[Paper Review] Quantum field theory for multipolar composite bosons with mass defect and relativistic corrections

Tobias Aßmann, Enno Giese+1 more|arXiv (Cornell University)|Jul 12, 2023

Cold Atom Physics and Bose-Einstein Condensates286 references4 citations

TL;DR

This paper develops a relativistic effective quantum field theory (EFT) for composite bosons (cobosons) formed by atomic nuclei and electrons, incorporating mass defect, relativistic corrections, and multipolar light-matter interactions. It derives a modified Gross-Pitaevskii equation, relativistic scattering potentials, and spin-orbit coupling from first principles in nonrelativistic QED, enabling precision studies in quantum clocks and interferometry.

ABSTRACT

Atomic high-precision measurements have become a competitive and essential technique for tests of fundamental physics, the Standard Model, and our theory of gravity. It is therefore self-evident that such measurements call for a consistent relativistic description of atoms that eventually originates from quantum field theories like quantum electrodynamics. Most quantum-metrological approaches even postulate effective field-theoretical treatments to describe a precision enhancement through techniques like squeezing. However, a consistent derivation of interacting atomic quantum gases from an elementary quantum field theory that includes both the internal structure as well as the center of mass of atoms, has not yet been addressed. We present such a subspace effective field theory for interacting, spin carrying, and possibly charged ensembles of atoms composed of nucleus and electron that form composite bosons called cobosons, where the interaction with light is included in a multipolar description. Relativistic corrections to the energy of a single coboson, light-matter interaction, and the scattering potential between cobosons arise in a consistent and natural manner. In particular, we obtain a relativistic coupling between the coboson's center-of-mass motion and internal structure encoded by the mass defect. We use these results to derive modified bound-state energies, including the motion of ions, modified scattering potentials, a relativistic extension of the Gross-Pitaevskii equation, and the mass defect applicable to atomic clocks or quantum clock interferometry.

Motivation & Objective

• To derive a consistent effective field theory for interacting, spin-carrying, charged atomic ensembles from quantum electrodynamics (QED), including relativistic corrections.
• To unify the description of center-of-mass motion with internal atomic structure via the relativistic mass defect and spin-orbit coupling.
• To include multipolar light-matter interactions and two-body scattering potentials in a relativistically corrected framework.
• To extend the Gross-Pitaevskii equation to include relativistic corrections and mass defect effects for many-body quantum systems.
• To provide a bridge between fundamental QED and effective field theories used in ultracold quantum gases and quantum metrology.

Proposed method

• Start from nonrelativistic QED (NRQED) as the fundamental theory for fermionic constituents (electrons and nuclei) and perform a systematic EFT derivation.
• Introduce coboson field operators to describe composite bosons formed by fermion pairs, ensuring proper statistics and many-body structure.
• Apply the Pekar-Peierls transformation (PZW) to map canonical momenta to physical fields, enabling consistent light-matter coupling in the multipolar gauge.
• Derive the single-coboson Hamiltonian including relativistic corrections at order $c^{-2}$, with contributions from electric multipole moments and self-energy.
• Construct the scattering potential via self-energy diagrams and relativistic corrections, incorporating spin-orbit and contact interactions.
• Use first-quantized wave functions in angular momentum eigenbasis to compute first-order energy shifts, including kinetic, Darwin, and spin-orbit terms.

Experimental results

Research questions

• RQ1 How can relativistic corrections, including the mass defect, be consistently derived from QED for composite bosons?
• RQ2 What is the form of the relativistic light-matter interaction Hamiltonian for cobosons in a multipolar description?
• RQ3 How do relativistic corrections modify the scattering potential between cobosons in a many-body system?
• RQ4 What is the structure of the modified Gross-Pitaevskii equation that includes mass defect and spin-orbit coupling?
• RQ5 How do relativistic corrections affect the energy levels of hydrogen-like cobosons, particularly in terms of quantum numbers $n, j, \ u, S$?

Key findings

• The first-order energy shift for hydrogen-like cobosons includes kinetic, orbit-orbit, Darwin, and spin-orbit contributions, explicitly depending on $n, j, \ell, S$.
• The spin-orbit coupling term is derived from relativistic corrections and encoded in the coefficient $C_{j,\ell}$, which depends on $j=\ell\pm1, \ell$ and includes contributions from magnetic dipole and spin-orbit interactions.
• The mass defect coupling between center-of-mass motion and internal structure emerges naturally from the $c^{-2}$ expansion of the Hamiltonian, enabling relativistic corrections to atomic energy levels.
• The scattering potential acquires a self-energy contribution $\hat{\mathcal{V}}_{\text{Self}}$, modifying two-body interactions in the many-body system.
• The modified Gross-Pitaevskii equation incorporates relativistic corrections and the mass defect, enabling accurate description of many-body ground states in ultracold quantum gases.
• The derived energy shift formula (Eq. 86) includes explicit dependence on reduced mass $m_r$, nuclear mass $m_n$, electron mass $m_e$, and Wilson coefficients $\alpha_v$, with terms scaling as $1/n^4$, $1/n^3$, and $1/\ell(\ell+1)$.