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Physics of Particles and Nuclei Letters

Quantum entanglement in second quantized fermion systems

O. D. Timofeevskaya

Published Nov 19, 2009• 1 citation
Publisher

Abstract

In this communication we consider the zero temperature properties of entanglement in free and interacting fermion systems following Bogoliubov’s excitation approach. We investigate spin biparticle entanglement in BCS superconductor ground state of electron gas and in EPS state of 3 He atoms. The relation between pair-distribution functions and biparticle quantum entanglement is discussed.

DOI
10.1134/s1547477109070255

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In this communication we consider the zero temperature properties of entanglement in free and interacting fermion systems following Bogoliubov’s excitation approach. We investigate spin biparticle entanglement in BCS superconductor ground state of electron gas and in EPS state of 3 He atoms. The relation between pair-distribution functions and biparticle quantum entanglement is discussed.
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.
We study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.
Abstract We study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin, providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous, non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.
In the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning