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arXiv:cond-mat/0503016 · New Journal of Physics

Quasi-particle entanglement: redefinition of the vacuum and reduced density matrix approach

Peter Samuelsson, Eugene V. Sukhorukov, Μ. Büttiker

Published Aug 26, 2005• 28 citations
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Abstract

A scattering approach to entanglement in mesoscopic conductors with independent fermionic quasi-particles is discussed. We focus on conductors in the tunnelling limit, where a redefinition of the quasi-particle vacuum transforms the wavefunction from a many-body product state of non-interacting particles to a state describing entangled two-particle excitations out of the new vacuum (Samuelsson, Sukhorukov and Büttiker 2003 Phys. Rev. Lett. 91 157002). The approach is illustrated with two examples: (i) a normal–superconducting system, where the transformation is made between Bogoliubov–de Gennes quasi-particles and Cooper pairs, and (ii) a normal system, where the transformation is made between electron quasi-particles and electron–hole pairs. This is compared to a scheme where an effective two-particle state is derived from the manybody scattering state by a reduced density matrix approach.

DOI
10.1088/1367-2630/7/1/176

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A scattering approach to entanglement in mesoscopic conductors with independent fermionic quasi-particles is discussed. We focus on conductors in the tunnelling limit, where a redefinition of the quasi-particle vacuum transforms the wavefunction from a many-body product state of non-interacting particles to a state describing entangled two-particle excitations out of the new vacuum (Samuelsson, Sukhorukov and Büttiker 2003 Phys. Rev. Lett. 91 157002). The approach is illustrated with two examples: (i) a normal–superconducting system, where the transformation is made between Bogoliubov–de Gennes quasi-particles and Cooper pairs, and (ii) a normal system, where the transformation is made between electron quasi-particles and electron–hole pairs. This is compared to a scheme where an effective two-particle state is derived from the manybody scattering state by a reduced density matrix approach.
We present a theory for two-particle entanglement production and detection in mesoscopic conductors at finite temperature. The entanglement of the density matrix projected out of the emitted many-body state differs from the entanglement of the reduced density matrix, detectable by current correlation measurements. Under general conditions reduced entanglement constitutes a witness for projected entanglement. Applied to the recent experiment [Neder et al., Nature (London) 448, 333 (2007)10.1038/nature05955] on a fermionic Hanbury Brown Twiss two-particle interferometer we find that despite an appreciable entanglement production in the experiment, the detectable entanglement is close to zero.
We analyze the role of resonances in two-fermion entanglement production for a quasi-one-dimensional two-channel scattering problem. We solve exactly for the problem of a two-fermion antisymmetric product state scattering off a double-$\ensuremath{\delta}$-well potential. It is shown that the two-particle concurrence of the post-selected state has an oscillatory behavior where the concurrence vanishes at the values of momenta for virtual bound states in the double well. These concurrence zeros are interpreted in terms of the uncertainty in the knowledge of the state of the one-particle subspace-reduced one-particle density matrix. Our results suggest the manipulation of fermion entanglement production through the resonance structure of quantum dots.
The entanglement between occupation numbers of different single particle basis states depends on coupling between different single particle basis states in the second-quantized Hamiltonian. Thus, in principle, interaction is not necessary for occupation-number entanglement to appear. However, in order to characterize quantum correlation caused by interaction , we use the eigenstates of the single-particle Hamiltonian as the single particle basis upon which the occupation-number entanglement is defined. Using this so-called proper single particle basis, if there is no interaction, the many-particle second-quantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be non-entangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on many-particle physics. The discussions on Fermi systems start with Fermi gas, the Hartree–Fock approximation and the electron–hole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as − f ln f − (1 − f )ln(1 − f ), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair g k , and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the Hartree–Gross–Pitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of two-particle correlation caused by the weak interaction. In these examples, the interaction-induced entanglement as calculated is directly related to the macroscopic physical properties.
The Bogolyubov transformation (1958) frequently used in dealing with pairing interactions of fermions, is extended to allow for binary coupling between any (even) number of fermions by using a matrix transformation 'coefficient'. The corresponding quasiparticles and quasi-vacuum state are derived. The normally ordered expansion of the unitary generalized Bogolyubov operator is obtained by application of the technique of integration within an ordered product (IWOP) for the fermion system.
We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean-field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.