In this work, three-body bound states were studied in finite volume using an Effective Field Theory framework. The finite volumes under consideration are cubic volumes with periodic boundary conditions. The Effective Field Theory framework used in this work employs only contact interactions and is particularly well suited for studies of universal properties, i.e. properties independent of the details of the interaction on short distances. A particular example in the three-body sector is the Efimov effect, the emergence of a geometrically spaced bound state spectrum. In the first part, systems of three identical bosons are investigated. As a consequence of the breakdown of the spherical symmetry to cubic symmetry, the partial waves of the bound state amplitude are coupled. An infinite set of coupled integral equations for these partial waves is derived. These equations have to be solved numerically in order to obtain the binding energies in the finite volume. The dependence of the energies on the box size is calculated and the results are explicitly verified to be renormalized. Results for positive and negative scattering lengths are shown. The effects of higher partial waves are investigated. The behavior of shallow trimers near the dimer energy as well as deeply bound trimers is studied. The shallowest state investigated crosses the dimer energy at a certain volume and behaves like a scattering state for smaller volumes. Numerical evidence for a universal scaling of the finite volume corrections is provided. Subsequently, the formalism is extended to systems of three nucleons. This case provides the main motivation for this work due to its applicability to Lattice Quantum Chromodynamics (QCD) calculations of the triton. Such calculations always take place inside a finite volume which makes control over the corresponding effects crucial for an understanding of results from the lattice. For the triton, there are two coupled channels already in the infinite volume corresponding to two different spin-isospin combinations. An infinite set of coupled integral equations for the partial waves of the bound state amplitudes is derived. The renormalization of all results is again explicitly verified. The physical triton inside a finite volume is investigated as well as the triton spectrum for unphysical pion masses. The former case qualitatively shows the same behavior as the three-boson case, and the volume dependence is calculated. The smallest volumes investigated are of the order of magnitude typical for present day Lattice calculations. The motivation for the latter part is twofold. On the one hand, Lattice QCD calculations are performed at pion masses larger than the physical one for computational reasons. On the other hand, it has been conjectured that QCD is close to the critical trajectory for an infrared renormalization group limit cycle, in which case the Efimov effect would occur for a critical pion mass. Close to this critical pion mass, the triton has excited states. The behavior of the ground state and of the excited states inside a finite volume is investigated for various pion masses around the critical one. The excited states cross the energy of the bound di-nucleon, as it was already observed for the shallowest bosonic trimer. The results for the ground state were used to provide strong numerical evidence for a universal scaling of the finite volume corrections.In this work, three-body bound states were studied in finite volume using an Effective Field Theory framework. The finite volumes under consideration are cubic volumes with periodic boundary conditions. The Effective Field Theory framework used in this work employs only contact interactions and is particularly well suited for studies of universal properties, i.e. properties independent of the details of the interaction on short distances. A particular example in the three-body sector is the Efimov effect, the emergence of a geometrically spaced bound state spectrum. In the first part, systems of three identical bosons are investigated. As a consequence of the breakdown of the spherical symmetry to cubic symmetry, the partial waves of the bound state amplitude are coupled. An infinite set of coupled integral equations for these partial waves is derived. These equations have to be solved numerically in order to obtain the binding energies in the finite volume. The dependence of the energies on the box size is calculated and the results are explicitly verified to be renormalized. Results for positive and negative scattering lengths are shown. The effects of higher partial waves are investigated. The behavior of shallow trimers near the dimer energy as well as deeply bound trimers is studied. The shallowest state investigated crosses the dimer energy at a certain volume and behaves like a scattering state for smaller volumes. Numerical evidence for a universal scaling of the finite volume corrections is provided. Subsequently, the formalism is extended to systems of three nucleons. This case provides the main motivation for this work due to its applicability to Lattice Quantum Chromodynamics (QCD) calculations of the triton. Such calculations always take place inside a finite volume which makes control over the corresponding effects crucial for an understanding of results from the lattice. For the triton, there are two coupled channels already in the infinite volume corresponding to two different spin-isospin combinations. An infinite set of coupled integral equations for the partial waves of the bound state amplitudes is derived. The renormalization of all results is again explicitly verified. The physical triton inside a finite volume is investigated as well as the triton spectrum for unphysical pion masses. The former case qualitatively shows the same behavior as the three-boson case, and the volume dependence is calculated. The smallest volumes investigated are of the order of magnitude typical for present day Lattice calculations. The motivation for the latter part is twofold. On the one hand, Lattice QCD calculations are performed at pion masses larger than the physical one for computational reasons. On the other hand, it has been conjectured that QCD is close to the critical trajectory for an infrared renormalization group limit cycle, in which case the Efimov effect would occur for a critical pion mass. Close to this critical pion mass, the triton has excited states. The behavior of the ground state and of the excited states inside a finite volume is investigated for various pion masses around the critical one. The excited states cross the energy of the bound di-nucleon, as it was already observed for the shallowest bosonic trimer. The results for the ground state were used to provide strong numerical evidence for a universal scaling of the finite volume corrections.

Nov 3, 20102 citations

Three-boson bound states in ﬁnite volume with EFT

Sebastian Kreuzer, H.‐W. Hammer

The universal properties of a three-boson system with large scattering length are well understood within the framework of Eﬀective Field Theory. They include a geometric spectrum of shallow three-body bound states called “Eﬁmov states” and log-periodic dependence of scattering observables on the scattering length. We investigate the modiﬁcation of this spectrum in a ﬁnite cubic box using a partial wave expansion. The dependence of the binding energies on the box size is calculated for systems with positive and negative two-body scattering length. We compare the full results to results obtained using an expansion around the inﬁnite volume binding energy. The renormalization of the Eﬀective Field Theory in the ﬁnite volume is veriﬁed explicitly.The universal properties of a three-boson system with large scattering length are well understood within the framework of Eﬀective Field Theory. They include a geometric spectrum of shallow three-body bound states called “Eﬁmov states” and log-periodic dependence of scattering observables on the scattering length. We investigate the modiﬁcation of this spectrum in a ﬁnite cubic box using a partial wave expansion. The dependence of the binding energies on the box size is calculated for systems with positive and negative two-body scattering length. We compare the full results to results obtained using an expansion around the inﬁnite volume binding energy. The renormalization of the Eﬀective Field Theory in the ﬁnite volume is veriﬁed explicitly.

EPJ Web of ConferencesJan 1, 20100 citations

The Determination of Atomic Wave Functions

Louis C. Green

Jan 1, 19550 citations

Generation of field-mediated three-qubit entangled state shared by Alice and Bob

Paulo José dos Reis, S. Shelly Sharma

A scheme to generate shared tripartite entangled states, with two-trapped atoms in a cavity held by Alice (qubits ${A}_{1}$ and ${A}_{2}$ ) entangled to a single-trapped atom in a remote laboratory owned by Bob $(B)$ , is proposed. The entanglement is generated through interaction of trapped atoms with two-mode squeezed light shared by the two cavities. The proposed scheme is an extension of the proposal of Son et al. [J. Mod. Opt. 49, 1739 (2002)], where the possibility of entangling two remote qubits using a bipartite continuous variable state was examined. While the global negativity detects the free entanglement of the three atom mixed state, the bound entanglement is detected by the negativity calculated from pure state decomposition of the state operator. The partial negativities calculated by selective partial transposition of the three atom mixed state detect the pairwise entanglement of qubit pairs ${A}_{1}B$ , ${A}_{2}B$ , and ${A}_{1}{A}_{2}$ . The entanglement of three atoms is found to be $W$ -like, no Greenberger-Horne-Zeilinger-like quantum correlations being generated.A scheme to generate shared tripartite entangled states, with two-trapped atoms in a cavity held by Alice (qubits ${A}_{1}$ and ${A}_{2}$) entangled to a single-trapped atom in a remote laboratory owned by Bob $(B)$, is proposed. The entanglement is generated through interaction of trapped atoms with two-mode squeezed light shared by the two cavities. The proposed scheme is an extension of the proposal of Son et al. [J. Mod. Opt. 49, 1739 (2002)], where the possibility of entangling two remote qubits using a bipartite continuous variable state was examined. While the global negativity detects the free entanglement of the three atom mixed state, the bound entanglement is detected by the negativity calculated from pure state decomposition of the state operator. The partial negativities calculated by selective partial transposition of the three atom mixed state detect the pairwise entanglement of qubit pairs ${A}_{1}B$, ${A}_{2}B$, and ${A}_{1}{A}_{2}$. The entanglement of three atoms is found to be $W$-like, no Greenberger-Horne-Zeilinger-like quantum correlations being generated.

arXiv:0807.2490Physical Review AJan 28, 20099 citations

Quantum Mechanical Formulas with Applications to Bound States

Michio Nishioka

Oct 1, 19800 citations

A scheme to generate shared tripartite entangled states, with two-trapped atoms in a cavity held by Alice (qubits

{A}_{1}

and

{A}_{2}

) entangled to a single-trapped atom in a remote laboratory owned by Bob

(B)

, is proposed. The entanglement is generated through interaction of trapped atoms with two-mode squeezed light shared by the two cavities. The proposed scheme is an extension of the proposal of Son et al. [J. Mod. Opt. 49, 1739 (2002)], where the possibility of entangling two remote qubits using a bipartite continuous variable state was examined. While the global negativity detects the free entanglement of the three atom mixed state, the bound entanglement is detected by the negativity calculated from pure state decomposition of the state operator. The partial negativities calculated by selective partial transposition of the three atom mixed state detect the pairwise entanglement of qubit pairs

{A}_{1}B

{A}_{2}B

, and

{A}_{1}{A}_{2}

. The entanglement of three atoms is found to be

W

-like, no Greenberger-Horne-Zeilinger-like quantum correlations being generated.