggesting the interesting possibility of `engineered' Luttinger liquids. In recent years, new experimental techniques have made it possible to grow nanostructures which are topologically one-dimensional, such as quantum wires and Carbon nanotubes. However, care must be taken when invoking existing models to discuss their electronic properties, since inhomogeneities must be taken into account in a fundamental way. Consider, for instance, the Luttinger liquid (LL), which is the standard model for low-energy phenomena involving interacting electrons in one dimension [1]. The absence of conductance renormalization in long high-mobility GaAs wires [2] has been explained in terms of a usual LL (representing the wire) in contact with a non-interacting LL at each of its ends (representing the Fermi liquid leads); that is, overall, the system can be thought of as an inhomogeneous Luttinger Liquid (ILL). LLs with different inhomogeneity profiles have also been used in the context of the fractional quantum Hall effect (FQHE), to describe transitions between edge states at different fillings [3, 4], or between an edge state connected to a Fermi liquid [5]. A different class of inhomogeneous systems is represented by superlattices (SLs) and multilayers. By varying the relative th...
The eect of electronic many-body interactions in one-dimensional (1D) systems is drastically enhanced compared to higher dimensions due to the restricted phasespace, often leading to strong electronic correlations. It can be shown that due to a breakdown of the Fermi liquid theory, the ground state at low temperatures becomes a Tomonaga-Luttinger liquid (TLL). In a TLL, the strength of electronic correlations is characterized by the Luttinger parameter K, ranging between K = 0 and K = 1, where K = 1 describes non-interacting electrons. Various systems, such as single-walled carbon nanotubes, 1D atomic metal chains, and MoS 2 mirror twin boundaries, have been reported to host TLLs, in which spin-degeneracy is intact.
A two-band fermion model with boundary fields describing the band structure of a single-walled carbon nanotube (SWNT) is proposed and solved exactly by the nested Bethe ansatz. The fermions, occupying two degenerate subbands that are shifted relative to each other, interact via inner- and interband on-site Coulomb interactions, and one-particle and correlated on-site hybridizations. The critical exponents of the correlation functions are calculated using the Bethe ansatz solution and conformal field theory. It is found that a two-component electron liquid state, one of which is defined by an attractive effective electron-electron interaction, is realized for strong hybridized interaction. The attractive interaction leads to the formation of a spinless bound state of Cooper-type pairs and dominating correlations of singlet pairs for arbitrary band fillings. We suggest that this electron pairing mechanism may be the key to resolving the nature of superconductivity in SWNT's.